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Any changes you make will be lost during the next clean build. + +// Dependent includes +#include + +// CUDA public interface, for type definitions and cu* function prototypes +#include "cudaVDPAU.h" + + +// ************************************************************************* +// Definitions of structs to hold parameters for each function +// ************************************************************************* + +typedef struct cuVDPAUGetDevice_params_st { + CUdevice *pDevice; + VdpDevice vdpDevice; + VdpGetProcAddress *vdpGetProcAddress; +} cuVDPAUGetDevice_params; + +typedef struct cuVDPAUCtxCreate_v2_params_st { + CUcontext *pCtx; + unsigned int flags; + CUdevice device; + VdpDevice vdpDevice; + VdpGetProcAddress *vdpGetProcAddress; +} cuVDPAUCtxCreate_v2_params; + +typedef struct cuGraphicsVDPAURegisterVideoSurface_params_st { + CUgraphicsResource *pCudaResource; + VdpVideoSurface vdpSurface; + unsigned int flags; +} cuGraphicsVDPAURegisterVideoSurface_params; + +typedef 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for musculotendon models.""" + +from dataclasses import dataclass + +from sympy.core.expr import UnevaluatedExpr +from sympy.core.function import ArgumentIndexError, Function +from sympy.core.numbers import Float, Integer +from sympy.functions.elementary.exponential import exp, log +from sympy.functions.elementary.hyperbolic import cosh, sinh +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.printing.precedence import PRECEDENCE + + +__all__ = [ + 'CharacteristicCurveCollection', + 'CharacteristicCurveFunction', + 'FiberForceLengthActiveDeGroote2016', + 'FiberForceLengthPassiveDeGroote2016', + 'FiberForceLengthPassiveInverseDeGroote2016', + 'FiberForceVelocityDeGroote2016', + 'FiberForceVelocityInverseDeGroote2016', + 'TendonForceLengthDeGroote2016', + 'TendonForceLengthInverseDeGroote2016', +] + + +class CharacteristicCurveFunction(Function): + """Base class for all musculotendon characteristic curve functions.""" + + @classmethod + def eval(cls): + msg = ( + f'Cannot directly instantiate {cls.__name__!r}, instances of ' + f'characteristic curves must be of a concrete subclass.' + + ) + raise TypeError(msg) + + def _print_code(self, printer): + """Print code for the function defining the curve using a printer. + + Explanation + =========== + + The order of operations may need to be controlled as constant folding + the numeric terms within the equations of a musculotendon + characteristic curve can sometimes results in a numerically-unstable + expression. + + Parameters + ========== + + printer : Printer + The printer to be used to print a string representation of the + characteristic curve as valid code in the target language. + + """ + return printer._print(printer.parenthesize( + self.doit(deep=False, evaluate=False), PRECEDENCE['Atom'], + )) + + _ccode = _print_code + _cupycode = _print_code + _cxxcode = _print_code + _fcode = _print_code + _jaxcode = _print_code + _lambdacode = _print_code + _mpmathcode = _print_code + _octave = _print_code + _pythoncode = _print_code + _numpycode = _print_code + _scipycode = _print_code + + +class TendonForceLengthDeGroote2016(CharacteristicCurveFunction): + r"""Tendon force-length curve based on De Groote et al., 2016 [1]_. + + Explanation + =========== + + Gives the normalized tendon force produced as a function of normalized + tendon length. + + The function is defined by the equation: + + $fl^T = c_0 \exp{c_3 \left( \tilde{l}^T - c_1 \right)} - c_2$ + + with constant values of $c_0 = 0.2$, $c_1 = 0.995$, $c_2 = 0.25$, and + $c_3 = 33.93669377311689$. + + While it is possible to change the constant values, these were carefully + selected in the original publication to give the characteristic curve + specific and required properties. For example, the function produces no + force when the tendon is in an unstrained state. It also produces a force + of 1 normalized unit when the tendon is under a 5% strain. + + Examples + ======== + + The preferred way to instantiate :class:`TendonForceLengthDeGroote2016` is using + the :meth:`~.with_defaults` constructor because this will automatically + populate the constants within the characteristic curve equation with the + floating point values from the original publication. This constructor takes + a single argument corresponding to normalized tendon length. We'll create a + :class:`~.Symbol` called ``l_T_tilde`` to represent this. + + >>> from sympy import Symbol + >>> from sympy.physics.biomechanics import TendonForceLengthDeGroote2016 + >>> l_T_tilde = Symbol('l_T_tilde') + >>> fl_T = TendonForceLengthDeGroote2016.with_defaults(l_T_tilde) + >>> fl_T + TendonForceLengthDeGroote2016(l_T_tilde, 0.2, 0.995, 0.25, + 33.93669377311689) + + It's also possible to populate the four constants with your own values too. + + >>> from sympy import symbols + >>> c0, c1, c2, c3 = symbols('c0 c1 c2 c3') + >>> fl_T = TendonForceLengthDeGroote2016(l_T_tilde, c0, c1, c2, c3) + >>> fl_T + TendonForceLengthDeGroote2016(l_T_tilde, c0, c1, c2, c3) + + You don't just have to use symbols as the arguments, it's also possible to + use expressions. Let's create a new pair of symbols, ``l_T`` and + ``l_T_slack``, representing tendon length and tendon slack length + respectively. We can then represent ``l_T_tilde`` as an expression, the + ratio of these. + + >>> l_T, l_T_slack = symbols('l_T l_T_slack') + >>> l_T_tilde = l_T/l_T_slack + >>> fl_T = TendonForceLengthDeGroote2016.with_defaults(l_T_tilde) + >>> fl_T + TendonForceLengthDeGroote2016(l_T/l_T_slack, 0.2, 0.995, 0.25, + 33.93669377311689) + + To inspect the actual symbolic expression that this function represents, + we can call the :meth:`~.doit` method on an instance. We'll use the keyword + argument ``evaluate=False`` as this will keep the expression in its + canonical form and won't simplify any constants. + + >>> fl_T.doit(evaluate=False) + -0.25 + 0.2*exp(33.93669377311689*(l_T/l_T_slack - 0.995)) + + The function can also be differentiated. We'll differentiate with respect + to l_T using the ``diff`` method on an instance with the single positional + argument ``l_T``. + + >>> fl_T.diff(l_T) + 6.787338754623378*exp(33.93669377311689*(l_T/l_T_slack - 0.995))/l_T_slack + + References + ========== + + .. [1] De Groote, F., Kinney, A. L., Rao, A. V., & Fregly, B. J., Evaluation + of direct collocation optimal control problem formulations for + solving the muscle redundancy problem, Annals of biomedical + engineering, 44(10), (2016) pp. 2922-2936 + + """ + + @classmethod + def with_defaults(cls, l_T_tilde): + r"""Recommended constructor that will use the published constants. + + Explanation + =========== + + Returns a new instance of the tendon force-length function using the + four constant values specified in the original publication. + + These have the values: + + $c_0 = 0.2$ + $c_1 = 0.995$ + $c_2 = 0.25$ + $c_3 = 33.93669377311689$ + + Parameters + ========== + + l_T_tilde : Any (sympifiable) + Normalized tendon length. + + """ + c0 = Float('0.2') + c1 = Float('0.995') + c2 = Float('0.25') + c3 = Float('33.93669377311689') + return cls(l_T_tilde, c0, c1, c2, c3) + + @classmethod + def eval(cls, l_T_tilde, c0, c1, c2, c3): + """Evaluation of basic inputs. + + Parameters + ========== + + l_T_tilde : Any (sympifiable) + Normalized tendon length. + c0 : Any (sympifiable) + The first constant in the characteristic equation. The published + value is ``0.2``. + c1 : Any (sympifiable) + The second constant in the characteristic equation. The published + value is ``0.995``. + c2 : Any (sympifiable) + The third constant in the characteristic equation. The published + value is ``0.25``. + c3 : Any (sympifiable) + The fourth constant in the characteristic equation. The published + value is ``33.93669377311689``. + + """ + pass + + def _eval_evalf(self, prec): + """Evaluate the expression numerically using ``evalf``.""" + return self.doit(deep=False, evaluate=False)._eval_evalf(prec) + + def doit(self, deep=True, evaluate=True, **hints): + """Evaluate the expression defining the function. + + Parameters + ========== + + deep : bool + Whether ``doit`` should be recursively called. Default is ``True``. + evaluate : bool. + Whether the SymPy expression should be evaluated as it is + constructed. If ``False``, then no constant folding will be + conducted which will leave the expression in a more numerically- + stable for values of ``l_T_tilde`` that correspond to a sensible + operating range for a musculotendon. Default is ``True``. + **kwargs : dict[str, Any] + Additional keyword argument pairs to be recursively passed to + ``doit``. + + """ + l_T_tilde, *constants = self.args + if deep: + hints['evaluate'] = evaluate + l_T_tilde = l_T_tilde.doit(deep=deep, **hints) + c0, c1, c2, c3 = [c.doit(deep=deep, **hints) for c in constants] + else: + c0, c1, c2, c3 = constants + + if evaluate: + return c0*exp(c3*(l_T_tilde - c1)) - c2 + + return c0*exp(c3*UnevaluatedExpr(l_T_tilde - c1)) - c2 + + def fdiff(self, argindex=1): + """Derivative of the function with respect to a single argument. + + Parameters + ========== + + argindex : int + The index of the function's arguments with respect to which the + derivative should be taken. Argument indexes start at ``1``. + Default is ``1``. + + """ + l_T_tilde, c0, c1, c2, c3 = self.args + if argindex == 1: + return c0*c3*exp(c3*UnevaluatedExpr(l_T_tilde - c1)) + elif argindex == 2: + return exp(c3*UnevaluatedExpr(l_T_tilde - c1)) + elif argindex == 3: + return -c0*c3*exp(c3*UnevaluatedExpr(l_T_tilde - c1)) + elif argindex == 4: + return Integer(-1) + elif argindex == 5: + return c0*(l_T_tilde - c1)*exp(c3*UnevaluatedExpr(l_T_tilde - c1)) + + raise ArgumentIndexError(self, argindex) + + def inverse(self, argindex=1): + """Inverse function. + + Parameters + ========== + + argindex : int + Value to start indexing the arguments at. Default is ``1``. + + """ + return TendonForceLengthInverseDeGroote2016 + + def _latex(self, printer): + """Print a LaTeX representation of the function defining the curve. + + Parameters + ========== + + printer : Printer + The printer to be used to print the LaTeX string representation. + + """ + l_T_tilde = self.args[0] + _l_T_tilde = printer._print(l_T_tilde) + return r'\operatorname{fl}^T \left( %s \right)' % _l_T_tilde + + +class TendonForceLengthInverseDeGroote2016(CharacteristicCurveFunction): + r"""Inverse tendon force-length curve based on De Groote et al., 2016 [1]_. + + Explanation + =========== + + Gives the normalized tendon length that produces a specific normalized + tendon force. + + The function is defined by the equation: + + ${fl^T}^{-1} = frac{\log{\frac{fl^T + c_2}{c_0}}}{c_3} + c_1$ + + with constant values of $c_0 = 0.2$, $c_1 = 0.995$, $c_2 = 0.25$, and + $c_3 = 33.93669377311689$. This function is the exact analytical inverse + of the related tendon force-length curve ``TendonForceLengthDeGroote2016``. + + While it is possible to change the constant values, these were carefully + selected in the original publication to give the characteristic curve + specific and required properties. For example, the function produces no + force when the tendon is in an unstrained state. It also produces a force + of 1 normalized unit when the tendon is under a 5% strain. + + Examples + ======== + + The preferred way to instantiate :class:`TendonForceLengthInverseDeGroote2016` is + using the :meth:`~.with_defaults` constructor because this will automatically + populate the constants within the characteristic curve equation with the + floating point values from the original publication. This constructor takes + a single argument corresponding to normalized tendon force-length, which is + equal to the tendon force. We'll create a :class:`~.Symbol` called ``fl_T`` to + represent this. + + >>> from sympy import Symbol + >>> from sympy.physics.biomechanics import TendonForceLengthInverseDeGroote2016 + >>> fl_T = Symbol('fl_T') + >>> l_T_tilde = TendonForceLengthInverseDeGroote2016.with_defaults(fl_T) + >>> l_T_tilde + TendonForceLengthInverseDeGroote2016(fl_T, 0.2, 0.995, 0.25, + 33.93669377311689) + + It's also possible to populate the four constants with your own values too. + + >>> from sympy import symbols + >>> c0, c1, c2, c3 = symbols('c0 c1 c2 c3') + >>> l_T_tilde = TendonForceLengthInverseDeGroote2016(fl_T, c0, c1, c2, c3) + >>> l_T_tilde + TendonForceLengthInverseDeGroote2016(fl_T, c0, c1, c2, c3) + + To inspect the actual symbolic expression that this function represents, + we can call the :meth:`~.doit` method on an instance. We'll use the keyword + argument ``evaluate=False`` as this will keep the expression in its + canonical form and won't simplify any constants. + + >>> l_T_tilde.doit(evaluate=False) + c1 + log((c2 + fl_T)/c0)/c3 + + The function can also be differentiated. We'll differentiate with respect + to l_T using the ``diff`` method on an instance with the single positional + argument ``l_T``. + + >>> l_T_tilde.diff(fl_T) + 1/(c3*(c2 + fl_T)) + + References + ========== + + .. [1] De Groote, F., Kinney, A. L., Rao, A. V., & Fregly, B. J., Evaluation + of direct collocation optimal control problem formulations for + solving the muscle redundancy problem, Annals of biomedical + engineering, 44(10), (2016) pp. 2922-2936 + + """ + + @classmethod + def with_defaults(cls, fl_T): + r"""Recommended constructor that will use the published constants. + + Explanation + =========== + + Returns a new instance of the inverse tendon force-length function + using the four constant values specified in the original publication. + + These have the values: + + $c_0 = 0.2$ + $c_1 = 0.995$ + $c_2 = 0.25$ + $c_3 = 33.93669377311689$ + + Parameters + ========== + + fl_T : Any (sympifiable) + Normalized tendon force as a function of tendon length. + + """ + c0 = Float('0.2') + c1 = Float('0.995') + c2 = Float('0.25') + c3 = Float('33.93669377311689') + return cls(fl_T, c0, c1, c2, c3) + + @classmethod + def eval(cls, fl_T, c0, c1, c2, c3): + """Evaluation of basic inputs. + + Parameters + ========== + + fl_T : Any (sympifiable) + Normalized tendon force as a function of tendon length. + c0 : Any (sympifiable) + The first constant in the characteristic equation. The published + value is ``0.2``. + c1 : Any (sympifiable) + The second constant in the characteristic equation. The published + value is ``0.995``. + c2 : Any (sympifiable) + The third constant in the characteristic equation. The published + value is ``0.25``. + c3 : Any (sympifiable) + The fourth constant in the characteristic equation. The published + value is ``33.93669377311689``. + + """ + pass + + def _eval_evalf(self, prec): + """Evaluate the expression numerically using ``evalf``.""" + return self.doit(deep=False, evaluate=False)._eval_evalf(prec) + + def doit(self, deep=True, evaluate=True, **hints): + """Evaluate the expression defining the function. + + Parameters + ========== + + deep : bool + Whether ``doit`` should be recursively called. Default is ``True``. + evaluate : bool. + Whether the SymPy expression should be evaluated as it is + constructed. If ``False``, then no constant folding will be + conducted which will leave the expression in a more numerically- + stable for values of ``l_T_tilde`` that correspond to a sensible + operating range for a musculotendon. Default is ``True``. + **kwargs : dict[str, Any] + Additional keyword argument pairs to be recursively passed to + ``doit``. + + """ + fl_T, *constants = self.args + if deep: + hints['evaluate'] = evaluate + fl_T = fl_T.doit(deep=deep, **hints) + c0, c1, c2, c3 = [c.doit(deep=deep, **hints) for c in constants] + else: + c0, c1, c2, c3 = constants + + if evaluate: + return log((fl_T + c2)/c0)/c3 + c1 + + return log(UnevaluatedExpr((fl_T + c2)/c0))/c3 + c1 + + def fdiff(self, argindex=1): + """Derivative of the function with respect to a single argument. + + Parameters + ========== + + argindex : int + The index of the function's arguments with respect to which the + derivative should be taken. Argument indexes start at ``1``. + Default is ``1``. + + """ + fl_T, c0, c1, c2, c3 = self.args + if argindex == 1: + return 1/(c3*(fl_T + c2)) + elif argindex == 2: + return -1/(c0*c3) + elif argindex == 3: + return Integer(1) + elif argindex == 4: + return 1/(c3*(fl_T + c2)) + elif argindex == 5: + return -log(UnevaluatedExpr((fl_T + c2)/c0))/c3**2 + + raise ArgumentIndexError(self, argindex) + + def inverse(self, argindex=1): + """Inverse function. + + Parameters + ========== + + argindex : int + Value to start indexing the arguments at. Default is ``1``. + + """ + return TendonForceLengthDeGroote2016 + + def _latex(self, printer): + """Print a LaTeX representation of the function defining the curve. + + Parameters + ========== + + printer : Printer + The printer to be used to print the LaTeX string representation. + + """ + fl_T = self.args[0] + _fl_T = printer._print(fl_T) + return r'\left( \operatorname{fl}^T \right)^{-1} \left( %s \right)' % _fl_T + + +class FiberForceLengthPassiveDeGroote2016(CharacteristicCurveFunction): + r"""Passive muscle fiber force-length curve based on De Groote et al., 2016 + [1]_. + + Explanation + =========== + + The function is defined by the equation: + + $fl^M_{pas} = \frac{\frac{\exp{c_1 \left(\tilde{l^M} - 1\right)}}{c_0} - 1}{\exp{c_1} - 1}$ + + with constant values of $c_0 = 0.6$ and $c_1 = 4.0$. + + While it is possible to change the constant values, these were carefully + selected in the original publication to give the characteristic curve + specific and required properties. For example, the function produces a + passive fiber force very close to 0 for all normalized fiber lengths + between 0 and 1. + + Examples + ======== + + The preferred way to instantiate :class:`FiberForceLengthPassiveDeGroote2016` is + using the :meth:`~.with_defaults` constructor because this will automatically + populate the constants within the characteristic curve equation with the + floating point values from the original publication. This constructor takes + a single argument corresponding to normalized muscle fiber length. We'll + create a :class:`~.Symbol` called ``l_M_tilde`` to represent this. + + >>> from sympy import Symbol + >>> from sympy.physics.biomechanics import FiberForceLengthPassiveDeGroote2016 + >>> l_M_tilde = Symbol('l_M_tilde') + >>> fl_M = FiberForceLengthPassiveDeGroote2016.with_defaults(l_M_tilde) + >>> fl_M + FiberForceLengthPassiveDeGroote2016(l_M_tilde, 0.6, 4.0) + + It's also possible to populate the two constants with your own values too. + + >>> from sympy import symbols + >>> c0, c1 = symbols('c0 c1') + >>> fl_M = FiberForceLengthPassiveDeGroote2016(l_M_tilde, c0, c1) + >>> fl_M + FiberForceLengthPassiveDeGroote2016(l_M_tilde, c0, c1) + + You don't just have to use symbols as the arguments, it's also possible to + use expressions. Let's create a new pair of symbols, ``l_M`` and + ``l_M_opt``, representing muscle fiber length and optimal muscle fiber + length respectively. We can then represent ``l_M_tilde`` as an expression, + the ratio of these. + + >>> l_M, l_M_opt = symbols('l_M l_M_opt') + >>> l_M_tilde = l_M/l_M_opt + >>> fl_M = FiberForceLengthPassiveDeGroote2016.with_defaults(l_M_tilde) + >>> fl_M + FiberForceLengthPassiveDeGroote2016(l_M/l_M_opt, 0.6, 4.0) + + To inspect the actual symbolic expression that this function represents, + we can call the :meth:`~.doit` method on an instance. We'll use the keyword + argument ``evaluate=False`` as this will keep the expression in its + canonical form and won't simplify any constants. + + >>> fl_M.doit(evaluate=False) + 0.0186573603637741*(-1 + exp(6.66666666666667*(l_M/l_M_opt - 1))) + + The function can also be differentiated. We'll differentiate with respect + to l_M using the ``diff`` method on an instance with the single positional + argument ``l_M``. + + >>> fl_M.diff(l_M) + 0.12438240242516*exp(6.66666666666667*(l_M/l_M_opt - 1))/l_M_opt + + References + ========== + + .. [1] De Groote, F., Kinney, A. L., Rao, A. V., & Fregly, B. J., Evaluation + of direct collocation optimal control problem formulations for + solving the muscle redundancy problem, Annals of biomedical + engineering, 44(10), (2016) pp. 2922-2936 + + """ + + @classmethod + def with_defaults(cls, l_M_tilde): + r"""Recommended constructor that will use the published constants. + + Explanation + =========== + + Returns a new instance of the muscle fiber passive force-length + function using the four constant values specified in the original + publication. + + These have the values: + + $c_0 = 0.6$ + $c_1 = 4.0$ + + Parameters + ========== + + l_M_tilde : Any (sympifiable) + Normalized muscle fiber length. + + """ + c0 = Float('0.6') + c1 = Float('4.0') + return cls(l_M_tilde, c0, c1) + + @classmethod + def eval(cls, l_M_tilde, c0, c1): + """Evaluation of basic inputs. + + Parameters + ========== + + l_M_tilde : Any (sympifiable) + Normalized muscle fiber length. + c0 : Any (sympifiable) + The first constant in the characteristic equation. The published + value is ``0.6``. + c1 : Any (sympifiable) + The second constant in the characteristic equation. The published + value is ``4.0``. + + """ + pass + + def _eval_evalf(self, prec): + """Evaluate the expression numerically using ``evalf``.""" + return self.doit(deep=False, evaluate=False)._eval_evalf(prec) + + def doit(self, deep=True, evaluate=True, **hints): + """Evaluate the expression defining the function. + + Parameters + ========== + + deep : bool + Whether ``doit`` should be recursively called. Default is ``True``. + evaluate : bool. + Whether the SymPy expression should be evaluated as it is + constructed. If ``False``, then no constant folding will be + conducted which will leave the expression in a more numerically- + stable for values of ``l_T_tilde`` that correspond to a sensible + operating range for a musculotendon. Default is ``True``. + **kwargs : dict[str, Any] + Additional keyword argument pairs to be recursively passed to + ``doit``. + + """ + l_M_tilde, *constants = self.args + if deep: + hints['evaluate'] = evaluate + l_M_tilde = l_M_tilde.doit(deep=deep, **hints) + c0, c1 = [c.doit(deep=deep, **hints) for c in constants] + else: + c0, c1 = constants + + if evaluate: + return (exp((c1*(l_M_tilde - 1))/c0) - 1)/(exp(c1) - 1) + + return (exp((c1*UnevaluatedExpr(l_M_tilde - 1))/c0) - 1)/(exp(c1) - 1) + + def fdiff(self, argindex=1): + """Derivative of the function with respect to a single argument. + + Parameters + ========== + + argindex : int + The index of the function's arguments with respect to which the + derivative should be taken. Argument indexes start at ``1``. + Default is ``1``. + + """ + l_M_tilde, c0, c1 = self.args + if argindex == 1: + return c1*exp(c1*UnevaluatedExpr(l_M_tilde - 1)/c0)/(c0*(exp(c1) - 1)) + elif argindex == 2: + return ( + -c1*exp(c1*UnevaluatedExpr(l_M_tilde - 1)/c0) + *UnevaluatedExpr(l_M_tilde - 1)/(c0**2*(exp(c1) - 1)) + ) + elif argindex == 3: + return ( + -exp(c1)*(-1 + exp(c1*UnevaluatedExpr(l_M_tilde - 1)/c0))/(exp(c1) - 1)**2 + + exp(c1*UnevaluatedExpr(l_M_tilde - 1)/c0)*(l_M_tilde - 1)/(c0*(exp(c1) - 1)) + ) + + raise ArgumentIndexError(self, argindex) + + def inverse(self, argindex=1): + """Inverse function. + + Parameters + ========== + + argindex : int + Value to start indexing the arguments at. Default is ``1``. + + """ + return FiberForceLengthPassiveInverseDeGroote2016 + + def _latex(self, printer): + """Print a LaTeX representation of the function defining the curve. + + Parameters + ========== + + printer : Printer + The printer to be used to print the LaTeX string representation. + + """ + l_M_tilde = self.args[0] + _l_M_tilde = printer._print(l_M_tilde) + return r'\operatorname{fl}^M_{pas} \left( %s \right)' % _l_M_tilde + + +class FiberForceLengthPassiveInverseDeGroote2016(CharacteristicCurveFunction): + r"""Inverse passive muscle fiber force-length curve based on De Groote et + al., 2016 [1]_. + + Explanation + =========== + + Gives the normalized muscle fiber length that produces a specific normalized + passive muscle fiber force. + + The function is defined by the equation: + + ${fl^M_{pas}}^{-1} = \frac{c_0 \log{\left(\exp{c_1} - 1\right)fl^M_pas + 1}}{c_1} + 1$ + + with constant values of $c_0 = 0.6$ and $c_1 = 4.0$. This function is the + exact analytical inverse of the related tendon force-length curve + ``FiberForceLengthPassiveDeGroote2016``. + + While it is possible to change the constant values, these were carefully + selected in the original publication to give the characteristic curve + specific and required properties. For example, the function produces a + passive fiber force very close to 0 for all normalized fiber lengths + between 0 and 1. + + Examples + ======== + + The preferred way to instantiate + :class:`FiberForceLengthPassiveInverseDeGroote2016` is using the + :meth:`~.with_defaults` constructor because this will automatically populate the + constants within the characteristic curve equation with the floating point + values from the original publication. This constructor takes a single + argument corresponding to the normalized passive muscle fiber length-force + component of the muscle fiber force. We'll create a :class:`~.Symbol` called + ``fl_M_pas`` to represent this. + + >>> from sympy import Symbol + >>> from sympy.physics.biomechanics import FiberForceLengthPassiveInverseDeGroote2016 + >>> fl_M_pas = Symbol('fl_M_pas') + >>> l_M_tilde = FiberForceLengthPassiveInverseDeGroote2016.with_defaults(fl_M_pas) + >>> l_M_tilde + FiberForceLengthPassiveInverseDeGroote2016(fl_M_pas, 0.6, 4.0) + + It's also possible to populate the two constants with your own values too. + + >>> from sympy import symbols + >>> c0, c1 = symbols('c0 c1') + >>> l_M_tilde = FiberForceLengthPassiveInverseDeGroote2016(fl_M_pas, c0, c1) + >>> l_M_tilde + FiberForceLengthPassiveInverseDeGroote2016(fl_M_pas, c0, c1) + + To inspect the actual symbolic expression that this function represents, + we can call the :meth:`~.doit` method on an instance. We'll use the keyword + argument ``evaluate=False`` as this will keep the expression in its + canonical form and won't simplify any constants. + + >>> l_M_tilde.doit(evaluate=False) + c0*log(1 + fl_M_pas*(exp(c1) - 1))/c1 + 1 + + The function can also be differentiated. We'll differentiate with respect + to fl_M_pas using the ``diff`` method on an instance with the single positional + argument ``fl_M_pas``. + + >>> l_M_tilde.diff(fl_M_pas) + c0*(exp(c1) - 1)/(c1*(fl_M_pas*(exp(c1) - 1) + 1)) + + References + ========== + + .. [1] De Groote, F., Kinney, A. L., Rao, A. V., & Fregly, B. J., Evaluation + of direct collocation optimal control problem formulations for + solving the muscle redundancy problem, Annals of biomedical + engineering, 44(10), (2016) pp. 2922-2936 + + """ + + @classmethod + def with_defaults(cls, fl_M_pas): + r"""Recommended constructor that will use the published constants. + + Explanation + =========== + + Returns a new instance of the inverse muscle fiber passive force-length + function using the four constant values specified in the original + publication. + + These have the values: + + $c_0 = 0.6$ + $c_1 = 4.0$ + + Parameters + ========== + + fl_M_pas : Any (sympifiable) + Normalized passive muscle fiber force as a function of muscle fiber + length. + + """ + c0 = Float('0.6') + c1 = Float('4.0') + return cls(fl_M_pas, c0, c1) + + @classmethod + def eval(cls, fl_M_pas, c0, c1): + """Evaluation of basic inputs. + + Parameters + ========== + + fl_M_pas : Any (sympifiable) + Normalized passive muscle fiber force. + c0 : Any (sympifiable) + The first constant in the characteristic equation. The published + value is ``0.6``. + c1 : Any (sympifiable) + The second constant in the characteristic equation. The published + value is ``4.0``. + + """ + pass + + def _eval_evalf(self, prec): + """Evaluate the expression numerically using ``evalf``.""" + return self.doit(deep=False, evaluate=False)._eval_evalf(prec) + + def doit(self, deep=True, evaluate=True, **hints): + """Evaluate the expression defining the function. + + Parameters + ========== + + deep : bool + Whether ``doit`` should be recursively called. Default is ``True``. + evaluate : bool. + Whether the SymPy expression should be evaluated as it is + constructed. If ``False``, then no constant folding will be + conducted which will leave the expression in a more numerically- + stable for values of ``l_T_tilde`` that correspond to a sensible + operating range for a musculotendon. Default is ``True``. + **kwargs : dict[str, Any] + Additional keyword argument pairs to be recursively passed to + ``doit``. + + """ + fl_M_pas, *constants = self.args + if deep: + hints['evaluate'] = evaluate + fl_M_pas = fl_M_pas.doit(deep=deep, **hints) + c0, c1 = [c.doit(deep=deep, **hints) for c in constants] + else: + c0, c1 = constants + + if evaluate: + return c0*log(fl_M_pas*(exp(c1) - 1) + 1)/c1 + 1 + + return c0*log(UnevaluatedExpr(fl_M_pas*(exp(c1) - 1)) + 1)/c1 + 1 + + def fdiff(self, argindex=1): + """Derivative of the function with respect to a single argument. + + Parameters + ========== + + argindex : int + The index of the function's arguments with respect to which the + derivative should be taken. Argument indexes start at ``1``. + Default is ``1``. + + """ + fl_M_pas, c0, c1 = self.args + if argindex == 1: + return c0*(exp(c1) - 1)/(c1*(fl_M_pas*(exp(c1) - 1) + 1)) + elif argindex == 2: + return log(fl_M_pas*(exp(c1) - 1) + 1)/c1 + elif argindex == 3: + return ( + c0*fl_M_pas*exp(c1)/(c1*(fl_M_pas*(exp(c1) - 1) + 1)) + - c0*log(fl_M_pas*(exp(c1) - 1) + 1)/c1**2 + ) + + raise ArgumentIndexError(self, argindex) + + def inverse(self, argindex=1): + """Inverse function. + + Parameters + ========== + + argindex : int + Value to start indexing the arguments at. Default is ``1``. + + """ + return FiberForceLengthPassiveDeGroote2016 + + def _latex(self, printer): + """Print a LaTeX representation of the function defining the curve. + + Parameters + ========== + + printer : Printer + The printer to be used to print the LaTeX string representation. + + """ + fl_M_pas = self.args[0] + _fl_M_pas = printer._print(fl_M_pas) + return r'\left( \operatorname{fl}^M_{pas} \right)^{-1} \left( %s \right)' % _fl_M_pas + + +class FiberForceLengthActiveDeGroote2016(CharacteristicCurveFunction): + r"""Active muscle fiber force-length curve based on De Groote et al., 2016 + [1]_. + + Explanation + =========== + + The function is defined by the equation: + + $fl_{\text{act}}^M = c_0 \exp\left(-\frac{1}{2}\left(\frac{\tilde{l}^M - c_1}{c_2 + c_3 \tilde{l}^M}\right)^2\right) + + c_4 \exp\left(-\frac{1}{2}\left(\frac{\tilde{l}^M - c_5}{c_6 + c_7 \tilde{l}^M}\right)^2\right) + + c_8 \exp\left(-\frac{1}{2}\left(\frac{\tilde{l}^M - c_9}{c_{10} + c_{11} \tilde{l}^M}\right)^2\right)$ + + with constant values of $c0 = 0.814$, $c1 = 1.06$, $c2 = 0.162$, + $c3 = 0.0633$, $c4 = 0.433$, $c5 = 0.717$, $c6 = -0.0299$, $c7 = 0.2$, + $c8 = 0.1$, $c9 = 1.0$, $c10 = 0.354$, and $c11 = 0.0$. + + While it is possible to change the constant values, these were carefully + selected in the original publication to give the characteristic curve + specific and required properties. For example, the function produces a + active fiber force of 1 at a normalized fiber length of 1, and an active + fiber force of 0 at normalized fiber lengths of 0 and 2. + + Examples + ======== + + The preferred way to instantiate :class:`FiberForceLengthActiveDeGroote2016` is + using the :meth:`~.with_defaults` constructor because this will automatically + populate the constants within the characteristic curve equation with the + floating point values from the original publication. This constructor takes + a single argument corresponding to normalized muscle fiber length. We'll + create a :class:`~.Symbol` called ``l_M_tilde`` to represent this. + + >>> from sympy import Symbol + >>> from sympy.physics.biomechanics import FiberForceLengthActiveDeGroote2016 + >>> l_M_tilde = Symbol('l_M_tilde') + >>> fl_M = FiberForceLengthActiveDeGroote2016.with_defaults(l_M_tilde) + >>> fl_M + FiberForceLengthActiveDeGroote2016(l_M_tilde, 0.814, 1.06, 0.162, 0.0633, + 0.433, 0.717, -0.0299, 0.2, 0.1, 1.0, 0.354, 0.0) + + It's also possible to populate the two constants with your own values too. + + >>> from sympy import symbols + >>> c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11 = symbols('c0:12') + >>> fl_M = FiberForceLengthActiveDeGroote2016(l_M_tilde, c0, c1, c2, c3, + ... c4, c5, c6, c7, c8, c9, c10, c11) + >>> fl_M + FiberForceLengthActiveDeGroote2016(l_M_tilde, c0, c1, c2, c3, c4, c5, c6, + c7, c8, c9, c10, c11) + + You don't just have to use symbols as the arguments, it's also possible to + use expressions. Let's create a new pair of symbols, ``l_M`` and + ``l_M_opt``, representing muscle fiber length and optimal muscle fiber + length respectively. We can then represent ``l_M_tilde`` as an expression, + the ratio of these. + + >>> l_M, l_M_opt = symbols('l_M l_M_opt') + >>> l_M_tilde = l_M/l_M_opt + >>> fl_M = FiberForceLengthActiveDeGroote2016.with_defaults(l_M_tilde) + >>> fl_M + FiberForceLengthActiveDeGroote2016(l_M/l_M_opt, 0.814, 1.06, 0.162, 0.0633, + 0.433, 0.717, -0.0299, 0.2, 0.1, 1.0, 0.354, 0.0) + + To inspect the actual symbolic expression that this function represents, + we can call the :meth:`~.doit` method on an instance. We'll use the keyword + argument ``evaluate=False`` as this will keep the expression in its + canonical form and won't simplify any constants. + + >>> fl_M.doit(evaluate=False) + 0.814*exp(-19.0519737844841*(l_M/l_M_opt + - 1.06)**2/(0.390740740740741*l_M/l_M_opt + 1)**2) + + 0.433*exp(-12.5*(l_M/l_M_opt - 0.717)**2/(l_M/l_M_opt - 0.1495)**2) + + 0.1*exp(-3.98991349867535*(l_M/l_M_opt - 1.0)**2) + + The function can also be differentiated. We'll differentiate with respect + to l_M using the ``diff`` method on an instance with the single positional + argument ``l_M``. + + >>> fl_M.diff(l_M) + ((-0.79798269973507*l_M/l_M_opt + + 0.79798269973507)*exp(-3.98991349867535*(l_M/l_M_opt - 1.0)**2) + + (10.825*(-l_M/l_M_opt + 0.717)/(l_M/l_M_opt - 0.1495)**2 + + 10.825*(l_M/l_M_opt - 0.717)**2/(l_M/l_M_opt + - 0.1495)**3)*exp(-12.5*(l_M/l_M_opt - 0.717)**2/(l_M/l_M_opt - 0.1495)**2) + + (31.0166133211401*(-l_M/l_M_opt + 1.06)/(0.390740740740741*l_M/l_M_opt + + 1)**2 + 13.6174190361677*(0.943396226415094*l_M/l_M_opt + - 1)**2/(0.390740740740741*l_M/l_M_opt + + 1)**3)*exp(-21.4067977442463*(0.943396226415094*l_M/l_M_opt + - 1)**2/(0.390740740740741*l_M/l_M_opt + 1)**2))/l_M_opt + + References + ========== + + .. [1] De Groote, F., Kinney, A. L., Rao, A. V., & Fregly, B. J., Evaluation + of direct collocation optimal control problem formulations for + solving the muscle redundancy problem, Annals of biomedical + engineering, 44(10), (2016) pp. 2922-2936 + + """ + + @classmethod + def with_defaults(cls, l_M_tilde): + r"""Recommended constructor that will use the published constants. + + Explanation + =========== + + Returns a new instance of the inverse muscle fiber act force-length + function using the four constant values specified in the original + publication. + + These have the values: + + $c0 = 0.814$ + $c1 = 1.06$ + $c2 = 0.162$ + $c3 = 0.0633$ + $c4 = 0.433$ + $c5 = 0.717$ + $c6 = -0.0299$ + $c7 = 0.2$ + $c8 = 0.1$ + $c9 = 1.0$ + $c10 = 0.354$ + $c11 = 0.0$ + + Parameters + ========== + + fl_M_act : Any (sympifiable) + Normalized passive muscle fiber force as a function of muscle fiber + length. + + """ + c0 = Float('0.814') + c1 = Float('1.06') + c2 = Float('0.162') + c3 = Float('0.0633') + c4 = Float('0.433') + c5 = Float('0.717') + c6 = Float('-0.0299') + c7 = Float('0.2') + c8 = Float('0.1') + c9 = Float('1.0') + c10 = Float('0.354') + c11 = Float('0.0') + return cls(l_M_tilde, c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11) + + @classmethod + def eval(cls, l_M_tilde, c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11): + """Evaluation of basic inputs. + + Parameters + ========== + + l_M_tilde : Any (sympifiable) + Normalized muscle fiber length. + c0 : Any (sympifiable) + The first constant in the characteristic equation. The published + value is ``0.814``. + c1 : Any (sympifiable) + The second constant in the characteristic equation. The published + value is ``1.06``. + c2 : Any (sympifiable) + The third constant in the characteristic equation. The published + value is ``0.162``. + c3 : Any (sympifiable) + The fourth constant in the characteristic equation. The published + value is ``0.0633``. + c4 : Any (sympifiable) + The fifth constant in the characteristic equation. The published + value is ``0.433``. + c5 : Any (sympifiable) + The sixth constant in the characteristic equation. The published + value is ``0.717``. + c6 : Any (sympifiable) + The seventh constant in the characteristic equation. The published + value is ``-0.0299``. + c7 : Any (sympifiable) + The eighth constant in the characteristic equation. The published + value is ``0.2``. + c8 : Any (sympifiable) + The ninth constant in the characteristic equation. The published + value is ``0.1``. + c9 : Any (sympifiable) + The tenth constant in the characteristic equation. The published + value is ``1.0``. + c10 : Any (sympifiable) + The eleventh constant in the characteristic equation. The published + value is ``0.354``. + c11 : Any (sympifiable) + The tweflth constant in the characteristic equation. The published + value is ``0.0``. + + """ + pass + + def _eval_evalf(self, prec): + """Evaluate the expression numerically using ``evalf``.""" + return self.doit(deep=False, evaluate=False)._eval_evalf(prec) + + def doit(self, deep=True, evaluate=True, **hints): + """Evaluate the expression defining the function. + + Parameters + ========== + + deep : bool + Whether ``doit`` should be recursively called. Default is ``True``. + evaluate : bool. + Whether the SymPy expression should be evaluated as it is + constructed. If ``False``, then no constant folding will be + conducted which will leave the expression in a more numerically- + stable for values of ``l_M_tilde`` that correspond to a sensible + operating range for a musculotendon. Default is ``True``. + **kwargs : dict[str, Any] + Additional keyword argument pairs to be recursively passed to + ``doit``. + + """ + l_M_tilde, *constants = self.args + if deep: + hints['evaluate'] = evaluate + l_M_tilde = l_M_tilde.doit(deep=deep, **hints) + constants = [c.doit(deep=deep, **hints) for c in constants] + c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11 = constants + + if evaluate: + return ( + c0*exp(-(((l_M_tilde - c1)/(c2 + c3*l_M_tilde))**2)/2) + + c4*exp(-(((l_M_tilde - c5)/(c6 + c7*l_M_tilde))**2)/2) + + c8*exp(-(((l_M_tilde - c9)/(c10 + c11*l_M_tilde))**2)/2) + ) + + return ( + c0*exp(-((UnevaluatedExpr(l_M_tilde - c1)/(c2 + c3*l_M_tilde))**2)/2) + + c4*exp(-((UnevaluatedExpr(l_M_tilde - c5)/(c6 + c7*l_M_tilde))**2)/2) + + c8*exp(-((UnevaluatedExpr(l_M_tilde - c9)/(c10 + c11*l_M_tilde))**2)/2) + ) + + def fdiff(self, argindex=1): + """Derivative of the function with respect to a single argument. + + Parameters + ========== + + argindex : int + The index of the function's arguments with respect to which the + derivative should be taken. Argument indexes start at ``1``. + Default is ``1``. + + """ + l_M_tilde, c0, c1, c2, c3, c4, c5, c6, c7, c8, c9, c10, c11 = self.args + if argindex == 1: + return ( + c0*( + c3*(l_M_tilde - c1)**2/(c2 + c3*l_M_tilde)**3 + + (c1 - l_M_tilde)/((c2 + c3*l_M_tilde)**2) + )*exp(-(l_M_tilde - c1)**2/(2*(c2 + c3*l_M_tilde)**2)) + + c4*( + c7*(l_M_tilde - c5)**2/(c6 + c7*l_M_tilde)**3 + + (c5 - l_M_tilde)/((c6 + c7*l_M_tilde)**2) + )*exp(-(l_M_tilde - c5)**2/(2*(c6 + c7*l_M_tilde)**2)) + + c8*( + c11*(l_M_tilde - c9)**2/(c10 + c11*l_M_tilde)**3 + + (c9 - l_M_tilde)/((c10 + c11*l_M_tilde)**2) + )*exp(-(l_M_tilde - c9)**2/(2*(c10 + c11*l_M_tilde)**2)) + ) + elif argindex == 2: + return exp(-(l_M_tilde - c1)**2/(2*(c2 + c3*l_M_tilde)**2)) + elif argindex == 3: + return ( + c0*(l_M_tilde - c1)/(c2 + c3*l_M_tilde)**2 + *exp(-(l_M_tilde - c1)**2 /(2*(c2 + c3*l_M_tilde)**2)) + ) + elif argindex == 4: + return ( + c0*(l_M_tilde - c1)**2/(c2 + c3*l_M_tilde)**3 + *exp(-(l_M_tilde - c1)**2/(2*(c2 + c3*l_M_tilde)**2)) + ) + elif argindex == 5: + return ( + c0*l_M_tilde*(l_M_tilde - c1)**2/(c2 + c3*l_M_tilde)**3 + *exp(-(l_M_tilde - c1)**2/(2*(c2 + c3*l_M_tilde)**2)) + ) + elif argindex == 6: + return exp(-(l_M_tilde - c5)**2/(2*(c6 + c7*l_M_tilde)**2)) + elif argindex == 7: + return ( + c4*(l_M_tilde - c5)/(c6 + c7*l_M_tilde)**2 + *exp(-(l_M_tilde - c5)**2 /(2*(c6 + c7*l_M_tilde)**2)) + ) + elif argindex == 8: + return ( + c4*(l_M_tilde - c5)**2/(c6 + c7*l_M_tilde)**3 + *exp(-(l_M_tilde - c5)**2/(2*(c6 + c7*l_M_tilde)**2)) + ) + elif argindex == 9: + return ( + c4*l_M_tilde*(l_M_tilde - c5)**2/(c6 + c7*l_M_tilde)**3 + *exp(-(l_M_tilde - c5)**2/(2*(c6 + c7*l_M_tilde)**2)) + ) + elif argindex == 10: + return exp(-(l_M_tilde - c9)**2/(2*(c10 + c11*l_M_tilde)**2)) + elif argindex == 11: + return ( + c8*(l_M_tilde - c9)/(c10 + c11*l_M_tilde)**2 + *exp(-(l_M_tilde - c9)**2 /(2*(c10 + c11*l_M_tilde)**2)) + ) + elif argindex == 12: + return ( + c8*(l_M_tilde - c9)**2/(c10 + c11*l_M_tilde)**3 + *exp(-(l_M_tilde - c9)**2/(2*(c10 + c11*l_M_tilde)**2)) + ) + elif argindex == 13: + return ( + c8*l_M_tilde*(l_M_tilde - c9)**2/(c10 + c11*l_M_tilde)**3 + *exp(-(l_M_tilde - c9)**2/(2*(c10 + c11*l_M_tilde)**2)) + ) + + raise ArgumentIndexError(self, argindex) + + def _latex(self, printer): + """Print a LaTeX representation of the function defining the curve. + + Parameters + ========== + + printer : Printer + The printer to be used to print the LaTeX string representation. + + """ + l_M_tilde = self.args[0] + _l_M_tilde = printer._print(l_M_tilde) + return r'\operatorname{fl}^M_{act} \left( %s \right)' % _l_M_tilde + + +class FiberForceVelocityDeGroote2016(CharacteristicCurveFunction): + r"""Muscle fiber force-velocity curve based on De Groote et al., 2016 [1]_. + + Explanation + =========== + + Gives the normalized muscle fiber force produced as a function of + normalized tendon velocity. + + The function is defined by the equation: + + $fv^M = c_0 \log{\left(c_1 \tilde{v}_m + c_2\right) + \sqrt{\left(c_1 \tilde{v}_m + c_2\right)^2 + 1}} + c_3$ + + with constant values of $c_0 = -0.318$, $c_1 = -8.149$, $c_2 = -0.374$, and + $c_3 = 0.886$. + + While it is possible to change the constant values, these were carefully + selected in the original publication to give the characteristic curve + specific and required properties. For example, the function produces a + normalized muscle fiber force of 1 when the muscle fibers are contracting + isometrically (they have an extension rate of 0). + + Examples + ======== + + The preferred way to instantiate :class:`FiberForceVelocityDeGroote2016` is using + the :meth:`~.with_defaults` constructor because this will automatically populate + the constants within the characteristic curve equation with the floating + point values from the original publication. This constructor takes a single + argument corresponding to normalized muscle fiber extension velocity. We'll + create a :class:`~.Symbol` called ``v_M_tilde`` to represent this. + + >>> from sympy import Symbol + >>> from sympy.physics.biomechanics import FiberForceVelocityDeGroote2016 + >>> v_M_tilde = Symbol('v_M_tilde') + >>> fv_M = FiberForceVelocityDeGroote2016.with_defaults(v_M_tilde) + >>> fv_M + FiberForceVelocityDeGroote2016(v_M_tilde, -0.318, -8.149, -0.374, 0.886) + + It's also possible to populate the four constants with your own values too. + + >>> from sympy import symbols + >>> c0, c1, c2, c3 = symbols('c0 c1 c2 c3') + >>> fv_M = FiberForceVelocityDeGroote2016(v_M_tilde, c0, c1, c2, c3) + >>> fv_M + FiberForceVelocityDeGroote2016(v_M_tilde, c0, c1, c2, c3) + + You don't just have to use symbols as the arguments, it's also possible to + use expressions. Let's create a new pair of symbols, ``v_M`` and + ``v_M_max``, representing muscle fiber extension velocity and maximum + muscle fiber extension velocity respectively. We can then represent + ``v_M_tilde`` as an expression, the ratio of these. + + >>> v_M, v_M_max = symbols('v_M v_M_max') + >>> v_M_tilde = v_M/v_M_max + >>> fv_M = FiberForceVelocityDeGroote2016.with_defaults(v_M_tilde) + >>> fv_M + FiberForceVelocityDeGroote2016(v_M/v_M_max, -0.318, -8.149, -0.374, 0.886) + + To inspect the actual symbolic expression that this function represents, + we can call the :meth:`~.doit` method on an instance. We'll use the keyword + argument ``evaluate=False`` as this will keep the expression in its + canonical form and won't simplify any constants. + + >>> fv_M.doit(evaluate=False) + 0.886 - 0.318*log(-8.149*v_M/v_M_max - 0.374 + sqrt(1 + (-8.149*v_M/v_M_max + - 0.374)**2)) + + The function can also be differentiated. We'll differentiate with respect + to v_M using the ``diff`` method on an instance with the single positional + argument ``v_M``. + + >>> fv_M.diff(v_M) + 2.591382*(1 + (-8.149*v_M/v_M_max - 0.374)**2)**(-1/2)/v_M_max + + References + ========== + + .. [1] De Groote, F., Kinney, A. L., Rao, A. V., & Fregly, B. J., Evaluation + of direct collocation optimal control problem formulations for + solving the muscle redundancy problem, Annals of biomedical + engineering, 44(10), (2016) pp. 2922-2936 + + """ + + @classmethod + def with_defaults(cls, v_M_tilde): + r"""Recommended constructor that will use the published constants. + + Explanation + =========== + + Returns a new instance of the muscle fiber force-velocity function + using the four constant values specified in the original publication. + + These have the values: + + $c_0 = -0.318$ + $c_1 = -8.149$ + $c_2 = -0.374$ + $c_3 = 0.886$ + + Parameters + ========== + + v_M_tilde : Any (sympifiable) + Normalized muscle fiber extension velocity. + + """ + c0 = Float('-0.318') + c1 = Float('-8.149') + c2 = Float('-0.374') + c3 = Float('0.886') + return cls(v_M_tilde, c0, c1, c2, c3) + + @classmethod + def eval(cls, v_M_tilde, c0, c1, c2, c3): + """Evaluation of basic inputs. + + Parameters + ========== + + v_M_tilde : Any (sympifiable) + Normalized muscle fiber extension velocity. + c0 : Any (sympifiable) + The first constant in the characteristic equation. The published + value is ``-0.318``. + c1 : Any (sympifiable) + The second constant in the characteristic equation. The published + value is ``-8.149``. + c2 : Any (sympifiable) + The third constant in the characteristic equation. The published + value is ``-0.374``. + c3 : Any (sympifiable) + The fourth constant in the characteristic equation. The published + value is ``0.886``. + + """ + pass + + def _eval_evalf(self, prec): + """Evaluate the expression numerically using ``evalf``.""" + return self.doit(deep=False, evaluate=False)._eval_evalf(prec) + + def doit(self, deep=True, evaluate=True, **hints): + """Evaluate the expression defining the function. + + Parameters + ========== + + deep : bool + Whether ``doit`` should be recursively called. Default is ``True``. + evaluate : bool. + Whether the SymPy expression should be evaluated as it is + constructed. If ``False``, then no constant folding will be + conducted which will leave the expression in a more numerically- + stable for values of ``v_M_tilde`` that correspond to a sensible + operating range for a musculotendon. Default is ``True``. + **kwargs : dict[str, Any] + Additional keyword argument pairs to be recursively passed to + ``doit``. + + """ + v_M_tilde, *constants = self.args + if deep: + hints['evaluate'] = evaluate + v_M_tilde = v_M_tilde.doit(deep=deep, **hints) + c0, c1, c2, c3 = [c.doit(deep=deep, **hints) for c in constants] + else: + c0, c1, c2, c3 = constants + + if evaluate: + return c0*log(c1*v_M_tilde + c2 + sqrt((c1*v_M_tilde + c2)**2 + 1)) + c3 + + return c0*log(c1*v_M_tilde + c2 + sqrt(UnevaluatedExpr(c1*v_M_tilde + c2)**2 + 1)) + c3 + + def fdiff(self, argindex=1): + """Derivative of the function with respect to a single argument. + + Parameters + ========== + + argindex : int + The index of the function's arguments with respect to which the + derivative should be taken. Argument indexes start at ``1``. + Default is ``1``. + + """ + v_M_tilde, c0, c1, c2, c3 = self.args + if argindex == 1: + return c0*c1/sqrt(UnevaluatedExpr(c1*v_M_tilde + c2)**2 + 1) + elif argindex == 2: + return log( + c1*v_M_tilde + c2 + + sqrt(UnevaluatedExpr(c1*v_M_tilde + c2)**2 + 1) + ) + elif argindex == 3: + return c0*v_M_tilde/sqrt(UnevaluatedExpr(c1*v_M_tilde + c2)**2 + 1) + elif argindex == 4: + return c0/sqrt(UnevaluatedExpr(c1*v_M_tilde + c2)**2 + 1) + elif argindex == 5: + return Integer(1) + + raise ArgumentIndexError(self, argindex) + + def inverse(self, argindex=1): + """Inverse function. + + Parameters + ========== + + argindex : int + Value to start indexing the arguments at. Default is ``1``. + + """ + return FiberForceVelocityInverseDeGroote2016 + + def _latex(self, printer): + """Print a LaTeX representation of the function defining the curve. + + Parameters + ========== + + printer : Printer + The printer to be used to print the LaTeX string representation. + + """ + v_M_tilde = self.args[0] + _v_M_tilde = printer._print(v_M_tilde) + return r'\operatorname{fv}^M \left( %s \right)' % _v_M_tilde + + +class FiberForceVelocityInverseDeGroote2016(CharacteristicCurveFunction): + r"""Inverse muscle fiber force-velocity curve based on De Groote et al., + 2016 [1]_. + + Explanation + =========== + + Gives the normalized muscle fiber velocity that produces a specific + normalized muscle fiber force. + + The function is defined by the equation: + + ${fv^M}^{-1} = \frac{\sinh{\frac{fv^M - c_3}{c_0}} - c_2}{c_1}$ + + with constant values of $c_0 = -0.318$, $c_1 = -8.149$, $c_2 = -0.374$, and + $c_3 = 0.886$. This function is the exact analytical inverse of the related + muscle fiber force-velocity curve ``FiberForceVelocityDeGroote2016``. + + While it is possible to change the constant values, these were carefully + selected in the original publication to give the characteristic curve + specific and required properties. For example, the function produces a + normalized muscle fiber force of 1 when the muscle fibers are contracting + isometrically (they have an extension rate of 0). + + Examples + ======== + + The preferred way to instantiate :class:`FiberForceVelocityInverseDeGroote2016` + is using the :meth:`~.with_defaults` constructor because this will automatically + populate the constants within the characteristic curve equation with the + floating point values from the original publication. This constructor takes + a single argument corresponding to normalized muscle fiber force-velocity + component of the muscle fiber force. We'll create a :class:`~.Symbol` called + ``fv_M`` to represent this. + + >>> from sympy import Symbol + >>> from sympy.physics.biomechanics import FiberForceVelocityInverseDeGroote2016 + >>> fv_M = Symbol('fv_M') + >>> v_M_tilde = FiberForceVelocityInverseDeGroote2016.with_defaults(fv_M) + >>> v_M_tilde + FiberForceVelocityInverseDeGroote2016(fv_M, -0.318, -8.149, -0.374, 0.886) + + It's also possible to populate the four constants with your own values too. + + >>> from sympy import symbols + >>> c0, c1, c2, c3 = symbols('c0 c1 c2 c3') + >>> v_M_tilde = FiberForceVelocityInverseDeGroote2016(fv_M, c0, c1, c2, c3) + >>> v_M_tilde + FiberForceVelocityInverseDeGroote2016(fv_M, c0, c1, c2, c3) + + To inspect the actual symbolic expression that this function represents, + we can call the :meth:`~.doit` method on an instance. We'll use the keyword + argument ``evaluate=False`` as this will keep the expression in its + canonical form and won't simplify any constants. + + >>> v_M_tilde.doit(evaluate=False) + (-c2 + sinh((-c3 + fv_M)/c0))/c1 + + The function can also be differentiated. We'll differentiate with respect + to fv_M using the ``diff`` method on an instance with the single positional + argument ``fv_M``. + + >>> v_M_tilde.diff(fv_M) + cosh((-c3 + fv_M)/c0)/(c0*c1) + + References + ========== + + .. [1] De Groote, F., Kinney, A. L., Rao, A. V., & Fregly, B. J., Evaluation + of direct collocation optimal control problem formulations for + solving the muscle redundancy problem, Annals of biomedical + engineering, 44(10), (2016) pp. 2922-2936 + + """ + + @classmethod + def with_defaults(cls, fv_M): + r"""Recommended constructor that will use the published constants. + + Explanation + =========== + + Returns a new instance of the inverse muscle fiber force-velocity + function using the four constant values specified in the original + publication. + + These have the values: + + $c_0 = -0.318$ + $c_1 = -8.149$ + $c_2 = -0.374$ + $c_3 = 0.886$ + + Parameters + ========== + + fv_M : Any (sympifiable) + Normalized muscle fiber extension velocity. + + """ + c0 = Float('-0.318') + c1 = Float('-8.149') + c2 = Float('-0.374') + c3 = Float('0.886') + return cls(fv_M, c0, c1, c2, c3) + + @classmethod + def eval(cls, fv_M, c0, c1, c2, c3): + """Evaluation of basic inputs. + + Parameters + ========== + + fv_M : Any (sympifiable) + Normalized muscle fiber force as a function of muscle fiber + extension velocity. + c0 : Any (sympifiable) + The first constant in the characteristic equation. The published + value is ``-0.318``. + c1 : Any (sympifiable) + The second constant in the characteristic equation. The published + value is ``-8.149``. + c2 : Any (sympifiable) + The third constant in the characteristic equation. The published + value is ``-0.374``. + c3 : Any (sympifiable) + The fourth constant in the characteristic equation. The published + value is ``0.886``. + + """ + pass + + def _eval_evalf(self, prec): + """Evaluate the expression numerically using ``evalf``.""" + return self.doit(deep=False, evaluate=False)._eval_evalf(prec) + + def doit(self, deep=True, evaluate=True, **hints): + """Evaluate the expression defining the function. + + Parameters + ========== + + deep : bool + Whether ``doit`` should be recursively called. Default is ``True``. + evaluate : bool. + Whether the SymPy expression should be evaluated as it is + constructed. If ``False``, then no constant folding will be + conducted which will leave the expression in a more numerically- + stable for values of ``fv_M`` that correspond to a sensible + operating range for a musculotendon. Default is ``True``. + **kwargs : dict[str, Any] + Additional keyword argument pairs to be recursively passed to + ``doit``. + + """ + fv_M, *constants = self.args + if deep: + hints['evaluate'] = evaluate + fv_M = fv_M.doit(deep=deep, **hints) + c0, c1, c2, c3 = [c.doit(deep=deep, **hints) for c in constants] + else: + c0, c1, c2, c3 = constants + + if evaluate: + return (sinh((fv_M - c3)/c0) - c2)/c1 + + return (sinh(UnevaluatedExpr(fv_M - c3)/c0) - c2)/c1 + + def fdiff(self, argindex=1): + """Derivative of the function with respect to a single argument. + + Parameters + ========== + + argindex : int + The index of the function's arguments with respect to which the + derivative should be taken. Argument indexes start at ``1``. + Default is ``1``. + + """ + fv_M, c0, c1, c2, c3 = self.args + if argindex == 1: + return cosh((fv_M - c3)/c0)/(c0*c1) + elif argindex == 2: + return (c3 - fv_M)*cosh((fv_M - c3)/c0)/(c0**2*c1) + elif argindex == 3: + return (c2 - sinh((fv_M - c3)/c0))/c1**2 + elif argindex == 4: + return -1/c1 + elif argindex == 5: + return -cosh((fv_M - c3)/c0)/(c0*c1) + + raise ArgumentIndexError(self, argindex) + + def inverse(self, argindex=1): + """Inverse function. + + Parameters + ========== + + argindex : int + Value to start indexing the arguments at. Default is ``1``. + + """ + return FiberForceVelocityDeGroote2016 + + def _latex(self, printer): + """Print a LaTeX representation of the function defining the curve. + + Parameters + ========== + + printer : Printer + The printer to be used to print the LaTeX string representation. + + """ + fv_M = self.args[0] + _fv_M = printer._print(fv_M) + return r'\left( \operatorname{fv}^M \right)^{-1} \left( %s \right)' % _fv_M + + +@dataclass(frozen=True) +class CharacteristicCurveCollection: + """Simple data container to group together related characteristic curves.""" + tendon_force_length: CharacteristicCurveFunction + tendon_force_length_inverse: CharacteristicCurveFunction + fiber_force_length_passive: CharacteristicCurveFunction + fiber_force_length_passive_inverse: CharacteristicCurveFunction + fiber_force_length_active: CharacteristicCurveFunction + fiber_force_velocity: CharacteristicCurveFunction + fiber_force_velocity_inverse: CharacteristicCurveFunction + + def __iter__(self): + """Iterator support for ``CharacteristicCurveCollection``.""" + yield self.tendon_force_length + yield self.tendon_force_length_inverse + yield self.fiber_force_length_passive + yield self.fiber_force_length_passive_inverse + yield self.fiber_force_length_active + yield self.fiber_force_velocity + yield self.fiber_force_velocity_inverse diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/biomechanics/tests/__init__.py b/wemm/lib/python3.10/site-packages/sympy/physics/biomechanics/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/biomechanics/tests/__pycache__/test_activation.cpython-310.pyc b/wemm/lib/python3.10/site-packages/sympy/physics/biomechanics/tests/__pycache__/test_activation.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..bbbbe638757bd64da24b24d8eed91d72d000b15f Binary files /dev/null and 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bode_magnitude_plot, + bode_phase_plot, bode_plot) + +__all__ = ['TransferFunction', 'Series', 'MIMOSeries', 'Parallel', + 'MIMOParallel', 'Feedback', 'MIMOFeedback', 'TransferFunctionMatrix', 'StateSpace', + 'gbt', 'bilinear', 'forward_diff', 'backward_diff', 'phase_margin', 'gain_margin', + 'pole_zero_numerical_data', 'pole_zero_plot', 'step_response_numerical_data', + 'step_response_plot', 'impulse_response_numerical_data', 'impulse_response_plot', + 'ramp_response_numerical_data', 'ramp_response_plot', + 'bode_magnitude_numerical_data', 'bode_phase_numerical_data', + 'bode_magnitude_plot', 'bode_phase_plot', 'bode_plot'] diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/control/__pycache__/__init__.cpython-310.pyc b/wemm/lib/python3.10/site-packages/sympy/physics/control/__pycache__/__init__.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..27e9b574e438070ca9592fbc4e4c17741e39ddfd Binary files /dev/null and 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sympy.functions.elementary.exponential import (exp, log) +from sympy.polys.partfrac import apart +from sympy.core.symbol import Dummy +from sympy.external import import_module +from sympy.functions import arg, Abs +from sympy.integrals.laplace import _fast_inverse_laplace +from sympy.physics.control.lti import SISOLinearTimeInvariant +from sympy.plotting.series import LineOver1DRangeSeries +from sympy.polys.polytools import Poly +from sympy.printing.latex import latex + +__all__ = ['pole_zero_numerical_data', 'pole_zero_plot', + 'step_response_numerical_data', 'step_response_plot', + 'impulse_response_numerical_data', 'impulse_response_plot', + 'ramp_response_numerical_data', 'ramp_response_plot', + 'bode_magnitude_numerical_data', 'bode_phase_numerical_data', + 'bode_magnitude_plot', 'bode_phase_plot', 'bode_plot'] + +matplotlib = import_module( + 'matplotlib', import_kwargs={'fromlist': ['pyplot']}, + catch=(RuntimeError,)) + +numpy = import_module('numpy') + +if matplotlib: + plt = matplotlib.pyplot + +if numpy: + np = numpy # Matplotlib already has numpy as a compulsory dependency. No need to install it separately. + + +def _check_system(system): + """Function to check whether the dynamical system passed for plots is + compatible or not.""" + if not isinstance(system, SISOLinearTimeInvariant): + raise NotImplementedError("Only SISO LTI systems are currently supported.") + sys = system.to_expr() + len_free_symbols = len(sys.free_symbols) + if len_free_symbols > 1: + raise ValueError("Extra degree of freedom found. Make sure" + " that there are no free symbols in the dynamical system other" + " than the variable of Laplace transform.") + if sys.has(exp): + # Should test that exp is not part of a constant, in which case + # no exception is required, compare exp(s) with s*exp(1) + raise NotImplementedError("Time delay terms are not supported.") + + +def pole_zero_numerical_data(system): + """ + Returns the numerical data of poles and zeros of the system. + It is internally used by ``pole_zero_plot`` to get the data + for plotting poles and zeros. Users can use this data to further + analyse the dynamics of the system or plot using a different + backend/plotting-module. + + Parameters + ========== + + system : SISOLinearTimeInvariant + The system for which the pole-zero data is to be computed. + + Returns + ======= + + tuple : (zeros, poles) + zeros = Zeros of the system. NumPy array of complex numbers. + poles = Poles of the system. NumPy array of complex numbers. + + Raises + ====== + + NotImplementedError + When a SISO LTI system is not passed. + + When time delay terms are present in the system. + + ValueError + When more than one free symbol is present in the system. + The only variable in the transfer function should be + the variable of the Laplace transform. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import pole_zero_numerical_data + >>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s) + >>> pole_zero_numerical_data(tf1) # doctest: +SKIP + ([-0.+1.j 0.-1.j], [-2. +0.j -0.5+0.8660254j -0.5-0.8660254j -1. +0.j ]) + + See Also + ======== + + pole_zero_plot + + """ + _check_system(system) + system = system.doit() # Get the equivalent TransferFunction object. + + num_poly = Poly(system.num, system.var).all_coeffs() + den_poly = Poly(system.den, system.var).all_coeffs() + + num_poly = np.array(num_poly, dtype=np.complex128) + den_poly = np.array(den_poly, dtype=np.complex128) + + zeros = np.roots(num_poly) + poles = np.roots(den_poly) + + return zeros, poles + + +def pole_zero_plot(system, pole_color='blue', pole_markersize=10, + zero_color='orange', zero_markersize=7, grid=True, show_axes=True, + show=True, **kwargs): + r""" + Returns the Pole-Zero plot (also known as PZ Plot or PZ Map) of a system. + + A Pole-Zero plot is a graphical representation of a system's poles and + zeros. It is plotted on a complex plane, with circular markers representing + the system's zeros and 'x' shaped markers representing the system's poles. + + Parameters + ========== + + system : SISOLinearTimeInvariant type systems + The system for which the pole-zero plot is to be computed. + pole_color : str, tuple, optional + The color of the pole points on the plot. Default color + is blue. The color can be provided as a matplotlib color string, + or a 3-tuple of floats each in the 0-1 range. + pole_markersize : Number, optional + The size of the markers used to mark the poles in the plot. + Default pole markersize is 10. + zero_color : str, tuple, optional + The color of the zero points on the plot. Default color + is orange. The color can be provided as a matplotlib color string, + or a 3-tuple of floats each in the 0-1 range. + zero_markersize : Number, optional + The size of the markers used to mark the zeros in the plot. + Default zero markersize is 7. + grid : boolean, optional + If ``True``, the plot will have a grid. Defaults to True. + show_axes : boolean, optional + If ``True``, the coordinate axes will be shown. Defaults to False. + show : boolean, optional + If ``True``, the plot will be displayed otherwise + the equivalent matplotlib ``plot`` object will be returned. + Defaults to True. + + Examples + ======== + + .. plot:: + :context: close-figs + :format: doctest + :include-source: True + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import pole_zero_plot + >>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s) + >>> pole_zero_plot(tf1) # doctest: +SKIP + + See Also + ======== + + pole_zero_numerical_data + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Pole%E2%80%93zero_plot + + """ + zeros, poles = pole_zero_numerical_data(system) + + zero_real = np.real(zeros) + zero_imag = np.imag(zeros) + + pole_real = np.real(poles) + pole_imag = np.imag(poles) + + plt.plot(pole_real, pole_imag, 'x', mfc='none', + markersize=pole_markersize, color=pole_color) + plt.plot(zero_real, zero_imag, 'o', markersize=zero_markersize, + color=zero_color) + plt.xlabel('Real Axis') + plt.ylabel('Imaginary Axis') + plt.title(f'Poles and Zeros of ${latex(system)}$', pad=20) + + if grid: + plt.grid() + if show_axes: + plt.axhline(0, color='black') + plt.axvline(0, color='black') + if show: + plt.show() + return + + return plt + + +def step_response_numerical_data(system, prec=8, lower_limit=0, + upper_limit=10, **kwargs): + """ + Returns the numerical values of the points in the step response plot + of a SISO continuous-time system. By default, adaptive sampling + is used. If the user wants to instead get an uniformly + sampled response, then ``adaptive`` kwarg should be passed ``False`` + and ``n`` must be passed as additional kwargs. + Refer to the parameters of class :class:`sympy.plotting.series.LineOver1DRangeSeries` + for more details. + + Parameters + ========== + + system : SISOLinearTimeInvariant + The system for which the unit step response data is to be computed. + prec : int, optional + The decimal point precision for the point coordinate values. + Defaults to 8. + lower_limit : Number, optional + The lower limit of the plot range. Defaults to 0. + upper_limit : Number, optional + The upper limit of the plot range. Defaults to 10. + kwargs : + Additional keyword arguments are passed to the underlying + :class:`sympy.plotting.series.LineOver1DRangeSeries` class. + + Returns + ======= + + tuple : (x, y) + x = Time-axis values of the points in the step response. NumPy array. + y = Amplitude-axis values of the points in the step response. NumPy array. + + Raises + ====== + + NotImplementedError + When a SISO LTI system is not passed. + + When time delay terms are present in the system. + + ValueError + When more than one free symbol is present in the system. + The only variable in the transfer function should be + the variable of the Laplace transform. + + When ``lower_limit`` parameter is less than 0. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import step_response_numerical_data + >>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s) + >>> step_response_numerical_data(tf1) # doctest: +SKIP + ([0.0, 0.025413462339411542, 0.0484508722725343, ... , 9.670250533855183, 9.844291913708725, 10.0], + [0.0, 0.023844582399907256, 0.042894276802320226, ..., 6.828770759094287e-12, 6.456457160755703e-12]) + + See Also + ======== + + step_response_plot + + """ + if lower_limit < 0: + raise ValueError("Lower limit of time must be greater " + "than or equal to zero.") + _check_system(system) + _x = Dummy("x") + expr = system.to_expr()/(system.var) + expr = apart(expr, system.var, full=True) + _y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec) + return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit), + **kwargs).get_points() + + +def step_response_plot(system, color='b', prec=8, lower_limit=0, + upper_limit=10, show_axes=False, grid=True, show=True, **kwargs): + r""" + Returns the unit step response of a continuous-time system. It is + the response of the system when the input signal is a step function. + + Parameters + ========== + + system : SISOLinearTimeInvariant type + The LTI SISO system for which the Step Response is to be computed. + color : str, tuple, optional + The color of the line. Default is Blue. + show : boolean, optional + If ``True``, the plot will be displayed otherwise + the equivalent matplotlib ``plot`` object will be returned. + Defaults to True. + lower_limit : Number, optional + The lower limit of the plot range. Defaults to 0. + upper_limit : Number, optional + The upper limit of the plot range. Defaults to 10. + prec : int, optional + The decimal point precision for the point coordinate values. + Defaults to 8. + show_axes : boolean, optional + If ``True``, the coordinate axes will be shown. Defaults to False. + grid : boolean, optional + If ``True``, the plot will have a grid. Defaults to True. + + Examples + ======== + + .. plot:: + :context: close-figs + :format: doctest + :include-source: True + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import step_response_plot + >>> tf1 = TransferFunction(8*s**2 + 18*s + 32, s**3 + 6*s**2 + 14*s + 24, s) + >>> step_response_plot(tf1) # doctest: +SKIP + + See Also + ======== + + impulse_response_plot, ramp_response_plot + + References + ========== + + .. [1] https://www.mathworks.com/help/control/ref/lti.step.html + + """ + x, y = step_response_numerical_data(system, prec=prec, + lower_limit=lower_limit, upper_limit=upper_limit, **kwargs) + plt.plot(x, y, color=color) + plt.xlabel('Time (s)') + plt.ylabel('Amplitude') + plt.title(f'Unit Step Response of ${latex(system)}$', pad=20) + + if grid: + plt.grid() + if show_axes: + plt.axhline(0, color='black') + plt.axvline(0, color='black') + if show: + plt.show() + return + + return plt + + +def impulse_response_numerical_data(system, prec=8, lower_limit=0, + upper_limit=10, **kwargs): + """ + Returns the numerical values of the points in the impulse response plot + of a SISO continuous-time system. By default, adaptive sampling + is used. If the user wants to instead get an uniformly + sampled response, then ``adaptive`` kwarg should be passed ``False`` + and ``n`` must be passed as additional kwargs. + Refer to the parameters of class :class:`sympy.plotting.series.LineOver1DRangeSeries` + for more details. + + Parameters + ========== + + system : SISOLinearTimeInvariant + The system for which the impulse response data is to be computed. + prec : int, optional + The decimal point precision for the point coordinate values. + Defaults to 8. + lower_limit : Number, optional + The lower limit of the plot range. Defaults to 0. + upper_limit : Number, optional + The upper limit of the plot range. Defaults to 10. + kwargs : + Additional keyword arguments are passed to the underlying + :class:`sympy.plotting.series.LineOver1DRangeSeries` class. + + Returns + ======= + + tuple : (x, y) + x = Time-axis values of the points in the impulse response. NumPy array. + y = Amplitude-axis values of the points in the impulse response. NumPy array. + + Raises + ====== + + NotImplementedError + When a SISO LTI system is not passed. + + When time delay terms are present in the system. + + ValueError + When more than one free symbol is present in the system. + The only variable in the transfer function should be + the variable of the Laplace transform. + + When ``lower_limit`` parameter is less than 0. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import impulse_response_numerical_data + >>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s) + >>> impulse_response_numerical_data(tf1) # doctest: +SKIP + ([0.0, 0.06616480200395854,... , 9.854500743565858, 10.0], + [0.9999999799999999, 0.7042848373025861,...,7.170748906965121e-13, -5.1901263495547205e-12]) + + See Also + ======== + + impulse_response_plot + + """ + if lower_limit < 0: + raise ValueError("Lower limit of time must be greater " + "than or equal to zero.") + _check_system(system) + _x = Dummy("x") + expr = system.to_expr() + expr = apart(expr, system.var, full=True) + _y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec) + return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit), + **kwargs).get_points() + + +def impulse_response_plot(system, color='b', prec=8, lower_limit=0, + upper_limit=10, show_axes=False, grid=True, show=True, **kwargs): + r""" + Returns the unit impulse response (Input is the Dirac-Delta Function) of a + continuous-time system. + + Parameters + ========== + + system : SISOLinearTimeInvariant type + The LTI SISO system for which the Impulse Response is to be computed. + color : str, tuple, optional + The color of the line. Default is Blue. + show : boolean, optional + If ``True``, the plot will be displayed otherwise + the equivalent matplotlib ``plot`` object will be returned. + Defaults to True. + lower_limit : Number, optional + The lower limit of the plot range. Defaults to 0. + upper_limit : Number, optional + The upper limit of the plot range. Defaults to 10. + prec : int, optional + The decimal point precision for the point coordinate values. + Defaults to 8. + show_axes : boolean, optional + If ``True``, the coordinate axes will be shown. Defaults to False. + grid : boolean, optional + If ``True``, the plot will have a grid. Defaults to True. + + Examples + ======== + + .. plot:: + :context: close-figs + :format: doctest + :include-source: True + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import impulse_response_plot + >>> tf1 = TransferFunction(8*s**2 + 18*s + 32, s**3 + 6*s**2 + 14*s + 24, s) + >>> impulse_response_plot(tf1) # doctest: +SKIP + + See Also + ======== + + step_response_plot, ramp_response_plot + + References + ========== + + .. [1] https://www.mathworks.com/help/control/ref/dynamicsystem.impulse.html + + """ + x, y = impulse_response_numerical_data(system, prec=prec, + lower_limit=lower_limit, upper_limit=upper_limit, **kwargs) + plt.plot(x, y, color=color) + plt.xlabel('Time (s)') + plt.ylabel('Amplitude') + plt.title(f'Impulse Response of ${latex(system)}$', pad=20) + + if grid: + plt.grid() + if show_axes: + plt.axhline(0, color='black') + plt.axvline(0, color='black') + if show: + plt.show() + return + + return plt + + +def ramp_response_numerical_data(system, slope=1, prec=8, + lower_limit=0, upper_limit=10, **kwargs): + """ + Returns the numerical values of the points in the ramp response plot + of a SISO continuous-time system. By default, adaptive sampling + is used. If the user wants to instead get an uniformly + sampled response, then ``adaptive`` kwarg should be passed ``False`` + and ``n`` must be passed as additional kwargs. + Refer to the parameters of class :class:`sympy.plotting.series.LineOver1DRangeSeries` + for more details. + + Parameters + ========== + + system : SISOLinearTimeInvariant + The system for which the ramp response data is to be computed. + slope : Number, optional + The slope of the input ramp function. Defaults to 1. + prec : int, optional + The decimal point precision for the point coordinate values. + Defaults to 8. + lower_limit : Number, optional + The lower limit of the plot range. Defaults to 0. + upper_limit : Number, optional + The upper limit of the plot range. Defaults to 10. + kwargs : + Additional keyword arguments are passed to the underlying + :class:`sympy.plotting.series.LineOver1DRangeSeries` class. + + Returns + ======= + + tuple : (x, y) + x = Time-axis values of the points in the ramp response plot. NumPy array. + y = Amplitude-axis values of the points in the ramp response plot. NumPy array. + + Raises + ====== + + NotImplementedError + When a SISO LTI system is not passed. + + When time delay terms are present in the system. + + ValueError + When more than one free symbol is present in the system. + The only variable in the transfer function should be + the variable of the Laplace transform. + + When ``lower_limit`` parameter is less than 0. + + When ``slope`` is negative. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import ramp_response_numerical_data + >>> tf1 = TransferFunction(s, s**2 + 5*s + 8, s) + >>> ramp_response_numerical_data(tf1) # doctest: +SKIP + (([0.0, 0.12166980856813935,..., 9.861246379582118, 10.0], + [1.4504508011325967e-09, 0.006046440489058766,..., 0.12499999999568202, 0.12499999999661349])) + + See Also + ======== + + ramp_response_plot + + """ + if slope < 0: + raise ValueError("Slope must be greater than or equal" + " to zero.") + if lower_limit < 0: + raise ValueError("Lower limit of time must be greater " + "than or equal to zero.") + _check_system(system) + _x = Dummy("x") + expr = (slope*system.to_expr())/((system.var)**2) + expr = apart(expr, system.var, full=True) + _y = _fast_inverse_laplace(expr, system.var, _x).evalf(prec) + return LineOver1DRangeSeries(_y, (_x, lower_limit, upper_limit), + **kwargs).get_points() + + +def ramp_response_plot(system, slope=1, color='b', prec=8, lower_limit=0, + upper_limit=10, show_axes=False, grid=True, show=True, **kwargs): + r""" + Returns the ramp response of a continuous-time system. + + Ramp function is defined as the straight line + passing through origin ($f(x) = mx$). The slope of + the ramp function can be varied by the user and + the default value is 1. + + Parameters + ========== + + system : SISOLinearTimeInvariant type + The LTI SISO system for which the Ramp Response is to be computed. + slope : Number, optional + The slope of the input ramp function. Defaults to 1. + color : str, tuple, optional + The color of the line. Default is Blue. + show : boolean, optional + If ``True``, the plot will be displayed otherwise + the equivalent matplotlib ``plot`` object will be returned. + Defaults to True. + lower_limit : Number, optional + The lower limit of the plot range. Defaults to 0. + upper_limit : Number, optional + The upper limit of the plot range. Defaults to 10. + prec : int, optional + The decimal point precision for the point coordinate values. + Defaults to 8. + show_axes : boolean, optional + If ``True``, the coordinate axes will be shown. Defaults to False. + grid : boolean, optional + If ``True``, the plot will have a grid. Defaults to True. + + Examples + ======== + + .. plot:: + :context: close-figs + :format: doctest + :include-source: True + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import ramp_response_plot + >>> tf1 = TransferFunction(s, (s+4)*(s+8), s) + >>> ramp_response_plot(tf1, upper_limit=2) # doctest: +SKIP + + See Also + ======== + + step_response_plot, impulse_response_plot + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Ramp_function + + """ + x, y = ramp_response_numerical_data(system, slope=slope, prec=prec, + lower_limit=lower_limit, upper_limit=upper_limit, **kwargs) + plt.plot(x, y, color=color) + plt.xlabel('Time (s)') + plt.ylabel('Amplitude') + plt.title(f'Ramp Response of ${latex(system)}$ [Slope = {slope}]', pad=20) + + if grid: + plt.grid() + if show_axes: + plt.axhline(0, color='black') + plt.axvline(0, color='black') + if show: + plt.show() + return + + return plt + + +def bode_magnitude_numerical_data(system, initial_exp=-5, final_exp=5, freq_unit='rad/sec', **kwargs): + """ + Returns the numerical data of the Bode magnitude plot of the system. + It is internally used by ``bode_magnitude_plot`` to get the data + for plotting Bode magnitude plot. Users can use this data to further + analyse the dynamics of the system or plot using a different + backend/plotting-module. + + Parameters + ========== + + system : SISOLinearTimeInvariant + The system for which the data is to be computed. + initial_exp : Number, optional + The initial exponent of 10 of the semilog plot. Defaults to -5. + final_exp : Number, optional + The final exponent of 10 of the semilog plot. Defaults to 5. + freq_unit : string, optional + User can choose between ``'rad/sec'`` (radians/second) and ``'Hz'`` (Hertz) as frequency units. + + Returns + ======= + + tuple : (x, y) + x = x-axis values of the Bode magnitude plot. + y = y-axis values of the Bode magnitude plot. + + Raises + ====== + + NotImplementedError + When a SISO LTI system is not passed. + + When time delay terms are present in the system. + + ValueError + When more than one free symbol is present in the system. + The only variable in the transfer function should be + the variable of the Laplace transform. + + When incorrect frequency units are given as input. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import bode_magnitude_numerical_data + >>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s) + >>> bode_magnitude_numerical_data(tf1) # doctest: +SKIP + ([1e-05, 1.5148378120533502e-05,..., 68437.36188804005, 100000.0], + [-6.020599914256786, -6.0205999155219505,..., -193.4117304087953, -200.00000000260573]) + + See Also + ======== + + bode_magnitude_plot, bode_phase_numerical_data + + """ + _check_system(system) + expr = system.to_expr() + freq_units = ('rad/sec', 'Hz') + if freq_unit not in freq_units: + raise ValueError('Only "rad/sec" and "Hz" are accepted frequency units.') + + _w = Dummy("w", real=True) + if freq_unit == 'Hz': + repl = I*_w*2*pi + else: + repl = I*_w + w_expr = expr.subs({system.var: repl}) + + mag = 20*log(Abs(w_expr), 10) + + x, y = LineOver1DRangeSeries(mag, + (_w, 10**initial_exp, 10**final_exp), xscale='log', **kwargs).get_points() + + return x, y + + +def bode_magnitude_plot(system, initial_exp=-5, final_exp=5, + color='b', show_axes=False, grid=True, show=True, freq_unit='rad/sec', **kwargs): + r""" + Returns the Bode magnitude plot of a continuous-time system. + + See ``bode_plot`` for all the parameters. + """ + x, y = bode_magnitude_numerical_data(system, initial_exp=initial_exp, + final_exp=final_exp, freq_unit=freq_unit) + plt.plot(x, y, color=color, **kwargs) + plt.xscale('log') + + + plt.xlabel('Frequency (%s) [Log Scale]' % freq_unit) + plt.ylabel('Magnitude (dB)') + plt.title(f'Bode Plot (Magnitude) of ${latex(system)}$', pad=20) + + if grid: + plt.grid(True) + if show_axes: + plt.axhline(0, color='black') + plt.axvline(0, color='black') + if show: + plt.show() + return + + return plt + + +def bode_phase_numerical_data(system, initial_exp=-5, final_exp=5, freq_unit='rad/sec', phase_unit='rad', phase_unwrap = True, **kwargs): + """ + Returns the numerical data of the Bode phase plot of the system. + It is internally used by ``bode_phase_plot`` to get the data + for plotting Bode phase plot. Users can use this data to further + analyse the dynamics of the system or plot using a different + backend/plotting-module. + + Parameters + ========== + + system : SISOLinearTimeInvariant + The system for which the Bode phase plot data is to be computed. + initial_exp : Number, optional + The initial exponent of 10 of the semilog plot. Defaults to -5. + final_exp : Number, optional + The final exponent of 10 of the semilog plot. Defaults to 5. + freq_unit : string, optional + User can choose between ``'rad/sec'`` (radians/second) and '``'Hz'`` (Hertz) as frequency units. + phase_unit : string, optional + User can choose between ``'rad'`` (radians) and ``'deg'`` (degree) as phase units. + phase_unwrap : bool, optional + Set to ``True`` by default. + + Returns + ======= + + tuple : (x, y) + x = x-axis values of the Bode phase plot. + y = y-axis values of the Bode phase plot. + + Raises + ====== + + NotImplementedError + When a SISO LTI system is not passed. + + When time delay terms are present in the system. + + ValueError + When more than one free symbol is present in the system. + The only variable in the transfer function should be + the variable of the Laplace transform. + + When incorrect frequency or phase units are given as input. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import bode_phase_numerical_data + >>> tf1 = TransferFunction(s**2 + 1, s**4 + 4*s**3 + 6*s**2 + 5*s + 2, s) + >>> bode_phase_numerical_data(tf1) # doctest: +SKIP + ([1e-05, 1.4472354033813751e-05, 2.035581932165858e-05,..., 47577.3248186011, 67884.09326036123, 100000.0], + [-2.5000000000291665e-05, -3.6180885085e-05, -5.08895483066e-05,...,-3.1415085799262523, -3.14155265358979]) + + See Also + ======== + + bode_magnitude_plot, bode_phase_numerical_data + + """ + _check_system(system) + expr = system.to_expr() + freq_units = ('rad/sec', 'Hz') + phase_units = ('rad', 'deg') + if freq_unit not in freq_units: + raise ValueError('Only "rad/sec" and "Hz" are accepted frequency units.') + if phase_unit not in phase_units: + raise ValueError('Only "rad" and "deg" are accepted phase units.') + + _w = Dummy("w", real=True) + if freq_unit == 'Hz': + repl = I*_w*2*pi + else: + repl = I*_w + w_expr = expr.subs({system.var: repl}) + + if phase_unit == 'deg': + phase = arg(w_expr)*180/pi + else: + phase = arg(w_expr) + + x, y = LineOver1DRangeSeries(phase, + (_w, 10**initial_exp, 10**final_exp), xscale='log', **kwargs).get_points() + + half = None + if phase_unwrap: + if(phase_unit == 'rad'): + half = pi + elif(phase_unit == 'deg'): + half = 180 + if half: + unit = 2*half + for i in range(1, len(y)): + diff = y[i] - y[i - 1] + if diff > half: # Jump from -half to half + y[i] = (y[i] - unit) + elif diff < -half: # Jump from half to -half + y[i] = (y[i] + unit) + + return x, y + + +def bode_phase_plot(system, initial_exp=-5, final_exp=5, + color='b', show_axes=False, grid=True, show=True, freq_unit='rad/sec', phase_unit='rad', phase_unwrap=True, **kwargs): + r""" + Returns the Bode phase plot of a continuous-time system. + + See ``bode_plot`` for all the parameters. + """ + x, y = bode_phase_numerical_data(system, initial_exp=initial_exp, + final_exp=final_exp, freq_unit=freq_unit, phase_unit=phase_unit, phase_unwrap=phase_unwrap) + plt.plot(x, y, color=color, **kwargs) + plt.xscale('log') + + plt.xlabel('Frequency (%s) [Log Scale]' % freq_unit) + plt.ylabel('Phase (%s)' % phase_unit) + plt.title(f'Bode Plot (Phase) of ${latex(system)}$', pad=20) + + if grid: + plt.grid(True) + if show_axes: + plt.axhline(0, color='black') + plt.axvline(0, color='black') + if show: + plt.show() + return + + return plt + + +def bode_plot(system, initial_exp=-5, final_exp=5, + grid=True, show_axes=False, show=True, freq_unit='rad/sec', phase_unit='rad', phase_unwrap=True, **kwargs): + r""" + Returns the Bode phase and magnitude plots of a continuous-time system. + + Parameters + ========== + + system : SISOLinearTimeInvariant type + The LTI SISO system for which the Bode Plot is to be computed. + initial_exp : Number, optional + The initial exponent of 10 of the semilog plot. Defaults to -5. + final_exp : Number, optional + The final exponent of 10 of the semilog plot. Defaults to 5. + show : boolean, optional + If ``True``, the plot will be displayed otherwise + the equivalent matplotlib ``plot`` object will be returned. + Defaults to True. + prec : int, optional + The decimal point precision for the point coordinate values. + Defaults to 8. + grid : boolean, optional + If ``True``, the plot will have a grid. Defaults to True. + show_axes : boolean, optional + If ``True``, the coordinate axes will be shown. Defaults to False. + freq_unit : string, optional + User can choose between ``'rad/sec'`` (radians/second) and ``'Hz'`` (Hertz) as frequency units. + phase_unit : string, optional + User can choose between ``'rad'`` (radians) and ``'deg'`` (degree) as phase units. + + Examples + ======== + + .. plot:: + :context: close-figs + :format: doctest + :include-source: True + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy.physics.control.control_plots import bode_plot + >>> tf1 = TransferFunction(1*s**2 + 0.1*s + 7.5, 1*s**4 + 0.12*s**3 + 9*s**2, s) + >>> bode_plot(tf1, initial_exp=0.2, final_exp=0.7) # doctest: +SKIP + + See Also + ======== + + bode_magnitude_plot, bode_phase_plot + + """ + plt.subplot(211) + mag = bode_magnitude_plot(system, initial_exp=initial_exp, final_exp=final_exp, + show=False, grid=grid, show_axes=show_axes, + freq_unit=freq_unit, **kwargs) + mag.title(f'Bode Plot of ${latex(system)}$', pad=20) + mag.xlabel(None) + plt.subplot(212) + bode_phase_plot(system, initial_exp=initial_exp, final_exp=final_exp, + show=False, grid=grid, show_axes=show_axes, freq_unit=freq_unit, phase_unit=phase_unit, phase_unwrap=phase_unwrap, **kwargs).title(None) + + if show: + plt.show() + return + + return plt diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/control/lti.py b/wemm/lib/python3.10/site-packages/sympy/physics/control/lti.py new file mode 100644 index 0000000000000000000000000000000000000000..54349e50e087077435ed2fcdf01c2aed23f0edea --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/control/lti.py @@ -0,0 +1,4304 @@ +from typing import Type +from sympy import Interval, numer, Rational, solveset +from sympy.core.add import Add +from sympy.core.basic import Basic +from sympy.core.containers import Tuple +from sympy.core.evalf import EvalfMixin +from sympy.core.expr import Expr +from sympy.core.function import expand +from sympy.core.logic import fuzzy_and +from sympy.core.mul import Mul +from sympy.core.numbers import I, pi, oo +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.core.symbol import Dummy, Symbol +from sympy.functions import Abs +from sympy.core.sympify import sympify, _sympify +from sympy.matrices import Matrix, ImmutableMatrix, ImmutableDenseMatrix, eye, ShapeError, zeros +from sympy.functions.elementary.exponential import (exp, log) +from sympy.matrices.expressions import MatMul, MatAdd +from sympy.polys import Poly, rootof +from sympy.polys.polyroots import roots +from sympy.polys.polytools import (cancel, degree) +from sympy.series import limit +from sympy.utilities.misc import filldedent + +from mpmath.libmp.libmpf import prec_to_dps + +__all__ = ['TransferFunction', 'Series', 'MIMOSeries', 'Parallel', 'MIMOParallel', + 'Feedback', 'MIMOFeedback', 'TransferFunctionMatrix', 'StateSpace', 'gbt', 'bilinear', 'forward_diff', 'backward_diff', + 'phase_margin', 'gain_margin'] + +def _roots(poly, var): + """ like roots, but works on higher-order polynomials. """ + r = roots(poly, var, multiple=True) + n = degree(poly) + if len(r) != n: + r = [rootof(poly, var, k) for k in range(n)] + return r + +def gbt(tf, sample_per, alpha): + r""" + Returns falling coefficients of H(z) from numerator and denominator. + + Explanation + =========== + + Where H(z) is the corresponding discretized transfer function, + discretized with the generalised bilinear transformation method. + H(z) is obtained from the continuous transfer function H(s) + by substituting $s(z) = \frac{z-1}{T(\alpha z + (1-\alpha))}$ into H(s), where T is the + sample period. + Coefficients are falling, i.e. $H(z) = \frac{az+b}{cz+d}$ is returned + as [a, b], [c, d]. + + Examples + ======== + + >>> from sympy.physics.control.lti import TransferFunction, gbt + >>> from sympy.abc import s, L, R, T + + >>> tf = TransferFunction(1, s*L + R, s) + >>> numZ, denZ = gbt(tf, T, 0.5) + >>> numZ + [T/(2*(L + R*T/2)), T/(2*(L + R*T/2))] + >>> denZ + [1, (-L + R*T/2)/(L + R*T/2)] + + >>> numZ, denZ = gbt(tf, T, 0) + >>> numZ + [T/L] + >>> denZ + [1, (-L + R*T)/L] + + >>> numZ, denZ = gbt(tf, T, 1) + >>> numZ + [T/(L + R*T), 0] + >>> denZ + [1, -L/(L + R*T)] + + >>> numZ, denZ = gbt(tf, T, 0.3) + >>> numZ + [3*T/(10*(L + 3*R*T/10)), 7*T/(10*(L + 3*R*T/10))] + >>> denZ + [1, (-L + 7*R*T/10)/(L + 3*R*T/10)] + + References + ========== + + .. [1] https://www.polyu.edu.hk/ama/profile/gfzhang/Research/ZCC09_IJC.pdf + """ + if not tf.is_SISO: + raise NotImplementedError("Not implemented for MIMO systems.") + + T = sample_per # and sample period T + s = tf.var + z = s # dummy discrete variable z + + np = tf.num.as_poly(s).all_coeffs() + dp = tf.den.as_poly(s).all_coeffs() + alpha = Rational(alpha).limit_denominator(1000) + + # The next line results from multiplying H(z) with z^N/z^N + N = max(len(np), len(dp)) - 1 + num = Add(*[ T**(N-i) * c * (z-1)**i * (alpha * z + 1 - alpha)**(N-i) for c, i in zip(np[::-1], range(len(np))) ]) + den = Add(*[ T**(N-i) * c * (z-1)**i * (alpha * z + 1 - alpha)**(N-i) for c, i in zip(dp[::-1], range(len(dp))) ]) + + num_coefs = num.as_poly(z).all_coeffs() + den_coefs = den.as_poly(z).all_coeffs() + + para = den_coefs[0] + num_coefs = [coef/para for coef in num_coefs] + den_coefs = [coef/para for coef in den_coefs] + + return num_coefs, den_coefs + +def bilinear(tf, sample_per): + r""" + Returns falling coefficients of H(z) from numerator and denominator. + + Explanation + =========== + + Where H(z) is the corresponding discretized transfer function, + discretized with the bilinear transform method. + H(z) is obtained from the continuous transfer function H(s) + by substituting $s(z) = \frac{2}{T}\frac{z-1}{z+1}$ into H(s), where T is the + sample period. + Coefficients are falling, i.e. $H(z) = \frac{az+b}{cz+d}$ is returned + as [a, b], [c, d]. + + Examples + ======== + + >>> from sympy.physics.control.lti import TransferFunction, bilinear + >>> from sympy.abc import s, L, R, T + + >>> tf = TransferFunction(1, s*L + R, s) + >>> numZ, denZ = bilinear(tf, T) + >>> numZ + [T/(2*(L + R*T/2)), T/(2*(L + R*T/2))] + >>> denZ + [1, (-L + R*T/2)/(L + R*T/2)] + """ + return gbt(tf, sample_per, S.Half) + +def forward_diff(tf, sample_per): + r""" + Returns falling coefficients of H(z) from numerator and denominator. + + Explanation + =========== + + Where H(z) is the corresponding discretized transfer function, + discretized with the forward difference transform method. + H(z) is obtained from the continuous transfer function H(s) + by substituting $s(z) = \frac{z-1}{T}$ into H(s), where T is the + sample period. + Coefficients are falling, i.e. $H(z) = \frac{az+b}{cz+d}$ is returned + as [a, b], [c, d]. + + Examples + ======== + + >>> from sympy.physics.control.lti import TransferFunction, forward_diff + >>> from sympy.abc import s, L, R, T + + >>> tf = TransferFunction(1, s*L + R, s) + >>> numZ, denZ = forward_diff(tf, T) + >>> numZ + [T/L] + >>> denZ + [1, (-L + R*T)/L] + """ + return gbt(tf, sample_per, S.Zero) + +def backward_diff(tf, sample_per): + r""" + Returns falling coefficients of H(z) from numerator and denominator. + + Explanation + =========== + + Where H(z) is the corresponding discretized transfer function, + discretized with the backward difference transform method. + H(z) is obtained from the continuous transfer function H(s) + by substituting $s(z) = \frac{z-1}{Tz}$ into H(s), where T is the + sample period. + Coefficients are falling, i.e. $H(z) = \frac{az+b}{cz+d}$ is returned + as [a, b], [c, d]. + + Examples + ======== + + >>> from sympy.physics.control.lti import TransferFunction, backward_diff + >>> from sympy.abc import s, L, R, T + + >>> tf = TransferFunction(1, s*L + R, s) + >>> numZ, denZ = backward_diff(tf, T) + >>> numZ + [T/(L + R*T), 0] + >>> denZ + [1, -L/(L + R*T)] + """ + return gbt(tf, sample_per, S.One) + +def phase_margin(system): + r""" + Returns the phase margin of a continuous time system. + Only applicable to Transfer Functions which can generate valid bode plots. + + Raises + ====== + + NotImplementedError + When time delay terms are present in the system. + + ValueError + When a SISO LTI system is not passed. + + When more than one free symbol is present in the system. + The only variable in the transfer function should be + the variable of the Laplace transform. + + Examples + ======== + + >>> from sympy.physics.control import TransferFunction, phase_margin + >>> from sympy.abc import s + + >>> tf = TransferFunction(1, s**3 + 2*s**2 + s, s) + >>> phase_margin(tf) + 180*(-pi + atan((-1 + (-2*18**(1/3)/(9 + sqrt(93))**(1/3) + 12**(1/3)*(9 + sqrt(93))**(1/3))**2/36)/(-12**(1/3)*(9 + sqrt(93))**(1/3)/3 + 2*18**(1/3)/(3*(9 + sqrt(93))**(1/3)))))/pi + 180 + >>> phase_margin(tf).n() + 21.3863897518751 + + >>> tf1 = TransferFunction(s**3, s**2 + 5*s, s) + >>> phase_margin(tf1) + -180 + 180*(atan(sqrt(2)*(-51/10 - sqrt(101)/10)*sqrt(1 + sqrt(101))/(2*(sqrt(101)/2 + 51/2))) + pi)/pi + >>> phase_margin(tf1).n() + -25.1783920627277 + + >>> tf2 = TransferFunction(1, s + 1, s) + >>> phase_margin(tf2) + -180 + + See Also + ======== + + gain_margin + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Phase_margin + + """ + from sympy.functions import arg + + if not isinstance(system, SISOLinearTimeInvariant): + raise ValueError("Margins are only applicable for SISO LTI systems.") + + _w = Dummy("w", real=True) + repl = I*_w + expr = system.to_expr() + len_free_symbols = len(expr.free_symbols) + if expr.has(exp): + raise NotImplementedError("Margins for systems with Time delay terms are not supported.") + elif len_free_symbols > 1: + raise ValueError("Extra degree of freedom found. Make sure" + " that there are no free symbols in the dynamical system other" + " than the variable of Laplace transform.") + + w_expr = expr.subs({system.var: repl}) + + mag = 20*log(Abs(w_expr), 10) + mag_sol = list(solveset(mag, _w, Interval(0, oo, left_open=True))) + + if (len(mag_sol) == 0): + pm = S(-180) + else: + wcp = mag_sol[0] + pm = ((arg(w_expr)*S(180)/pi).subs({_w:wcp}) + S(180)) % 360 + + if(pm >= 180): + pm = pm - 360 + + return pm + +def gain_margin(system): + r""" + Returns the gain margin of a continuous time system. + Only applicable to Transfer Functions which can generate valid bode plots. + + Raises + ====== + + NotImplementedError + When time delay terms are present in the system. + + ValueError + When a SISO LTI system is not passed. + + When more than one free symbol is present in the system. + The only variable in the transfer function should be + the variable of the Laplace transform. + + Examples + ======== + + >>> from sympy.physics.control import TransferFunction, gain_margin + >>> from sympy.abc import s + + >>> tf = TransferFunction(1, s**3 + 2*s**2 + s, s) + >>> gain_margin(tf) + 20*log(2)/log(10) + >>> gain_margin(tf).n() + 6.02059991327962 + + >>> tf1 = TransferFunction(s**3, s**2 + 5*s, s) + >>> gain_margin(tf1) + oo + + See Also + ======== + + phase_margin + + References + ========== + + https://en.wikipedia.org/wiki/Bode_plot + + """ + if not isinstance(system, SISOLinearTimeInvariant): + raise ValueError("Margins are only applicable for SISO LTI systems.") + + _w = Dummy("w", real=True) + repl = I*_w + expr = system.to_expr() + len_free_symbols = len(expr.free_symbols) + if expr.has(exp): + raise NotImplementedError("Margins for systems with Time delay terms are not supported.") + elif len_free_symbols > 1: + raise ValueError("Extra degree of freedom found. Make sure" + " that there are no free symbols in the dynamical system other" + " than the variable of Laplace transform.") + + w_expr = expr.subs({system.var: repl}) + + mag = 20*log(Abs(w_expr), 10) + phase = w_expr + phase_sol = list(solveset(numer(phase.as_real_imag()[1].cancel()),_w, Interval(0, oo, left_open = True))) + + if (len(phase_sol) == 0): + gm = oo + else: + wcg = phase_sol[0] + gm = -mag.subs({_w:wcg}) + + return gm + +class LinearTimeInvariant(Basic, EvalfMixin): + """A common class for all the Linear Time-Invariant Dynamical Systems.""" + + _clstype: Type + + # Users should not directly interact with this class. + def __new__(cls, *system, **kwargs): + if cls is LinearTimeInvariant: + raise NotImplementedError('The LTICommon class is not meant to be used directly.') + return super(LinearTimeInvariant, cls).__new__(cls, *system, **kwargs) + + @classmethod + def _check_args(cls, args): + if not args: + raise ValueError("At least 1 argument must be passed.") + if not all(isinstance(arg, cls._clstype) for arg in args): + raise TypeError(f"All arguments must be of type {cls._clstype}.") + var_set = {arg.var for arg in args} + if len(var_set) != 1: + raise ValueError(filldedent(f""" + All transfer functions should use the same complex variable + of the Laplace transform. {len(var_set)} different + values found.""")) + + @property + def is_SISO(self): + """Returns `True` if the passed LTI system is SISO else returns False.""" + return self._is_SISO + + +class SISOLinearTimeInvariant(LinearTimeInvariant): + """A common class for all the SISO Linear Time-Invariant Dynamical Systems.""" + # Users should not directly interact with this class. + _is_SISO = True + + +class MIMOLinearTimeInvariant(LinearTimeInvariant): + """A common class for all the MIMO Linear Time-Invariant Dynamical Systems.""" + # Users should not directly interact with this class. + _is_SISO = False + + +SISOLinearTimeInvariant._clstype = SISOLinearTimeInvariant +MIMOLinearTimeInvariant._clstype = MIMOLinearTimeInvariant + + +def _check_other_SISO(func): + def wrapper(*args, **kwargs): + if not isinstance(args[-1], SISOLinearTimeInvariant): + return NotImplemented + else: + return func(*args, **kwargs) + return wrapper + + +def _check_other_MIMO(func): + def wrapper(*args, **kwargs): + if not isinstance(args[-1], MIMOLinearTimeInvariant): + return NotImplemented + else: + return func(*args, **kwargs) + return wrapper + + +class TransferFunction(SISOLinearTimeInvariant): + r""" + A class for representing LTI (Linear, time-invariant) systems that can be strictly described + by ratio of polynomials in the Laplace transform complex variable. The arguments + are ``num``, ``den``, and ``var``, where ``num`` and ``den`` are numerator and + denominator polynomials of the ``TransferFunction`` respectively, and the third argument is + a complex variable of the Laplace transform used by these polynomials of the transfer function. + ``num`` and ``den`` can be either polynomials or numbers, whereas ``var`` + has to be a :py:class:`~.Symbol`. + + Explanation + =========== + + Generally, a dynamical system representing a physical model can be described in terms of Linear + Ordinary Differential Equations like - + + $\small{b_{m}y^{\left(m\right)}+b_{m-1}y^{\left(m-1\right)}+\dots+b_{1}y^{\left(1\right)}+b_{0}y= + a_{n}x^{\left(n\right)}+a_{n-1}x^{\left(n-1\right)}+\dots+a_{1}x^{\left(1\right)}+a_{0}x}$ + + Here, $x$ is the input signal and $y$ is the output signal and superscript on both is the order of derivative + (not exponent). Derivative is taken with respect to the independent variable, $t$. Also, generally $m$ is greater + than $n$. + + It is not feasible to analyse the properties of such systems in their native form therefore, we use + mathematical tools like Laplace transform to get a better perspective. Taking the Laplace transform + of both the sides in the equation (at zero initial conditions), we get - + + $\small{\mathcal{L}[b_{m}y^{\left(m\right)}+b_{m-1}y^{\left(m-1\right)}+\dots+b_{1}y^{\left(1\right)}+b_{0}y]= + \mathcal{L}[a_{n}x^{\left(n\right)}+a_{n-1}x^{\left(n-1\right)}+\dots+a_{1}x^{\left(1\right)}+a_{0}x]}$ + + Using the linearity property of Laplace transform and also considering zero initial conditions + (i.e. $\small{y(0^{-}) = 0}$, $\small{y'(0^{-}) = 0}$ and so on), the equation + above gets translated to - + + $\small{b_{m}\mathcal{L}[y^{\left(m\right)}]+\dots+b_{1}\mathcal{L}[y^{\left(1\right)}]+b_{0}\mathcal{L}[y]= + a_{n}\mathcal{L}[x^{\left(n\right)}]+\dots+a_{1}\mathcal{L}[x^{\left(1\right)}]+a_{0}\mathcal{L}[x]}$ + + Now, applying Derivative property of Laplace transform, + + $\small{b_{m}s^{m}\mathcal{L}[y]+\dots+b_{1}s\mathcal{L}[y]+b_{0}\mathcal{L}[y]= + a_{n}s^{n}\mathcal{L}[x]+\dots+a_{1}s\mathcal{L}[x]+a_{0}\mathcal{L}[x]}$ + + Here, the superscript on $s$ is **exponent**. Note that the zero initial conditions assumption, mentioned above, is very important + and cannot be ignored otherwise the dynamical system cannot be considered time-independent and the simplified equation above + cannot be reached. + + Collecting $\mathcal{L}[y]$ and $\mathcal{L}[x]$ terms from both the sides and taking the ratio + $\frac{ \mathcal{L}\left\{y\right\} }{ \mathcal{L}\left\{x\right\} }$, we get the typical rational form of transfer + function. + + The numerator of the transfer function is, therefore, the Laplace transform of the output signal + (The signals are represented as functions of time) and similarly, the denominator + of the transfer function is the Laplace transform of the input signal. It is also a convention + to denote the input and output signal's Laplace transform with capital alphabets like shown below. + + $H(s) = \frac{Y(s)}{X(s)} = \frac{ \mathcal{L}\left\{y(t)\right\} }{ \mathcal{L}\left\{x(t)\right\} }$ + + $s$, also known as complex frequency, is a complex variable in the Laplace domain. It corresponds to the + equivalent variable $t$, in the time domain. Transfer functions are sometimes also referred to as the Laplace + transform of the system's impulse response. Transfer function, $H$, is represented as a rational + function in $s$ like, + + $H(s) =\ \frac{a_{n}s^{n}+a_{n-1}s^{n-1}+\dots+a_{1}s+a_{0}}{b_{m}s^{m}+b_{m-1}s^{m-1}+\dots+b_{1}s+b_{0}}$ + + Parameters + ========== + + num : Expr, Number + The numerator polynomial of the transfer function. + den : Expr, Number + The denominator polynomial of the transfer function. + var : Symbol + Complex variable of the Laplace transform used by the + polynomials of the transfer function. + + Raises + ====== + + TypeError + When ``var`` is not a Symbol or when ``num`` or ``den`` is not a + number or a polynomial. + ValueError + When ``den`` is zero. + + Examples + ======== + + >>> from sympy.abc import s, p, a + >>> from sympy.physics.control.lti import TransferFunction + >>> tf1 = TransferFunction(s + a, s**2 + s + 1, s) + >>> tf1 + TransferFunction(a + s, s**2 + s + 1, s) + >>> tf1.num + a + s + >>> tf1.den + s**2 + s + 1 + >>> tf1.var + s + >>> tf1.args + (a + s, s**2 + s + 1, s) + + Any complex variable can be used for ``var``. + + >>> tf2 = TransferFunction(a*p**3 - a*p**2 + s*p, p + a**2, p) + >>> tf2 + TransferFunction(a*p**3 - a*p**2 + p*s, a**2 + p, p) + >>> tf3 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p) + >>> tf3 + TransferFunction((p - 1)*(p + 3), (p - 1)*(p + 5), p) + + To negate a transfer function the ``-`` operator can be prepended: + + >>> tf4 = TransferFunction(-a + s, p**2 + s, p) + >>> -tf4 + TransferFunction(a - s, p**2 + s, p) + >>> tf5 = TransferFunction(s**4 - 2*s**3 + 5*s + 4, s + 4, s) + >>> -tf5 + TransferFunction(-s**4 + 2*s**3 - 5*s - 4, s + 4, s) + + You can use a float or an integer (or other constants) as numerator and denominator: + + >>> tf6 = TransferFunction(1/2, 4, s) + >>> tf6.num + 0.500000000000000 + >>> tf6.den + 4 + >>> tf6.var + s + >>> tf6.args + (0.5, 4, s) + + You can take the integer power of a transfer function using the ``**`` operator: + + >>> tf7 = TransferFunction(s + a, s - a, s) + >>> tf7**3 + TransferFunction((a + s)**3, (-a + s)**3, s) + >>> tf7**0 + TransferFunction(1, 1, s) + >>> tf8 = TransferFunction(p + 4, p - 3, p) + >>> tf8**-1 + TransferFunction(p - 3, p + 4, p) + + Addition, subtraction, and multiplication of transfer functions can form + unevaluated ``Series`` or ``Parallel`` objects. + + >>> tf9 = TransferFunction(s + 1, s**2 + s + 1, s) + >>> tf10 = TransferFunction(s - p, s + 3, s) + >>> tf11 = TransferFunction(4*s**2 + 2*s - 4, s - 1, s) + >>> tf12 = TransferFunction(1 - s, s**2 + 4, s) + >>> tf9 + tf10 + Parallel(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(-p + s, s + 3, s)) + >>> tf10 - tf11 + Parallel(TransferFunction(-p + s, s + 3, s), TransferFunction(-4*s**2 - 2*s + 4, s - 1, s)) + >>> tf9 * tf10 + Series(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(-p + s, s + 3, s)) + >>> tf10 - (tf9 + tf12) + Parallel(TransferFunction(-p + s, s + 3, s), TransferFunction(-s - 1, s**2 + s + 1, s), TransferFunction(s - 1, s**2 + 4, s)) + >>> tf10 - (tf9 * tf12) + Parallel(TransferFunction(-p + s, s + 3, s), Series(TransferFunction(-1, 1, s), TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(1 - s, s**2 + 4, s))) + >>> tf11 * tf10 * tf9 + Series(TransferFunction(4*s**2 + 2*s - 4, s - 1, s), TransferFunction(-p + s, s + 3, s), TransferFunction(s + 1, s**2 + s + 1, s)) + >>> tf9 * tf11 + tf10 * tf12 + Parallel(Series(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(4*s**2 + 2*s - 4, s - 1, s)), Series(TransferFunction(-p + s, s + 3, s), TransferFunction(1 - s, s**2 + 4, s))) + >>> (tf9 + tf12) * (tf10 + tf11) + Series(Parallel(TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(1 - s, s**2 + 4, s)), Parallel(TransferFunction(-p + s, s + 3, s), TransferFunction(4*s**2 + 2*s - 4, s - 1, s))) + + These unevaluated ``Series`` or ``Parallel`` objects can convert into the + resultant transfer function using ``.doit()`` method or by ``.rewrite(TransferFunction)``. + + >>> ((tf9 + tf10) * tf12).doit() + TransferFunction((1 - s)*((-p + s)*(s**2 + s + 1) + (s + 1)*(s + 3)), (s + 3)*(s**2 + 4)*(s**2 + s + 1), s) + >>> (tf9 * tf10 - tf11 * tf12).rewrite(TransferFunction) + TransferFunction(-(1 - s)*(s + 3)*(s**2 + s + 1)*(4*s**2 + 2*s - 4) + (-p + s)*(s - 1)*(s + 1)*(s**2 + 4), (s - 1)*(s + 3)*(s**2 + 4)*(s**2 + s + 1), s) + + See Also + ======== + + Feedback, Series, Parallel + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Transfer_function + .. [2] https://en.wikipedia.org/wiki/Laplace_transform + + """ + def __new__(cls, num, den, var): + num, den = _sympify(num), _sympify(den) + + if not isinstance(var, Symbol): + raise TypeError("Variable input must be a Symbol.") + + if den == 0: + raise ValueError("TransferFunction cannot have a zero denominator.") + + if (((isinstance(num, (Expr, TransferFunction, Series, Parallel)) and num.has(Symbol)) or num.is_number) and + ((isinstance(den, (Expr, TransferFunction, Series, Parallel)) and den.has(Symbol)) or den.is_number)): + return super(TransferFunction, cls).__new__(cls, num, den, var) + + else: + raise TypeError("Unsupported type for numerator or denominator of TransferFunction.") + + @classmethod + def from_rational_expression(cls, expr, var=None): + r""" + Creates a new ``TransferFunction`` efficiently from a rational expression. + + Parameters + ========== + + expr : Expr, Number + The rational expression representing the ``TransferFunction``. + var : Symbol, optional + Complex variable of the Laplace transform used by the + polynomials of the transfer function. + + Raises + ====== + + ValueError + When ``expr`` is of type ``Number`` and optional parameter ``var`` + is not passed. + + When ``expr`` has more than one variables and an optional parameter + ``var`` is not passed. + ZeroDivisionError + When denominator of ``expr`` is zero or it has ``ComplexInfinity`` + in its numerator. + + Examples + ======== + + >>> from sympy.abc import s, p, a + >>> from sympy.physics.control.lti import TransferFunction + >>> expr1 = (s + 5)/(3*s**2 + 2*s + 1) + >>> tf1 = TransferFunction.from_rational_expression(expr1) + >>> tf1 + TransferFunction(s + 5, 3*s**2 + 2*s + 1, s) + >>> expr2 = (a*p**3 - a*p**2 + s*p)/(p + a**2) # Expr with more than one variables + >>> tf2 = TransferFunction.from_rational_expression(expr2, p) + >>> tf2 + TransferFunction(a*p**3 - a*p**2 + p*s, a**2 + p, p) + + In case of conflict between two or more variables in a expression, SymPy will + raise a ``ValueError``, if ``var`` is not passed by the user. + + >>> tf = TransferFunction.from_rational_expression((a + a*s)/(s**2 + s + 1)) + Traceback (most recent call last): + ... + ValueError: Conflicting values found for positional argument `var` ({a, s}). Specify it manually. + + This can be corrected by specifying the ``var`` parameter manually. + + >>> tf = TransferFunction.from_rational_expression((a + a*s)/(s**2 + s + 1), s) + >>> tf + TransferFunction(a*s + a, s**2 + s + 1, s) + + ``var`` also need to be specified when ``expr`` is a ``Number`` + + >>> tf3 = TransferFunction.from_rational_expression(10, s) + >>> tf3 + TransferFunction(10, 1, s) + + """ + expr = _sympify(expr) + if var is None: + _free_symbols = expr.free_symbols + _len_free_symbols = len(_free_symbols) + if _len_free_symbols == 1: + var = list(_free_symbols)[0] + elif _len_free_symbols == 0: + raise ValueError(filldedent(""" + Positional argument `var` not found in the + TransferFunction defined. Specify it manually.""")) + else: + raise ValueError(filldedent(""" + Conflicting values found for positional argument `var` ({}). + Specify it manually.""".format(_free_symbols))) + + _num, _den = expr.as_numer_denom() + if _den == 0 or _num.has(S.ComplexInfinity): + raise ZeroDivisionError("TransferFunction cannot have a zero denominator.") + return cls(_num, _den, var) + + @classmethod + def from_coeff_lists(cls, num_list, den_list, var): + r""" + Creates a new ``TransferFunction`` efficiently from a list of coefficients. + + Parameters + ========== + + num_list : Sequence + Sequence comprising of numerator coefficients. + den_list : Sequence + Sequence comprising of denominator coefficients. + var : Symbol + Complex variable of the Laplace transform used by the + polynomials of the transfer function. + + Raises + ====== + + ZeroDivisionError + When the constructed denominator is zero. + + Examples + ======== + + >>> from sympy.abc import s, p + >>> from sympy.physics.control.lti import TransferFunction + >>> num = [1, 0, 2] + >>> den = [3, 2, 2, 1] + >>> tf = TransferFunction.from_coeff_lists(num, den, s) + >>> tf + TransferFunction(s**2 + 2, 3*s**3 + 2*s**2 + 2*s + 1, s) + + # Create a Transfer Function with more than one variable + >>> tf1 = TransferFunction.from_coeff_lists([p, 1], [2*p, 0, 4], s) + >>> tf1 + TransferFunction(p*s + 1, 2*p*s**2 + 4, s) + + """ + num_list = num_list[::-1] + den_list = den_list[::-1] + num_var_powers = [var**i for i in range(len(num_list))] + den_var_powers = [var**i for i in range(len(den_list))] + + _num = sum(coeff * var_power for coeff, var_power in zip(num_list, num_var_powers)) + _den = sum(coeff * var_power for coeff, var_power in zip(den_list, den_var_powers)) + + if _den == 0: + raise ZeroDivisionError("TransferFunction cannot have a zero denominator.") + + return cls(_num, _den, var) + + @classmethod + def from_zpk(cls, zeros, poles, gain, var): + r""" + Creates a new ``TransferFunction`` from given zeros, poles and gain. + + Parameters + ========== + + zeros : Sequence + Sequence comprising of zeros of transfer function. + poles : Sequence + Sequence comprising of poles of transfer function. + gain : Number, Symbol, Expression + A scalar value specifying gain of the model. + var : Symbol + Complex variable of the Laplace transform used by the + polynomials of the transfer function. + + Examples + ======== + + >>> from sympy.abc import s, p, k + >>> from sympy.physics.control.lti import TransferFunction + >>> zeros = [1, 2, 3] + >>> poles = [6, 5, 4] + >>> gain = 7 + >>> tf = TransferFunction.from_zpk(zeros, poles, gain, s) + >>> tf + TransferFunction(7*(s - 3)*(s - 2)*(s - 1), (s - 6)*(s - 5)*(s - 4), s) + + # Create a Transfer Function with variable poles and zeros + >>> tf1 = TransferFunction.from_zpk([p, k], [p + k, p - k], 2, s) + >>> tf1 + TransferFunction(2*(-k + s)*(-p + s), (-k - p + s)*(k - p + s), s) + + # Complex poles or zeros are acceptable + >>> tf2 = TransferFunction.from_zpk([0], [1-1j, 1+1j, 2], -2, s) + >>> tf2 + TransferFunction(-2*s, (s - 2)*(s - 1.0 - 1.0*I)*(s - 1.0 + 1.0*I), s) + + """ + num_poly = 1 + den_poly = 1 + for zero in zeros: + num_poly *= var - zero + for pole in poles: + den_poly *= var - pole + + return cls(gain*num_poly, den_poly, var) + + @property + def num(self): + """ + Returns the numerator polynomial of the transfer function. + + Examples + ======== + + >>> from sympy.abc import s, p + >>> from sympy.physics.control.lti import TransferFunction + >>> G1 = TransferFunction(s**2 + p*s + 3, s - 4, s) + >>> G1.num + p*s + s**2 + 3 + >>> G2 = TransferFunction((p + 5)*(p - 3), (p - 3)*(p + 1), p) + >>> G2.num + (p - 3)*(p + 5) + + """ + return self.args[0] + + @property + def den(self): + """ + Returns the denominator polynomial of the transfer function. + + Examples + ======== + + >>> from sympy.abc import s, p + >>> from sympy.physics.control.lti import TransferFunction + >>> G1 = TransferFunction(s + 4, p**3 - 2*p + 4, s) + >>> G1.den + p**3 - 2*p + 4 + >>> G2 = TransferFunction(3, 4, s) + >>> G2.den + 4 + + """ + return self.args[1] + + @property + def var(self): + """ + Returns the complex variable of the Laplace transform used by the polynomials of + the transfer function. + + Examples + ======== + + >>> from sympy.abc import s, p + >>> from sympy.physics.control.lti import TransferFunction + >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) + >>> G1.var + p + >>> G2 = TransferFunction(0, s - 5, s) + >>> G2.var + s + + """ + return self.args[2] + + def _eval_subs(self, old, new): + arg_num = self.num.subs(old, new) + arg_den = self.den.subs(old, new) + argnew = TransferFunction(arg_num, arg_den, self.var) + return self if old == self.var else argnew + + def _eval_evalf(self, prec): + return TransferFunction( + self.num._eval_evalf(prec), + self.den._eval_evalf(prec), + self.var) + + def _eval_simplify(self, **kwargs): + tf = cancel(Mul(self.num, 1/self.den, evaluate=False), expand=False).as_numer_denom() + num_, den_ = tf[0], tf[1] + return TransferFunction(num_, den_, self.var) + + def _eval_rewrite_as_StateSpace(self, *args): + """ + Returns the equivalent space space model of the transfer function model. + The state space model will be returned in the controllable cannonical form. + + Unlike the space state to transfer function model conversion, the transfer function + to state space model conversion is not unique. There can be multiple state space + representations of a given transfer function model. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control import TransferFunction, StateSpace + >>> tf = TransferFunction(s**2 + 1, s**3 + 2*s + 10, s) + >>> tf.rewrite(StateSpace) + StateSpace(Matrix([ + [ 0, 1, 0], + [ 0, 0, 1], + [-10, -2, 0]]), Matrix([ + [0], + [0], + [1]]), Matrix([[1, 0, 1]]), Matrix([[0]])) + + """ + if not self.is_proper: + raise ValueError("Transfer Function must be proper.") + + num_poly = Poly(self.num, self.var) + den_poly = Poly(self.den, self.var) + n = den_poly.degree() + + num_coeffs = num_poly.all_coeffs() + den_coeffs = den_poly.all_coeffs() + diff = n - num_poly.degree() + num_coeffs = [0]*diff + num_coeffs + + a = den_coeffs[1:] + a_mat = Matrix([[(-1)*coefficient/den_coeffs[0] for coefficient in reversed(a)]]) + vert = zeros(n-1, 1) + mat = eye(n-1) + A = vert.row_join(mat) + A = A.col_join(a_mat) + + B = zeros(n, 1) + B[n-1] = 1 + + i = n + C = [] + while(i > 0): + C.append(num_coeffs[i] - den_coeffs[i]*num_coeffs[0]) + i -= 1 + C = Matrix([C]) + + D = Matrix([num_coeffs[0]]) + + return StateSpace(A, B, C, D) + + def expand(self): + """ + Returns the transfer function with numerator and denominator + in expanded form. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction + >>> G1 = TransferFunction((a - s)**2, (s**2 + a)**2, s) + >>> G1.expand() + TransferFunction(a**2 - 2*a*s + s**2, a**2 + 2*a*s**2 + s**4, s) + >>> G2 = TransferFunction((p + 3*b)*(p - b), (p - b)*(p + 2*b), p) + >>> G2.expand() + TransferFunction(-3*b**2 + 2*b*p + p**2, -2*b**2 + b*p + p**2, p) + + """ + return TransferFunction(expand(self.num), expand(self.den), self.var) + + def dc_gain(self): + """ + Computes the gain of the response as the frequency approaches zero. + + The DC gain is infinite for systems with pure integrators. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction + >>> tf1 = TransferFunction(s + 3, s**2 - 9, s) + >>> tf1.dc_gain() + -1/3 + >>> tf2 = TransferFunction(p**2, p - 3 + p**3, p) + >>> tf2.dc_gain() + 0 + >>> tf3 = TransferFunction(a*p**2 - b, s + b, s) + >>> tf3.dc_gain() + (a*p**2 - b)/b + >>> tf4 = TransferFunction(1, s, s) + >>> tf4.dc_gain() + oo + + """ + m = Mul(self.num, Pow(self.den, -1, evaluate=False), evaluate=False) + return limit(m, self.var, 0) + + def poles(self): + """ + Returns the poles of a transfer function. + + Examples + ======== + + >>> from sympy.abc import s, p, a + >>> from sympy.physics.control.lti import TransferFunction + >>> tf1 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p) + >>> tf1.poles() + [-5, 1] + >>> tf2 = TransferFunction((1 - s)**2, (s**2 + 1)**2, s) + >>> tf2.poles() + [I, I, -I, -I] + >>> tf3 = TransferFunction(s**2, a*s + p, s) + >>> tf3.poles() + [-p/a] + + """ + return _roots(Poly(self.den, self.var), self.var) + + def zeros(self): + """ + Returns the zeros of a transfer function. + + Examples + ======== + + >>> from sympy.abc import s, p, a + >>> from sympy.physics.control.lti import TransferFunction + >>> tf1 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p) + >>> tf1.zeros() + [-3, 1] + >>> tf2 = TransferFunction((1 - s)**2, (s**2 + 1)**2, s) + >>> tf2.zeros() + [1, 1] + >>> tf3 = TransferFunction(s**2, a*s + p, s) + >>> tf3.zeros() + [0, 0] + + """ + return _roots(Poly(self.num, self.var), self.var) + + def eval_frequency(self, other): + """ + Returns the system response at any point in the real or complex plane. + + Examples + ======== + + >>> from sympy.abc import s, p, a + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy import I + >>> tf1 = TransferFunction(1, s**2 + 2*s + 1, s) + >>> omega = 0.1 + >>> tf1.eval_frequency(I*omega) + 1/(0.99 + 0.2*I) + >>> tf2 = TransferFunction(s**2, a*s + p, s) + >>> tf2.eval_frequency(2) + 4/(2*a + p) + >>> tf2.eval_frequency(I*2) + -4/(2*I*a + p) + """ + arg_num = self.num.subs(self.var, other) + arg_den = self.den.subs(self.var, other) + argnew = TransferFunction(arg_num, arg_den, self.var).to_expr() + return argnew.expand() + + def is_stable(self): + """ + Returns True if the transfer function is asymptotically stable; else False. + + This would not check the marginal or conditional stability of the system. + + Examples + ======== + + >>> from sympy.abc import s, p, a + >>> from sympy import symbols + >>> from sympy.physics.control.lti import TransferFunction + >>> q, r = symbols('q, r', negative=True) + >>> tf1 = TransferFunction((1 - s)**2, (s + 1)**2, s) + >>> tf1.is_stable() + True + >>> tf2 = TransferFunction((1 - p)**2, (s**2 + 1)**2, s) + >>> tf2.is_stable() + False + >>> tf3 = TransferFunction(4, q*s - r, s) + >>> tf3.is_stable() + False + >>> tf4 = TransferFunction(p + 1, a*p - s**2, p) + >>> tf4.is_stable() is None # Not enough info about the symbols to determine stability + True + + """ + return fuzzy_and(pole.as_real_imag()[0].is_negative for pole in self.poles()) + + def __add__(self, other): + if isinstance(other, (TransferFunction, Series)): + if not self.var == other.var: + raise ValueError(filldedent(""" + All the transfer functions should use the same complex variable + of the Laplace transform.""")) + return Parallel(self, other) + elif isinstance(other, Parallel): + if not self.var == other.var: + raise ValueError(filldedent(""" + All the transfer functions should use the same complex variable + of the Laplace transform.""")) + arg_list = list(other.args) + return Parallel(self, *arg_list) + else: + raise ValueError("TransferFunction cannot be added with {}.". + format(type(other))) + + def __radd__(self, other): + return self + other + + def __sub__(self, other): + if isinstance(other, (TransferFunction, Series)): + if not self.var == other.var: + raise ValueError(filldedent(""" + All the transfer functions should use the same complex variable + of the Laplace transform.""")) + return Parallel(self, -other) + elif isinstance(other, Parallel): + if not self.var == other.var: + raise ValueError(filldedent(""" + All the transfer functions should use the same complex variable + of the Laplace transform.""")) + arg_list = [-i for i in list(other.args)] + return Parallel(self, *arg_list) + else: + raise ValueError("{} cannot be subtracted from a TransferFunction." + .format(type(other))) + + def __rsub__(self, other): + return -self + other + + def __mul__(self, other): + if isinstance(other, (TransferFunction, Parallel)): + if not self.var == other.var: + raise ValueError(filldedent(""" + All the transfer functions should use the same complex variable + of the Laplace transform.""")) + return Series(self, other) + elif isinstance(other, Series): + if not self.var == other.var: + raise ValueError(filldedent(""" + All the transfer functions should use the same complex variable + of the Laplace transform.""")) + arg_list = list(other.args) + return Series(self, *arg_list) + else: + raise ValueError("TransferFunction cannot be multiplied with {}." + .format(type(other))) + + __rmul__ = __mul__ + + def __truediv__(self, other): + if isinstance(other, TransferFunction): + if not self.var == other.var: + raise ValueError(filldedent(""" + All the transfer functions should use the same complex variable + of the Laplace transform.""")) + return Series(self, TransferFunction(other.den, other.num, self.var)) + elif (isinstance(other, Parallel) and len(other.args + ) == 2 and isinstance(other.args[0], TransferFunction) + and isinstance(other.args[1], (Series, TransferFunction))): + + if not self.var == other.var: + raise ValueError(filldedent(""" + Both TransferFunction and Parallel should use the + same complex variable of the Laplace transform.""")) + if other.args[1] == self: + # plant and controller with unit feedback. + return Feedback(self, other.args[0]) + other_arg_list = list(other.args[1].args) if isinstance( + other.args[1], Series) else other.args[1] + if other_arg_list == other.args[1]: + return Feedback(self, other_arg_list) + elif self in other_arg_list: + other_arg_list.remove(self) + else: + return Feedback(self, Series(*other_arg_list)) + + if len(other_arg_list) == 1: + return Feedback(self, *other_arg_list) + else: + return Feedback(self, Series(*other_arg_list)) + else: + raise ValueError("TransferFunction cannot be divided by {}.". + format(type(other))) + + __rtruediv__ = __truediv__ + + def __pow__(self, p): + p = sympify(p) + if not p.is_Integer: + raise ValueError("Exponent must be an integer.") + if p is S.Zero: + return TransferFunction(1, 1, self.var) + elif p > 0: + num_, den_ = self.num**p, self.den**p + else: + p = abs(p) + num_, den_ = self.den**p, self.num**p + + return TransferFunction(num_, den_, self.var) + + def __neg__(self): + return TransferFunction(-self.num, self.den, self.var) + + @property + def is_proper(self): + """ + Returns True if degree of the numerator polynomial is less than + or equal to degree of the denominator polynomial, else False. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction + >>> tf1 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s) + >>> tf1.is_proper + False + >>> tf2 = TransferFunction(p**2 - 4*p, p**3 + 3*p + 2, p) + >>> tf2.is_proper + True + + """ + return degree(self.num, self.var) <= degree(self.den, self.var) + + @property + def is_strictly_proper(self): + """ + Returns True if degree of the numerator polynomial is strictly less + than degree of the denominator polynomial, else False. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf1.is_strictly_proper + False + >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) + >>> tf2.is_strictly_proper + True + + """ + return degree(self.num, self.var) < degree(self.den, self.var) + + @property + def is_biproper(self): + """ + Returns True if degree of the numerator polynomial is equal to + degree of the denominator polynomial, else False. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf1.is_biproper + True + >>> tf2 = TransferFunction(p**2, p + a, p) + >>> tf2.is_biproper + False + + """ + return degree(self.num, self.var) == degree(self.den, self.var) + + def to_expr(self): + """ + Converts a ``TransferFunction`` object to SymPy Expr. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction + >>> from sympy import Expr + >>> tf1 = TransferFunction(s, a*s**2 + 1, s) + >>> tf1.to_expr() + s/(a*s**2 + 1) + >>> isinstance(_, Expr) + True + >>> tf2 = TransferFunction(1, (p + 3*b)*(b - p), p) + >>> tf2.to_expr() + 1/((b - p)*(3*b + p)) + >>> tf3 = TransferFunction((s - 2)*(s - 3), (s - 1)*(s - 2)*(s - 3), s) + >>> tf3.to_expr() + ((s - 3)*(s - 2))/(((s - 3)*(s - 2)*(s - 1))) + + """ + + if self.num != 1: + return Mul(self.num, Pow(self.den, -1, evaluate=False), evaluate=False) + else: + return Pow(self.den, -1, evaluate=False) + + +def _flatten_args(args, _cls): + temp_args = [] + for arg in args: + if isinstance(arg, _cls): + temp_args.extend(arg.args) + else: + temp_args.append(arg) + return tuple(temp_args) + + +def _dummify_args(_arg, var): + dummy_dict = {} + dummy_arg_list = [] + + for arg in _arg: + _s = Dummy() + dummy_dict[_s] = var + dummy_arg = arg.subs({var: _s}) + dummy_arg_list.append(dummy_arg) + + return dummy_arg_list, dummy_dict + + +class Series(SISOLinearTimeInvariant): + r""" + A class for representing a series configuration of SISO systems. + + Parameters + ========== + + args : SISOLinearTimeInvariant + SISO systems in a series configuration. + evaluate : Boolean, Keyword + When passed ``True``, returns the equivalent + ``Series(*args).doit()``. Set to ``False`` by default. + + Raises + ====== + + ValueError + When no argument is passed. + + ``var`` attribute is not same for every system. + TypeError + Any of the passed ``*args`` has unsupported type + + A combination of SISO and MIMO systems is + passed. There should be homogeneity in the + type of systems passed, SISO in this case. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, Series, Parallel + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) + >>> tf3 = TransferFunction(p**2, p + s, s) + >>> S1 = Series(tf1, tf2) + >>> S1 + Series(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)) + >>> S1.var + s + >>> S2 = Series(tf2, Parallel(tf3, -tf1)) + >>> S2 + Series(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), Parallel(TransferFunction(p**2, p + s, s), TransferFunction(-a*p**2 - b*s, -p + s, s))) + >>> S2.var + s + >>> S3 = Series(Parallel(tf1, tf2), Parallel(tf2, tf3)) + >>> S3 + Series(Parallel(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)), Parallel(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), TransferFunction(p**2, p + s, s))) + >>> S3.var + s + + You can get the resultant transfer function by using ``.doit()`` method: + + >>> S3 = Series(tf1, tf2, -tf3) + >>> S3.doit() + TransferFunction(-p**2*(s**3 - 2)*(a*p**2 + b*s), (-p + s)*(p + s)*(s**4 + 5*s + 6), s) + >>> S4 = Series(tf2, Parallel(tf1, -tf3)) + >>> S4.doit() + TransferFunction((s**3 - 2)*(-p**2*(-p + s) + (p + s)*(a*p**2 + b*s)), (-p + s)*(p + s)*(s**4 + 5*s + 6), s) + + Notes + ===== + + All the transfer functions should use the same complex variable + ``var`` of the Laplace transform. + + See Also + ======== + + MIMOSeries, Parallel, TransferFunction, Feedback + + """ + def __new__(cls, *args, evaluate=False): + + args = _flatten_args(args, Series) + cls._check_args(args) + obj = super().__new__(cls, *args) + + return obj.doit() if evaluate else obj + + @property + def var(self): + """ + Returns the complex variable used by all the transfer functions. + + Examples + ======== + + >>> from sympy.abc import p + >>> from sympy.physics.control.lti import TransferFunction, Series, Parallel + >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) + >>> G2 = TransferFunction(p, 4 - p, p) + >>> G3 = TransferFunction(0, p**4 - 1, p) + >>> Series(G1, G2).var + p + >>> Series(-G3, Parallel(G1, G2)).var + p + + """ + return self.args[0].var + + def doit(self, **hints): + """ + Returns the resultant transfer function obtained after evaluating + the transfer functions in series configuration. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, Series + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) + >>> Series(tf2, tf1).doit() + TransferFunction((s**3 - 2)*(a*p**2 + b*s), (-p + s)*(s**4 + 5*s + 6), s) + >>> Series(-tf1, -tf2).doit() + TransferFunction((2 - s**3)*(-a*p**2 - b*s), (-p + s)*(s**4 + 5*s + 6), s) + + """ + + _num_arg = (arg.doit().num for arg in self.args) + _den_arg = (arg.doit().den for arg in self.args) + res_num = Mul(*_num_arg, evaluate=True) + res_den = Mul(*_den_arg, evaluate=True) + return TransferFunction(res_num, res_den, self.var) + + def _eval_rewrite_as_TransferFunction(self, *args, **kwargs): + return self.doit() + + @_check_other_SISO + def __add__(self, other): + + if isinstance(other, Parallel): + arg_list = list(other.args) + return Parallel(self, *arg_list) + + return Parallel(self, other) + + __radd__ = __add__ + + @_check_other_SISO + def __sub__(self, other): + return self + (-other) + + def __rsub__(self, other): + return -self + other + + @_check_other_SISO + def __mul__(self, other): + + arg_list = list(self.args) + return Series(*arg_list, other) + + def __truediv__(self, other): + if isinstance(other, TransferFunction): + return Series(*self.args, TransferFunction(other.den, other.num, other.var)) + elif isinstance(other, Series): + tf_self = self.rewrite(TransferFunction) + tf_other = other.rewrite(TransferFunction) + return tf_self / tf_other + elif (isinstance(other, Parallel) and len(other.args) == 2 + and isinstance(other.args[0], TransferFunction) and isinstance(other.args[1], Series)): + + if not self.var == other.var: + raise ValueError(filldedent(""" + All the transfer functions should use the same complex variable + of the Laplace transform.""")) + self_arg_list = set(self.args) + other_arg_list = set(other.args[1].args) + res = list(self_arg_list ^ other_arg_list) + if len(res) == 0: + return Feedback(self, other.args[0]) + elif len(res) == 1: + return Feedback(self, *res) + else: + return Feedback(self, Series(*res)) + else: + raise ValueError("This transfer function expression is invalid.") + + def __neg__(self): + return Series(TransferFunction(-1, 1, self.var), self) + + def to_expr(self): + """Returns the equivalent ``Expr`` object.""" + return Mul(*(arg.to_expr() for arg in self.args), evaluate=False) + + @property + def is_proper(self): + """ + Returns True if degree of the numerator polynomial of the resultant transfer + function is less than or equal to degree of the denominator polynomial of + the same, else False. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, Series + >>> tf1 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s) + >>> tf2 = TransferFunction(p**2 - 4*p, p**3 + 3*s + 2, s) + >>> tf3 = TransferFunction(s, s**2 + s + 1, s) + >>> S1 = Series(-tf2, tf1) + >>> S1.is_proper + False + >>> S2 = Series(tf1, tf2, tf3) + >>> S2.is_proper + True + + """ + return self.doit().is_proper + + @property + def is_strictly_proper(self): + """ + Returns True if degree of the numerator polynomial of the resultant transfer + function is strictly less than degree of the denominator polynomial of + the same, else False. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, Series + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf2 = TransferFunction(s**3 - 2, s**2 + 5*s + 6, s) + >>> tf3 = TransferFunction(1, s**2 + s + 1, s) + >>> S1 = Series(tf1, tf2) + >>> S1.is_strictly_proper + False + >>> S2 = Series(tf1, tf2, tf3) + >>> S2.is_strictly_proper + True + + """ + return self.doit().is_strictly_proper + + @property + def is_biproper(self): + r""" + Returns True if degree of the numerator polynomial of the resultant transfer + function is equal to degree of the denominator polynomial of + the same, else False. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, Series + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf2 = TransferFunction(p, s**2, s) + >>> tf3 = TransferFunction(s**2, 1, s) + >>> S1 = Series(tf1, -tf2) + >>> S1.is_biproper + False + >>> S2 = Series(tf2, tf3) + >>> S2.is_biproper + True + + """ + return self.doit().is_biproper + + +def _mat_mul_compatible(*args): + """To check whether shapes are compatible for matrix mul.""" + return all(args[i].num_outputs == args[i+1].num_inputs for i in range(len(args)-1)) + + +class MIMOSeries(MIMOLinearTimeInvariant): + r""" + A class for representing a series configuration of MIMO systems. + + Parameters + ========== + + args : MIMOLinearTimeInvariant + MIMO systems in a series configuration. + evaluate : Boolean, Keyword + When passed ``True``, returns the equivalent + ``MIMOSeries(*args).doit()``. Set to ``False`` by default. + + Raises + ====== + + ValueError + When no argument is passed. + + ``var`` attribute is not same for every system. + + ``num_outputs`` of the MIMO system is not equal to the + ``num_inputs`` of its adjacent MIMO system. (Matrix + multiplication constraint, basically) + TypeError + Any of the passed ``*args`` has unsupported type + + A combination of SISO and MIMO systems is + passed. There should be homogeneity in the + type of systems passed, MIMO in this case. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import MIMOSeries, TransferFunctionMatrix + >>> from sympy import Matrix, pprint + >>> mat_a = Matrix([[5*s], [5]]) # 2 Outputs 1 Input + >>> mat_b = Matrix([[5, 1/(6*s**2)]]) # 1 Output 2 Inputs + >>> mat_c = Matrix([[1, s], [5/s, 1]]) # 2 Outputs 2 Inputs + >>> tfm_a = TransferFunctionMatrix.from_Matrix(mat_a, s) + >>> tfm_b = TransferFunctionMatrix.from_Matrix(mat_b, s) + >>> tfm_c = TransferFunctionMatrix.from_Matrix(mat_c, s) + >>> MIMOSeries(tfm_c, tfm_b, tfm_a) + MIMOSeries(TransferFunctionMatrix(((TransferFunction(1, 1, s), TransferFunction(s, 1, s)), (TransferFunction(5, s, s), TransferFunction(1, 1, s)))), TransferFunctionMatrix(((TransferFunction(5, 1, s), TransferFunction(1, 6*s**2, s)),)), TransferFunctionMatrix(((TransferFunction(5*s, 1, s),), (TransferFunction(5, 1, s),)))) + >>> pprint(_, use_unicode=False) # For Better Visualization + [5*s] [1 s] + [---] [5 1 ] [- -] + [ 1 ] [- ----] [1 1] + [ ] *[1 2] *[ ] + [ 5 ] [ 6*s ]{t} [5 1] + [ - ] [- -] + [ 1 ]{t} [s 1]{t} + >>> MIMOSeries(tfm_c, tfm_b, tfm_a).doit() + TransferFunctionMatrix(((TransferFunction(150*s**4 + 25*s, 6*s**3, s), TransferFunction(150*s**4 + 5*s, 6*s**2, s)), (TransferFunction(150*s**3 + 25, 6*s**3, s), TransferFunction(150*s**3 + 5, 6*s**2, s)))) + >>> pprint(_, use_unicode=False) # (2 Inputs -A-> 2 Outputs) -> (2 Inputs -B-> 1 Output) -> (1 Input -C-> 2 Outputs) is equivalent to (2 Inputs -Series Equivalent-> 2 Outputs). + [ 4 4 ] + [150*s + 25*s 150*s + 5*s] + [------------- ------------] + [ 3 2 ] + [ 6*s 6*s ] + [ ] + [ 3 3 ] + [ 150*s + 25 150*s + 5 ] + [ ----------- ---------- ] + [ 3 2 ] + [ 6*s 6*s ]{t} + + Notes + ===== + + All the transfer function matrices should use the same complex variable ``var`` of the Laplace transform. + + ``MIMOSeries(A, B)`` is not equivalent to ``A*B``. It is always in the reverse order, that is ``B*A``. + + See Also + ======== + + Series, MIMOParallel + + """ + def __new__(cls, *args, evaluate=False): + + cls._check_args(args) + + if _mat_mul_compatible(*args): + obj = super().__new__(cls, *args) + + else: + raise ValueError(filldedent(""" + Number of input signals do not match the number + of output signals of adjacent systems for some args.""")) + + return obj.doit() if evaluate else obj + + @property + def var(self): + """ + Returns the complex variable used by all the transfer functions. + + Examples + ======== + + >>> from sympy.abc import p + >>> from sympy.physics.control.lti import TransferFunction, MIMOSeries, TransferFunctionMatrix + >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) + >>> G2 = TransferFunction(p, 4 - p, p) + >>> G3 = TransferFunction(0, p**4 - 1, p) + >>> tfm_1 = TransferFunctionMatrix([[G1, G2, G3]]) + >>> tfm_2 = TransferFunctionMatrix([[G1], [G2], [G3]]) + >>> MIMOSeries(tfm_2, tfm_1).var + p + + """ + return self.args[0].var + + @property + def num_inputs(self): + """Returns the number of input signals of the series system.""" + return self.args[0].num_inputs + + @property + def num_outputs(self): + """Returns the number of output signals of the series system.""" + return self.args[-1].num_outputs + + @property + def shape(self): + """Returns the shape of the equivalent MIMO system.""" + return self.num_outputs, self.num_inputs + + def doit(self, cancel=False, **kwargs): + """ + Returns the resultant transfer function matrix obtained after evaluating + the MIMO systems arranged in a series configuration. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, MIMOSeries, TransferFunctionMatrix + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) + >>> tfm1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf2]]) + >>> tfm2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf1]]) + >>> MIMOSeries(tfm2, tfm1).doit() + TransferFunctionMatrix(((TransferFunction(2*(-p + s)*(s**3 - 2)*(a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)**2*(s**4 + 5*s + 6)**2, s), TransferFunction((-p + s)**2*(s**3 - 2)*(a*p**2 + b*s) + (-p + s)*(a*p**2 + b*s)**2*(s**4 + 5*s + 6), (-p + s)**3*(s**4 + 5*s + 6), s)), (TransferFunction((-p + s)*(s**3 - 2)**2*(s**4 + 5*s + 6) + (s**3 - 2)*(a*p**2 + b*s)*(s**4 + 5*s + 6)**2, (-p + s)*(s**4 + 5*s + 6)**3, s), TransferFunction(2*(s**3 - 2)*(a*p**2 + b*s), (-p + s)*(s**4 + 5*s + 6), s)))) + + """ + _arg = (arg.doit()._expr_mat for arg in reversed(self.args)) + + if cancel: + res = MatMul(*_arg, evaluate=True) + return TransferFunctionMatrix.from_Matrix(res, self.var) + + _dummy_args, _dummy_dict = _dummify_args(_arg, self.var) + res = MatMul(*_dummy_args, evaluate=True) + temp_tfm = TransferFunctionMatrix.from_Matrix(res, self.var) + return temp_tfm.subs(_dummy_dict) + + def _eval_rewrite_as_TransferFunctionMatrix(self, *args, **kwargs): + return self.doit() + + @_check_other_MIMO + def __add__(self, other): + + if isinstance(other, MIMOParallel): + arg_list = list(other.args) + return MIMOParallel(self, *arg_list) + + return MIMOParallel(self, other) + + __radd__ = __add__ + + @_check_other_MIMO + def __sub__(self, other): + return self + (-other) + + def __rsub__(self, other): + return -self + other + + @_check_other_MIMO + def __mul__(self, other): + + if isinstance(other, MIMOSeries): + self_arg_list = list(self.args) + other_arg_list = list(other.args) + return MIMOSeries(*other_arg_list, *self_arg_list) # A*B = MIMOSeries(B, A) + + arg_list = list(self.args) + return MIMOSeries(other, *arg_list) + + def __neg__(self): + arg_list = list(self.args) + arg_list[0] = -arg_list[0] + return MIMOSeries(*arg_list) + + +class Parallel(SISOLinearTimeInvariant): + r""" + A class for representing a parallel configuration of SISO systems. + + Parameters + ========== + + args : SISOLinearTimeInvariant + SISO systems in a parallel arrangement. + evaluate : Boolean, Keyword + When passed ``True``, returns the equivalent + ``Parallel(*args).doit()``. Set to ``False`` by default. + + Raises + ====== + + ValueError + When no argument is passed. + + ``var`` attribute is not same for every system. + TypeError + Any of the passed ``*args`` has unsupported type + + A combination of SISO and MIMO systems is + passed. There should be homogeneity in the + type of systems passed. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, Parallel, Series + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) + >>> tf3 = TransferFunction(p**2, p + s, s) + >>> P1 = Parallel(tf1, tf2) + >>> P1 + Parallel(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)) + >>> P1.var + s + >>> P2 = Parallel(tf2, Series(tf3, -tf1)) + >>> P2 + Parallel(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), Series(TransferFunction(p**2, p + s, s), TransferFunction(-a*p**2 - b*s, -p + s, s))) + >>> P2.var + s + >>> P3 = Parallel(Series(tf1, tf2), Series(tf2, tf3)) + >>> P3 + Parallel(Series(TransferFunction(a*p**2 + b*s, -p + s, s), TransferFunction(s**3 - 2, s**4 + 5*s + 6, s)), Series(TransferFunction(s**3 - 2, s**4 + 5*s + 6, s), TransferFunction(p**2, p + s, s))) + >>> P3.var + s + + You can get the resultant transfer function by using ``.doit()`` method: + + >>> Parallel(tf1, tf2, -tf3).doit() + TransferFunction(-p**2*(-p + s)*(s**4 + 5*s + 6) + (-p + s)*(p + s)*(s**3 - 2) + (p + s)*(a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(p + s)*(s**4 + 5*s + 6), s) + >>> Parallel(tf2, Series(tf1, -tf3)).doit() + TransferFunction(-p**2*(a*p**2 + b*s)*(s**4 + 5*s + 6) + (-p + s)*(p + s)*(s**3 - 2), (-p + s)*(p + s)*(s**4 + 5*s + 6), s) + + Notes + ===== + + All the transfer functions should use the same complex variable + ``var`` of the Laplace transform. + + See Also + ======== + + Series, TransferFunction, Feedback + + """ + def __new__(cls, *args, evaluate=False): + + args = _flatten_args(args, Parallel) + cls._check_args(args) + obj = super().__new__(cls, *args) + + return obj.doit() if evaluate else obj + + @property + def var(self): + """ + Returns the complex variable used by all the transfer functions. + + Examples + ======== + + >>> from sympy.abc import p + >>> from sympy.physics.control.lti import TransferFunction, Parallel, Series + >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) + >>> G2 = TransferFunction(p, 4 - p, p) + >>> G3 = TransferFunction(0, p**4 - 1, p) + >>> Parallel(G1, G2).var + p + >>> Parallel(-G3, Series(G1, G2)).var + p + + """ + return self.args[0].var + + def doit(self, **hints): + """ + Returns the resultant transfer function obtained after evaluating + the transfer functions in parallel configuration. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, Parallel + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) + >>> Parallel(tf2, tf1).doit() + TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s) + >>> Parallel(-tf1, -tf2).doit() + TransferFunction((2 - s**3)*(-p + s) + (-a*p**2 - b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s) + + """ + + _arg = (arg.doit().to_expr() for arg in self.args) + res = Add(*_arg).as_numer_denom() + return TransferFunction(*res, self.var) + + def _eval_rewrite_as_TransferFunction(self, *args, **kwargs): + return self.doit() + + @_check_other_SISO + def __add__(self, other): + + self_arg_list = list(self.args) + return Parallel(*self_arg_list, other) + + __radd__ = __add__ + + @_check_other_SISO + def __sub__(self, other): + return self + (-other) + + def __rsub__(self, other): + return -self + other + + @_check_other_SISO + def __mul__(self, other): + + if isinstance(other, Series): + arg_list = list(other.args) + return Series(self, *arg_list) + + return Series(self, other) + + def __neg__(self): + return Series(TransferFunction(-1, 1, self.var), self) + + def to_expr(self): + """Returns the equivalent ``Expr`` object.""" + return Add(*(arg.to_expr() for arg in self.args), evaluate=False) + + @property + def is_proper(self): + """ + Returns True if degree of the numerator polynomial of the resultant transfer + function is less than or equal to degree of the denominator polynomial of + the same, else False. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, Parallel + >>> tf1 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s) + >>> tf2 = TransferFunction(p**2 - 4*p, p**3 + 3*s + 2, s) + >>> tf3 = TransferFunction(s, s**2 + s + 1, s) + >>> P1 = Parallel(-tf2, tf1) + >>> P1.is_proper + False + >>> P2 = Parallel(tf2, tf3) + >>> P2.is_proper + True + + """ + return self.doit().is_proper + + @property + def is_strictly_proper(self): + """ + Returns True if degree of the numerator polynomial of the resultant transfer + function is strictly less than degree of the denominator polynomial of + the same, else False. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, Parallel + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) + >>> tf3 = TransferFunction(s, s**2 + s + 1, s) + >>> P1 = Parallel(tf1, tf2) + >>> P1.is_strictly_proper + False + >>> P2 = Parallel(tf2, tf3) + >>> P2.is_strictly_proper + True + + """ + return self.doit().is_strictly_proper + + @property + def is_biproper(self): + """ + Returns True if degree of the numerator polynomial of the resultant transfer + function is equal to degree of the denominator polynomial of + the same, else False. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, Parallel + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf2 = TransferFunction(p**2, p + s, s) + >>> tf3 = TransferFunction(s, s**2 + s + 1, s) + >>> P1 = Parallel(tf1, -tf2) + >>> P1.is_biproper + True + >>> P2 = Parallel(tf2, tf3) + >>> P2.is_biproper + False + + """ + return self.doit().is_biproper + + +class MIMOParallel(MIMOLinearTimeInvariant): + r""" + A class for representing a parallel configuration of MIMO systems. + + Parameters + ========== + + args : MIMOLinearTimeInvariant + MIMO Systems in a parallel arrangement. + evaluate : Boolean, Keyword + When passed ``True``, returns the equivalent + ``MIMOParallel(*args).doit()``. Set to ``False`` by default. + + Raises + ====== + + ValueError + When no argument is passed. + + ``var`` attribute is not same for every system. + + All MIMO systems passed do not have same shape. + TypeError + Any of the passed ``*args`` has unsupported type + + A combination of SISO and MIMO systems is + passed. There should be homogeneity in the + type of systems passed, MIMO in this case. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunctionMatrix, MIMOParallel + >>> from sympy import Matrix, pprint + >>> expr_1 = 1/s + >>> expr_2 = s/(s**2-1) + >>> expr_3 = (2 + s)/(s**2 - 1) + >>> expr_4 = 5 + >>> tfm_a = TransferFunctionMatrix.from_Matrix(Matrix([[expr_1, expr_2], [expr_3, expr_4]]), s) + >>> tfm_b = TransferFunctionMatrix.from_Matrix(Matrix([[expr_2, expr_1], [expr_4, expr_3]]), s) + >>> tfm_c = TransferFunctionMatrix.from_Matrix(Matrix([[expr_3, expr_4], [expr_1, expr_2]]), s) + >>> MIMOParallel(tfm_a, tfm_b, tfm_c) + MIMOParallel(TransferFunctionMatrix(((TransferFunction(1, s, s), TransferFunction(s, s**2 - 1, s)), (TransferFunction(s + 2, s**2 - 1, s), TransferFunction(5, 1, s)))), TransferFunctionMatrix(((TransferFunction(s, s**2 - 1, s), TransferFunction(1, s, s)), (TransferFunction(5, 1, s), TransferFunction(s + 2, s**2 - 1, s)))), TransferFunctionMatrix(((TransferFunction(s + 2, s**2 - 1, s), TransferFunction(5, 1, s)), (TransferFunction(1, s, s), TransferFunction(s, s**2 - 1, s))))) + >>> pprint(_, use_unicode=False) # For Better Visualization + [ 1 s ] [ s 1 ] [s + 2 5 ] + [ - ------] [------ - ] [------ - ] + [ s 2 ] [ 2 s ] [ 2 1 ] + [ s - 1] [s - 1 ] [s - 1 ] + [ ] + [ ] + [ ] + [s + 2 5 ] [ 5 s + 2 ] [ 1 s ] + [------ - ] [ - ------] [ - ------] + [ 2 1 ] [ 1 2 ] [ s 2 ] + [s - 1 ]{t} [ s - 1]{t} [ s - 1]{t} + >>> MIMOParallel(tfm_a, tfm_b, tfm_c).doit() + TransferFunctionMatrix(((TransferFunction(s**2 + s*(2*s + 2) - 1, s*(s**2 - 1), s), TransferFunction(2*s**2 + 5*s*(s**2 - 1) - 1, s*(s**2 - 1), s)), (TransferFunction(s**2 + s*(s + 2) + 5*s*(s**2 - 1) - 1, s*(s**2 - 1), s), TransferFunction(5*s**2 + 2*s - 3, s**2 - 1, s)))) + >>> pprint(_, use_unicode=False) + [ 2 2 / 2 \ ] + [ s + s*(2*s + 2) - 1 2*s + 5*s*\s - 1/ - 1] + [ -------------------- -----------------------] + [ / 2 \ / 2 \ ] + [ s*\s - 1/ s*\s - 1/ ] + [ ] + [ 2 / 2 \ 2 ] + [s + s*(s + 2) + 5*s*\s - 1/ - 1 5*s + 2*s - 3 ] + [--------------------------------- -------------- ] + [ / 2 \ 2 ] + [ s*\s - 1/ s - 1 ]{t} + + Notes + ===== + + All the transfer function matrices should use the same complex variable + ``var`` of the Laplace transform. + + See Also + ======== + + Parallel, MIMOSeries + + """ + def __new__(cls, *args, evaluate=False): + + args = _flatten_args(args, MIMOParallel) + + cls._check_args(args) + + if any(arg.shape != args[0].shape for arg in args): + raise TypeError("Shape of all the args is not equal.") + + obj = super().__new__(cls, *args) + + return obj.doit() if evaluate else obj + + @property + def var(self): + """ + Returns the complex variable used by all the systems. + + Examples + ======== + + >>> from sympy.abc import p + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOParallel + >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) + >>> G2 = TransferFunction(p, 4 - p, p) + >>> G3 = TransferFunction(0, p**4 - 1, p) + >>> G4 = TransferFunction(p**2, p**2 - 1, p) + >>> tfm_a = TransferFunctionMatrix([[G1, G2], [G3, G4]]) + >>> tfm_b = TransferFunctionMatrix([[G2, G1], [G4, G3]]) + >>> MIMOParallel(tfm_a, tfm_b).var + p + + """ + return self.args[0].var + + @property + def num_inputs(self): + """Returns the number of input signals of the parallel system.""" + return self.args[0].num_inputs + + @property + def num_outputs(self): + """Returns the number of output signals of the parallel system.""" + return self.args[0].num_outputs + + @property + def shape(self): + """Returns the shape of the equivalent MIMO system.""" + return self.num_outputs, self.num_inputs + + def doit(self, **hints): + """ + Returns the resultant transfer function matrix obtained after evaluating + the MIMO systems arranged in a parallel configuration. + + Examples + ======== + + >>> from sympy.abc import s, p, a, b + >>> from sympy.physics.control.lti import TransferFunction, MIMOParallel, TransferFunctionMatrix + >>> tf1 = TransferFunction(a*p**2 + b*s, s - p, s) + >>> tf2 = TransferFunction(s**3 - 2, s**4 + 5*s + 6, s) + >>> tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) + >>> tfm_2 = TransferFunctionMatrix([[tf2, tf1], [tf1, tf2]]) + >>> MIMOParallel(tfm_1, tfm_2).doit() + TransferFunctionMatrix(((TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s), TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s)), (TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s), TransferFunction((-p + s)*(s**3 - 2) + (a*p**2 + b*s)*(s**4 + 5*s + 6), (-p + s)*(s**4 + 5*s + 6), s)))) + + """ + _arg = (arg.doit()._expr_mat for arg in self.args) + res = MatAdd(*_arg, evaluate=True) + return TransferFunctionMatrix.from_Matrix(res, self.var) + + def _eval_rewrite_as_TransferFunctionMatrix(self, *args, **kwargs): + return self.doit() + + @_check_other_MIMO + def __add__(self, other): + + self_arg_list = list(self.args) + return MIMOParallel(*self_arg_list, other) + + __radd__ = __add__ + + @_check_other_MIMO + def __sub__(self, other): + return self + (-other) + + def __rsub__(self, other): + return -self + other + + @_check_other_MIMO + def __mul__(self, other): + + if isinstance(other, MIMOSeries): + arg_list = list(other.args) + return MIMOSeries(*arg_list, self) + + return MIMOSeries(other, self) + + def __neg__(self): + arg_list = [-arg for arg in list(self.args)] + return MIMOParallel(*arg_list) + + +class Feedback(TransferFunction): + r""" + A class for representing closed-loop feedback interconnection between two + SISO input/output systems. + + The first argument, ``sys1``, is the feedforward part of the closed-loop + system or in simple words, the dynamical model representing the process + to be controlled. The second argument, ``sys2``, is the feedback system + and controls the fed back signal to ``sys1``. Both ``sys1`` and ``sys2`` + can either be ``Series`` or ``TransferFunction`` objects. + + Parameters + ========== + + sys1 : Series, TransferFunction + The feedforward path system. + sys2 : Series, TransferFunction, optional + The feedback path system (often a feedback controller). + It is the model sitting on the feedback path. + + If not specified explicitly, the sys2 is + assumed to be unit (1.0) transfer function. + sign : int, optional + The sign of feedback. Can either be ``1`` + (for positive feedback) or ``-1`` (for negative feedback). + Default value is `-1`. + + Raises + ====== + + ValueError + When ``sys1`` and ``sys2`` are not using the + same complex variable of the Laplace transform. + + When a combination of ``sys1`` and ``sys2`` yields + zero denominator. + + TypeError + When either ``sys1`` or ``sys2`` is not a ``Series`` or a + ``TransferFunction`` object. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction, Feedback + >>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) + >>> controller = TransferFunction(5*s - 10, s + 7, s) + >>> F1 = Feedback(plant, controller) + >>> F1 + Feedback(TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s), TransferFunction(5*s - 10, s + 7, s), -1) + >>> F1.var + s + >>> F1.args + (TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s), TransferFunction(5*s - 10, s + 7, s), -1) + + You can get the feedforward and feedback path systems by using ``.sys1`` and ``.sys2`` respectively. + + >>> F1.sys1 + TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) + >>> F1.sys2 + TransferFunction(5*s - 10, s + 7, s) + + You can get the resultant closed loop transfer function obtained by negative feedback + interconnection using ``.doit()`` method. + + >>> F1.doit() + TransferFunction((s + 7)*(s**2 - 4*s + 2)*(3*s**2 + 7*s - 3), ((s + 7)*(s**2 - 4*s + 2) + (5*s - 10)*(3*s**2 + 7*s - 3))*(s**2 - 4*s + 2), s) + >>> G = TransferFunction(2*s**2 + 5*s + 1, s**2 + 2*s + 3, s) + >>> C = TransferFunction(5*s + 10, s + 10, s) + >>> F2 = Feedback(G*C, TransferFunction(1, 1, s)) + >>> F2.doit() + TransferFunction((s + 10)*(5*s + 10)*(s**2 + 2*s + 3)*(2*s**2 + 5*s + 1), (s + 10)*((s + 10)*(s**2 + 2*s + 3) + (5*s + 10)*(2*s**2 + 5*s + 1))*(s**2 + 2*s + 3), s) + + To negate a ``Feedback`` object, the ``-`` operator can be prepended: + + >>> -F1 + Feedback(TransferFunction(-3*s**2 - 7*s + 3, s**2 - 4*s + 2, s), TransferFunction(10 - 5*s, s + 7, s), -1) + >>> -F2 + Feedback(Series(TransferFunction(-1, 1, s), TransferFunction(2*s**2 + 5*s + 1, s**2 + 2*s + 3, s), TransferFunction(5*s + 10, s + 10, s)), TransferFunction(-1, 1, s), -1) + + See Also + ======== + + MIMOFeedback, Series, Parallel + + """ + def __new__(cls, sys1, sys2=None, sign=-1): + if not sys2: + sys2 = TransferFunction(1, 1, sys1.var) + + if not (isinstance(sys1, (TransferFunction, Series, Feedback)) + and isinstance(sys2, (TransferFunction, Series, Feedback))): + raise TypeError("Unsupported type for `sys1` or `sys2` of Feedback.") + + if sign not in [-1, 1]: + raise ValueError(filldedent(""" + Unsupported type for feedback. `sign` arg should + either be 1 (positive feedback loop) or -1 + (negative feedback loop).""")) + + if Mul(sys1.to_expr(), sys2.to_expr()).simplify() == sign: + raise ValueError("The equivalent system will have zero denominator.") + + if sys1.var != sys2.var: + raise ValueError(filldedent(""" + Both `sys1` and `sys2` should be using the + same complex variable.""")) + + return super(TransferFunction, cls).__new__(cls, sys1, sys2, _sympify(sign)) + + @property + def sys1(self): + """ + Returns the feedforward system of the feedback interconnection. + + Examples + ======== + + >>> from sympy.abc import s, p + >>> from sympy.physics.control.lti import TransferFunction, Feedback + >>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) + >>> controller = TransferFunction(5*s - 10, s + 7, s) + >>> F1 = Feedback(plant, controller) + >>> F1.sys1 + TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) + >>> G = TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p) + >>> C = TransferFunction(5*p + 10, p + 10, p) + >>> P = TransferFunction(1 - s, p + 2, p) + >>> F2 = Feedback(TransferFunction(1, 1, p), G*C*P) + >>> F2.sys1 + TransferFunction(1, 1, p) + + """ + return self.args[0] + + @property + def sys2(self): + """ + Returns the feedback controller of the feedback interconnection. + + Examples + ======== + + >>> from sympy.abc import s, p + >>> from sympy.physics.control.lti import TransferFunction, Feedback + >>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) + >>> controller = TransferFunction(5*s - 10, s + 7, s) + >>> F1 = Feedback(plant, controller) + >>> F1.sys2 + TransferFunction(5*s - 10, s + 7, s) + >>> G = TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p) + >>> C = TransferFunction(5*p + 10, p + 10, p) + >>> P = TransferFunction(1 - s, p + 2, p) + >>> F2 = Feedback(TransferFunction(1, 1, p), G*C*P) + >>> F2.sys2 + Series(TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p), TransferFunction(5*p + 10, p + 10, p), TransferFunction(1 - s, p + 2, p)) + + """ + return self.args[1] + + @property + def var(self): + """ + Returns the complex variable of the Laplace transform used by all + the transfer functions involved in the feedback interconnection. + + Examples + ======== + + >>> from sympy.abc import s, p + >>> from sympy.physics.control.lti import TransferFunction, Feedback + >>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) + >>> controller = TransferFunction(5*s - 10, s + 7, s) + >>> F1 = Feedback(plant, controller) + >>> F1.var + s + >>> G = TransferFunction(2*s**2 + 5*s + 1, p**2 + 2*p + 3, p) + >>> C = TransferFunction(5*p + 10, p + 10, p) + >>> P = TransferFunction(1 - s, p + 2, p) + >>> F2 = Feedback(TransferFunction(1, 1, p), G*C*P) + >>> F2.var + p + + """ + return self.sys1.var + + @property + def sign(self): + """ + Returns the type of MIMO Feedback model. ``1`` + for Positive and ``-1`` for Negative. + """ + return self.args[2] + + @property + def num(self): + """ + Returns the numerator of the closed loop feedback system. + """ + return self.sys1 + + @property + def den(self): + """ + Returns the denominator of the closed loop feedback model. + """ + unit = TransferFunction(1, 1, self.var) + arg_list = list(self.sys1.args) if isinstance(self.sys1, Series) else [self.sys1] + if self.sign == 1: + return Parallel(unit, -Series(self.sys2, *arg_list)) + return Parallel(unit, Series(self.sys2, *arg_list)) + + @property + def sensitivity(self): + """ + Returns the sensitivity function of the feedback loop. + + Sensitivity of a Feedback system is the ratio + of change in the open loop gain to the change in + the closed loop gain. + + .. note:: + This method would not return the complementary + sensitivity function. + + Examples + ======== + + >>> from sympy.abc import p + >>> from sympy.physics.control.lti import TransferFunction, Feedback + >>> C = TransferFunction(5*p + 10, p + 10, p) + >>> P = TransferFunction(1 - p, p + 2, p) + >>> F_1 = Feedback(P, C) + >>> F_1.sensitivity + 1/((1 - p)*(5*p + 10)/((p + 2)*(p + 10)) + 1) + + """ + + return 1/(1 - self.sign*self.sys1.to_expr()*self.sys2.to_expr()) + + def doit(self, cancel=False, expand=False, **hints): + """ + Returns the resultant transfer function obtained by the + feedback interconnection. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction, Feedback + >>> plant = TransferFunction(3*s**2 + 7*s - 3, s**2 - 4*s + 2, s) + >>> controller = TransferFunction(5*s - 10, s + 7, s) + >>> F1 = Feedback(plant, controller) + >>> F1.doit() + TransferFunction((s + 7)*(s**2 - 4*s + 2)*(3*s**2 + 7*s - 3), ((s + 7)*(s**2 - 4*s + 2) + (5*s - 10)*(3*s**2 + 7*s - 3))*(s**2 - 4*s + 2), s) + >>> G = TransferFunction(2*s**2 + 5*s + 1, s**2 + 2*s + 3, s) + >>> F2 = Feedback(G, TransferFunction(1, 1, s)) + >>> F2.doit() + TransferFunction((s**2 + 2*s + 3)*(2*s**2 + 5*s + 1), (s**2 + 2*s + 3)*(3*s**2 + 7*s + 4), s) + + Use kwarg ``expand=True`` to expand the resultant transfer function. + Use ``cancel=True`` to cancel out the common terms in numerator and + denominator. + + >>> F2.doit(cancel=True, expand=True) + TransferFunction(2*s**2 + 5*s + 1, 3*s**2 + 7*s + 4, s) + >>> F2.doit(expand=True) + TransferFunction(2*s**4 + 9*s**3 + 17*s**2 + 17*s + 3, 3*s**4 + 13*s**3 + 27*s**2 + 29*s + 12, s) + + """ + arg_list = list(self.sys1.args) if isinstance(self.sys1, Series) else [self.sys1] + # F_n and F_d are resultant TFs of num and den of Feedback. + F_n, unit = self.sys1.doit(), TransferFunction(1, 1, self.sys1.var) + if self.sign == -1: + F_d = Parallel(unit, Series(self.sys2, *arg_list)).doit() + else: + F_d = Parallel(unit, -Series(self.sys2, *arg_list)).doit() + + _resultant_tf = TransferFunction(F_n.num * F_d.den, F_n.den * F_d.num, F_n.var) + + if cancel: + _resultant_tf = _resultant_tf.simplify() + + if expand: + _resultant_tf = _resultant_tf.expand() + + return _resultant_tf + + def _eval_rewrite_as_TransferFunction(self, num, den, sign, **kwargs): + return self.doit() + + def to_expr(self): + """ + Converts a ``Feedback`` object to SymPy Expr. + + Examples + ======== + + >>> from sympy.abc import s, a, b + >>> from sympy.physics.control.lti import TransferFunction, Feedback + >>> from sympy import Expr + >>> tf1 = TransferFunction(a+s, 1, s) + >>> tf2 = TransferFunction(b+s, 1, s) + >>> fd1 = Feedback(tf1, tf2) + >>> fd1.to_expr() + (a + s)/((a + s)*(b + s) + 1) + >>> isinstance(_, Expr) + True + """ + + return self.doit().to_expr() + + def __neg__(self): + return Feedback(-self.sys1, -self.sys2, self.sign) + + +def _is_invertible(a, b, sign): + """ + Checks whether a given pair of MIMO + systems passed is invertible or not. + """ + _mat = eye(a.num_outputs) - sign*(a.doit()._expr_mat)*(b.doit()._expr_mat) + _det = _mat.det() + + return _det != 0 + + +class MIMOFeedback(MIMOLinearTimeInvariant): + r""" + A class for representing closed-loop feedback interconnection between two + MIMO input/output systems. + + Parameters + ========== + + sys1 : MIMOSeries, TransferFunctionMatrix + The MIMO system placed on the feedforward path. + sys2 : MIMOSeries, TransferFunctionMatrix + The system placed on the feedback path + (often a feedback controller). + sign : int, optional + The sign of feedback. Can either be ``1`` + (for positive feedback) or ``-1`` (for negative feedback). + Default value is `-1`. + + Raises + ====== + + ValueError + When ``sys1`` and ``sys2`` are not using the + same complex variable of the Laplace transform. + + Forward path model should have an equal number of inputs/outputs + to the feedback path outputs/inputs. + + When product of ``sys1`` and ``sys2`` is not a square matrix. + + When the equivalent MIMO system is not invertible. + + TypeError + When either ``sys1`` or ``sys2`` is not a ``MIMOSeries`` or a + ``TransferFunctionMatrix`` object. + + Examples + ======== + + >>> from sympy import Matrix, pprint + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunctionMatrix, MIMOFeedback + >>> plant_mat = Matrix([[1, 1/s], [0, 1]]) + >>> controller_mat = Matrix([[10, 0], [0, 10]]) # Constant Gain + >>> plant = TransferFunctionMatrix.from_Matrix(plant_mat, s) + >>> controller = TransferFunctionMatrix.from_Matrix(controller_mat, s) + >>> feedback = MIMOFeedback(plant, controller) # Negative Feedback (default) + >>> pprint(feedback, use_unicode=False) + / [1 1] [10 0 ] \-1 [1 1] + | [- -] [-- - ] | [- -] + | [1 s] [1 1 ] | [1 s] + |I + [ ] *[ ] | * [ ] + | [0 1] [0 10] | [0 1] + | [- -] [- --] | [- -] + \ [1 1]{t} [1 1 ]{t}/ [1 1]{t} + + To get the equivalent system matrix, use either ``doit`` or ``rewrite`` method. + + >>> pprint(feedback.doit(), use_unicode=False) + [1 1 ] + [-- -----] + [11 121*s] + [ ] + [0 1 ] + [- -- ] + [1 11 ]{t} + + To negate the ``MIMOFeedback`` object, use ``-`` operator. + + >>> neg_feedback = -feedback + >>> pprint(neg_feedback.doit(), use_unicode=False) + [-1 -1 ] + [--- -----] + [11 121*s] + [ ] + [ 0 -1 ] + [ - --- ] + [ 1 11 ]{t} + + See Also + ======== + + Feedback, MIMOSeries, MIMOParallel + + """ + def __new__(cls, sys1, sys2, sign=-1): + if not (isinstance(sys1, (TransferFunctionMatrix, MIMOSeries)) + and isinstance(sys2, (TransferFunctionMatrix, MIMOSeries))): + raise TypeError("Unsupported type for `sys1` or `sys2` of MIMO Feedback.") + + if sys1.num_inputs != sys2.num_outputs or \ + sys1.num_outputs != sys2.num_inputs: + raise ValueError(filldedent(""" + Product of `sys1` and `sys2` must + yield a square matrix.""")) + + if sign not in (-1, 1): + raise ValueError(filldedent(""" + Unsupported type for feedback. `sign` arg should + either be 1 (positive feedback loop) or -1 + (negative feedback loop).""")) + + if not _is_invertible(sys1, sys2, sign): + raise ValueError("Non-Invertible system inputted.") + if sys1.var != sys2.var: + raise ValueError(filldedent(""" + Both `sys1` and `sys2` should be using the + same complex variable.""")) + + return super().__new__(cls, sys1, sys2, _sympify(sign)) + + @property + def sys1(self): + r""" + Returns the system placed on the feedforward path of the MIMO feedback interconnection. + + Examples + ======== + + >>> from sympy import pprint + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback + >>> tf1 = TransferFunction(s**2 + s + 1, s**2 - s + 1, s) + >>> tf2 = TransferFunction(1, s, s) + >>> tf3 = TransferFunction(1, 1, s) + >>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) + >>> sys2 = TransferFunctionMatrix([[tf3, tf3], [tf3, tf2]]) + >>> F_1 = MIMOFeedback(sys1, sys2, 1) + >>> F_1.sys1 + TransferFunctionMatrix(((TransferFunction(s**2 + s + 1, s**2 - s + 1, s), TransferFunction(1, s, s)), (TransferFunction(1, s, s), TransferFunction(s**2 + s + 1, s**2 - s + 1, s)))) + >>> pprint(_, use_unicode=False) + [ 2 ] + [s + s + 1 1 ] + [---------- - ] + [ 2 s ] + [s - s + 1 ] + [ ] + [ 2 ] + [ 1 s + s + 1] + [ - ----------] + [ s 2 ] + [ s - s + 1]{t} + + """ + return self.args[0] + + @property + def sys2(self): + r""" + Returns the feedback controller of the MIMO feedback interconnection. + + Examples + ======== + + >>> from sympy import pprint + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback + >>> tf1 = TransferFunction(s**2, s**3 - s + 1, s) + >>> tf2 = TransferFunction(1, s, s) + >>> tf3 = TransferFunction(1, 1, s) + >>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) + >>> sys2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf2]]) + >>> F_1 = MIMOFeedback(sys1, sys2) + >>> F_1.sys2 + TransferFunctionMatrix(((TransferFunction(s**2, s**3 - s + 1, s), TransferFunction(1, 1, s)), (TransferFunction(1, 1, s), TransferFunction(1, s, s)))) + >>> pprint(_, use_unicode=False) + [ 2 ] + [ s 1] + [---------- -] + [ 3 1] + [s - s + 1 ] + [ ] + [ 1 1] + [ - -] + [ 1 s]{t} + + """ + return self.args[1] + + @property + def var(self): + r""" + Returns the complex variable of the Laplace transform used by all + the transfer functions involved in the MIMO feedback loop. + + Examples + ======== + + >>> from sympy.abc import p + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback + >>> tf1 = TransferFunction(p, 1 - p, p) + >>> tf2 = TransferFunction(1, p, p) + >>> tf3 = TransferFunction(1, 1, p) + >>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) + >>> sys2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf2]]) + >>> F_1 = MIMOFeedback(sys1, sys2, 1) # Positive feedback + >>> F_1.var + p + + """ + return self.sys1.var + + @property + def sign(self): + r""" + Returns the type of feedback interconnection of two models. ``1`` + for Positive and ``-1`` for Negative. + """ + return self.args[2] + + @property + def sensitivity(self): + r""" + Returns the sensitivity function matrix of the feedback loop. + + Sensitivity of a closed-loop system is the ratio of change + in the open loop gain to the change in the closed loop gain. + + .. note:: + This method would not return the complementary + sensitivity function. + + Examples + ======== + + >>> from sympy import pprint + >>> from sympy.abc import p + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback + >>> tf1 = TransferFunction(p, 1 - p, p) + >>> tf2 = TransferFunction(1, p, p) + >>> tf3 = TransferFunction(1, 1, p) + >>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf2, tf1]]) + >>> sys2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf2]]) + >>> F_1 = MIMOFeedback(sys1, sys2, 1) # Positive feedback + >>> F_2 = MIMOFeedback(sys1, sys2) # Negative feedback + >>> pprint(F_1.sensitivity, use_unicode=False) + [ 4 3 2 5 4 2 ] + [- p + 3*p - 4*p + 3*p - 1 p - 2*p + 3*p - 3*p + 1 ] + [---------------------------- -----------------------------] + [ 4 3 2 5 4 3 2 ] + [ p + 3*p - 8*p + 8*p - 3 p + 3*p - 8*p + 8*p - 3*p] + [ ] + [ 4 3 2 3 2 ] + [ p - p - p + p 3*p - 6*p + 4*p - 1 ] + [ -------------------------- -------------------------- ] + [ 4 3 2 4 3 2 ] + [ p + 3*p - 8*p + 8*p - 3 p + 3*p - 8*p + 8*p - 3 ] + >>> pprint(F_2.sensitivity, use_unicode=False) + [ 4 3 2 5 4 2 ] + [p - 3*p + 2*p + p - 1 p - 2*p + 3*p - 3*p + 1] + [------------------------ --------------------------] + [ 4 3 5 4 2 ] + [ p - 3*p + 2*p - 1 p - 3*p + 2*p - p ] + [ ] + [ 4 3 2 4 3 ] + [ p - p - p + p 2*p - 3*p + 2*p - 1 ] + [ ------------------- --------------------- ] + [ 4 3 4 3 ] + [ p - 3*p + 2*p - 1 p - 3*p + 2*p - 1 ] + + """ + _sys1_mat = self.sys1.doit()._expr_mat + _sys2_mat = self.sys2.doit()._expr_mat + + return (eye(self.sys1.num_inputs) - \ + self.sign*_sys1_mat*_sys2_mat).inv() + + def doit(self, cancel=True, expand=False, **hints): + r""" + Returns the resultant transfer function matrix obtained by the + feedback interconnection. + + Examples + ======== + + >>> from sympy import pprint + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, MIMOFeedback + >>> tf1 = TransferFunction(s, 1 - s, s) + >>> tf2 = TransferFunction(1, s, s) + >>> tf3 = TransferFunction(5, 1, s) + >>> tf4 = TransferFunction(s - 1, s, s) + >>> tf5 = TransferFunction(0, 1, s) + >>> sys1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]]) + >>> sys2 = TransferFunctionMatrix([[tf3, tf5], [tf5, tf5]]) + >>> F_1 = MIMOFeedback(sys1, sys2, 1) + >>> pprint(F_1, use_unicode=False) + / [ s 1 ] [5 0] \-1 [ s 1 ] + | [----- - ] [- -] | [----- - ] + | [1 - s s ] [1 1] | [1 - s s ] + |I - [ ] *[ ] | * [ ] + | [ 5 s - 1] [0 0] | [ 5 s - 1] + | [ - -----] [- -] | [ - -----] + \ [ 1 s ]{t} [1 1]{t}/ [ 1 s ]{t} + >>> pprint(F_1.doit(), use_unicode=False) + [ -s s - 1 ] + [------- ----------- ] + [6*s - 1 s*(6*s - 1) ] + [ ] + [5*s - 5 (s - 1)*(6*s + 24)] + [------- ------------------] + [6*s - 1 s*(6*s - 1) ]{t} + + If the user wants the resultant ``TransferFunctionMatrix`` object without + canceling the common factors then the ``cancel`` kwarg should be passed ``False``. + + >>> pprint(F_1.doit(cancel=False), use_unicode=False) + [ s*(s - 1) s - 1 ] + [ ----------------- ----------- ] + [ (1 - s)*(6*s - 1) s*(6*s - 1) ] + [ ] + [s*(25*s - 25) + 5*(1 - s)*(6*s - 1) s*(s - 1)*(6*s - 1) + s*(25*s - 25)] + [----------------------------------- -----------------------------------] + [ (1 - s)*(6*s - 1) 2 ] + [ s *(6*s - 1) ]{t} + + If the user wants the expanded form of the resultant transfer function matrix, + the ``expand`` kwarg should be passed as ``True``. + + >>> pprint(F_1.doit(expand=True), use_unicode=False) + [ -s s - 1 ] + [------- -------- ] + [6*s - 1 2 ] + [ 6*s - s ] + [ ] + [ 2 ] + [5*s - 5 6*s + 18*s - 24] + [------- ----------------] + [6*s - 1 2 ] + [ 6*s - s ]{t} + + """ + _mat = self.sensitivity * self.sys1.doit()._expr_mat + + _resultant_tfm = _to_TFM(_mat, self.var) + + if cancel: + _resultant_tfm = _resultant_tfm.simplify() + + if expand: + _resultant_tfm = _resultant_tfm.expand() + + return _resultant_tfm + + def _eval_rewrite_as_TransferFunctionMatrix(self, sys1, sys2, sign, **kwargs): + return self.doit() + + def __neg__(self): + return MIMOFeedback(-self.sys1, -self.sys2, self.sign) + + +def _to_TFM(mat, var): + """Private method to convert ImmutableMatrix to TransferFunctionMatrix efficiently""" + to_tf = lambda expr: TransferFunction.from_rational_expression(expr, var) + arg = [[to_tf(expr) for expr in row] for row in mat.tolist()] + return TransferFunctionMatrix(arg) + + +class TransferFunctionMatrix(MIMOLinearTimeInvariant): + r""" + A class for representing the MIMO (multiple-input and multiple-output) + generalization of the SISO (single-input and single-output) transfer function. + + It is a matrix of transfer functions (``TransferFunction``, SISO-``Series`` or SISO-``Parallel``). + There is only one argument, ``arg`` which is also the compulsory argument. + ``arg`` is expected to be strictly of the type list of lists + which holds the transfer functions or reducible to transfer functions. + + Parameters + ========== + + arg : Nested ``List`` (strictly). + Users are expected to input a nested list of ``TransferFunction``, ``Series`` + and/or ``Parallel`` objects. + + Examples + ======== + + .. note:: + ``pprint()`` can be used for better visualization of ``TransferFunctionMatrix`` objects. + + >>> from sympy.abc import s, p, a + >>> from sympy import pprint + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, Series, Parallel + >>> tf_1 = TransferFunction(s + a, s**2 + s + 1, s) + >>> tf_2 = TransferFunction(p**4 - 3*p + 2, s + p, s) + >>> tf_3 = TransferFunction(3, s + 2, s) + >>> tf_4 = TransferFunction(-a + p, 9*s - 9, s) + >>> tfm_1 = TransferFunctionMatrix([[tf_1], [tf_2], [tf_3]]) + >>> tfm_1 + TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(3, s + 2, s),))) + >>> tfm_1.var + s + >>> tfm_1.num_inputs + 1 + >>> tfm_1.num_outputs + 3 + >>> tfm_1.shape + (3, 1) + >>> tfm_1.args + (((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(3, s + 2, s),)),) + >>> tfm_2 = TransferFunctionMatrix([[tf_1, -tf_3], [tf_2, -tf_1], [tf_3, -tf_2]]) + >>> tfm_2 + TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s), TransferFunction(-3, s + 2, s)), (TransferFunction(p**4 - 3*p + 2, p + s, s), TransferFunction(-a - s, s**2 + s + 1, s)), (TransferFunction(3, s + 2, s), TransferFunction(-p**4 + 3*p - 2, p + s, s)))) + >>> pprint(tfm_2, use_unicode=False) # pretty-printing for better visualization + [ a + s -3 ] + [ ---------- ----- ] + [ 2 s + 2 ] + [ s + s + 1 ] + [ ] + [ 4 ] + [p - 3*p + 2 -a - s ] + [------------ ---------- ] + [ p + s 2 ] + [ s + s + 1 ] + [ ] + [ 4 ] + [ 3 - p + 3*p - 2] + [ ----- --------------] + [ s + 2 p + s ]{t} + + TransferFunctionMatrix can be transposed, if user wants to switch the input and output transfer functions + + >>> tfm_2.transpose() + TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s), TransferFunction(p**4 - 3*p + 2, p + s, s), TransferFunction(3, s + 2, s)), (TransferFunction(-3, s + 2, s), TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(-p**4 + 3*p - 2, p + s, s)))) + >>> pprint(_, use_unicode=False) + [ 4 ] + [ a + s p - 3*p + 2 3 ] + [---------- ------------ ----- ] + [ 2 p + s s + 2 ] + [s + s + 1 ] + [ ] + [ 4 ] + [ -3 -a - s - p + 3*p - 2] + [ ----- ---------- --------------] + [ s + 2 2 p + s ] + [ s + s + 1 ]{t} + + >>> tf_5 = TransferFunction(5, s, s) + >>> tf_6 = TransferFunction(5*s, (2 + s**2), s) + >>> tf_7 = TransferFunction(5, (s*(2 + s**2)), s) + >>> tf_8 = TransferFunction(5, 1, s) + >>> tfm_3 = TransferFunctionMatrix([[tf_5, tf_6], [tf_7, tf_8]]) + >>> tfm_3 + TransferFunctionMatrix(((TransferFunction(5, s, s), TransferFunction(5*s, s**2 + 2, s)), (TransferFunction(5, s*(s**2 + 2), s), TransferFunction(5, 1, s)))) + >>> pprint(tfm_3, use_unicode=False) + [ 5 5*s ] + [ - ------] + [ s 2 ] + [ s + 2] + [ ] + [ 5 5 ] + [---------- - ] + [ / 2 \ 1 ] + [s*\s + 2/ ]{t} + >>> tfm_3.var + s + >>> tfm_3.shape + (2, 2) + >>> tfm_3.num_outputs + 2 + >>> tfm_3.num_inputs + 2 + >>> tfm_3.args + (((TransferFunction(5, s, s), TransferFunction(5*s, s**2 + 2, s)), (TransferFunction(5, s*(s**2 + 2), s), TransferFunction(5, 1, s))),) + + To access the ``TransferFunction`` at any index in the ``TransferFunctionMatrix``, use the index notation. + + >>> tfm_3[1, 0] # gives the TransferFunction present at 2nd Row and 1st Col. Similar to that in Matrix classes + TransferFunction(5, s*(s**2 + 2), s) + >>> tfm_3[0, 0] # gives the TransferFunction present at 1st Row and 1st Col. + TransferFunction(5, s, s) + >>> tfm_3[:, 0] # gives the first column + TransferFunctionMatrix(((TransferFunction(5, s, s),), (TransferFunction(5, s*(s**2 + 2), s),))) + >>> pprint(_, use_unicode=False) + [ 5 ] + [ - ] + [ s ] + [ ] + [ 5 ] + [----------] + [ / 2 \] + [s*\s + 2/]{t} + >>> tfm_3[0, :] # gives the first row + TransferFunctionMatrix(((TransferFunction(5, s, s), TransferFunction(5*s, s**2 + 2, s)),)) + >>> pprint(_, use_unicode=False) + [5 5*s ] + [- ------] + [s 2 ] + [ s + 2]{t} + + To negate a transfer function matrix, ``-`` operator can be prepended: + + >>> tfm_4 = TransferFunctionMatrix([[tf_2], [-tf_1], [tf_3]]) + >>> -tfm_4 + TransferFunctionMatrix(((TransferFunction(-p**4 + 3*p - 2, p + s, s),), (TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(-3, s + 2, s),))) + >>> tfm_5 = TransferFunctionMatrix([[tf_1, tf_2], [tf_3, -tf_1]]) + >>> -tfm_5 + TransferFunctionMatrix(((TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(-p**4 + 3*p - 2, p + s, s)), (TransferFunction(-3, s + 2, s), TransferFunction(a + s, s**2 + s + 1, s)))) + + ``subs()`` returns the ``TransferFunctionMatrix`` object with the value substituted in the expression. This will not + mutate your original ``TransferFunctionMatrix``. + + >>> tfm_2.subs(p, 2) # substituting p everywhere in tfm_2 with 2. + TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s), TransferFunction(-3, s + 2, s)), (TransferFunction(12, s + 2, s), TransferFunction(-a - s, s**2 + s + 1, s)), (TransferFunction(3, s + 2, s), TransferFunction(-12, s + 2, s)))) + >>> pprint(_, use_unicode=False) + [ a + s -3 ] + [---------- ----- ] + [ 2 s + 2 ] + [s + s + 1 ] + [ ] + [ 12 -a - s ] + [ ----- ----------] + [ s + 2 2 ] + [ s + s + 1] + [ ] + [ 3 -12 ] + [ ----- ----- ] + [ s + 2 s + 2 ]{t} + >>> pprint(tfm_2, use_unicode=False) # State of tfm_2 is unchanged after substitution + [ a + s -3 ] + [ ---------- ----- ] + [ 2 s + 2 ] + [ s + s + 1 ] + [ ] + [ 4 ] + [p - 3*p + 2 -a - s ] + [------------ ---------- ] + [ p + s 2 ] + [ s + s + 1 ] + [ ] + [ 4 ] + [ 3 - p + 3*p - 2] + [ ----- --------------] + [ s + 2 p + s ]{t} + + ``subs()`` also supports multiple substitutions. + + >>> tfm_2.subs({p: 2, a: 1}) # substituting p with 2 and a with 1 + TransferFunctionMatrix(((TransferFunction(s + 1, s**2 + s + 1, s), TransferFunction(-3, s + 2, s)), (TransferFunction(12, s + 2, s), TransferFunction(-s - 1, s**2 + s + 1, s)), (TransferFunction(3, s + 2, s), TransferFunction(-12, s + 2, s)))) + >>> pprint(_, use_unicode=False) + [ s + 1 -3 ] + [---------- ----- ] + [ 2 s + 2 ] + [s + s + 1 ] + [ ] + [ 12 -s - 1 ] + [ ----- ----------] + [ s + 2 2 ] + [ s + s + 1] + [ ] + [ 3 -12 ] + [ ----- ----- ] + [ s + 2 s + 2 ]{t} + + Users can reduce the ``Series`` and ``Parallel`` elements of the matrix to ``TransferFunction`` by using + ``doit()``. + + >>> tfm_6 = TransferFunctionMatrix([[Series(tf_3, tf_4), Parallel(tf_3, tf_4)]]) + >>> tfm_6 + TransferFunctionMatrix(((Series(TransferFunction(3, s + 2, s), TransferFunction(-a + p, 9*s - 9, s)), Parallel(TransferFunction(3, s + 2, s), TransferFunction(-a + p, 9*s - 9, s))),)) + >>> pprint(tfm_6, use_unicode=False) + [-a + p 3 -a + p 3 ] + [-------*----- ------- + -----] + [9*s - 9 s + 2 9*s - 9 s + 2]{t} + >>> tfm_6.doit() + TransferFunctionMatrix(((TransferFunction(-3*a + 3*p, (s + 2)*(9*s - 9), s), TransferFunction(27*s + (-a + p)*(s + 2) - 27, (s + 2)*(9*s - 9), s)),)) + >>> pprint(_, use_unicode=False) + [ -3*a + 3*p 27*s + (-a + p)*(s + 2) - 27] + [----------------- ----------------------------] + [(s + 2)*(9*s - 9) (s + 2)*(9*s - 9) ]{t} + >>> tf_9 = TransferFunction(1, s, s) + >>> tf_10 = TransferFunction(1, s**2, s) + >>> tfm_7 = TransferFunctionMatrix([[Series(tf_9, tf_10), tf_9], [tf_10, Parallel(tf_9, tf_10)]]) + >>> tfm_7 + TransferFunctionMatrix(((Series(TransferFunction(1, s, s), TransferFunction(1, s**2, s)), TransferFunction(1, s, s)), (TransferFunction(1, s**2, s), Parallel(TransferFunction(1, s, s), TransferFunction(1, s**2, s))))) + >>> pprint(tfm_7, use_unicode=False) + [ 1 1 ] + [---- - ] + [ 2 s ] + [s*s ] + [ ] + [ 1 1 1] + [ -- -- + -] + [ 2 2 s] + [ s s ]{t} + >>> tfm_7.doit() + TransferFunctionMatrix(((TransferFunction(1, s**3, s), TransferFunction(1, s, s)), (TransferFunction(1, s**2, s), TransferFunction(s**2 + s, s**3, s)))) + >>> pprint(_, use_unicode=False) + [1 1 ] + [-- - ] + [ 3 s ] + [s ] + [ ] + [ 2 ] + [1 s + s] + [-- ------] + [ 2 3 ] + [s s ]{t} + + Addition, subtraction, and multiplication of transfer function matrices can form + unevaluated ``Series`` or ``Parallel`` objects. + + - For addition and subtraction: + All the transfer function matrices must have the same shape. + + - For multiplication (C = A * B): + The number of inputs of the first transfer function matrix (A) must be equal to the + number of outputs of the second transfer function matrix (B). + + Also, use pretty-printing (``pprint``) to analyse better. + + >>> tfm_8 = TransferFunctionMatrix([[tf_3], [tf_2], [-tf_1]]) + >>> tfm_9 = TransferFunctionMatrix([[-tf_3]]) + >>> tfm_10 = TransferFunctionMatrix([[tf_1], [tf_2], [tf_4]]) + >>> tfm_11 = TransferFunctionMatrix([[tf_4], [-tf_1]]) + >>> tfm_12 = TransferFunctionMatrix([[tf_4, -tf_1, tf_3], [-tf_2, -tf_4, -tf_3]]) + >>> tfm_8 + tfm_10 + MIMOParallel(TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))), TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a + p, 9*s - 9, s),)))) + >>> pprint(_, use_unicode=False) + [ 3 ] [ a + s ] + [ ----- ] [ ---------- ] + [ s + 2 ] [ 2 ] + [ ] [ s + s + 1 ] + [ 4 ] [ ] + [p - 3*p + 2] [ 4 ] + [------------] + [p - 3*p + 2] + [ p + s ] [------------] + [ ] [ p + s ] + [ -a - s ] [ ] + [ ---------- ] [ -a + p ] + [ 2 ] [ ------- ] + [ s + s + 1 ]{t} [ 9*s - 9 ]{t} + >>> -tfm_10 - tfm_8 + MIMOParallel(TransferFunctionMatrix(((TransferFunction(-a - s, s**2 + s + 1, s),), (TransferFunction(-p**4 + 3*p - 2, p + s, s),), (TransferFunction(a - p, 9*s - 9, s),))), TransferFunctionMatrix(((TransferFunction(-3, s + 2, s),), (TransferFunction(-p**4 + 3*p - 2, p + s, s),), (TransferFunction(a + s, s**2 + s + 1, s),)))) + >>> pprint(_, use_unicode=False) + [ -a - s ] [ -3 ] + [ ---------- ] [ ----- ] + [ 2 ] [ s + 2 ] + [ s + s + 1 ] [ ] + [ ] [ 4 ] + [ 4 ] [- p + 3*p - 2] + [- p + 3*p - 2] + [--------------] + [--------------] [ p + s ] + [ p + s ] [ ] + [ ] [ a + s ] + [ a - p ] [ ---------- ] + [ ------- ] [ 2 ] + [ 9*s - 9 ]{t} [ s + s + 1 ]{t} + >>> tfm_12 * tfm_8 + MIMOSeries(TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))), TransferFunctionMatrix(((TransferFunction(-a + p, 9*s - 9, s), TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(3, s + 2, s)), (TransferFunction(-p**4 + 3*p - 2, p + s, s), TransferFunction(a - p, 9*s - 9, s), TransferFunction(-3, s + 2, s))))) + >>> pprint(_, use_unicode=False) + [ 3 ] + [ ----- ] + [ -a + p -a - s 3 ] [ s + 2 ] + [ ------- ---------- -----] [ ] + [ 9*s - 9 2 s + 2] [ 4 ] + [ s + s + 1 ] [p - 3*p + 2] + [ ] *[------------] + [ 4 ] [ p + s ] + [- p + 3*p - 2 a - p -3 ] [ ] + [-------------- ------- -----] [ -a - s ] + [ p + s 9*s - 9 s + 2]{t} [ ---------- ] + [ 2 ] + [ s + s + 1 ]{t} + >>> tfm_12 * tfm_8 * tfm_9 + MIMOSeries(TransferFunctionMatrix(((TransferFunction(-3, s + 2, s),),)), TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))), TransferFunctionMatrix(((TransferFunction(-a + p, 9*s - 9, s), TransferFunction(-a - s, s**2 + s + 1, s), TransferFunction(3, s + 2, s)), (TransferFunction(-p**4 + 3*p - 2, p + s, s), TransferFunction(a - p, 9*s - 9, s), TransferFunction(-3, s + 2, s))))) + >>> pprint(_, use_unicode=False) + [ 3 ] + [ ----- ] + [ -a + p -a - s 3 ] [ s + 2 ] + [ ------- ---------- -----] [ ] + [ 9*s - 9 2 s + 2] [ 4 ] + [ s + s + 1 ] [p - 3*p + 2] [ -3 ] + [ ] *[------------] *[-----] + [ 4 ] [ p + s ] [s + 2]{t} + [- p + 3*p - 2 a - p -3 ] [ ] + [-------------- ------- -----] [ -a - s ] + [ p + s 9*s - 9 s + 2]{t} [ ---------- ] + [ 2 ] + [ s + s + 1 ]{t} + >>> tfm_10 + tfm_8*tfm_9 + MIMOParallel(TransferFunctionMatrix(((TransferFunction(a + s, s**2 + s + 1, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a + p, 9*s - 9, s),))), MIMOSeries(TransferFunctionMatrix(((TransferFunction(-3, s + 2, s),),)), TransferFunctionMatrix(((TransferFunction(3, s + 2, s),), (TransferFunction(p**4 - 3*p + 2, p + s, s),), (TransferFunction(-a - s, s**2 + s + 1, s),))))) + >>> pprint(_, use_unicode=False) + [ a + s ] [ 3 ] + [ ---------- ] [ ----- ] + [ 2 ] [ s + 2 ] + [ s + s + 1 ] [ ] + [ ] [ 4 ] + [ 4 ] [p - 3*p + 2] [ -3 ] + [p - 3*p + 2] + [------------] *[-----] + [------------] [ p + s ] [s + 2]{t} + [ p + s ] [ ] + [ ] [ -a - s ] + [ -a + p ] [ ---------- ] + [ ------- ] [ 2 ] + [ 9*s - 9 ]{t} [ s + s + 1 ]{t} + + These unevaluated ``Series`` or ``Parallel`` objects can convert into the + resultant transfer function matrix using ``.doit()`` method or by + ``.rewrite(TransferFunctionMatrix)``. + + >>> (-tfm_8 + tfm_10 + tfm_8*tfm_9).doit() + TransferFunctionMatrix(((TransferFunction((a + s)*(s + 2)**3 - 3*(s + 2)**2*(s**2 + s + 1) - 9*(s + 2)*(s**2 + s + 1), (s + 2)**3*(s**2 + s + 1), s),), (TransferFunction((p + s)*(-3*p**4 + 9*p - 6), (p + s)**2*(s + 2), s),), (TransferFunction((-a + p)*(s + 2)*(s**2 + s + 1)**2 + (a + s)*(s + 2)*(9*s - 9)*(s**2 + s + 1) + (3*a + 3*s)*(9*s - 9)*(s**2 + s + 1), (s + 2)*(9*s - 9)*(s**2 + s + 1)**2, s),))) + >>> (-tfm_12 * -tfm_8 * -tfm_9).rewrite(TransferFunctionMatrix) + TransferFunctionMatrix(((TransferFunction(3*(-3*a + 3*p)*(p + s)*(s + 2)*(s**2 + s + 1)**2 + 3*(-3*a - 3*s)*(p + s)*(s + 2)*(9*s - 9)*(s**2 + s + 1) + 3*(a + s)*(s + 2)**2*(9*s - 9)*(-p**4 + 3*p - 2)*(s**2 + s + 1), (p + s)*(s + 2)**3*(9*s - 9)*(s**2 + s + 1)**2, s),), (TransferFunction(3*(-a + p)*(p + s)*(s + 2)**2*(-p**4 + 3*p - 2)*(s**2 + s + 1) + 3*(3*a + 3*s)*(p + s)**2*(s + 2)*(9*s - 9) + 3*(p + s)*(s + 2)*(9*s - 9)*(-3*p**4 + 9*p - 6)*(s**2 + s + 1), (p + s)**2*(s + 2)**3*(9*s - 9)*(s**2 + s + 1), s),))) + + See Also + ======== + + TransferFunction, MIMOSeries, MIMOParallel, Feedback + + """ + def __new__(cls, arg): + + expr_mat_arg = [] + try: + var = arg[0][0].var + except TypeError: + raise ValueError(filldedent(""" + `arg` param in TransferFunctionMatrix should + strictly be a nested list containing TransferFunction + objects.""")) + for row in arg: + temp = [] + for element in row: + if not isinstance(element, SISOLinearTimeInvariant): + raise TypeError(filldedent(""" + Each element is expected to be of + type `SISOLinearTimeInvariant`.""")) + + if var != element.var: + raise ValueError(filldedent(""" + Conflicting value(s) found for `var`. All TransferFunction + instances in TransferFunctionMatrix should use the same + complex variable in Laplace domain.""")) + + temp.append(element.to_expr()) + expr_mat_arg.append(temp) + + if isinstance(arg, (tuple, list, Tuple)): + # Making nested Tuple (sympy.core.containers.Tuple) from nested list or nested Python tuple + arg = Tuple(*(Tuple(*r, sympify=False) for r in arg), sympify=False) + + obj = super(TransferFunctionMatrix, cls).__new__(cls, arg) + obj._expr_mat = ImmutableMatrix(expr_mat_arg) + return obj + + @classmethod + def from_Matrix(cls, matrix, var): + """ + Creates a new ``TransferFunctionMatrix`` efficiently from a SymPy Matrix of ``Expr`` objects. + + Parameters + ========== + + matrix : ``ImmutableMatrix`` having ``Expr``/``Number`` elements. + var : Symbol + Complex variable of the Laplace transform which will be used by the + all the ``TransferFunction`` objects in the ``TransferFunctionMatrix``. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunctionMatrix + >>> from sympy import Matrix, pprint + >>> M = Matrix([[s, 1/s], [1/(s+1), s]]) + >>> M_tf = TransferFunctionMatrix.from_Matrix(M, s) + >>> pprint(M_tf, use_unicode=False) + [ s 1] + [ - -] + [ 1 s] + [ ] + [ 1 s] + [----- -] + [s + 1 1]{t} + >>> M_tf.elem_poles() + [[[], [0]], [[-1], []]] + >>> M_tf.elem_zeros() + [[[0], []], [[], [0]]] + + """ + return _to_TFM(matrix, var) + + @property + def var(self): + """ + Returns the complex variable used by all the transfer functions or + ``Series``/``Parallel`` objects in a transfer function matrix. + + Examples + ======== + + >>> from sympy.abc import p, s + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix, Series, Parallel + >>> G1 = TransferFunction(p**2 + 2*p + 4, p - 6, p) + >>> G2 = TransferFunction(p, 4 - p, p) + >>> G3 = TransferFunction(0, p**4 - 1, p) + >>> G4 = TransferFunction(s + 1, s**2 + s + 1, s) + >>> S1 = Series(G1, G2) + >>> S2 = Series(-G3, Parallel(G2, -G1)) + >>> tfm1 = TransferFunctionMatrix([[G1], [G2], [G3]]) + >>> tfm1.var + p + >>> tfm2 = TransferFunctionMatrix([[-S1, -S2], [S1, S2]]) + >>> tfm2.var + p + >>> tfm3 = TransferFunctionMatrix([[G4]]) + >>> tfm3.var + s + + """ + return self.args[0][0][0].var + + @property + def num_inputs(self): + """ + Returns the number of inputs of the system. + + Examples + ======== + + >>> from sympy.abc import s, p + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix + >>> G1 = TransferFunction(s + 3, s**2 - 3, s) + >>> G2 = TransferFunction(4, s**2, s) + >>> G3 = TransferFunction(p**2 + s**2, p - 3, s) + >>> tfm_1 = TransferFunctionMatrix([[G2, -G1, G3], [-G2, -G1, -G3]]) + >>> tfm_1.num_inputs + 3 + + See Also + ======== + + num_outputs + + """ + return self._expr_mat.shape[1] + + @property + def num_outputs(self): + """ + Returns the number of outputs of the system. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunctionMatrix + >>> from sympy import Matrix + >>> M_1 = Matrix([[s], [1/s]]) + >>> TFM = TransferFunctionMatrix.from_Matrix(M_1, s) + >>> print(TFM) + TransferFunctionMatrix(((TransferFunction(s, 1, s),), (TransferFunction(1, s, s),))) + >>> TFM.num_outputs + 2 + + See Also + ======== + + num_inputs + + """ + return self._expr_mat.shape[0] + + @property + def shape(self): + """ + Returns the shape of the transfer function matrix, that is, ``(# of outputs, # of inputs)``. + + Examples + ======== + + >>> from sympy.abc import s, p + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix + >>> tf1 = TransferFunction(p**2 - 1, s**4 + s**3 - p, p) + >>> tf2 = TransferFunction(1 - p, p**2 - 3*p + 7, p) + >>> tf3 = TransferFunction(3, 4, p) + >>> tfm1 = TransferFunctionMatrix([[tf1, -tf2]]) + >>> tfm1.shape + (1, 2) + >>> tfm2 = TransferFunctionMatrix([[-tf2, tf3], [tf1, -tf1]]) + >>> tfm2.shape + (2, 2) + + """ + return self._expr_mat.shape + + def __neg__(self): + neg = -self._expr_mat + return _to_TFM(neg, self.var) + + @_check_other_MIMO + def __add__(self, other): + + if not isinstance(other, MIMOParallel): + return MIMOParallel(self, other) + other_arg_list = list(other.args) + return MIMOParallel(self, *other_arg_list) + + @_check_other_MIMO + def __sub__(self, other): + return self + (-other) + + @_check_other_MIMO + def __mul__(self, other): + + if not isinstance(other, MIMOSeries): + return MIMOSeries(other, self) + other_arg_list = list(other.args) + return MIMOSeries(*other_arg_list, self) + + def __getitem__(self, key): + trunc = self._expr_mat.__getitem__(key) + if isinstance(trunc, ImmutableMatrix): + return _to_TFM(trunc, self.var) + return TransferFunction.from_rational_expression(trunc, self.var) + + def transpose(self): + """Returns the transpose of the ``TransferFunctionMatrix`` (switched input and output layers).""" + transposed_mat = self._expr_mat.transpose() + return _to_TFM(transposed_mat, self.var) + + def elem_poles(self): + """ + Returns the poles of each element of the ``TransferFunctionMatrix``. + + .. note:: + Actual poles of a MIMO system are NOT the poles of individual elements. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix + >>> tf_1 = TransferFunction(3, (s + 1), s) + >>> tf_2 = TransferFunction(s + 6, (s + 1)*(s + 2), s) + >>> tf_3 = TransferFunction(s + 3, s**2 + 3*s + 2, s) + >>> tf_4 = TransferFunction(s + 2, s**2 + 5*s - 10, s) + >>> tfm_1 = TransferFunctionMatrix([[tf_1, tf_2], [tf_3, tf_4]]) + >>> tfm_1 + TransferFunctionMatrix(((TransferFunction(3, s + 1, s), TransferFunction(s + 6, (s + 1)*(s + 2), s)), (TransferFunction(s + 3, s**2 + 3*s + 2, s), TransferFunction(s + 2, s**2 + 5*s - 10, s)))) + >>> tfm_1.elem_poles() + [[[-1], [-2, -1]], [[-2, -1], [-5/2 + sqrt(65)/2, -sqrt(65)/2 - 5/2]]] + + See Also + ======== + + elem_zeros + + """ + return [[element.poles() for element in row] for row in self.doit().args[0]] + + def elem_zeros(self): + """ + Returns the zeros of each element of the ``TransferFunctionMatrix``. + + .. note:: + Actual zeros of a MIMO system are NOT the zeros of individual elements. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix + >>> tf_1 = TransferFunction(3, (s + 1), s) + >>> tf_2 = TransferFunction(s + 6, (s + 1)*(s + 2), s) + >>> tf_3 = TransferFunction(s + 3, s**2 + 3*s + 2, s) + >>> tf_4 = TransferFunction(s**2 - 9*s + 20, s**2 + 5*s - 10, s) + >>> tfm_1 = TransferFunctionMatrix([[tf_1, tf_2], [tf_3, tf_4]]) + >>> tfm_1 + TransferFunctionMatrix(((TransferFunction(3, s + 1, s), TransferFunction(s + 6, (s + 1)*(s + 2), s)), (TransferFunction(s + 3, s**2 + 3*s + 2, s), TransferFunction(s**2 - 9*s + 20, s**2 + 5*s - 10, s)))) + >>> tfm_1.elem_zeros() + [[[], [-6]], [[-3], [4, 5]]] + + See Also + ======== + + elem_poles + + """ + return [[element.zeros() for element in row] for row in self.doit().args[0]] + + def eval_frequency(self, other): + """ + Evaluates system response of each transfer function in the ``TransferFunctionMatrix`` at any point in the real or complex plane. + + Examples + ======== + + >>> from sympy.abc import s + >>> from sympy.physics.control.lti import TransferFunction, TransferFunctionMatrix + >>> from sympy import I + >>> tf_1 = TransferFunction(3, (s + 1), s) + >>> tf_2 = TransferFunction(s + 6, (s + 1)*(s + 2), s) + >>> tf_3 = TransferFunction(s + 3, s**2 + 3*s + 2, s) + >>> tf_4 = TransferFunction(s**2 - 9*s + 20, s**2 + 5*s - 10, s) + >>> tfm_1 = TransferFunctionMatrix([[tf_1, tf_2], [tf_3, tf_4]]) + >>> tfm_1 + TransferFunctionMatrix(((TransferFunction(3, s + 1, s), TransferFunction(s + 6, (s + 1)*(s + 2), s)), (TransferFunction(s + 3, s**2 + 3*s + 2, s), TransferFunction(s**2 - 9*s + 20, s**2 + 5*s - 10, s)))) + >>> tfm_1.eval_frequency(2) + Matrix([ + [ 1, 2/3], + [5/12, 3/2]]) + >>> tfm_1.eval_frequency(I*2) + Matrix([ + [ 3/5 - 6*I/5, -I], + [3/20 - 11*I/20, -101/74 + 23*I/74]]) + """ + mat = self._expr_mat.subs(self.var, other) + return mat.expand() + + def _flat(self): + """Returns flattened list of args in TransferFunctionMatrix""" + return [elem for tup in self.args[0] for elem in tup] + + def _eval_evalf(self, prec): + """Calls evalf() on each transfer function in the transfer function matrix""" + dps = prec_to_dps(prec) + mat = self._expr_mat.applyfunc(lambda a: a.evalf(n=dps)) + return _to_TFM(mat, self.var) + + def _eval_simplify(self, **kwargs): + """Simplifies the transfer function matrix""" + simp_mat = self._expr_mat.applyfunc(lambda a: cancel(a, expand=False)) + return _to_TFM(simp_mat, self.var) + + def expand(self, **hints): + """Expands the transfer function matrix""" + expand_mat = self._expr_mat.expand(**hints) + return _to_TFM(expand_mat, self.var) + +class StateSpace(LinearTimeInvariant): + r""" + State space model (ssm) of a linear, time invariant control system. + + Represents the standard state-space model with A, B, C, D as state-space matrices. + This makes the linear control system: + (1) x'(t) = A * x(t) + B * u(t); x in R^n , u in R^k + (2) y(t) = C * x(t) + D * u(t); y in R^m + where u(t) is any input signal, y(t) the corresponding output, and x(t) the system's state. + + Parameters + ========== + + A : Matrix + The State matrix of the state space model. + B : Matrix + The Input-to-State matrix of the state space model. + C : Matrix + The State-to-Output matrix of the state space model. + D : Matrix + The Feedthrough matrix of the state space model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + + The easiest way to create a StateSpaceModel is via four matrices: + + >>> A = Matrix([[1, 2], [1, 0]]) + >>> B = Matrix([1, 1]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([0]) + >>> StateSpace(A, B, C, D) + StateSpace(Matrix([ + [1, 2], + [1, 0]]), Matrix([ + [1], + [1]]), Matrix([[0, 1]]), Matrix([[0]])) + + + One can use less matrices. The rest will be filled with a minimum of zeros: + + >>> StateSpace(A, B) + StateSpace(Matrix([ + [1, 2], + [1, 0]]), Matrix([ + [1], + [1]]), Matrix([[0, 0]]), Matrix([[0]])) + + + See Also + ======== + + TransferFunction, TransferFunctionMatrix + + References + ========== + .. [1] https://en.wikipedia.org/wiki/State-space_representation + .. [2] https://in.mathworks.com/help/control/ref/ss.html + + """ + def __new__(cls, A=None, B=None, C=None, D=None): + if A is None: + A = zeros(1) + if B is None: + B = zeros(A.rows, 1) + if C is None: + C = zeros(1, A.cols) + if D is None: + D = zeros(C.rows, B.cols) + + A = _sympify(A) + B = _sympify(B) + C = _sympify(C) + D = _sympify(D) + + if (isinstance(A, ImmutableDenseMatrix) and isinstance(B, ImmutableDenseMatrix) and + isinstance(C, ImmutableDenseMatrix) and isinstance(D, ImmutableDenseMatrix)): + # Check State Matrix is square + if A.rows != A.cols: + raise ShapeError("Matrix A must be a square matrix.") + + # Check State and Input matrices have same rows + if A.rows != B.rows: + raise ShapeError("Matrices A and B must have the same number of rows.") + + # Check Ouput and Feedthrough matrices have same rows + if C.rows != D.rows: + raise ShapeError("Matrices C and D must have the same number of rows.") + + # Check State and Ouput matrices have same columns + if A.cols != C.cols: + raise ShapeError("Matrices A and C must have the same number of columns.") + + # Check Input and Feedthrough matrices have same columns + if B.cols != D.cols: + raise ShapeError("Matrices B and D must have the same number of columns.") + + obj = super(StateSpace, cls).__new__(cls, A, B, C, D) + obj._A = A + obj._B = B + obj._C = C + obj._D = D + + # Determine if the system is SISO or MIMO + num_outputs = D.rows + num_inputs = D.cols + if num_inputs == 1 and num_outputs == 1: + obj._is_SISO = True + obj._clstype = SISOLinearTimeInvariant + else: + obj._is_SISO = False + obj._clstype = MIMOLinearTimeInvariant + + return obj + + else: + raise TypeError("A, B, C and D inputs must all be sympy Matrices.") + + @property + def state_matrix(self): + """ + Returns the state matrix of the model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[1, 2], [1, 0]]) + >>> B = Matrix([1, 1]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.state_matrix + Matrix([ + [1, 2], + [1, 0]]) + + """ + return self._A + + @property + def input_matrix(self): + """ + Returns the input matrix of the model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[1, 2], [1, 0]]) + >>> B = Matrix([1, 1]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.input_matrix + Matrix([ + [1], + [1]]) + + """ + return self._B + + @property + def output_matrix(self): + """ + Returns the output matrix of the model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[1, 2], [1, 0]]) + >>> B = Matrix([1, 1]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.output_matrix + Matrix([[0, 1]]) + + """ + return self._C + + @property + def feedforward_matrix(self): + """ + Returns the feedforward matrix of the model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[1, 2], [1, 0]]) + >>> B = Matrix([1, 1]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.feedforward_matrix + Matrix([[0]]) + + """ + return self._D + + @property + def num_states(self): + """ + Returns the number of states of the model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[1, 2], [1, 0]]) + >>> B = Matrix([1, 1]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.num_states + 2 + + """ + return self._A.rows + + @property + def num_inputs(self): + """ + Returns the number of inputs of the model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[1, 2], [1, 0]]) + >>> B = Matrix([1, 1]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.num_inputs + 1 + + """ + return self._D.cols + + @property + def num_outputs(self): + """ + Returns the number of outputs of the model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[1, 2], [1, 0]]) + >>> B = Matrix([1, 1]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.num_outputs + 1 + + """ + return self._D.rows + + def _eval_evalf(self, prec): + """ + Returns state space model where numerical expressions are evaluated into floating point numbers. + """ + dps = prec_to_dps(prec) + return StateSpace( + self._A.evalf(n = dps), + self._B.evalf(n = dps), + self._C.evalf(n = dps), + self._D.evalf(n = dps)) + + def _eval_rewrite_as_TransferFunction(self, *args): + """ + Returns the equivalent Transfer Function of the state space model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import TransferFunction, StateSpace + >>> A = Matrix([[-5, -1], [3, -1]]) + >>> B = Matrix([2, 5]) + >>> C = Matrix([[1, 2]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.rewrite(TransferFunction) + [[TransferFunction(12*s + 59, s**2 + 6*s + 8, s)]] + + """ + s = Symbol('s') + n = self._A.shape[0] + I = eye(n) + G = self._C*(s*I - self._A).solve(self._B) + self._D + G = G.simplify() + to_tf = lambda expr: TransferFunction.from_rational_expression(expr, s) + tf_mat = [[to_tf(expr) for expr in sublist] for sublist in G.tolist()] + return tf_mat + + def __add__(self, other): + """ + Add two State Space systems (parallel connection). + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A1 = Matrix([[1]]) + >>> B1 = Matrix([[2]]) + >>> C1 = Matrix([[-1]]) + >>> D1 = Matrix([[-2]]) + >>> A2 = Matrix([[-1]]) + >>> B2 = Matrix([[-2]]) + >>> C2 = Matrix([[1]]) + >>> D2 = Matrix([[2]]) + >>> ss1 = StateSpace(A1, B1, C1, D1) + >>> ss2 = StateSpace(A2, B2, C2, D2) + >>> ss1 + ss2 + StateSpace(Matrix([ + [1, 0], + [0, -1]]), Matrix([ + [ 2], + [-2]]), Matrix([[-1, 1]]), Matrix([[0]])) + + """ + # Check for scalars + if isinstance(other, (int, float, complex, Symbol)): + A = self._A + B = self._B + C = self._C + D = self._D.applyfunc(lambda element: element + other) + + else: + # Check nature of system + if not isinstance(other, StateSpace): + raise ValueError("Addition is only supported for 2 State Space models.") + # Check dimensions of system + elif ((self.num_inputs != other.num_inputs) or (self.num_outputs != other.num_outputs)): + raise ShapeError("Systems with incompatible inputs and outputs cannot be added.") + + m1 = (self._A).row_join(zeros(self._A.shape[0], other._A.shape[-1])) + m2 = zeros(other._A.shape[0], self._A.shape[-1]).row_join(other._A) + + A = m1.col_join(m2) + B = self._B.col_join(other._B) + C = self._C.row_join(other._C) + D = self._D + other._D + + return StateSpace(A, B, C, D) + + def __radd__(self, other): + """ + Right add two State Space systems. + + Examples + ======== + + >>> from sympy.physics.control import StateSpace + >>> s = StateSpace() + >>> 5 + s + StateSpace(Matrix([[0]]), Matrix([[0]]), Matrix([[0]]), Matrix([[5]])) + + """ + return self + other + + def __sub__(self, other): + """ + Subtract two State Space systems. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A1 = Matrix([[1]]) + >>> B1 = Matrix([[2]]) + >>> C1 = Matrix([[-1]]) + >>> D1 = Matrix([[-2]]) + >>> A2 = Matrix([[-1]]) + >>> B2 = Matrix([[-2]]) + >>> C2 = Matrix([[1]]) + >>> D2 = Matrix([[2]]) + >>> ss1 = StateSpace(A1, B1, C1, D1) + >>> ss2 = StateSpace(A2, B2, C2, D2) + >>> ss1 - ss2 + StateSpace(Matrix([ + [1, 0], + [0, -1]]), Matrix([ + [ 2], + [-2]]), Matrix([[-1, -1]]), Matrix([[-4]])) + + """ + return self + (-other) + + def __rsub__(self, other): + """ + Right subtract two tate Space systems. + + Examples + ======== + + >>> from sympy.physics.control import StateSpace + >>> s = StateSpace() + >>> 5 - s + StateSpace(Matrix([[0]]), Matrix([[0]]), Matrix([[0]]), Matrix([[5]])) + + """ + return other + (-self) + + def __neg__(self): + """ + Returns the negation of the state space model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[-5, -1], [3, -1]]) + >>> B = Matrix([2, 5]) + >>> C = Matrix([[1, 2]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> -ss + StateSpace(Matrix([ + [-5, -1], + [ 3, -1]]), Matrix([ + [2], + [5]]), Matrix([[-1, -2]]), Matrix([[0]])) + + """ + return StateSpace(self._A, self._B, -self._C, -self._D) + + def __mul__(self, other): + """ + Multiplication of two State Space systems (serial connection). + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[-5, -1], [3, -1]]) + >>> B = Matrix([2, 5]) + >>> C = Matrix([[1, 2]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> ss*5 + StateSpace(Matrix([ + [-5, -1], + [ 3, -1]]), Matrix([ + [2], + [5]]), Matrix([[5, 10]]), Matrix([[0]])) + + """ + # Check for scalars + if isinstance(other, (int, float, complex, Symbol)): + A = self._A + B = self._B + C = self._C.applyfunc(lambda element: element*other) + D = self._D.applyfunc(lambda element: element*other) + + else: + # Check nature of system + if not isinstance(other, StateSpace): + raise ValueError("Multiplication is only supported for 2 State Space models.") + # Check dimensions of system + elif self.num_inputs != other.num_outputs: + raise ShapeError("Systems with incompatible inputs and outputs cannot be multiplied.") + + m1 = (other._A).row_join(zeros(other._A.shape[0], self._A.shape[1])) + m2 = (self._B * other._C).row_join(self._A) + + A = m1.col_join(m2) + B = (other._B).col_join(self._B * other._D) + C = (self._D * other._C).row_join(self._C) + D = self._D * other._D + + return StateSpace(A, B, C, D) + + def __rmul__(self, other): + """ + Right multiply two tate Space systems. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[-5, -1], [3, -1]]) + >>> B = Matrix([2, 5]) + >>> C = Matrix([[1, 2]]) + >>> D = Matrix([0]) + >>> ss = StateSpace(A, B, C, D) + >>> 5*ss + StateSpace(Matrix([ + [-5, -1], + [ 3, -1]]), Matrix([ + [10], + [25]]), Matrix([[1, 2]]), Matrix([[0]])) + + """ + if isinstance(other, (int, float, complex, Symbol)): + A = self._A + C = self._C + B = self._B.applyfunc(lambda element: element*other) + D = self._D.applyfunc(lambda element: element*other) + return StateSpace(A, B, C, D) + else: + return self*other + + def __repr__(self): + A_str = self._A.__repr__() + B_str = self._B.__repr__() + C_str = self._C.__repr__() + D_str = self._D.__repr__() + + return f"StateSpace(\n{A_str},\n\n{B_str},\n\n{C_str},\n\n{D_str})" + + + def append(self, other): + """ + Returns the first model appended with the second model. The order is preserved. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A1 = Matrix([[1]]) + >>> B1 = Matrix([[2]]) + >>> C1 = Matrix([[-1]]) + >>> D1 = Matrix([[-2]]) + >>> A2 = Matrix([[-1]]) + >>> B2 = Matrix([[-2]]) + >>> C2 = Matrix([[1]]) + >>> D2 = Matrix([[2]]) + >>> ss1 = StateSpace(A1, B1, C1, D1) + >>> ss2 = StateSpace(A2, B2, C2, D2) + >>> ss1.append(ss2) + StateSpace(Matrix([ + [1, 0], + [0, -1]]), Matrix([ + [2, 0], + [0, -2]]), Matrix([ + [-1, 0], + [ 0, 1]]), Matrix([ + [-2, 0], + [ 0, 2]])) + + """ + n = self.num_states + other.num_states + m = self.num_inputs + other.num_inputs + p = self.num_outputs + other.num_outputs + + A = zeros(n, n) + B = zeros(n, m) + C = zeros(p, n) + D = zeros(p, m) + + A[:self.num_states, :self.num_states] = self._A + A[self.num_states:, self.num_states:] = other._A + B[:self.num_states, :self.num_inputs] = self._B + B[self.num_states:, self.num_inputs:] = other._B + C[:self.num_outputs, :self.num_states] = self._C + C[self.num_outputs:, self.num_states:] = other._C + D[:self.num_outputs, :self.num_inputs] = self._D + D[self.num_outputs:, self.num_inputs:] = other._D + return StateSpace(A, B, C, D) + + def observability_matrix(self): + """ + Returns the observability matrix of the state space model: + [C, C * A^1, C * A^2, .. , C * A^(n-1)]; A in R^(n x n), C in R^(m x k) + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[-1.5, -2], [1, 0]]) + >>> B = Matrix([0.5, 0]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([1]) + >>> ss = StateSpace(A, B, C, D) + >>> ob = ss.observability_matrix() + >>> ob + Matrix([ + [0, 1], + [1, 0]]) + + References + ========== + .. [1] https://in.mathworks.com/help/control/ref/statespacemodel.obsv.html + + """ + n = self.num_states + ob = self._C + for i in range(1,n): + ob = ob.col_join(self._C * self._A**i) + + return ob + + def observable_subspace(self): + """ + Returns the observable subspace of the state space model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[-1.5, -2], [1, 0]]) + >>> B = Matrix([0.5, 0]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([1]) + >>> ss = StateSpace(A, B, C, D) + >>> ob_subspace = ss.observable_subspace() + >>> ob_subspace + [Matrix([ + [0], + [1]]), Matrix([ + [1], + [0]])] + + """ + return self.observability_matrix().columnspace() + + def is_observable(self): + """ + Returns if the state space model is observable. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[-1.5, -2], [1, 0]]) + >>> B = Matrix([0.5, 0]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([1]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.is_observable() + True + + """ + return self.observability_matrix().rank() == self.num_states + + def controllability_matrix(self): + """ + Returns the controllability matrix of the system: + [B, A * B, A^2 * B, .. , A^(n-1) * B]; A in R^(n x n), B in R^(n x m) + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[-1.5, -2], [1, 0]]) + >>> B = Matrix([0.5, 0]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([1]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.controllability_matrix() + Matrix([ + [0.5, -0.75], + [ 0, 0.5]]) + + References + ========== + .. [1] https://in.mathworks.com/help/control/ref/statespacemodel.ctrb.html + + """ + co = self._B + n = self._A.shape[0] + for i in range(1, n): + co = co.row_join(((self._A)**i) * self._B) + + return co + + def controllable_subspace(self): + """ + Returns the controllable subspace of the state space model. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[-1.5, -2], [1, 0]]) + >>> B = Matrix([0.5, 0]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([1]) + >>> ss = StateSpace(A, B, C, D) + >>> co_subspace = ss.controllable_subspace() + >>> co_subspace + [Matrix([ + [0.5], + [ 0]]), Matrix([ + [-0.75], + [ 0.5]])] + + """ + return self.controllability_matrix().columnspace() + + def is_controllable(self): + """ + Returns if the state space model is controllable. + + Examples + ======== + + >>> from sympy import Matrix + >>> from sympy.physics.control import StateSpace + >>> A = Matrix([[-1.5, -2], [1, 0]]) + >>> B = Matrix([0.5, 0]) + >>> C = Matrix([[0, 1]]) + >>> D = Matrix([1]) + >>> ss = StateSpace(A, B, C, D) + >>> ss.is_controllable() + True + + """ + return self.controllability_matrix().rank() == self.num_states diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/__init__.py b/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/__pycache__/__init__.cpython-310.pyc b/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/__pycache__/__init__.cpython-310.pyc new file mode 100644 index 0000000000000000000000000000000000000000..e65d89090e62ef71dc3d65d5fc241926a07c9acf Binary files /dev/null and b/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/__pycache__/__init__.cpython-310.pyc differ diff --git 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a/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/test_control_plots.py b/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/test_control_plots.py new file mode 100644 index 0000000000000000000000000000000000000000..673fcee6cfdbde67ab691d2fbe2f8c36d86c9443 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/test_control_plots.py @@ -0,0 +1,299 @@ +from math import isclose +from sympy.core.numbers import I +from sympy.core.symbol import Dummy +from sympy.functions.elementary.complexes import (Abs, arg) +from sympy.functions.elementary.exponential import log +from sympy.abc import s, p, a +from sympy.external import import_module +from sympy.physics.control.control_plots import \ + (pole_zero_numerical_data, pole_zero_plot, step_response_numerical_data, + step_response_plot, impulse_response_numerical_data, + impulse_response_plot, ramp_response_numerical_data, + ramp_response_plot, bode_magnitude_numerical_data, + bode_phase_numerical_data, bode_plot) +from sympy.physics.control.lti import (TransferFunction, + Series, Parallel, TransferFunctionMatrix) +from sympy.testing.pytest import raises, skip + +matplotlib = import_module( + 'matplotlib', import_kwargs={'fromlist': ['pyplot']}, + catch=(RuntimeError,)) + +numpy = import_module('numpy') + +tf1 = TransferFunction(1, p**2 + 0.5*p + 2, p) +tf2 = TransferFunction(p, 6*p**2 + 3*p + 1, p) +tf3 = TransferFunction(p, p**3 - 1, p) +tf4 = TransferFunction(10, p**3, p) +tf5 = TransferFunction(5, s**2 + 2*s + 10, s) +tf6 = TransferFunction(1, 1, s) +tf7 = TransferFunction(4*s*3 + 9*s**2 + 0.1*s + 11, 8*s**6 + 9*s**4 + 11, s) +tf8 = TransferFunction(5, s**2 + (2+I)*s + 10, s) + +ser1 = Series(tf4, TransferFunction(1, p - 5, p)) +ser2 = Series(tf3, TransferFunction(p, p + 2, p)) + +par1 = Parallel(tf1, tf2) + + +def _to_tuple(a, b): + return tuple(a), tuple(b) + +def _trim_tuple(a, b): + a, b = _to_tuple(a, b) + return tuple(a[0: 2] + a[len(a)//2 : len(a)//2 + 1] + a[-2:]), \ + tuple(b[0: 2] + b[len(b)//2 : len(b)//2 + 1] + b[-2:]) + +def y_coordinate_equality(plot_data_func, evalf_func, system): + """Checks whether the y-coordinate value of the plotted + data point is equal to the value of the function at a + particular x.""" + x, y = plot_data_func(system) + x, y = _trim_tuple(x, y) + y_exp = tuple(evalf_func(system, x_i) for x_i in x) + return all(Abs(y_exp_i - y_i) < 1e-8 for y_exp_i, y_i in zip(y_exp, y)) + + +def test_errors(): + if not matplotlib: + skip("Matplotlib not the default backend") + + # Invalid `system` check + tfm = TransferFunctionMatrix([[tf6, tf5], [tf5, tf6]]) + expr = 1/(s**2 - 1) + raises(NotImplementedError, lambda: pole_zero_plot(tfm)) + raises(NotImplementedError, lambda: pole_zero_numerical_data(expr)) + raises(NotImplementedError, lambda: impulse_response_plot(expr)) + raises(NotImplementedError, lambda: impulse_response_numerical_data(tfm)) + raises(NotImplementedError, lambda: step_response_plot(tfm)) + raises(NotImplementedError, lambda: step_response_numerical_data(expr)) + raises(NotImplementedError, lambda: ramp_response_plot(expr)) + raises(NotImplementedError, lambda: ramp_response_numerical_data(tfm)) + raises(NotImplementedError, lambda: bode_plot(tfm)) + + # More than 1 variables + tf_a = TransferFunction(a, s + 1, s) + raises(ValueError, lambda: pole_zero_plot(tf_a)) + raises(ValueError, lambda: pole_zero_numerical_data(tf_a)) + raises(ValueError, lambda: impulse_response_plot(tf_a)) + raises(ValueError, lambda: impulse_response_numerical_data(tf_a)) + raises(ValueError, lambda: step_response_plot(tf_a)) + raises(ValueError, lambda: step_response_numerical_data(tf_a)) + raises(ValueError, lambda: ramp_response_plot(tf_a)) + raises(ValueError, lambda: ramp_response_numerical_data(tf_a)) + raises(ValueError, lambda: bode_plot(tf_a)) + + # lower_limit > 0 for response plots + raises(ValueError, lambda: impulse_response_plot(tf1, lower_limit=-1)) + raises(ValueError, lambda: step_response_plot(tf1, lower_limit=-0.1)) + raises(ValueError, lambda: ramp_response_plot(tf1, lower_limit=-4/3)) + + # slope in ramp_response_plot() is negative + raises(ValueError, lambda: ramp_response_plot(tf1, slope=-0.1)) + + # incorrect frequency or phase unit + raises(ValueError, lambda: bode_plot(tf1,freq_unit = 'hz')) + raises(ValueError, lambda: bode_plot(tf1,phase_unit = 'degree')) + + +def test_pole_zero(): + if not numpy: + skip("NumPy is required for this test") + + def pz_tester(sys, expected_value): + z, p = pole_zero_numerical_data(sys) + z_check = numpy.allclose(z, expected_value[0]) + p_check = numpy.allclose(p, expected_value[1]) + return p_check and z_check + + exp1 = [[], [-0.24999999999999994+1.3919410907075054j, -0.24999999999999994-1.3919410907075054j]] + exp2 = [[0.0], [-0.25+0.3227486121839514j, -0.25-0.3227486121839514j]] + exp3 = [[0.0], [-0.5000000000000004+0.8660254037844395j, + -0.5000000000000004-0.8660254037844395j, 0.9999999999999998+0j]] + exp4 = [[], [5.0, 0.0, 0.0, 0.0]] + exp5 = [[-5.645751311064592, -0.5000000000000008, -0.3542486889354093], + [-0.24999999999999986+1.3919410907075052j, + -0.24999999999999986-1.3919410907075052j, -0.2499999999999998+0.32274861218395134j, + -0.2499999999999998-0.32274861218395134j]] + exp6 = [[], [-1.1641600331447917-3.545808351896439j, + -0.8358399668552097+2.5458083518964383j]] + + assert pz_tester(tf1, exp1) + assert pz_tester(tf2, exp2) + assert pz_tester(tf3, exp3) + assert pz_tester(ser1, exp4) + assert pz_tester(par1, exp5) + assert pz_tester(tf8, exp6) + + +def test_bode(): + if not numpy: + skip("NumPy is required for this test") + + def bode_phase_evalf(system, point): + expr = system.to_expr() + _w = Dummy("w", real=True) + w_expr = expr.subs({system.var: I*_w}) + return arg(w_expr).subs({_w: point}).evalf() + + def bode_mag_evalf(system, point): + expr = system.to_expr() + _w = Dummy("w", real=True) + w_expr = expr.subs({system.var: I*_w}) + return 20*log(Abs(w_expr), 10).subs({_w: point}).evalf() + + def test_bode_data(sys): + return y_coordinate_equality(bode_magnitude_numerical_data, bode_mag_evalf, sys) \ + and y_coordinate_equality(bode_phase_numerical_data, bode_phase_evalf, sys) + + assert test_bode_data(tf1) + assert test_bode_data(tf2) + assert test_bode_data(tf3) + assert test_bode_data(tf4) + assert test_bode_data(tf5) + + +def check_point_accuracy(a, b): + return all(isclose(*_, rel_tol=1e-1, abs_tol=1e-6 + ) for _ in zip(a, b)) + + +def test_impulse_response(): + if not numpy: + skip("NumPy is required for this test") + + def impulse_res_tester(sys, expected_value): + x, y = _to_tuple(*impulse_response_numerical_data(sys, + adaptive=False, n=10)) + x_check = check_point_accuracy(x, expected_value[0]) + y_check = check_point_accuracy(y, expected_value[1]) + return x_check and y_check + + exp1 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, + 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), + (0.0, 0.544019738507865, 0.01993849743234938, -0.31140243360893216, -0.022852779906491996, 0.1778306498155759, + 0.01962941084328499, -0.1013115194573652, -0.014975541213105696, 0.0575789724730714)) + exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, + 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.1666666675, 0.08389223412935855, + 0.02338051973475047, -0.014966807776379383, -0.034645954223054234, -0.040560075735512804, + -0.037658628907103885, -0.030149507719590022, -0.021162090730736834, -0.012721292737437523)) + exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, + 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (4.369893391586999e-09, 1.1750333000630964, + 3.2922404058312473, 9.432290008148343, 28.37098083007151, 86.18577464367974, 261.90356653762115, + 795.6538758627842, 2416.9920942096983, 7342.159505206647)) + exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, + 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 6.17283950617284, 24.69135802469136, + 55.555555555555564, 98.76543209876544, 154.320987654321, 222.22222222222226, 302.46913580246917, + 395.0617283950618, 500.0)) + exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, + 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, -0.10455606138085417, + 0.06757671513476461, -0.03234567568833768, 0.013582514927757873, -0.005273419510705473, + 0.0019364083003354075, -0.000680070134067832, 0.00022969845960406913, -7.476094359583917e-05)) + exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, + 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), + (-6.016699583000218e-09, 0.35039802056107394, 3.3728423827689884, 12.119846079276684, + 25.86101014293389, 29.352480635282088, -30.49475907497664, -273.8717189554019, -863.2381702029659, + -1747.0262164682233)) + exp7 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, + 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, + 8.88888888888889, 10.0), (0.0, 18.934638095560974, 5346.93244680907, 1384609.8718249386, + 358161126.65801865, 92645770015.70108, 23964739753087.42, 6198974342083139.0, 1.603492601616059e+18, + 4.147764422869658e+20)) + + assert impulse_res_tester(tf1, exp1) + assert impulse_res_tester(tf2, exp2) + assert impulse_res_tester(tf3, exp3) + assert impulse_res_tester(tf4, exp4) + assert impulse_res_tester(tf5, exp5) + assert impulse_res_tester(tf7, exp6) + assert impulse_res_tester(ser1, exp7) + + +def test_step_response(): + if not numpy: + skip("NumPy is required for this test") + + def step_res_tester(sys, expected_value): + x, y = _to_tuple(*step_response_numerical_data(sys, + adaptive=False, n=10)) + x_check = check_point_accuracy(x, expected_value[0]) + y_check = check_point_accuracy(y, expected_value[1]) + return x_check and y_check + + exp1 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, + 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), + (-1.9193285738516863e-08, 0.42283495488246126, 0.7840485977945262, 0.5546841805655717, + 0.33903033806932087, 0.4627251747410237, 0.5909907598988051, 0.5247213989553071, + 0.4486997874319281, 0.4839358435839171)) + exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, + 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), + (0.0, 0.13728409095645816, 0.19474559355325086, 0.1974909129243011, 0.16841657696573073, + 0.12559777736159378, 0.08153828016664713, 0.04360471317348958, 0.015072994568868221, + -0.003636420058445484)) + exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, + 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), + (0.0, 0.6314542141914303, 2.9356520038101035, 9.37731009663807, 28.452300356688376, + 86.25721933273988, 261.9236645044672, 795.6435410577224, 2416.9786984578764, 7342.154119725917)) + exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, + 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), + (0.0, 2.286236899862826, 18.28989519890261, 61.72839629629631, 146.31916159122088, 285.7796124828532, + 493.8271703703705, 784.1792566529494, 1170.553292729767, 1666.6667)) + exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, + 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), + (-3.999999997894577e-09, 0.6720357068882895, 0.4429938256137113, 0.5182010838004518, + 0.4944139147159695, 0.5016379853883338, 0.4995466896527733, 0.5001154784851325, + 0.49997448824584123, 0.5000039745919259)) + exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, + 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), + (-1.5433688493882158e-09, 0.3428705539937336, 1.1253619102202777, 3.1849962651016517, + 9.47532757182671, 28.727231099148135, 87.29426924860557, 265.2138681048606, 805.6636260007757, + 2447.387582370878)) + + assert step_res_tester(tf1, exp1) + assert step_res_tester(tf2, exp2) + assert step_res_tester(tf3, exp3) + assert step_res_tester(tf4, exp4) + assert step_res_tester(tf5, exp5) + assert step_res_tester(ser2, exp6) + + +def test_ramp_response(): + if not numpy: + skip("NumPy is required for this test") + + def ramp_res_tester(sys, num_points, expected_value, slope=1): + x, y = _to_tuple(*ramp_response_numerical_data(sys, + slope=slope, adaptive=False, n=num_points)) + x_check = check_point_accuracy(x, expected_value[0]) + y_check = check_point_accuracy(y, expected_value[1]) + return x_check and y_check + + exp1 = ((0.0, 2.0, 4.0, 6.0, 8.0, 10.0), (0.0, 0.7324667795033895, 1.9909720978650398, + 2.7956587704217783, 3.9224897567931514, 4.85022655284895)) + exp2 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, + 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), + (2.4360213402019326e-08, 0.10175320182493253, 0.33057612497658406, 0.5967937263298935, + 0.8431511866718248, 1.0398805391471613, 1.1776043125035738, 1.2600994825747305, 1.2981042689274653, + 1.304684417610106)) + exp3 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, + 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (-3.9329040468771836e-08, + 0.34686634635794555, 2.9998828170537903, 12.33303690737476, 40.993913948137795, 127.84145222317912, + 391.41713691996, 1192.0006858708389, 3623.9808672503405, 11011.728034546572)) + exp4 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, + 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 1.9051973784484078, 30.483158055174524, + 154.32098765432104, 487.7305288827924, 1190.7483615302544, 2469.1358024691367, 4574.3789056546275, + 7803.688462124678, 12500.0)) + exp5 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, + 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 3.8844361856975635, 9.141792069209865, + 14.096349157657231, 19.09783068994694, 24.10179770390321, 29.09907319114121, 34.10040420185154, + 39.09983919254265, 44.10006013058409)) + exp6 = ((0.0, 1.1111111111111112, 2.2222222222222223, 3.3333333333333335, 4.444444444444445, 5.555555555555555, + 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0), (0.0, 1.1111111111111112, 2.2222222222222223, + 3.3333333333333335, 4.444444444444445, 5.555555555555555, 6.666666666666667, 7.777777777777779, 8.88888888888889, 10.0)) + + assert ramp_res_tester(tf1, 6, exp1) + assert ramp_res_tester(tf2, 10, exp2, 1.2) + assert ramp_res_tester(tf3, 10, exp3, 1.5) + assert ramp_res_tester(tf4, 10, exp4, 3) + assert ramp_res_tester(tf5, 10, exp5, 9) + assert ramp_res_tester(tf6, 10, exp6) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/test_lti.py b/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/test_lti.py new file mode 100644 index 0000000000000000000000000000000000000000..9a3f599cbe240827cf24c7473782f7b71ad9a562 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/control/tests/test_lti.py @@ -0,0 +1,1750 @@ +from sympy.core.add import Add +from sympy.core.function import Function +from sympy.core.mul import Mul +from sympy.core.numbers import (I, pi, Rational, oo) +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.elementary.exponential import (exp, log) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import atan +from sympy.matrices.dense import eye +from sympy.polys.polytools import factor +from sympy.polys.rootoftools import CRootOf +from sympy.simplify.simplify import simplify +from sympy.core.containers import Tuple +from sympy.matrices import ImmutableMatrix, Matrix, ShapeError +from sympy.physics.control import (TransferFunction, Series, Parallel, + Feedback, TransferFunctionMatrix, MIMOSeries, MIMOParallel, MIMOFeedback, + StateSpace, gbt, bilinear, forward_diff, backward_diff, phase_margin, gain_margin) +from sympy.testing.pytest import raises + +a, x, b, c, s, g, d, p, k, tau, zeta, wn, T = symbols('a, x, b, c, s, g, d, p, k,\ + tau, zeta, wn, T') +a0, a1, a2, a3, b0, b1, b2, b3, c0, c1, c2, c3, d0, d1, d2, d3 = symbols('a0:4,\ + b0:4, c0:4, d0:4') +TF1 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s) +TF2 = TransferFunction(k, 1, s) +TF3 = TransferFunction(a2*p - s, a2*s + p, s) + + +def test_TransferFunction_construction(): + tf = TransferFunction(s + 1, s**2 + s + 1, s) + assert tf.num == (s + 1) + assert tf.den == (s**2 + s + 1) + assert tf.args == (s + 1, s**2 + s + 1, s) + + tf1 = TransferFunction(s + 4, s - 5, s) + assert tf1.num == (s + 4) + assert tf1.den == (s - 5) + assert tf1.args == (s + 4, s - 5, s) + + # using different polynomial variables. + tf2 = TransferFunction(p + 3, p**2 - 9, p) + assert tf2.num == (p + 3) + assert tf2.den == (p**2 - 9) + assert tf2.args == (p + 3, p**2 - 9, p) + + tf3 = TransferFunction(p**3 + 5*p**2 + 4, p**4 + 3*p + 1, p) + assert tf3.args == (p**3 + 5*p**2 + 4, p**4 + 3*p + 1, p) + + # no pole-zero cancellation on its own. + tf4 = TransferFunction((s + 3)*(s - 1), (s - 1)*(s + 5), s) + assert tf4.den == (s - 1)*(s + 5) + assert tf4.args == ((s + 3)*(s - 1), (s - 1)*(s + 5), s) + + tf4_ = TransferFunction(p + 2, p + 2, p) + assert tf4_.args == (p + 2, p + 2, p) + + tf5 = TransferFunction(s - 1, 4 - p, s) + assert tf5.args == (s - 1, 4 - p, s) + + tf5_ = TransferFunction(s - 1, s - 1, s) + assert tf5_.args == (s - 1, s - 1, s) + + tf6 = TransferFunction(5, 6, s) + assert tf6.num == 5 + assert tf6.den == 6 + assert tf6.args == (5, 6, s) + + tf6_ = TransferFunction(1/2, 4, s) + assert tf6_.num == 0.5 + assert tf6_.den == 4 + assert tf6_.args == (0.500000000000000, 4, s) + + tf7 = TransferFunction(3*s**2 + 2*p + 4*s, 8*p**2 + 7*s, s) + tf8 = TransferFunction(3*s**2 + 2*p + 4*s, 8*p**2 + 7*s, p) + assert not tf7 == tf8 + + tf7_ = TransferFunction(a0*s + a1*s**2 + a2*s**3, b0*p - b1*s, s) + tf8_ = TransferFunction(a0*s + a1*s**2 + a2*s**3, b0*p - b1*s, s) + assert tf7_ == tf8_ + assert -(-tf7_) == tf7_ == -(-(-(-tf7_))) + + tf9 = TransferFunction(a*s**3 + b*s**2 + g*s + d, d*p + g*p**2 + g*s, s) + assert tf9.args == (a*s**3 + b*s**2 + d + g*s, d*p + g*p**2 + g*s, s) + + tf10 = TransferFunction(p**3 + d, g*s**2 + d*s + a, p) + tf10_ = TransferFunction(p**3 + d, g*s**2 + d*s + a, p) + assert tf10.args == (d + p**3, a + d*s + g*s**2, p) + assert tf10_ == tf10 + + tf11 = TransferFunction(a1*s + a0, b2*s**2 + b1*s + b0, s) + assert tf11.num == (a0 + a1*s) + assert tf11.den == (b0 + b1*s + b2*s**2) + assert tf11.args == (a0 + a1*s, b0 + b1*s + b2*s**2, s) + + # when just the numerator is 0, leave the denominator alone. + tf12 = TransferFunction(0, p**2 - p + 1, p) + assert tf12.args == (0, p**2 - p + 1, p) + + tf13 = TransferFunction(0, 1, s) + assert tf13.args == (0, 1, s) + + # float exponents + tf14 = TransferFunction(a0*s**0.5 + a2*s**0.6 - a1, a1*p**(-8.7), s) + assert tf14.args == (a0*s**0.5 - a1 + a2*s**0.6, a1*p**(-8.7), s) + + tf15 = TransferFunction(a2**2*p**(1/4) + a1*s**(-4/5), a0*s - p, p) + assert tf15.args == (a1*s**(-0.8) + a2**2*p**0.25, a0*s - p, p) + + omega_o, k_p, k_o, k_i = symbols('omega_o, k_p, k_o, k_i') + tf18 = TransferFunction((k_p + k_o*s + k_i/s), s**2 + 2*omega_o*s + omega_o**2, s) + assert tf18.num == k_i/s + k_o*s + k_p + assert tf18.args == (k_i/s + k_o*s + k_p, omega_o**2 + 2*omega_o*s + s**2, s) + + # ValueError when denominator is zero. + raises(ValueError, lambda: TransferFunction(4, 0, s)) + raises(ValueError, lambda: TransferFunction(s, 0, s)) + raises(ValueError, lambda: TransferFunction(0, 0, s)) + + raises(TypeError, lambda: TransferFunction(Matrix([1, 2, 3]), s, s)) + + raises(TypeError, lambda: TransferFunction(s**2 + 2*s - 1, s + 3, 3)) + raises(TypeError, lambda: TransferFunction(p + 1, 5 - p, 4)) + raises(TypeError, lambda: TransferFunction(3, 4, 8)) + + +def test_TransferFunction_functions(): + # classmethod from_rational_expression + expr_1 = Mul(0, Pow(s, -1, evaluate=False), evaluate=False) + expr_2 = s/0 + expr_3 = (p*s**2 + 5*s)/(s + 1)**3 + expr_4 = 6 + expr_5 = ((2 + 3*s)*(5 + 2*s))/((9 + 3*s)*(5 + 2*s**2)) + expr_6 = (9*s**4 + 4*s**2 + 8)/((s + 1)*(s + 9)) + tf = TransferFunction(s + 1, s**2 + 2, s) + delay = exp(-s/tau) + expr_7 = delay*tf.to_expr() + H1 = TransferFunction.from_rational_expression(expr_7, s) + H2 = TransferFunction(s + 1, (s**2 + 2)*exp(s/tau), s) + expr_8 = Add(2, 3*s/(s**2 + 1), evaluate=False) + + assert TransferFunction.from_rational_expression(expr_1) == TransferFunction(0, s, s) + raises(ZeroDivisionError, lambda: TransferFunction.from_rational_expression(expr_2)) + raises(ValueError, lambda: TransferFunction.from_rational_expression(expr_3)) + assert TransferFunction.from_rational_expression(expr_3, s) == TransferFunction((p*s**2 + 5*s), (s + 1)**3, s) + assert TransferFunction.from_rational_expression(expr_3, p) == TransferFunction((p*s**2 + 5*s), (s + 1)**3, p) + raises(ValueError, lambda: TransferFunction.from_rational_expression(expr_4)) + assert TransferFunction.from_rational_expression(expr_4, s) == TransferFunction(6, 1, s) + assert TransferFunction.from_rational_expression(expr_5, s) == \ + TransferFunction((2 + 3*s)*(5 + 2*s), (9 + 3*s)*(5 + 2*s**2), s) + assert TransferFunction.from_rational_expression(expr_6, s) == \ + TransferFunction((9*s**4 + 4*s**2 + 8), (s + 1)*(s + 9), s) + assert H1 == H2 + assert TransferFunction.from_rational_expression(expr_8, s) == \ + TransferFunction(2*s**2 + 3*s + 2, s**2 + 1, s) + + # classmethod from_coeff_lists + tf1 = TransferFunction.from_coeff_lists([1, 2], [3, 4, 5], s) + num2 = [p**2, 2*p] + den2 = [p**3, p + 1, 4] + tf2 = TransferFunction.from_coeff_lists(num2, den2, s) + num3 = [1, 2, 3] + den3 = [0, 0] + + assert tf1 == TransferFunction(s + 2, 3*s**2 + 4*s + 5, s) + assert tf2 == TransferFunction(p**2*s + 2*p, p**3*s**2 + s*(p + 1) + 4, s) + raises(ZeroDivisionError, lambda: TransferFunction.from_coeff_lists(num3, den3, s)) + + # classmethod from_zpk + zeros = [4] + poles = [-1+2j, -1-2j] + gain = 3 + tf1 = TransferFunction.from_zpk(zeros, poles, gain, s) + + assert tf1 == TransferFunction(3*s - 12, (s + 1.0 - 2.0*I)*(s + 1.0 + 2.0*I), s) + + # explicitly cancel poles and zeros. + tf0 = TransferFunction(s**5 + s**3 + s, s - s**2, s) + a = TransferFunction(-(s**4 + s**2 + 1), s - 1, s) + assert tf0.simplify() == simplify(tf0) == a + + tf1 = TransferFunction((p + 3)*(p - 1), (p - 1)*(p + 5), p) + b = TransferFunction(p + 3, p + 5, p) + assert tf1.simplify() == simplify(tf1) == b + + # expand the numerator and the denominator. + G1 = TransferFunction((1 - s)**2, (s**2 + 1)**2, s) + G2 = TransferFunction(1, -3, p) + c = (a2*s**p + a1*s**s + a0*p**p)*(p**s + s**p) + d = (b0*s**s + b1*p**s)*(b2*s*p + p**p) + e = a0*p**p*p**s + a0*p**p*s**p + a1*p**s*s**s + a1*s**p*s**s + a2*p**s*s**p + a2*s**(2*p) + f = b0*b2*p*s*s**s + b0*p**p*s**s + b1*b2*p*p**s*s + b1*p**p*p**s + g = a1*a2*s*s**p + a1*p*s + a2*b1*p*s*s**p + b1*p**2*s + G3 = TransferFunction(c, d, s) + G4 = TransferFunction(a0*s**s - b0*p**p, (a1*s + b1*s*p)*(a2*s**p + p), p) + + assert G1.expand() == TransferFunction(s**2 - 2*s + 1, s**4 + 2*s**2 + 1, s) + assert tf1.expand() == TransferFunction(p**2 + 2*p - 3, p**2 + 4*p - 5, p) + assert G2.expand() == G2 + assert G3.expand() == TransferFunction(e, f, s) + assert G4.expand() == TransferFunction(a0*s**s - b0*p**p, g, p) + + # purely symbolic polynomials. + p1 = a1*s + a0 + p2 = b2*s**2 + b1*s + b0 + SP1 = TransferFunction(p1, p2, s) + expect1 = TransferFunction(2.0*s + 1.0, 5.0*s**2 + 4.0*s + 3.0, s) + expect1_ = TransferFunction(2*s + 1, 5*s**2 + 4*s + 3, s) + assert SP1.subs({a0: 1, a1: 2, b0: 3, b1: 4, b2: 5}) == expect1_ + assert SP1.subs({a0: 1, a1: 2, b0: 3, b1: 4, b2: 5}).evalf() == expect1 + assert expect1_.evalf() == expect1 + + c1, d0, d1, d2 = symbols('c1, d0:3') + p3, p4 = c1*p, d2*p**3 + d1*p**2 - d0 + SP2 = TransferFunction(p3, p4, p) + expect2 = TransferFunction(2.0*p, 5.0*p**3 + 2.0*p**2 - 3.0, p) + expect2_ = TransferFunction(2*p, 5*p**3 + 2*p**2 - 3, p) + assert SP2.subs({c1: 2, d0: 3, d1: 2, d2: 5}) == expect2_ + assert SP2.subs({c1: 2, d0: 3, d1: 2, d2: 5}).evalf() == expect2 + assert expect2_.evalf() == expect2 + + SP3 = TransferFunction(a0*p**3 + a1*s**2 - b0*s + b1, a1*s + p, s) + expect3 = TransferFunction(2.0*p**3 + 4.0*s**2 - s + 5.0, p + 4.0*s, s) + expect3_ = TransferFunction(2*p**3 + 4*s**2 - s + 5, p + 4*s, s) + assert SP3.subs({a0: 2, a1: 4, b0: 1, b1: 5}) == expect3_ + assert SP3.subs({a0: 2, a1: 4, b0: 1, b1: 5}).evalf() == expect3 + assert expect3_.evalf() == expect3 + + SP4 = TransferFunction(s - a1*p**3, a0*s + p, p) + expect4 = TransferFunction(7.0*p**3 + s, p - s, p) + expect4_ = TransferFunction(7*p**3 + s, p - s, p) + assert SP4.subs({a0: -1, a1: -7}) == expect4_ + assert SP4.subs({a0: -1, a1: -7}).evalf() == expect4 + assert expect4_.evalf() == expect4 + + # evaluate the transfer function at particular frequencies. + assert tf1.eval_frequency(wn) == wn**2/(wn**2 + 4*wn - 5) + 2*wn/(wn**2 + 4*wn - 5) - 3/(wn**2 + 4*wn - 5) + assert G1.eval_frequency(1 + I) == S(3)/25 + S(4)*I/25 + assert G4.eval_frequency(S(5)/3) == \ + a0*s**s/(a1*a2*s**(S(8)/3) + S(5)*a1*s/3 + 5*a2*b1*s**(S(8)/3)/3 + S(25)*b1*s/9) - 5*3**(S(1)/3)*5**(S(2)/3)*b0/(9*a1*a2*s**(S(8)/3) + 15*a1*s + 15*a2*b1*s**(S(8)/3) + 25*b1*s) + + # Low-frequency (or DC) gain. + assert tf0.dc_gain() == 1 + assert tf1.dc_gain() == Rational(3, 5) + assert SP2.dc_gain() == 0 + assert expect4.dc_gain() == -1 + assert expect2_.dc_gain() == 0 + assert TransferFunction(1, s, s).dc_gain() == oo + + # Poles of a transfer function. + tf_ = TransferFunction(x**3 - k, k, x) + _tf = TransferFunction(k, x**4 - k, x) + TF_ = TransferFunction(x**2, x**10 + x + x**2, x) + _TF = TransferFunction(x**10 + x + x**2, x**2, x) + assert G1.poles() == [I, I, -I, -I] + assert G2.poles() == [] + assert tf1.poles() == [-5, 1] + assert expect4_.poles() == [s] + assert SP4.poles() == [-a0*s] + assert expect3.poles() == [-0.25*p] + assert str(expect2.poles()) == str([0.729001428685125, -0.564500714342563 - 0.710198984796332*I, -0.564500714342563 + 0.710198984796332*I]) + assert str(expect1.poles()) == str([-0.4 - 0.66332495807108*I, -0.4 + 0.66332495807108*I]) + assert _tf.poles() == [k**(Rational(1, 4)), -k**(Rational(1, 4)), I*k**(Rational(1, 4)), -I*k**(Rational(1, 4))] + assert TF_.poles() == [CRootOf(x**9 + x + 1, 0), 0, CRootOf(x**9 + x + 1, 1), CRootOf(x**9 + x + 1, 2), + CRootOf(x**9 + x + 1, 3), CRootOf(x**9 + x + 1, 4), CRootOf(x**9 + x + 1, 5), CRootOf(x**9 + x + 1, 6), + CRootOf(x**9 + x + 1, 7), CRootOf(x**9 + x + 1, 8)] + raises(NotImplementedError, lambda: TransferFunction(x**2, a0*x**10 + x + x**2, x).poles()) + + # Stability of a transfer function. + q, r = symbols('q, r', negative=True) + t = symbols('t', positive=True) + TF_ = TransferFunction(s**2 + a0 - a1*p, q*s - r, s) + stable_tf = TransferFunction(s**2 + a0 - a1*p, q*s - 1, s) + stable_tf_ = TransferFunction(s**2 + a0 - a1*p, q*s - t, s) + + assert G1.is_stable() is False + assert G2.is_stable() is True + assert tf1.is_stable() is False # as one pole is +ve, and the other is -ve. + assert expect2.is_stable() is False + assert expect1.is_stable() is True + assert stable_tf.is_stable() is True + assert stable_tf_.is_stable() is True + assert TF_.is_stable() is False + assert expect4_.is_stable() is None # no assumption provided for the only pole 's'. + assert SP4.is_stable() is None + + # Zeros of a transfer function. + assert G1.zeros() == [1, 1] + assert G2.zeros() == [] + assert tf1.zeros() == [-3, 1] + assert expect4_.zeros() == [7**(Rational(2, 3))*(-s)**(Rational(1, 3))/7, -7**(Rational(2, 3))*(-s)**(Rational(1, 3))/14 - + sqrt(3)*7**(Rational(2, 3))*I*(-s)**(Rational(1, 3))/14, -7**(Rational(2, 3))*(-s)**(Rational(1, 3))/14 + sqrt(3)*7**(Rational(2, 3))*I*(-s)**(Rational(1, 3))/14] + assert SP4.zeros() == [(s/a1)**(Rational(1, 3)), -(s/a1)**(Rational(1, 3))/2 - sqrt(3)*I*(s/a1)**(Rational(1, 3))/2, + -(s/a1)**(Rational(1, 3))/2 + sqrt(3)*I*(s/a1)**(Rational(1, 3))/2] + assert str(expect3.zeros()) == str([0.125 - 1.11102430216445*sqrt(-0.405063291139241*p**3 - 1.0), + 1.11102430216445*sqrt(-0.405063291139241*p**3 - 1.0) + 0.125]) + assert tf_.zeros() == [k**(Rational(1, 3)), -k**(Rational(1, 3))/2 - sqrt(3)*I*k**(Rational(1, 3))/2, + -k**(Rational(1, 3))/2 + sqrt(3)*I*k**(Rational(1, 3))/2] + assert _TF.zeros() == [CRootOf(x**9 + x + 1, 0), 0, CRootOf(x**9 + x + 1, 1), CRootOf(x**9 + x + 1, 2), + CRootOf(x**9 + x + 1, 3), CRootOf(x**9 + x + 1, 4), CRootOf(x**9 + x + 1, 5), CRootOf(x**9 + x + 1, 6), + CRootOf(x**9 + x + 1, 7), CRootOf(x**9 + x + 1, 8)] + raises(NotImplementedError, lambda: TransferFunction(a0*x**10 + x + x**2, x**2, x).zeros()) + + # negation of TF. + tf2 = TransferFunction(s + 3, s**2 - s**3 + 9, s) + tf3 = TransferFunction(-3*p + 3, 1 - p, p) + assert -tf2 == TransferFunction(-s - 3, s**2 - s**3 + 9, s) + assert -tf3 == TransferFunction(3*p - 3, 1 - p, p) + + # taking power of a TF. + tf4 = TransferFunction(p + 4, p - 3, p) + tf5 = TransferFunction(s**2 + 1, 1 - s, s) + expect2 = TransferFunction((s**2 + 1)**3, (1 - s)**3, s) + expect1 = TransferFunction((p + 4)**2, (p - 3)**2, p) + assert (tf4*tf4).doit() == tf4**2 == pow(tf4, 2) == expect1 + assert (tf5*tf5*tf5).doit() == tf5**3 == pow(tf5, 3) == expect2 + assert tf5**0 == pow(tf5, 0) == TransferFunction(1, 1, s) + assert Series(tf4).doit()**-1 == tf4**-1 == pow(tf4, -1) == TransferFunction(p - 3, p + 4, p) + assert (tf5*tf5).doit()**-1 == tf5**-2 == pow(tf5, -2) == TransferFunction((1 - s)**2, (s**2 + 1)**2, s) + + raises(ValueError, lambda: tf4**(s**2 + s - 1)) + raises(ValueError, lambda: tf5**s) + raises(ValueError, lambda: tf4**tf5) + + # SymPy's own functions. + tf = TransferFunction(s - 1, s**2 - 2*s + 1, s) + tf6 = TransferFunction(s + p, p**2 - 5, s) + assert factor(tf) == TransferFunction(s - 1, (s - 1)**2, s) + assert tf.num.subs(s, 2) == tf.den.subs(s, 2) == 1 + # subs & xreplace + assert tf.subs(s, 2) == TransferFunction(s - 1, s**2 - 2*s + 1, s) + assert tf6.subs(p, 3) == TransferFunction(s + 3, 4, s) + assert tf3.xreplace({p: s}) == TransferFunction(-3*s + 3, 1 - s, s) + raises(TypeError, lambda: tf3.xreplace({p: exp(2)})) + assert tf3.subs(p, exp(2)) == tf3 + + tf7 = TransferFunction(a0*s**p + a1*p**s, a2*p - s, s) + assert tf7.xreplace({s: k}) == TransferFunction(a0*k**p + a1*p**k, a2*p - k, k) + assert tf7.subs(s, k) == TransferFunction(a0*s**p + a1*p**s, a2*p - s, s) + + # Conversion to Expr with to_expr() + tf8 = TransferFunction(a0*s**5 + 5*s**2 + 3, s**6 - 3, s) + tf9 = TransferFunction((5 + s), (5 + s)*(6 + s), s) + tf10 = TransferFunction(0, 1, s) + tf11 = TransferFunction(1, 1, s) + assert tf8.to_expr() == Mul((a0*s**5 + 5*s**2 + 3), Pow((s**6 - 3), -1, evaluate=False), evaluate=False) + assert tf9.to_expr() == Mul((s + 5), Pow((5 + s)*(6 + s), -1, evaluate=False), evaluate=False) + assert tf10.to_expr() == Mul(S(0), Pow(1, -1, evaluate=False), evaluate=False) + assert tf11.to_expr() == Pow(1, -1, evaluate=False) + +def test_TransferFunction_addition_and_subtraction(): + tf1 = TransferFunction(s + 6, s - 5, s) + tf2 = TransferFunction(s + 3, s + 1, s) + tf3 = TransferFunction(s + 1, s**2 + s + 1, s) + tf4 = TransferFunction(p, 2 - p, p) + + # addition + assert tf1 + tf2 == Parallel(tf1, tf2) + assert tf3 + tf1 == Parallel(tf3, tf1) + assert -tf1 + tf2 + tf3 == Parallel(-tf1, tf2, tf3) + assert tf1 + (tf2 + tf3) == Parallel(tf1, tf2, tf3) + + c = symbols("c", commutative=False) + raises(ValueError, lambda: tf1 + Matrix([1, 2, 3])) + raises(ValueError, lambda: tf2 + c) + raises(ValueError, lambda: tf3 + tf4) + raises(ValueError, lambda: tf1 + (s - 1)) + raises(ValueError, lambda: tf1 + 8) + raises(ValueError, lambda: (1 - p**3) + tf1) + + # subtraction + assert tf1 - tf2 == Parallel(tf1, -tf2) + assert tf3 - tf2 == Parallel(tf3, -tf2) + assert -tf1 - tf3 == Parallel(-tf1, -tf3) + assert tf1 - tf2 + tf3 == Parallel(tf1, -tf2, tf3) + + raises(ValueError, lambda: tf1 - Matrix([1, 2, 3])) + raises(ValueError, lambda: tf3 - tf4) + raises(ValueError, lambda: tf1 - (s - 1)) + raises(ValueError, lambda: tf1 - 8) + raises(ValueError, lambda: (s + 5) - tf2) + raises(ValueError, lambda: (1 + p**4) - tf1) + + +def test_TransferFunction_multiplication_and_division(): + G1 = TransferFunction(s + 3, -s**3 + 9, s) + G2 = TransferFunction(s + 1, s - 5, s) + G3 = TransferFunction(p, p**4 - 6, p) + G4 = TransferFunction(p + 4, p - 5, p) + G5 = TransferFunction(s + 6, s - 5, s) + G6 = TransferFunction(s + 3, s + 1, s) + G7 = TransferFunction(1, 1, s) + + # multiplication + assert G1*G2 == Series(G1, G2) + assert -G1*G5 == Series(-G1, G5) + assert -G2*G5*-G6 == Series(-G2, G5, -G6) + assert -G1*-G2*-G5*-G6 == Series(-G1, -G2, -G5, -G6) + assert G3*G4 == Series(G3, G4) + assert (G1*G2)*-(G5*G6) == \ + Series(G1, G2, TransferFunction(-1, 1, s), Series(G5, G6)) + assert G1*G2*(G5 + G6) == Series(G1, G2, Parallel(G5, G6)) + + # division - See ``test_Feedback_functions()`` for division by Parallel objects. + assert G5/G6 == Series(G5, pow(G6, -1)) + assert -G3/G4 == Series(-G3, pow(G4, -1)) + assert (G5*G6)/G7 == Series(G5, G6, pow(G7, -1)) + + c = symbols("c", commutative=False) + raises(ValueError, lambda: G3 * Matrix([1, 2, 3])) + raises(ValueError, lambda: G1 * c) + raises(ValueError, lambda: G3 * G5) + raises(ValueError, lambda: G5 * (s - 1)) + raises(ValueError, lambda: 9 * G5) + + raises(ValueError, lambda: G3 / Matrix([1, 2, 3])) + raises(ValueError, lambda: G6 / 0) + raises(ValueError, lambda: G3 / G5) + raises(ValueError, lambda: G5 / 2) + raises(ValueError, lambda: G5 / s**2) + raises(ValueError, lambda: (s - 4*s**2) / G2) + raises(ValueError, lambda: 0 / G4) + raises(ValueError, lambda: G7 / (1 + G6)) + raises(ValueError, lambda: G7 / (G5 * G6)) + raises(ValueError, lambda: G7 / (G7 + (G5 + G6))) + + +def test_TransferFunction_is_proper(): + omega_o, zeta, tau = symbols('omega_o, zeta, tau') + G1 = TransferFunction(omega_o**2, s**2 + p*omega_o*zeta*s + omega_o**2, omega_o) + G2 = TransferFunction(tau - s**3, tau + p**4, tau) + G3 = TransferFunction(a*b*s**3 + s**2 - a*p + s, b - s*p**2, p) + G4 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s) + assert G1.is_proper + assert G2.is_proper + assert G3.is_proper + assert not G4.is_proper + + +def test_TransferFunction_is_strictly_proper(): + omega_o, zeta, tau = symbols('omega_o, zeta, tau') + tf1 = TransferFunction(omega_o**2, s**2 + p*omega_o*zeta*s + omega_o**2, omega_o) + tf2 = TransferFunction(tau - s**3, tau + p**4, tau) + tf3 = TransferFunction(a*b*s**3 + s**2 - a*p + s, b - s*p**2, p) + tf4 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s) + assert not tf1.is_strictly_proper + assert not tf2.is_strictly_proper + assert tf3.is_strictly_proper + assert not tf4.is_strictly_proper + + +def test_TransferFunction_is_biproper(): + tau, omega_o, zeta = symbols('tau, omega_o, zeta') + tf1 = TransferFunction(omega_o**2, s**2 + p*omega_o*zeta*s + omega_o**2, omega_o) + tf2 = TransferFunction(tau - s**3, tau + p**4, tau) + tf3 = TransferFunction(a*b*s**3 + s**2 - a*p + s, b - s*p**2, p) + tf4 = TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s) + assert tf1.is_biproper + assert tf2.is_biproper + assert not tf3.is_biproper + assert not tf4.is_biproper + + +def test_Series_construction(): + tf = TransferFunction(a0*s**3 + a1*s**2 - a2*s, b0*p**4 + b1*p**3 - b2*s*p, s) + tf2 = TransferFunction(a2*p - s, a2*s + p, s) + tf3 = TransferFunction(a0*p + p**a1 - s, p, p) + tf4 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s) + inp = Function('X_d')(s) + out = Function('X')(s) + + s0 = Series(tf, tf2) + assert s0.args == (tf, tf2) + assert s0.var == s + + s1 = Series(Parallel(tf, -tf2), tf2) + assert s1.args == (Parallel(tf, -tf2), tf2) + assert s1.var == s + + tf3_ = TransferFunction(inp, 1, s) + tf4_ = TransferFunction(-out, 1, s) + s2 = Series(tf, Parallel(tf3_, tf4_), tf2) + assert s2.args == (tf, Parallel(tf3_, tf4_), tf2) + + s3 = Series(tf, tf2, tf4) + assert s3.args == (tf, tf2, tf4) + + s4 = Series(tf3_, tf4_) + assert s4.args == (tf3_, tf4_) + assert s4.var == s + + s6 = Series(tf2, tf4, Parallel(tf2, -tf), tf4) + assert s6.args == (tf2, tf4, Parallel(tf2, -tf), tf4) + + s7 = Series(tf, tf2) + assert s0 == s7 + assert not s0 == s2 + + raises(ValueError, lambda: Series(tf, tf3)) + raises(ValueError, lambda: Series(tf, tf2, tf3, tf4)) + raises(ValueError, lambda: Series(-tf3, tf2)) + raises(TypeError, lambda: Series(2, tf, tf4)) + raises(TypeError, lambda: Series(s**2 + p*s, tf3, tf2)) + raises(TypeError, lambda: Series(tf3, Matrix([1, 2, 3, 4]))) + + +def test_MIMOSeries_construction(): + tf_1 = TransferFunction(a0*s**3 + a1*s**2 - a2*s, b0*p**4 + b1*p**3 - b2*s*p, s) + tf_2 = TransferFunction(a2*p - s, a2*s + p, s) + tf_3 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s) + + tfm_1 = TransferFunctionMatrix([[tf_1, tf_2, tf_3], [-tf_3, -tf_2, tf_1]]) + tfm_2 = TransferFunctionMatrix([[-tf_2], [-tf_2], [-tf_3]]) + tfm_3 = TransferFunctionMatrix([[-tf_3]]) + tfm_4 = TransferFunctionMatrix([[TF3], [TF2], [-TF1]]) + tfm_5 = TransferFunctionMatrix.from_Matrix(Matrix([1/p]), p) + + s8 = MIMOSeries(tfm_2, tfm_1) + assert s8.args == (tfm_2, tfm_1) + assert s8.var == s + assert s8.shape == (s8.num_outputs, s8.num_inputs) == (2, 1) + + s9 = MIMOSeries(tfm_3, tfm_2, tfm_1) + assert s9.args == (tfm_3, tfm_2, tfm_1) + assert s9.var == s + assert s9.shape == (s9.num_outputs, s9.num_inputs) == (2, 1) + + s11 = MIMOSeries(tfm_3, MIMOParallel(-tfm_2, -tfm_4), tfm_1) + assert s11.args == (tfm_3, MIMOParallel(-tfm_2, -tfm_4), tfm_1) + assert s11.shape == (s11.num_outputs, s11.num_inputs) == (2, 1) + + # arg cannot be empty tuple. + raises(ValueError, lambda: MIMOSeries()) + + # arg cannot contain SISO as well as MIMO systems. + raises(TypeError, lambda: MIMOSeries(tfm_1, tf_1)) + + # for all the adjacent transfer function matrices: + # no. of inputs of first TFM must be equal to the no. of outputs of the second TFM. + raises(ValueError, lambda: MIMOSeries(tfm_1, tfm_2, -tfm_1)) + + # all the TFMs must use the same complex variable. + raises(ValueError, lambda: MIMOSeries(tfm_3, tfm_5)) + + # Number or expression not allowed in the arguments. + raises(TypeError, lambda: MIMOSeries(2, tfm_2, tfm_3)) + raises(TypeError, lambda: MIMOSeries(s**2 + p*s, -tfm_2, tfm_3)) + raises(TypeError, lambda: MIMOSeries(Matrix([1/p]), tfm_3)) + + +def test_Series_functions(): + tf1 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s) + tf2 = TransferFunction(k, 1, s) + tf3 = TransferFunction(a2*p - s, a2*s + p, s) + tf4 = TransferFunction(a0*p + p**a1 - s, p, p) + tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s) + + assert tf1*tf2*tf3 == Series(tf1, tf2, tf3) == Series(Series(tf1, tf2), tf3) \ + == Series(tf1, Series(tf2, tf3)) + assert tf1*(tf2 + tf3) == Series(tf1, Parallel(tf2, tf3)) + assert tf1*tf2 + tf5 == Parallel(Series(tf1, tf2), tf5) + assert tf1*tf2 - tf5 == Parallel(Series(tf1, tf2), -tf5) + assert tf1*tf2 + tf3 + tf5 == Parallel(Series(tf1, tf2), tf3, tf5) + assert tf1*tf2 - tf3 - tf5 == Parallel(Series(tf1, tf2), -tf3, -tf5) + assert tf1*tf2 - tf3 + tf5 == Parallel(Series(tf1, tf2), -tf3, tf5) + assert tf1*tf2 + tf3*tf5 == Parallel(Series(tf1, tf2), Series(tf3, tf5)) + assert tf1*tf2 - tf3*tf5 == Parallel(Series(tf1, tf2), Series(TransferFunction(-1, 1, s), Series(tf3, tf5))) + assert tf2*tf3*(tf2 - tf1)*tf3 == Series(tf2, tf3, Parallel(tf2, -tf1), tf3) + assert -tf1*tf2 == Series(-tf1, tf2) + assert -(tf1*tf2) == Series(TransferFunction(-1, 1, s), Series(tf1, tf2)) + raises(ValueError, lambda: tf1*tf2*tf4) + raises(ValueError, lambda: tf1*(tf2 - tf4)) + raises(ValueError, lambda: tf3*Matrix([1, 2, 3])) + + # evaluate=True -> doit() + assert Series(tf1, tf2, evaluate=True) == Series(tf1, tf2).doit() == \ + TransferFunction(k, s**2 + 2*s*wn*zeta + wn**2, s) + assert Series(tf1, tf2, Parallel(tf1, -tf3), evaluate=True) == Series(tf1, tf2, Parallel(tf1, -tf3)).doit() == \ + TransferFunction(k*(a2*s + p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2)), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2)**2, s) + assert Series(tf2, tf1, -tf3, evaluate=True) == Series(tf2, tf1, -tf3).doit() == \ + TransferFunction(k*(-a2*p + s), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + assert not Series(tf1, -tf2, evaluate=False) == Series(tf1, -tf2).doit() + + assert Series(Parallel(tf1, tf2), Parallel(tf2, -tf3)).doit() == \ + TransferFunction((k*(s**2 + 2*s*wn*zeta + wn**2) + 1)*(-a2*p + k*(a2*s + p) + s), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + assert Series(-tf1, -tf2, -tf3).doit() == \ + TransferFunction(k*(-a2*p + s), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + assert -Series(tf1, tf2, tf3).doit() == \ + TransferFunction(-k*(a2*p - s), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + assert Series(tf2, tf3, Parallel(tf2, -tf1), tf3).doit() == \ + TransferFunction(k*(a2*p - s)**2*(k*(s**2 + 2*s*wn*zeta + wn**2) - 1), (a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2), s) + + assert Series(tf1, tf2).rewrite(TransferFunction) == TransferFunction(k, s**2 + 2*s*wn*zeta + wn**2, s) + assert Series(tf2, tf1, -tf3).rewrite(TransferFunction) == \ + TransferFunction(k*(-a2*p + s), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + + S1 = Series(Parallel(tf1, tf2), Parallel(tf2, -tf3)) + assert S1.is_proper + assert not S1.is_strictly_proper + assert S1.is_biproper + + S2 = Series(tf1, tf2, tf3) + assert S2.is_proper + assert S2.is_strictly_proper + assert not S2.is_biproper + + S3 = Series(tf1, -tf2, Parallel(tf1, -tf3)) + assert S3.is_proper + assert S3.is_strictly_proper + assert not S3.is_biproper + + +def test_MIMOSeries_functions(): + tfm1 = TransferFunctionMatrix([[TF1, TF2, TF3], [-TF3, -TF2, TF1]]) + tfm2 = TransferFunctionMatrix([[-TF1], [-TF2], [-TF3]]) + tfm3 = TransferFunctionMatrix([[-TF1]]) + tfm4 = TransferFunctionMatrix([[-TF2, -TF3], [-TF1, TF2]]) + tfm5 = TransferFunctionMatrix([[TF2, -TF2], [-TF3, -TF2]]) + tfm6 = TransferFunctionMatrix([[-TF3], [TF1]]) + tfm7 = TransferFunctionMatrix([[TF1], [-TF2]]) + + assert tfm1*tfm2 + tfm6 == MIMOParallel(MIMOSeries(tfm2, tfm1), tfm6) + assert tfm1*tfm2 + tfm7 + tfm6 == MIMOParallel(MIMOSeries(tfm2, tfm1), tfm7, tfm6) + assert tfm1*tfm2 - tfm6 - tfm7 == MIMOParallel(MIMOSeries(tfm2, tfm1), -tfm6, -tfm7) + assert tfm4*tfm5 + (tfm4 - tfm5) == MIMOParallel(MIMOSeries(tfm5, tfm4), tfm4, -tfm5) + assert tfm4*-tfm6 + (-tfm4*tfm6) == MIMOParallel(MIMOSeries(-tfm6, tfm4), MIMOSeries(tfm6, -tfm4)) + + raises(ValueError, lambda: tfm1*tfm2 + TF1) + raises(TypeError, lambda: tfm1*tfm2 + a0) + raises(TypeError, lambda: tfm4*tfm6 - (s - 1)) + raises(TypeError, lambda: tfm4*-tfm6 - 8) + raises(TypeError, lambda: (-1 + p**5) + tfm1*tfm2) + + # Shape criteria. + + raises(TypeError, lambda: -tfm1*tfm2 + tfm4) + raises(TypeError, lambda: tfm1*tfm2 - tfm4 + tfm5) + raises(TypeError, lambda: tfm1*tfm2 - tfm4*tfm5) + + assert tfm1*tfm2*-tfm3 == MIMOSeries(-tfm3, tfm2, tfm1) + assert (tfm1*-tfm2)*tfm3 == MIMOSeries(tfm3, -tfm2, tfm1) + + # Multiplication of a Series object with a SISO TF not allowed. + + raises(ValueError, lambda: tfm4*tfm5*TF1) + raises(TypeError, lambda: tfm4*tfm5*a1) + raises(TypeError, lambda: tfm4*-tfm5*(s - 2)) + raises(TypeError, lambda: tfm5*tfm4*9) + raises(TypeError, lambda: (-p**3 + 1)*tfm5*tfm4) + + # Transfer function matrix in the arguments. + assert (MIMOSeries(tfm2, tfm1, evaluate=True) == MIMOSeries(tfm2, tfm1).doit() + == TransferFunctionMatrix(((TransferFunction(-k**2*(a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2 + (-a2*p + s)*(a2*p - s)*(s**2 + 2*s*wn*zeta + wn**2)**2 - (a2*s + p)**2, + (a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2, s),), + (TransferFunction(k**2*(a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2 + (-a2*p + s)*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) + (a2*p - s)*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), + (a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2, s),)))) + + # doit() should not cancel poles and zeros. + mat_1 = Matrix([[1/(1+s), (1+s)/(1+s**2+2*s)**3]]) + mat_2 = Matrix([[(1+s)], [(1+s**2+2*s)**3/(1+s)]]) + tm_1, tm_2 = TransferFunctionMatrix.from_Matrix(mat_1, s), TransferFunctionMatrix.from_Matrix(mat_2, s) + assert (MIMOSeries(tm_2, tm_1).doit() + == TransferFunctionMatrix(((TransferFunction(2*(s + 1)**2*(s**2 + 2*s + 1)**3, (s + 1)**2*(s**2 + 2*s + 1)**3, s),),))) + assert MIMOSeries(tm_2, tm_1).doit().simplify() == TransferFunctionMatrix(((TransferFunction(2, 1, s),),)) + + # calling doit() will expand the internal Series and Parallel objects. + assert (MIMOSeries(-tfm3, -tfm2, tfm1, evaluate=True) + == MIMOSeries(-tfm3, -tfm2, tfm1).doit() + == TransferFunctionMatrix(((TransferFunction(k**2*(a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2 + (a2*p - s)**2*(s**2 + 2*s*wn*zeta + wn**2)**2 + (a2*s + p)**2, + (a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**3, s),), + (TransferFunction(-k**2*(a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**2 + (-a2*p + s)*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) + (a2*p - s)*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), + (a2*s + p)**2*(s**2 + 2*s*wn*zeta + wn**2)**3, s),)))) + assert (MIMOSeries(MIMOParallel(tfm4, tfm5), tfm5, evaluate=True) + == MIMOSeries(MIMOParallel(tfm4, tfm5), tfm5).doit() + == TransferFunctionMatrix(((TransferFunction(-k*(-a2*s - p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2)), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s), TransferFunction(k*(-a2*p - \ + k*(a2*s + p) + s), a2*s + p, s)), (TransferFunction(-k*(-a2*s - p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2)), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s), \ + TransferFunction((-a2*p + s)*(-a2*p - k*(a2*s + p) + s), (a2*s + p)**2, s)))) == MIMOSeries(MIMOParallel(tfm4, tfm5), tfm5).rewrite(TransferFunctionMatrix)) + + +def test_Parallel_construction(): + tf = TransferFunction(a0*s**3 + a1*s**2 - a2*s, b0*p**4 + b1*p**3 - b2*s*p, s) + tf2 = TransferFunction(a2*p - s, a2*s + p, s) + tf3 = TransferFunction(a0*p + p**a1 - s, p, p) + tf4 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s) + inp = Function('X_d')(s) + out = Function('X')(s) + + p0 = Parallel(tf, tf2) + assert p0.args == (tf, tf2) + assert p0.var == s + + p1 = Parallel(Series(tf, -tf2), tf2) + assert p1.args == (Series(tf, -tf2), tf2) + assert p1.var == s + + tf3_ = TransferFunction(inp, 1, s) + tf4_ = TransferFunction(-out, 1, s) + p2 = Parallel(tf, Series(tf3_, -tf4_), tf2) + assert p2.args == (tf, Series(tf3_, -tf4_), tf2) + + p3 = Parallel(tf, tf2, tf4) + assert p3.args == (tf, tf2, tf4) + + p4 = Parallel(tf3_, tf4_) + assert p4.args == (tf3_, tf4_) + assert p4.var == s + + p5 = Parallel(tf, tf2) + assert p0 == p5 + assert not p0 == p1 + + p6 = Parallel(tf2, tf4, Series(tf2, -tf4)) + assert p6.args == (tf2, tf4, Series(tf2, -tf4)) + + p7 = Parallel(tf2, tf4, Series(tf2, -tf), tf4) + assert p7.args == (tf2, tf4, Series(tf2, -tf), tf4) + + raises(ValueError, lambda: Parallel(tf, tf3)) + raises(ValueError, lambda: Parallel(tf, tf2, tf3, tf4)) + raises(ValueError, lambda: Parallel(-tf3, tf4)) + raises(TypeError, lambda: Parallel(2, tf, tf4)) + raises(TypeError, lambda: Parallel(s**2 + p*s, tf3, tf2)) + raises(TypeError, lambda: Parallel(tf3, Matrix([1, 2, 3, 4]))) + + +def test_MIMOParallel_construction(): + tfm1 = TransferFunctionMatrix([[TF1], [TF2], [TF3]]) + tfm2 = TransferFunctionMatrix([[-TF3], [TF2], [TF1]]) + tfm3 = TransferFunctionMatrix([[TF1]]) + tfm4 = TransferFunctionMatrix([[TF2], [TF1], [TF3]]) + tfm5 = TransferFunctionMatrix([[TF1, TF2], [TF2, TF1]]) + tfm6 = TransferFunctionMatrix([[TF2, TF1], [TF1, TF2]]) + tfm7 = TransferFunctionMatrix.from_Matrix(Matrix([[1/p]]), p) + + p8 = MIMOParallel(tfm1, tfm2) + assert p8.args == (tfm1, tfm2) + assert p8.var == s + assert p8.shape == (p8.num_outputs, p8.num_inputs) == (3, 1) + + p9 = MIMOParallel(MIMOSeries(tfm3, tfm1), tfm2) + assert p9.args == (MIMOSeries(tfm3, tfm1), tfm2) + assert p9.var == s + assert p9.shape == (p9.num_outputs, p9.num_inputs) == (3, 1) + + p10 = MIMOParallel(tfm1, MIMOSeries(tfm3, tfm4), tfm2) + assert p10.args == (tfm1, MIMOSeries(tfm3, tfm4), tfm2) + assert p10.var == s + assert p10.shape == (p10.num_outputs, p10.num_inputs) == (3, 1) + + p11 = MIMOParallel(tfm2, tfm1, tfm4) + assert p11.args == (tfm2, tfm1, tfm4) + assert p11.shape == (p11.num_outputs, p11.num_inputs) == (3, 1) + + p12 = MIMOParallel(tfm6, tfm5) + assert p12.args == (tfm6, tfm5) + assert p12.shape == (p12.num_outputs, p12.num_inputs) == (2, 2) + + p13 = MIMOParallel(tfm2, tfm4, MIMOSeries(-tfm3, tfm4), -tfm4) + assert p13.args == (tfm2, tfm4, MIMOSeries(-tfm3, tfm4), -tfm4) + assert p13.shape == (p13.num_outputs, p13.num_inputs) == (3, 1) + + # arg cannot be empty tuple. + raises(TypeError, lambda: MIMOParallel(())) + + # arg cannot contain SISO as well as MIMO systems. + raises(TypeError, lambda: MIMOParallel(tfm1, tfm2, TF1)) + + # all TFMs must have same shapes. + raises(TypeError, lambda: MIMOParallel(tfm1, tfm3, tfm4)) + + # all TFMs must be using the same complex variable. + raises(ValueError, lambda: MIMOParallel(tfm3, tfm7)) + + # Number or expression not allowed in the arguments. + raises(TypeError, lambda: MIMOParallel(2, tfm1, tfm4)) + raises(TypeError, lambda: MIMOParallel(s**2 + p*s, -tfm4, tfm2)) + + +def test_Parallel_functions(): + tf1 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s) + tf2 = TransferFunction(k, 1, s) + tf3 = TransferFunction(a2*p - s, a2*s + p, s) + tf4 = TransferFunction(a0*p + p**a1 - s, p, p) + tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s) + + assert tf1 + tf2 + tf3 == Parallel(tf1, tf2, tf3) + assert tf1 + tf2 + tf3 + tf5 == Parallel(tf1, tf2, tf3, tf5) + assert tf1 + tf2 - tf3 - tf5 == Parallel(tf1, tf2, -tf3, -tf5) + assert tf1 + tf2*tf3 == Parallel(tf1, Series(tf2, tf3)) + assert tf1 - tf2*tf3 == Parallel(tf1, -Series(tf2,tf3)) + assert -tf1 - tf2 == Parallel(-tf1, -tf2) + assert -(tf1 + tf2) == Series(TransferFunction(-1, 1, s), Parallel(tf1, tf2)) + assert (tf2 + tf3)*tf1 == Series(Parallel(tf2, tf3), tf1) + assert (tf1 + tf2)*(tf3*tf5) == Series(Parallel(tf1, tf2), tf3, tf5) + assert -(tf2 + tf3)*-tf5 == Series(TransferFunction(-1, 1, s), Parallel(tf2, tf3), -tf5) + assert tf2 + tf3 + tf2*tf1 + tf5 == Parallel(tf2, tf3, Series(tf2, tf1), tf5) + assert tf2 + tf3 + tf2*tf1 - tf3 == Parallel(tf2, tf3, Series(tf2, tf1), -tf3) + assert (tf1 + tf2 + tf5)*(tf3 + tf5) == Series(Parallel(tf1, tf2, tf5), Parallel(tf3, tf5)) + raises(ValueError, lambda: tf1 + tf2 + tf4) + raises(ValueError, lambda: tf1 - tf2*tf4) + raises(ValueError, lambda: tf3 + Matrix([1, 2, 3])) + + # evaluate=True -> doit() + assert Parallel(tf1, tf2, evaluate=True) == Parallel(tf1, tf2).doit() == \ + TransferFunction(k*(s**2 + 2*s*wn*zeta + wn**2) + 1, s**2 + 2*s*wn*zeta + wn**2, s) + assert Parallel(tf1, tf2, Series(-tf1, tf3), evaluate=True) == \ + Parallel(tf1, tf2, Series(-tf1, tf3)).doit() == TransferFunction(k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2)**2 + \ + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2) + (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), (a2*s + p)*(s**2 + \ + 2*s*wn*zeta + wn**2)**2, s) + assert Parallel(tf2, tf1, -tf3, evaluate=True) == Parallel(tf2, tf1, -tf3).doit() == \ + TransferFunction(a2*s + k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) + p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2) \ + , (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + assert not Parallel(tf1, -tf2, evaluate=False) == Parallel(tf1, -tf2).doit() + + assert Parallel(Series(tf1, tf2), Series(tf2, tf3)).doit() == \ + TransferFunction(k*(a2*p - s)*(s**2 + 2*s*wn*zeta + wn**2) + k*(a2*s + p), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + assert Parallel(-tf1, -tf2, -tf3).doit() == \ + TransferFunction(-a2*s - k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) - p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + wn**2), \ + (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + assert -Parallel(tf1, tf2, tf3).doit() == \ + TransferFunction(-a2*s - k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) - p - (a2*p - s)*(s**2 + 2*s*wn*zeta + wn**2), \ + (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + assert Parallel(tf2, tf3, Series(tf2, -tf1), tf3).doit() == \ + TransferFunction(k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) - k*(a2*s + p) + (2*a2*p - 2*s)*(s**2 + 2*s*wn*zeta \ + + wn**2), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + + assert Parallel(tf1, tf2).rewrite(TransferFunction) == \ + TransferFunction(k*(s**2 + 2*s*wn*zeta + wn**2) + 1, s**2 + 2*s*wn*zeta + wn**2, s) + assert Parallel(tf2, tf1, -tf3).rewrite(TransferFunction) == \ + TransferFunction(a2*s + k*(a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2) + p + (-a2*p + s)*(s**2 + 2*s*wn*zeta + \ + wn**2), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + + assert Parallel(tf1, Parallel(tf2, tf3)) == Parallel(tf1, tf2, tf3) == Parallel(Parallel(tf1, tf2), tf3) + + P1 = Parallel(Series(tf1, tf2), Series(tf2, tf3)) + assert P1.is_proper + assert not P1.is_strictly_proper + assert P1.is_biproper + + P2 = Parallel(tf1, -tf2, -tf3) + assert P2.is_proper + assert not P2.is_strictly_proper + assert P2.is_biproper + + P3 = Parallel(tf1, -tf2, Series(tf1, tf3)) + assert P3.is_proper + assert not P3.is_strictly_proper + assert P3.is_biproper + + +def test_MIMOParallel_functions(): + tf4 = TransferFunction(a0*p + p**a1 - s, p, p) + tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s) + + tfm1 = TransferFunctionMatrix([[TF1], [TF2], [TF3]]) + tfm2 = TransferFunctionMatrix([[-TF2], [tf5], [-TF1]]) + tfm3 = TransferFunctionMatrix([[tf5], [-tf5], [TF2]]) + tfm4 = TransferFunctionMatrix([[TF2, -tf5], [TF1, tf5]]) + tfm5 = TransferFunctionMatrix([[TF1, TF2], [TF3, -tf5]]) + tfm6 = TransferFunctionMatrix([[-TF2]]) + tfm7 = TransferFunctionMatrix([[tf4], [-tf4], [tf4]]) + + assert tfm1 + tfm2 + tfm3 == MIMOParallel(tfm1, tfm2, tfm3) == MIMOParallel(MIMOParallel(tfm1, tfm2), tfm3) + assert tfm2 - tfm1 - tfm3 == MIMOParallel(tfm2, -tfm1, -tfm3) + assert tfm2 - tfm3 + (-tfm1*tfm6*-tfm6) == MIMOParallel(tfm2, -tfm3, MIMOSeries(-tfm6, tfm6, -tfm1)) + assert tfm1 + tfm1 - (-tfm1*tfm6) == MIMOParallel(tfm1, tfm1, -MIMOSeries(tfm6, -tfm1)) + assert tfm2 - tfm3 - tfm1 + tfm2 == MIMOParallel(tfm2, -tfm3, -tfm1, tfm2) + assert tfm1 + tfm2 - tfm3 - tfm1 == MIMOParallel(tfm1, tfm2, -tfm3, -tfm1) + raises(ValueError, lambda: tfm1 + tfm2 + TF2) + raises(TypeError, lambda: tfm1 - tfm2 - a1) + raises(TypeError, lambda: tfm2 - tfm3 - (s - 1)) + raises(TypeError, lambda: -tfm3 - tfm2 - 9) + raises(TypeError, lambda: (1 - p**3) - tfm3 - tfm2) + # All TFMs must use the same complex var. tfm7 uses 'p'. + raises(ValueError, lambda: tfm3 - tfm2 - tfm7) + raises(ValueError, lambda: tfm2 - tfm1 + tfm7) + # (tfm1 +/- tfm2) has (3, 1) shape while tfm4 has (2, 2) shape. + raises(TypeError, lambda: tfm1 + tfm2 + tfm4) + raises(TypeError, lambda: (tfm1 - tfm2) - tfm4) + + assert (tfm1 + tfm2)*tfm6 == MIMOSeries(tfm6, MIMOParallel(tfm1, tfm2)) + assert (tfm2 - tfm3)*tfm6*-tfm6 == MIMOSeries(-tfm6, tfm6, MIMOParallel(tfm2, -tfm3)) + assert (tfm2 - tfm1 - tfm3)*(tfm6 + tfm6) == MIMOSeries(MIMOParallel(tfm6, tfm6), MIMOParallel(tfm2, -tfm1, -tfm3)) + raises(ValueError, lambda: (tfm4 + tfm5)*TF1) + raises(TypeError, lambda: (tfm2 - tfm3)*a2) + raises(TypeError, lambda: (tfm3 + tfm2)*(s - 6)) + raises(TypeError, lambda: (tfm1 + tfm2 + tfm3)*0) + raises(TypeError, lambda: (1 - p**3)*(tfm1 + tfm3)) + + # (tfm3 - tfm2) has (3, 1) shape while tfm4*tfm5 has (2, 2) shape. + raises(ValueError, lambda: (tfm3 - tfm2)*tfm4*tfm5) + # (tfm1 - tfm2) has (3, 1) shape while tfm5 has (2, 2) shape. + raises(ValueError, lambda: (tfm1 - tfm2)*tfm5) + + # TFM in the arguments. + assert (MIMOParallel(tfm1, tfm2, evaluate=True) == MIMOParallel(tfm1, tfm2).doit() + == MIMOParallel(tfm1, tfm2).rewrite(TransferFunctionMatrix) + == TransferFunctionMatrix(((TransferFunction(-k*(s**2 + 2*s*wn*zeta + wn**2) + 1, s**2 + 2*s*wn*zeta + wn**2, s),), \ + (TransferFunction(-a0 + a1*s**2 + a2*s + k*(a0 + s), a0 + s, s),), (TransferFunction(-a2*s - p + (a2*p - s)* \ + (s**2 + 2*s*wn*zeta + wn**2), (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s),)))) + + +def test_Feedback_construction(): + tf1 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s) + tf2 = TransferFunction(k, 1, s) + tf3 = TransferFunction(a2*p - s, a2*s + p, s) + tf4 = TransferFunction(a0*p + p**a1 - s, p, p) + tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s) + tf6 = TransferFunction(s - p, p + s, p) + + f1 = Feedback(TransferFunction(1, 1, s), tf1*tf2*tf3) + assert f1.args == (TransferFunction(1, 1, s), Series(tf1, tf2, tf3), -1) + assert f1.sys1 == TransferFunction(1, 1, s) + assert f1.sys2 == Series(tf1, tf2, tf3) + assert f1.var == s + + f2 = Feedback(tf1, tf2*tf3) + assert f2.args == (tf1, Series(tf2, tf3), -1) + assert f2.sys1 == tf1 + assert f2.sys2 == Series(tf2, tf3) + assert f2.var == s + + f3 = Feedback(tf1*tf2, tf5) + assert f3.args == (Series(tf1, tf2), tf5, -1) + assert f3.sys1 == Series(tf1, tf2) + + f4 = Feedback(tf4, tf6) + assert f4.args == (tf4, tf6, -1) + assert f4.sys1 == tf4 + assert f4.var == p + + f5 = Feedback(tf5, TransferFunction(1, 1, s)) + assert f5.args == (tf5, TransferFunction(1, 1, s), -1) + assert f5.var == s + assert f5 == Feedback(tf5) # When sys2 is not passed explicitly, it is assumed to be unit tf. + + f6 = Feedback(TransferFunction(1, 1, p), tf4) + assert f6.args == (TransferFunction(1, 1, p), tf4, -1) + assert f6.var == p + + f7 = -Feedback(tf4*tf6, TransferFunction(1, 1, p)) + assert f7.args == (Series(TransferFunction(-1, 1, p), Series(tf4, tf6)), -TransferFunction(1, 1, p), -1) + assert f7.sys1 == Series(TransferFunction(-1, 1, p), Series(tf4, tf6)) + + # denominator can't be a Parallel instance + raises(TypeError, lambda: Feedback(tf1, tf2 + tf3)) + raises(TypeError, lambda: Feedback(tf1, Matrix([1, 2, 3]))) + raises(TypeError, lambda: Feedback(TransferFunction(1, 1, s), s - 1)) + raises(TypeError, lambda: Feedback(1, 1)) + # raises(ValueError, lambda: Feedback(TransferFunction(1, 1, s), TransferFunction(1, 1, s))) + raises(ValueError, lambda: Feedback(tf2, tf4*tf5)) + raises(ValueError, lambda: Feedback(tf2, tf1, 1.5)) # `sign` can only be -1 or 1 + raises(ValueError, lambda: Feedback(tf1, -tf1**-1)) # denominator can't be zero + raises(ValueError, lambda: Feedback(tf4, tf5)) # Both systems should use the same `var` + + +def test_Feedback_functions(): + tf = TransferFunction(1, 1, s) + tf1 = TransferFunction(1, s**2 + 2*zeta*wn*s + wn**2, s) + tf2 = TransferFunction(k, 1, s) + tf3 = TransferFunction(a2*p - s, a2*s + p, s) + tf4 = TransferFunction(a0*p + p**a1 - s, p, p) + tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s) + tf6 = TransferFunction(s - p, p + s, p) + + assert (tf1*tf2*tf3 / tf3*tf5) == Series(tf1, tf2, tf3, pow(tf3, -1), tf5) + assert (tf1*tf2*tf3) / (tf3*tf5) == Series((tf1*tf2*tf3).doit(), pow((tf3*tf5).doit(),-1)) + assert tf / (tf + tf1) == Feedback(tf, tf1) + assert tf / (tf + tf1*tf2*tf3) == Feedback(tf, tf1*tf2*tf3) + assert tf1 / (tf + tf1*tf2*tf3) == Feedback(tf1, tf2*tf3) + assert (tf1*tf2) / (tf + tf1*tf2) == Feedback(tf1*tf2, tf) + assert (tf1*tf2) / (tf + tf1*tf2*tf5) == Feedback(tf1*tf2, tf5) + assert (tf1*tf2) / (tf + tf1*tf2*tf5*tf3) in (Feedback(tf1*tf2, tf5*tf3), Feedback(tf1*tf2, tf3*tf5)) + assert tf4 / (TransferFunction(1, 1, p) + tf4*tf6) == Feedback(tf4, tf6) + assert tf5 / (tf + tf5) == Feedback(tf5, tf) + + raises(TypeError, lambda: tf1*tf2*tf3 / (1 + tf1*tf2*tf3)) + raises(ValueError, lambda: tf2*tf3 / (tf + tf2*tf3*tf4)) + + assert Feedback(tf, tf1*tf2*tf3).doit() == \ + TransferFunction((a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), k*(a2*p - s) + \ + (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), s) + assert Feedback(tf, tf1*tf2*tf3).sensitivity == \ + 1/(k*(a2*p - s)/((a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2)) + 1) + assert Feedback(tf1, tf2*tf3).doit() == \ + TransferFunction((a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2), (k*(a2*p - s) + \ + (a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2))*(s**2 + 2*s*wn*zeta + wn**2), s) + assert Feedback(tf1, tf2*tf3).sensitivity == \ + 1/(k*(a2*p - s)/((a2*s + p)*(s**2 + 2*s*wn*zeta + wn**2)) + 1) + assert Feedback(tf1*tf2, tf5).doit() == \ + TransferFunction(k*(a0 + s)*(s**2 + 2*s*wn*zeta + wn**2), (k*(-a0 + a1*s**2 + a2*s) + \ + (a0 + s)*(s**2 + 2*s*wn*zeta + wn**2))*(s**2 + 2*s*wn*zeta + wn**2), s) + assert Feedback(tf1*tf2, tf5, 1).sensitivity == \ + 1/(-k*(-a0 + a1*s**2 + a2*s)/((a0 + s)*(s**2 + 2*s*wn*zeta + wn**2)) + 1) + assert Feedback(tf4, tf6).doit() == \ + TransferFunction(p*(p + s)*(a0*p + p**a1 - s), p*(p*(p + s) + (-p + s)*(a0*p + p**a1 - s)), p) + assert -Feedback(tf4*tf6, TransferFunction(1, 1, p)).doit() == \ + TransferFunction(-p*(-p + s)*(p + s)*(a0*p + p**a1 - s), p*(p + s)*(p*(p + s) + (-p + s)*(a0*p + p**a1 - s)), p) + assert Feedback(tf, tf).doit() == TransferFunction(1, 2, s) + + assert Feedback(tf1, tf2*tf5).rewrite(TransferFunction) == \ + TransferFunction((a0 + s)*(s**2 + 2*s*wn*zeta + wn**2), (k*(-a0 + a1*s**2 + a2*s) + \ + (a0 + s)*(s**2 + 2*s*wn*zeta + wn**2))*(s**2 + 2*s*wn*zeta + wn**2), s) + assert Feedback(TransferFunction(1, 1, p), tf4).rewrite(TransferFunction) == \ + TransferFunction(p, a0*p + p + p**a1 - s, p) + + +def test_Feedback_as_TransferFunction(): + # Solves issue https://github.com/sympy/sympy/issues/26161 + tf1 = TransferFunction(s+1, 1, s) + tf2 = TransferFunction(s+2, 1, s) + fd1 = Feedback(tf1, tf2, -1) # Negative Feedback system + fd2 = Feedback(tf1, tf2, 1) # Positive Feedback system + unit = TransferFunction(1, 1, s) + + # Checking the type + assert isinstance(fd1, TransferFunction) + assert isinstance(fd1, Feedback) + + # Testing the numerator and denominator + assert fd1.num == tf1 + assert fd2.num == tf1 + assert fd1.den == Parallel(unit, Series(tf2, tf1)) + assert fd2.den == Parallel(unit, -Series(tf2, tf1)) + + # Testing the Series and Parallel Combination with Feedback and TransferFunction + s1 = Series(tf1, fd1) + p1 = Parallel(tf1, fd1) + assert tf1 * fd1 == s1 + assert tf1 + fd1 == p1 + assert s1.doit() == TransferFunction((s + 1)**2, (s + 1)*(s + 2) + 1, s) + assert p1.doit() == TransferFunction(s + (s + 1)*((s + 1)*(s + 2) + 1) + 1, (s + 1)*(s + 2) + 1, s) + + # Testing the use of Feedback and TransferFunction with Feedback + fd3 = Feedback(tf1*fd1, tf2, -1) + assert fd3 == Feedback(Series(tf1, fd1), tf2) + assert fd3.num == tf1 * fd1 + assert fd3.den == Parallel(unit, Series(tf2, Series(tf1, fd1))) + + # Testing the use of Feedback and TransferFunction with TransferFunction + tf3 = TransferFunction(tf1*fd1, tf2, s) + assert tf3 == TransferFunction(Series(tf1, fd1), tf2, s) + assert tf3.num == tf1*fd1 + +def test_issue_26161(): + # Issue https://github.com/sympy/sympy/issues/26161 + Ib, Is, m, h, l2, l1 = symbols('I_b, I_s, m, h, l2, l1', + real=True, nonnegative=True) + KD, KP, v = symbols('K_D, K_P, v', real=True) + + tau1_sq = (Ib + m * h ** 2) / m / g / h + tau2 = l2 / v + tau3 = v / (l1 + l2) + K = v ** 2 / g / (l1 + l2) + + Gtheta = TransferFunction(-K * (tau2 * s + 1), tau1_sq * s ** 2 - 1, s) + Gdelta = TransferFunction(1, Is * s ** 2 + c * s, s) + Gpsi = TransferFunction(1, tau3 * s, s) + Dcont = TransferFunction(KD * s, 1, s) + PIcont = TransferFunction(KP, s, s) + Gunity = TransferFunction(1, 1, s) + + Ginner = Feedback(Dcont * Gdelta, Gtheta) + Gouter = Feedback(PIcont * Ginner * Gpsi, Gunity) + assert Gouter == Feedback(Series(PIcont, Series(Ginner, Gpsi)), Gunity) + assert Gouter.num == Series(PIcont, Series(Ginner, Gpsi)) + assert Gouter.den == Parallel(Gunity, Series(Gunity, Series(PIcont, Series(Ginner, Gpsi)))) + expr = (KD*KP*g*s**3*v**2*(l1 + l2)*(Is*s**2 + c*s)**2*(-g*h*m + s**2*(Ib + h**2*m))*(-KD*g*h*m*s*v**2*(l2*s + v) + \ + g*v*(l1 + l2)*(Is*s**2 + c*s)*(-g*h*m + s**2*(Ib + h**2*m))))/((s**2*v*(Is*s**2 + c*s)*(-KD*g*h*m*s*v**2* \ + (l2*s + v) + g*v*(l1 + l2)*(Is*s**2 + c*s)*(-g*h*m + s**2*(Ib + h**2*m)))*(KD*KP*g*s*v*(l1 + l2)**2* \ + (Is*s**2 + c*s)*(-g*h*m + s**2*(Ib + h**2*m)) + s**2*v*(Is*s**2 + c*s)*(-KD*g*h*m*s*v**2*(l2*s + v) + \ + g*v*(l1 + l2)*(Is*s**2 + c*s)*(-g*h*m + s**2*(Ib + h**2*m))))/(l1 + l2))) + + assert (Gouter.to_expr() - expr).simplify() == 0 + + +def test_MIMOFeedback_construction(): + tf1 = TransferFunction(1, s, s) + tf2 = TransferFunction(s, s**3 - 1, s) + tf3 = TransferFunction(s, s + 1, s) + tf4 = TransferFunction(s, s**2 + 1, s) + + tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]]) + tfm_2 = TransferFunctionMatrix([[tf2, tf3], [tf4, tf1]]) + tfm_3 = TransferFunctionMatrix([[tf3, tf4], [tf1, tf2]]) + + f1 = MIMOFeedback(tfm_1, tfm_2) + assert f1.args == (tfm_1, tfm_2, -1) + assert f1.sys1 == tfm_1 + assert f1.sys2 == tfm_2 + assert f1.var == s + assert f1.sign == -1 + assert -(-f1) == f1 + + f2 = MIMOFeedback(tfm_2, tfm_1, 1) + assert f2.args == (tfm_2, tfm_1, 1) + assert f2.sys1 == tfm_2 + assert f2.sys2 == tfm_1 + assert f2.var == s + assert f2.sign == 1 + + f3 = MIMOFeedback(tfm_1, MIMOSeries(tfm_3, tfm_2)) + assert f3.args == (tfm_1, MIMOSeries(tfm_3, tfm_2), -1) + assert f3.sys1 == tfm_1 + assert f3.sys2 == MIMOSeries(tfm_3, tfm_2) + assert f3.var == s + assert f3.sign == -1 + + mat = Matrix([[1, 1/s], [0, 1]]) + sys1 = controller = TransferFunctionMatrix.from_Matrix(mat, s) + f4 = MIMOFeedback(sys1, controller) + assert f4.args == (sys1, controller, -1) + assert f4.sys1 == f4.sys2 == sys1 + + +def test_MIMOFeedback_errors(): + tf1 = TransferFunction(1, s, s) + tf2 = TransferFunction(s, s**3 - 1, s) + tf3 = TransferFunction(s, s - 1, s) + tf4 = TransferFunction(s, s**2 + 1, s) + tf5 = TransferFunction(1, 1, s) + tf6 = TransferFunction(-1, s - 1, s) + + tfm_1 = TransferFunctionMatrix([[tf1, tf2], [tf3, tf4]]) + tfm_2 = TransferFunctionMatrix([[tf2, tf3], [tf4, tf1]]) + tfm_3 = TransferFunctionMatrix.from_Matrix(eye(2), var=s) + tfm_4 = TransferFunctionMatrix([[tf1, tf5], [tf5, tf5]]) + tfm_5 = TransferFunctionMatrix([[-tf3, tf3], [tf3, tf6]]) + # tfm_4 is inverse of tfm_5. Therefore tfm_5*tfm_4 = I + tfm_6 = TransferFunctionMatrix([[-tf3]]) + tfm_7 = TransferFunctionMatrix([[tf3, tf4]]) + + # Unsupported Types + raises(TypeError, lambda: MIMOFeedback(tf1, tf2)) + raises(TypeError, lambda: MIMOFeedback(MIMOParallel(tfm_1, tfm_2), tfm_3)) + # Shape Errors + raises(ValueError, lambda: MIMOFeedback(tfm_1, tfm_6, 1)) + raises(ValueError, lambda: MIMOFeedback(tfm_7, tfm_7)) + # sign not 1/-1 + raises(ValueError, lambda: MIMOFeedback(tfm_1, tfm_2, -2)) + # Non-Invertible Systems + raises(ValueError, lambda: MIMOFeedback(tfm_5, tfm_4, 1)) + raises(ValueError, lambda: MIMOFeedback(tfm_4, -tfm_5)) + raises(ValueError, lambda: MIMOFeedback(tfm_3, tfm_3, 1)) + # Variable not same in both the systems + tfm_8 = TransferFunctionMatrix.from_Matrix(eye(2), var=p) + raises(ValueError, lambda: MIMOFeedback(tfm_1, tfm_8, 1)) + + +def test_MIMOFeedback_functions(): + tf1 = TransferFunction(1, s, s) + tf2 = TransferFunction(s, s - 1, s) + tf3 = TransferFunction(1, 1, s) + tf4 = TransferFunction(-1, s - 1, s) + + tfm_1 = TransferFunctionMatrix.from_Matrix(eye(2), var=s) + tfm_2 = TransferFunctionMatrix([[tf1, tf3], [tf3, tf3]]) + tfm_3 = TransferFunctionMatrix([[-tf2, tf2], [tf2, tf4]]) + tfm_4 = TransferFunctionMatrix([[tf1, tf2], [-tf2, tf1]]) + + # sensitivity, doit(), rewrite() + F_1 = MIMOFeedback(tfm_2, tfm_3) + F_2 = MIMOFeedback(tfm_2, MIMOSeries(tfm_4, -tfm_1), 1) + + assert F_1.sensitivity == Matrix([[S.Half, 0], [0, S.Half]]) + assert F_2.sensitivity == Matrix([[(-2*s**4 + s**2)/(s**2 - s + 1), + (2*s**3 - s**2)/(s**2 - s + 1)], [-s**2, s]]) + + assert F_1.doit() == \ + TransferFunctionMatrix(((TransferFunction(1, 2*s, s), + TransferFunction(1, 2, s)), (TransferFunction(1, 2, s), + TransferFunction(1, 2, s)))) == F_1.rewrite(TransferFunctionMatrix) + assert F_2.doit(cancel=False, expand=True) == \ + TransferFunctionMatrix(((TransferFunction(-s**5 + 2*s**4 - 2*s**3 + s**2, s**5 - 2*s**4 + 3*s**3 - 2*s**2 + s, s), + TransferFunction(-2*s**4 + 2*s**3, s**2 - s + 1, s)), (TransferFunction(0, 1, s), TransferFunction(-s**2 + s, 1, s)))) + assert F_2.doit(cancel=False) == \ + TransferFunctionMatrix(((TransferFunction(s*(2*s**3 - s**2)*(s**2 - s + 1) + \ + (-2*s**4 + s**2)*(s**2 - s + 1), s*(s**2 - s + 1)**2, s), TransferFunction(-2*s**4 + 2*s**3, s**2 - s + 1, s)), + (TransferFunction(0, 1, s), TransferFunction(-s**2 + s, 1, s)))) + assert F_2.doit() == \ + TransferFunctionMatrix(((TransferFunction(s*(-2*s**2 + s*(2*s - 1) + 1), s**2 - s + 1, s), + TransferFunction(-2*s**3*(s - 1), s**2 - s + 1, s)), (TransferFunction(0, 1, s), TransferFunction(s*(1 - s), 1, s)))) + assert F_2.doit(expand=True) == \ + TransferFunctionMatrix(((TransferFunction(-s**2 + s, s**2 - s + 1, s), TransferFunction(-2*s**4 + 2*s**3, s**2 - s + 1, s)), + (TransferFunction(0, 1, s), TransferFunction(-s**2 + s, 1, s)))) + + assert -(F_1.doit()) == (-F_1).doit() # First negating then calculating vs calculating then negating. + + +def test_TransferFunctionMatrix_construction(): + tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s) + tf4 = TransferFunction(a0*p + p**a1 - s, p, p) + + tfm3_ = TransferFunctionMatrix([[-TF3]]) + assert tfm3_.shape == (tfm3_.num_outputs, tfm3_.num_inputs) == (1, 1) + assert tfm3_.args == Tuple(Tuple(Tuple(-TF3))) + assert tfm3_.var == s + + tfm5 = TransferFunctionMatrix([[TF1, -TF2], [TF3, tf5]]) + assert tfm5.shape == (tfm5.num_outputs, tfm5.num_inputs) == (2, 2) + assert tfm5.args == Tuple(Tuple(Tuple(TF1, -TF2), Tuple(TF3, tf5))) + assert tfm5.var == s + + tfm7 = TransferFunctionMatrix([[TF1, TF2], [TF3, -tf5], [-tf5, TF2]]) + assert tfm7.shape == (tfm7.num_outputs, tfm7.num_inputs) == (3, 2) + assert tfm7.args == Tuple(Tuple(Tuple(TF1, TF2), Tuple(TF3, -tf5), Tuple(-tf5, TF2))) + assert tfm7.var == s + + # all transfer functions will use the same complex variable. tf4 uses 'p'. + raises(ValueError, lambda: TransferFunctionMatrix([[TF1], [TF2], [tf4]])) + raises(ValueError, lambda: TransferFunctionMatrix([[TF1, tf4], [TF3, tf5]])) + + # length of all the lists in the TFM should be equal. + raises(ValueError, lambda: TransferFunctionMatrix([[TF1], [TF3, tf5]])) + raises(ValueError, lambda: TransferFunctionMatrix([[TF1, TF3], [tf5]])) + + # lists should only support transfer functions in them. + raises(TypeError, lambda: TransferFunctionMatrix([[TF1, TF2], [TF3, Matrix([1, 2])]])) + raises(TypeError, lambda: TransferFunctionMatrix([[TF1, Matrix([1, 2])], [TF3, TF2]])) + + # `arg` should strictly be nested list of TransferFunction + raises(ValueError, lambda: TransferFunctionMatrix([TF1, TF2, tf5])) + raises(ValueError, lambda: TransferFunctionMatrix([TF1])) + +def test_TransferFunctionMatrix_functions(): + tf5 = TransferFunction(a1*s**2 + a2*s - a0, s + a0, s) + + # Classmethod (from_matrix) + + mat_1 = ImmutableMatrix([ + [s*(s + 1)*(s - 3)/(s**4 + 1), 2], + [p, p*(s + 1)/(s*(s**1 + 1))] + ]) + mat_2 = ImmutableMatrix([[(2*s + 1)/(s**2 - 9)]]) + mat_3 = ImmutableMatrix([[1, 2], [3, 4]]) + assert TransferFunctionMatrix.from_Matrix(mat_1, s) == \ + TransferFunctionMatrix([[TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(2, 1, s)], + [TransferFunction(p, 1, s), TransferFunction(p, s, s)]]) + assert TransferFunctionMatrix.from_Matrix(mat_2, s) == \ + TransferFunctionMatrix([[TransferFunction(2*s + 1, s**2 - 9, s)]]) + assert TransferFunctionMatrix.from_Matrix(mat_3, p) == \ + TransferFunctionMatrix([[TransferFunction(1, 1, p), TransferFunction(2, 1, p)], + [TransferFunction(3, 1, p), TransferFunction(4, 1, p)]]) + + # Negating a TFM + + tfm1 = TransferFunctionMatrix([[TF1], [TF2]]) + assert -tfm1 == TransferFunctionMatrix([[-TF1], [-TF2]]) + + tfm2 = TransferFunctionMatrix([[TF1, TF2, TF3], [tf5, -TF1, -TF3]]) + assert -tfm2 == TransferFunctionMatrix([[-TF1, -TF2, -TF3], [-tf5, TF1, TF3]]) + + # subs() + + H_1 = TransferFunctionMatrix.from_Matrix(mat_1, s) + H_2 = TransferFunctionMatrix([[TransferFunction(a*p*s, k*s**2, s), TransferFunction(p*s, k*(s**2 - a), s)]]) + assert H_1.subs(p, 1) == TransferFunctionMatrix([[TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(2, 1, s)], [TransferFunction(1, 1, s), TransferFunction(1, s, s)]]) + assert H_1.subs({p: 1}) == TransferFunctionMatrix([[TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(2, 1, s)], [TransferFunction(1, 1, s), TransferFunction(1, s, s)]]) + assert H_1.subs({p: 1, s: 1}) == TransferFunctionMatrix([[TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(2, 1, s)], [TransferFunction(1, 1, s), TransferFunction(1, s, s)]]) # This should ignore `s` as it is `var` + assert H_2.subs(p, 2) == TransferFunctionMatrix([[TransferFunction(2*a*s, k*s**2, s), TransferFunction(2*s, k*(-a + s**2), s)]]) + assert H_2.subs(k, 1) == TransferFunctionMatrix([[TransferFunction(a*p*s, s**2, s), TransferFunction(p*s, -a + s**2, s)]]) + assert H_2.subs(a, 0) == TransferFunctionMatrix([[TransferFunction(0, k*s**2, s), TransferFunction(p*s, k*s**2, s)]]) + assert H_2.subs({p: 1, k: 1, a: a0}) == TransferFunctionMatrix([[TransferFunction(a0*s, s**2, s), TransferFunction(s, -a0 + s**2, s)]]) + + # eval_frequency() + assert H_2.eval_frequency(S(1)/2 + I) == Matrix([[2*a*p/(5*k) - 4*I*a*p/(5*k), I*p/(-a*k - 3*k/4 + I*k) + p/(-2*a*k - 3*k/2 + 2*I*k)]]) + + # transpose() + + assert H_1.transpose() == TransferFunctionMatrix([[TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(p, 1, s)], [TransferFunction(2, 1, s), TransferFunction(p, s, s)]]) + assert H_2.transpose() == TransferFunctionMatrix([[TransferFunction(a*p*s, k*s**2, s)], [TransferFunction(p*s, k*(-a + s**2), s)]]) + assert H_1.transpose().transpose() == H_1 + assert H_2.transpose().transpose() == H_2 + + # elem_poles() + + assert H_1.elem_poles() == [[[-sqrt(2)/2 - sqrt(2)*I/2, -sqrt(2)/2 + sqrt(2)*I/2, sqrt(2)/2 - sqrt(2)*I/2, sqrt(2)/2 + sqrt(2)*I/2], []], + [[], [0]]] + assert H_2.elem_poles() == [[[0, 0], [sqrt(a), -sqrt(a)]]] + assert tfm2.elem_poles() == [[[wn*(-zeta + sqrt((zeta - 1)*(zeta + 1))), wn*(-zeta - sqrt((zeta - 1)*(zeta + 1)))], [], [-p/a2]], + [[-a0], [wn*(-zeta + sqrt((zeta - 1)*(zeta + 1))), wn*(-zeta - sqrt((zeta - 1)*(zeta + 1)))], [-p/a2]]] + + # elem_zeros() + + assert H_1.elem_zeros() == [[[-1, 0, 3], []], [[], []]] + assert H_2.elem_zeros() == [[[0], [0]]] + assert tfm2.elem_zeros() == [[[], [], [a2*p]], + [[-a2/(2*a1) - sqrt(4*a0*a1 + a2**2)/(2*a1), -a2/(2*a1) + sqrt(4*a0*a1 + a2**2)/(2*a1)], [], [a2*p]]] + + # doit() + + H_3 = TransferFunctionMatrix([[Series(TransferFunction(1, s**3 - 3, s), TransferFunction(s**2 - 2*s + 5, 1, s), TransferFunction(1, s, s))]]) + H_4 = TransferFunctionMatrix([[Parallel(TransferFunction(s**3 - 3, 4*s**4 - s**2 - 2*s + 5, s), TransferFunction(4 - s**3, 4*s**4 - s**2 - 2*s + 5, s))]]) + + assert H_3.doit() == TransferFunctionMatrix([[TransferFunction(s**2 - 2*s + 5, s*(s**3 - 3), s)]]) + assert H_4.doit() == TransferFunctionMatrix([[TransferFunction(1, 4*s**4 - s**2 - 2*s + 5, s)]]) + + # _flat() + + assert H_1._flat() == [TransferFunction(s*(s - 3)*(s + 1), s**4 + 1, s), TransferFunction(2, 1, s), TransferFunction(p, 1, s), TransferFunction(p, s, s)] + assert H_2._flat() == [TransferFunction(a*p*s, k*s**2, s), TransferFunction(p*s, k*(-a + s**2), s)] + assert H_3._flat() == [Series(TransferFunction(1, s**3 - 3, s), TransferFunction(s**2 - 2*s + 5, 1, s), TransferFunction(1, s, s))] + assert H_4._flat() == [Parallel(TransferFunction(s**3 - 3, 4*s**4 - s**2 - 2*s + 5, s), TransferFunction(4 - s**3, 4*s**4 - s**2 - 2*s + 5, s))] + + # evalf() + + assert H_1.evalf() == \ + TransferFunctionMatrix(((TransferFunction(s*(s - 3.0)*(s + 1.0), s**4 + 1.0, s), TransferFunction(2.0, 1, s)), (TransferFunction(1.0*p, 1, s), TransferFunction(p, s, s)))) + assert H_2.subs({a:3.141, p:2.88, k:2}).evalf() == \ + TransferFunctionMatrix(((TransferFunction(4.5230399999999999494093572138808667659759521484375, s, s), + TransferFunction(2.87999999999999989341858963598497211933135986328125*s, 2.0*s**2 - 6.282000000000000028421709430404007434844970703125, s)),)) + + # simplify() + + H_5 = TransferFunctionMatrix([[TransferFunction(s**5 + s**3 + s, s - s**2, s), + TransferFunction((s + 3)*(s - 1), (s - 1)*(s + 5), s)]]) + + assert H_5.simplify() == simplify(H_5) == \ + TransferFunctionMatrix(((TransferFunction(-s**4 - s**2 - 1, s - 1, s), TransferFunction(s + 3, s + 5, s)),)) + + # expand() + + assert (H_1.expand() + == TransferFunctionMatrix(((TransferFunction(s**3 - 2*s**2 - 3*s, s**4 + 1, s), TransferFunction(2, 1, s)), + (TransferFunction(p, 1, s), TransferFunction(p, s, s))))) + assert H_5.expand() == \ + TransferFunctionMatrix(((TransferFunction(s**5 + s**3 + s, -s**2 + s, s), TransferFunction(s**2 + 2*s - 3, s**2 + 4*s - 5, s)),)) + +def test_TransferFunction_gbt(): + # simple transfer function, e.g. ohms law + tf = TransferFunction(1, a*s+b, s) + numZ, denZ = gbt(tf, T, 0.5) + # discretized transfer function with coefs from tf.gbt() + tf_test_bilinear = TransferFunction(s * numZ[0] + numZ[1], s * denZ[0] + denZ[1], s) + # corresponding tf with manually calculated coefs + tf_test_manual = TransferFunction(s * T/(2*(a + b*T/2)) + T/(2*(a + b*T/2)), s + (-a + b*T/2)/(a + b*T/2), s) + + assert S.Zero == (tf_test_bilinear.simplify()-tf_test_manual.simplify()).simplify().num + + tf = TransferFunction(1, a*s+b, s) + numZ, denZ = gbt(tf, T, 0) + # discretized transfer function with coefs from tf.gbt() + tf_test_forward = TransferFunction(numZ[0], s*denZ[0]+denZ[1], s) + # corresponding tf with manually calculated coefs + tf_test_manual = TransferFunction(T/a, s + (-a + b*T)/a, s) + + assert S.Zero == (tf_test_forward.simplify()-tf_test_manual.simplify()).simplify().num + + tf = TransferFunction(1, a*s+b, s) + numZ, denZ = gbt(tf, T, 1) + # discretized transfer function with coefs from tf.gbt() + tf_test_backward = TransferFunction(s*numZ[0], s*denZ[0]+denZ[1], s) + # corresponding tf with manually calculated coefs + tf_test_manual = TransferFunction(s * T/(a + b*T), s - a/(a + b*T), s) + + assert S.Zero == (tf_test_backward.simplify()-tf_test_manual.simplify()).simplify().num + + tf = TransferFunction(1, a*s+b, s) + numZ, denZ = gbt(tf, T, 0.3) + # discretized transfer function with coefs from tf.gbt() + tf_test_gbt = TransferFunction(s*numZ[0]+numZ[1], s*denZ[0]+denZ[1], s) + # corresponding tf with manually calculated coefs + tf_test_manual = TransferFunction(s*3*T/(10*(a + 3*b*T/10)) + 7*T/(10*(a + 3*b*T/10)), s + (-a + 7*b*T/10)/(a + 3*b*T/10), s) + + assert S.Zero == (tf_test_gbt.simplify()-tf_test_manual.simplify()).simplify().num + +def test_TransferFunction_bilinear(): + # simple transfer function, e.g. ohms law + tf = TransferFunction(1, a*s+b, s) + numZ, denZ = bilinear(tf, T) + # discretized transfer function with coefs from tf.bilinear() + tf_test_bilinear = TransferFunction(s*numZ[0]+numZ[1], s*denZ[0]+denZ[1], s) + # corresponding tf with manually calculated coefs + tf_test_manual = TransferFunction(s * T/(2*(a + b*T/2)) + T/(2*(a + b*T/2)), s + (-a + b*T/2)/(a + b*T/2), s) + + assert S.Zero == (tf_test_bilinear.simplify()-tf_test_manual.simplify()).simplify().num + +def test_TransferFunction_forward_diff(): + # simple transfer function, e.g. ohms law + tf = TransferFunction(1, a*s+b, s) + numZ, denZ = forward_diff(tf, T) + # discretized transfer function with coefs from tf.forward_diff() + tf_test_forward = TransferFunction(numZ[0], s*denZ[0]+denZ[1], s) + # corresponding tf with manually calculated coefs + tf_test_manual = TransferFunction(T/a, s + (-a + b*T)/a, s) + + assert S.Zero == (tf_test_forward.simplify()-tf_test_manual.simplify()).simplify().num + +def test_TransferFunction_backward_diff(): + # simple transfer function, e.g. ohms law + tf = TransferFunction(1, a*s+b, s) + numZ, denZ = backward_diff(tf, T) + # discretized transfer function with coefs from tf.backward_diff() + tf_test_backward = TransferFunction(s*numZ[0]+numZ[1], s*denZ[0]+denZ[1], s) + # corresponding tf with manually calculated coefs + tf_test_manual = TransferFunction(s * T/(a + b*T), s - a/(a + b*T), s) + + assert S.Zero == (tf_test_backward.simplify()-tf_test_manual.simplify()).simplify().num + +def test_TransferFunction_phase_margin(): + # Test for phase margin + tf1 = TransferFunction(10, p**3 + 1, p) + tf2 = TransferFunction(s**2, 10, s) + tf3 = TransferFunction(1, a*s+b, s) + tf4 = TransferFunction((s + 1)*exp(s/tau), s**2 + 2, s) + tf_m = TransferFunctionMatrix([[tf2],[tf3]]) + + assert phase_margin(tf1) == -180 + 180*atan(3*sqrt(11))/pi + assert phase_margin(tf2) == 0 + + raises(NotImplementedError, lambda: phase_margin(tf4)) + raises(ValueError, lambda: phase_margin(tf3)) + raises(ValueError, lambda: phase_margin(MIMOSeries(tf_m))) + +def test_TransferFunction_gain_margin(): + # Test for gain margin + tf1 = TransferFunction(s**2, 5*(s+1)*(s-5)*(s-10), s) + tf2 = TransferFunction(s**2 + 2*s + 1, 1, s) + tf3 = TransferFunction(1, a*s+b, s) + tf4 = TransferFunction((s + 1)*exp(s/tau), s**2 + 2, s) + tf_m = TransferFunctionMatrix([[tf2],[tf3]]) + + assert gain_margin(tf1) == -20*log(S(7)/540)/log(10) + assert gain_margin(tf2) == oo + + raises(NotImplementedError, lambda: gain_margin(tf4)) + raises(ValueError, lambda: gain_margin(tf3)) + raises(ValueError, lambda: gain_margin(MIMOSeries(tf_m))) + + +def test_StateSpace_construction(): + # using different numbers for a SISO system. + A1 = Matrix([[0, 1], [1, 0]]) + B1 = Matrix([1, 0]) + C1 = Matrix([[0, 1]]) + D1 = Matrix([0]) + ss1 = StateSpace(A1, B1, C1, D1) + + assert ss1.state_matrix == Matrix([[0, 1], [1, 0]]) + assert ss1.input_matrix == Matrix([1, 0]) + assert ss1.output_matrix == Matrix([[0, 1]]) + assert ss1.feedforward_matrix == Matrix([0]) + assert ss1.args == (Matrix([[0, 1], [1, 0]]), Matrix([[1], [0]]), Matrix([[0, 1]]), Matrix([[0]])) + + # using different symbols for a SISO system. + ss2 = StateSpace(Matrix([a0]), Matrix([a1]), + Matrix([a2]), Matrix([a3])) + + assert ss2.state_matrix == Matrix([[a0]]) + assert ss2.input_matrix == Matrix([[a1]]) + assert ss2.output_matrix == Matrix([[a2]]) + assert ss2.feedforward_matrix == Matrix([[a3]]) + assert ss2.args == (Matrix([[a0]]), Matrix([[a1]]), Matrix([[a2]]), Matrix([[a3]])) + + # using different numbers for a MIMO system. + ss3 = StateSpace(Matrix([[-1.5, -2], [1, 0]]), + Matrix([[0.5, 0], [0, 1]]), + Matrix([[0, 1], [0, 2]]), + Matrix([[2, 2], [1, 1]])) + + assert ss3.state_matrix == Matrix([[-1.5, -2], [1, 0]]) + assert ss3.input_matrix == Matrix([[0.5, 0], [0, 1]]) + assert ss3.output_matrix == Matrix([[0, 1], [0, 2]]) + assert ss3.feedforward_matrix == Matrix([[2, 2], [1, 1]]) + assert ss3.args == (Matrix([[-1.5, -2], + [1, 0]]), + Matrix([[0.5, 0], + [0, 1]]), + Matrix([[0, 1], + [0, 2]]), + Matrix([[2, 2], + [1, 1]])) + + # using different symbols for a MIMO system. + A4 = Matrix([[a0, a1], [a2, a3]]) + B4 = Matrix([[b0, b1], [b2, b3]]) + C4 = Matrix([[c0, c1], [c2, c3]]) + D4 = Matrix([[d0, d1], [d2, d3]]) + ss4 = StateSpace(A4, B4, C4, D4) + + assert ss4.state_matrix == Matrix([[a0, a1], [a2, a3]]) + assert ss4.input_matrix == Matrix([[b0, b1], [b2, b3]]) + assert ss4.output_matrix == Matrix([[c0, c1], [c2, c3]]) + assert ss4.feedforward_matrix == Matrix([[d0, d1], [d2, d3]]) + assert ss4.args == (Matrix([[a0, a1], + [a2, a3]]), + Matrix([[b0, b1], + [b2, b3]]), + Matrix([[c0, c1], + [c2, c3]]), + Matrix([[d0, d1], + [d2, d3]])) + + # using less matrices. Rest will be filled with a minimum of zeros. + ss5 = StateSpace() + assert ss5.args == (Matrix([[0]]), Matrix([[0]]), Matrix([[0]]), Matrix([[0]])) + + A6 = Matrix([[0, 1], [1, 0]]) + B6 = Matrix([1, 1]) + ss6 = StateSpace(A6, B6) + + assert ss6.state_matrix == Matrix([[0, 1], [1, 0]]) + assert ss6.input_matrix == Matrix([1, 1]) + assert ss6.output_matrix == Matrix([[0, 0]]) + assert ss6.feedforward_matrix == Matrix([[0]]) + assert ss6.args == (Matrix([[0, 1], + [1, 0]]), + Matrix([[1], + [1]]), + Matrix([[0, 0]]), + Matrix([[0]])) + + # Check if the system is SISO or MIMO. + # If system is not SISO, then it is definitely MIMO. + + assert ss1.is_SISO == True + assert ss2.is_SISO == True + assert ss3.is_SISO == False + assert ss4.is_SISO == False + assert ss5.is_SISO == True + assert ss6.is_SISO == True + + # ShapeError if matrices do not fit. + raises(ShapeError, lambda: StateSpace(Matrix([s, (s+1)**2]), Matrix([s+1]), + Matrix([s**2 - 1]), Matrix([2*s]))) + raises(ShapeError, lambda: StateSpace(Matrix([s]), Matrix([s+1, s**3 + 1]), + Matrix([s**2 - 1]), Matrix([2*s]))) + raises(ShapeError, lambda: StateSpace(Matrix([s]), Matrix([s+1]), + Matrix([[s**2 - 1], [s**2 + 2*s + 1]]), Matrix([2*s]))) + raises(ShapeError, lambda: StateSpace(Matrix([[-s, -s], [s, 0]]), + Matrix([[s/2, 0], [0, s]]), + Matrix([[0, s]]), + Matrix([[2*s, 2*s], [s, s]]))) + + # TypeError if arguments are not sympy matrices. + raises(TypeError, lambda: StateSpace(s**2, s+1, 2*s, 1)) + raises(TypeError, lambda: StateSpace(Matrix([2, 0.5]), Matrix([-1]), + Matrix([1]), 0)) +def test_StateSpace_add(): + A1 = Matrix([[4, 1],[2, -3]]) + B1 = Matrix([[5, 2],[-3, -3]]) + C1 = Matrix([[2, -4],[0, 1]]) + D1 = Matrix([[3, 2],[1, -1]]) + ss1 = StateSpace(A1, B1, C1, D1) + + A2 = Matrix([[-3, 4, 2],[-1, -3, 0],[2, 5, 3]]) + B2 = Matrix([[1, 4],[-3, -3],[-2, 1]]) + C2 = Matrix([[4, 2, -3],[1, 4, 3]]) + D2 = Matrix([[-2, 4],[0, 1]]) + ss2 = StateSpace(A2, B2, C2, D2) + ss3 = StateSpace() + ss4 = StateSpace(Matrix([1]), Matrix([2]), Matrix([3]), Matrix([4])) + + expected_add = \ + StateSpace( + Matrix([ + [4, 1, 0, 0, 0], + [2, -3, 0, 0, 0], + [0, 0, -3, 4, 2], + [0, 0, -1, -3, 0], + [0, 0, 2, 5, 3]]), + Matrix([ + [ 5, 2], + [-3, -3], + [ 1, 4], + [-3, -3], + [-2, 1]]), + Matrix([ + [2, -4, 4, 2, -3], + [0, 1, 1, 4, 3]]), + Matrix([ + [1, 6], + [1, 0]])) + + expected_mul = \ + StateSpace( + Matrix([ + [ -3, 4, 2, 0, 0], + [ -1, -3, 0, 0, 0], + [ 2, 5, 3, 0, 0], + [ 22, 18, -9, 4, 1], + [-15, -18, 0, 2, -3]]), + Matrix([ + [ 1, 4], + [ -3, -3], + [ -2, 1], + [-10, 22], + [ 6, -15]]), + Matrix([ + [14, 14, -3, 2, -4], + [ 3, -2, -6, 0, 1]]), + Matrix([ + [-6, 14], + [-2, 3]])) + + assert ss1 + ss2 == expected_add + assert ss1*ss2 == expected_mul + assert ss3 + 1/2 == StateSpace(Matrix([[0]]), Matrix([[0]]), Matrix([[0]]), Matrix([[0.5]])) + assert ss4*1.5 == StateSpace(Matrix([[1]]), Matrix([[2]]), Matrix([[4.5]]), Matrix([[6.0]])) + assert 1.5*ss4 == StateSpace(Matrix([[1]]), Matrix([[3.0]]), Matrix([[3]]), Matrix([[6.0]])) + raises(ShapeError, lambda: ss1 + ss3) + raises(ShapeError, lambda: ss2*ss4) + +def test_StateSpace_negation(): + A = Matrix([[a0, a1], [a2, a3]]) + B = Matrix([[b0, b1], [b2, b3]]) + C = Matrix([[c0, c1], [c1, c2], [c2, c3]]) + D = Matrix([[d0, d1], [d1, d2], [d2, d3]]) + SS = StateSpace(A, B, C, D) + SS_neg = -SS + + state_mat = Matrix([[-1, 1], [1, -1]]) + input_mat = Matrix([1, -1]) + output_mat = Matrix([[-1, 1]]) + feedforward_mat = Matrix([1]) + system = StateSpace(state_mat, input_mat, output_mat, feedforward_mat) + + assert SS_neg == \ + StateSpace(Matrix([[a0, a1], + [a2, a3]]), + Matrix([[b0, b1], + [b2, b3]]), + Matrix([[-c0, -c1], + [-c1, -c2], + [-c2, -c3]]), + Matrix([[-d0, -d1], + [-d1, -d2], + [-d2, -d3]])) + assert -system == \ + StateSpace(Matrix([[-1, 1], + [ 1, -1]]), + Matrix([[ 1],[-1]]), + Matrix([[1, -1]]), + Matrix([[-1]])) + assert -SS_neg == SS + assert -(-(-(-system))) == system + +def test_SymPy_substitution_functions(): + # subs + ss1 = StateSpace(Matrix([s]), Matrix([(s + 1)**2]), Matrix([s**2 - 1]), Matrix([2*s])) + ss2 = StateSpace(Matrix([s + p]), Matrix([(s + 1)*(p - 1)]), Matrix([p**3 - s**3]), Matrix([s - p])) + + assert ss1.subs({s:5}) == StateSpace(Matrix([[5]]), Matrix([[36]]), Matrix([[24]]), Matrix([[10]])) + assert ss2.subs({p:1}) == StateSpace(Matrix([[s + 1]]), Matrix([[0]]), Matrix([[1 - s**3]]), Matrix([[s - 1]])) + + # xreplace + assert ss1.xreplace({s:p}) == \ + StateSpace(Matrix([[p]]), Matrix([[(p + 1)**2]]), Matrix([[p**2 - 1]]), Matrix([[2*p]])) + assert ss2.xreplace({s:a, p:b}) == \ + StateSpace(Matrix([[a + b]]), Matrix([[(a + 1)*(b - 1)]]), Matrix([[-a**3 + b**3]]), Matrix([[a - b]])) + + # evalf + p1 = a1*s + a0 + p2 = b2*s**2 + b1*s + b0 + G = StateSpace(Matrix([p1]), Matrix([p2])) + expect = StateSpace(Matrix([[2*s + 1]]), Matrix([[5*s**2 + 4*s + 3]]), Matrix([[0]]), Matrix([[0]])) + expect_ = StateSpace(Matrix([[2.0*s + 1.0]]), Matrix([[5.0*s**2 + 4.0*s + 3.0]]), Matrix([[0]]), Matrix([[0]])) + assert G.subs({a0: 1, a1: 2, b0: 3, b1: 4, b2: 5}) == expect + assert G.subs({a0: 1, a1: 2, b0: 3, b1: 4, b2: 5}).evalf() == expect_ + assert expect.evalf() == expect_ + +def test_conversion(): + # StateSpace to TransferFunction for SISO + A1 = Matrix([[-5, -1], [3, -1]]) + B1 = Matrix([2, 5]) + C1 = Matrix([[1, 2]]) + D1 = Matrix([0]) + H1 = StateSpace(A1, B1, C1, D1) + tm1 = H1.rewrite(TransferFunction) + tm2 = (-H1).rewrite(TransferFunction) + + tf1 = tm1[0][0] + tf2 = tm2[0][0] + + assert tf1 == TransferFunction(12*s + 59, s**2 + 6*s + 8, s) + assert tf2.num == -tf1.num + assert tf2.den == tf1.den + + # StateSpace to TransferFunction for MIMO + A2 = Matrix([[-1.5, -2, 3], [1, 0, 1], [2, 1, 1]]) + B2 = Matrix([[0.5, 0, 1], [0, 1, 2], [2, 2, 3]]) + C2 = Matrix([[0, 1, 0], [0, 2, 1], [1, 0, 2]]) + D2 = Matrix([[2, 2, 0], [1, 1, 1], [3, 2, 1]]) + H2 = StateSpace(A2, B2, C2, D2) + tm3 = H2.rewrite(TransferFunction) + + # outputs for input i obtained at Index i-1. Consider input 1 + assert tm3[0][0] == TransferFunction(2.0*s**3 + 1.0*s**2 - 10.5*s + 4.5, 1.0*s**3 + 0.5*s**2 - 6.5*s - 2.5, s) + assert tm3[0][1] == TransferFunction(2.0*s**3 + 2.0*s**2 - 10.5*s - 3.5, 1.0*s**3 + 0.5*s**2 - 6.5*s - 2.5, s) + assert tm3[0][2] == TransferFunction(2.0*s**2 + 5.0*s - 0.5, 1.0*s**3 + 0.5*s**2 - 6.5*s - 2.5, s) + + # TransferFunction to StateSpace + SS = TF1.rewrite(StateSpace) + assert SS == \ + StateSpace(Matrix([[ 0, 1], + [-wn**2, -2*wn*zeta]]), + Matrix([[0], + [1]]), + Matrix([[1, 0]]), + Matrix([[0]])) + assert SS.rewrite(TransferFunction)[0][0] == TF1 + + # Transfer function has to be proper + raises(ValueError, lambda: TransferFunction(b*s**2 + p**2 - a*p + s, b - p**2, s).rewrite(StateSpace)) + + +def test_StateSpace_functions(): + # https://in.mathworks.com/help/control/ref/statespacemodel.obsv.html + + A_mat = Matrix([[-1.5, -2], [1, 0]]) + B_mat = Matrix([0.5, 0]) + C_mat = Matrix([[0, 1]]) + D_mat = Matrix([1]) + SS1 = StateSpace(A_mat, B_mat, C_mat, D_mat) + SS2 = StateSpace(Matrix([[1, 1], [4, -2]]),Matrix([[0, 1], [0, 2]]),Matrix([[-1, 1], [1, -1]])) + SS3 = StateSpace(Matrix([[1, 1], [4, -2]]),Matrix([[1, -1], [1, -1]])) + + # Observability + assert SS1.is_observable() == True + assert SS2.is_observable() == False + assert SS1.observability_matrix() == Matrix([[0, 1], [1, 0]]) + assert SS2.observability_matrix() == Matrix([[-1, 1], [ 1, -1], [ 3, -3], [-3, 3]]) + assert SS1.observable_subspace() == [Matrix([[0], [1]]), Matrix([[1], [0]])] + assert SS2.observable_subspace() == [Matrix([[-1], [ 1], [ 3], [-3]])] + + # Controllability + assert SS1.is_controllable() == True + assert SS3.is_controllable() == False + assert SS1.controllability_matrix() == Matrix([[0.5, -0.75], [ 0, 0.5]]) + assert SS3.controllability_matrix() == Matrix([[1, -1, 2, -2], [1, -1, 2, -2]]) + assert SS1.controllable_subspace() == [Matrix([[0.5], [ 0]]), Matrix([[-0.75], [ 0.5]])] + assert SS3.controllable_subspace() == [Matrix([[1], [1]])] + + # Append + A1 = Matrix([[0, 1], [1, 0]]) + B1 = Matrix([[0], [1]]) + C1 = Matrix([[0, 1]]) + D1 = Matrix([[0]]) + ss1 = StateSpace(A1, B1, C1, D1) + ss2 = StateSpace(Matrix([[1, 0], [0, 1]]), Matrix([[1], [0]]), Matrix([[1, 0]]), Matrix([[1]])) + ss3 = ss1.append(ss2) + + assert ss3.num_states == ss1.num_states + ss2.num_states + assert ss3.num_inputs == ss1.num_inputs + ss2.num_inputs + assert ss3.num_outputs == ss1.num_outputs + ss2.num_outputs + assert ss3.state_matrix == Matrix([[0, 1, 0, 0], [1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]) + assert ss3.input_matrix == Matrix([[0, 0], [1, 0], [0, 1], [0, 0]]) + assert ss3.output_matrix == Matrix([[0, 1, 0, 0], [0, 0, 1, 0]]) + assert ss3.feedforward_matrix == Matrix([[0, 0], [0, 1]]) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/optics/gaussopt.py b/wemm/lib/python3.10/site-packages/sympy/physics/optics/gaussopt.py new file mode 100644 index 0000000000000000000000000000000000000000..d9e8ef555d60e3204341cdc65cdd05fb02b2f196 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/optics/gaussopt.py @@ -0,0 +1,923 @@ +""" +Gaussian optics. + +The module implements: + +- Ray transfer matrices for geometrical and gaussian optics. + + See RayTransferMatrix, GeometricRay and BeamParameter + +- Conjugation relations for geometrical and gaussian optics. + + See geometric_conj*, gauss_conj and conjugate_gauss_beams + +The conventions for the distances are as follows: + +focal distance + positive for convergent lenses +object distance + positive for real objects +image distance + positive for real images +""" + +__all__ = [ + 'RayTransferMatrix', + 'FreeSpace', + 'FlatRefraction', + 'CurvedRefraction', + 'FlatMirror', + 'CurvedMirror', + 'ThinLens', + 'GeometricRay', + 'BeamParameter', + 'waist2rayleigh', + 'rayleigh2waist', + 'geometric_conj_ab', + 'geometric_conj_af', + 'geometric_conj_bf', + 'gaussian_conj', + 'conjugate_gauss_beams', +] + + +from sympy.core.expr import Expr +from sympy.core.numbers import (I, pi) +from sympy.core.sympify import sympify +from sympy.functions.elementary.complexes import (im, re) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import atan2 +from sympy.matrices.dense import Matrix, MutableDenseMatrix +from sympy.polys.rationaltools import together +from sympy.utilities.misc import filldedent + +### +# A, B, C, D matrices +### + + +class RayTransferMatrix(MutableDenseMatrix): + """ + Base class for a Ray Transfer Matrix. + + It should be used if there is not already a more specific subclass mentioned + in See Also. + + Parameters + ========== + + parameters : + A, B, C and D or 2x2 matrix (Matrix(2, 2, [A, B, C, D])) + + Examples + ======== + + >>> from sympy.physics.optics import RayTransferMatrix, ThinLens + >>> from sympy import Symbol, Matrix + + >>> mat = RayTransferMatrix(1, 2, 3, 4) + >>> mat + Matrix([ + [1, 2], + [3, 4]]) + + >>> RayTransferMatrix(Matrix([[1, 2], [3, 4]])) + Matrix([ + [1, 2], + [3, 4]]) + + >>> mat.A + 1 + + >>> f = Symbol('f') + >>> lens = ThinLens(f) + >>> lens + Matrix([ + [ 1, 0], + [-1/f, 1]]) + + >>> lens.C + -1/f + + See Also + ======== + + GeometricRay, BeamParameter, + FreeSpace, FlatRefraction, CurvedRefraction, + FlatMirror, CurvedMirror, ThinLens + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis + """ + + def __new__(cls, *args): + + if len(args) == 4: + temp = ((args[0], args[1]), (args[2], args[3])) + elif len(args) == 1 \ + and isinstance(args[0], Matrix) \ + and args[0].shape == (2, 2): + temp = args[0] + else: + raise ValueError(filldedent(''' + Expecting 2x2 Matrix or the 4 elements of + the Matrix but got %s''' % str(args))) + return Matrix.__new__(cls, temp) + + def __mul__(self, other): + if isinstance(other, RayTransferMatrix): + return RayTransferMatrix(Matrix(self)*Matrix(other)) + elif isinstance(other, GeometricRay): + return GeometricRay(Matrix(self)*Matrix(other)) + elif isinstance(other, BeamParameter): + temp = Matrix(self)*Matrix(((other.q,), (1,))) + q = (temp[0]/temp[1]).expand(complex=True) + return BeamParameter(other.wavelen, + together(re(q)), + z_r=together(im(q))) + else: + return Matrix.__mul__(self, other) + + @property + def A(self): + """ + The A parameter of the Matrix. + + Examples + ======== + + >>> from sympy.physics.optics import RayTransferMatrix + >>> mat = RayTransferMatrix(1, 2, 3, 4) + >>> mat.A + 1 + """ + return self[0, 0] + + @property + def B(self): + """ + The B parameter of the Matrix. + + Examples + ======== + + >>> from sympy.physics.optics import RayTransferMatrix + >>> mat = RayTransferMatrix(1, 2, 3, 4) + >>> mat.B + 2 + """ + return self[0, 1] + + @property + def C(self): + """ + The C parameter of the Matrix. + + Examples + ======== + + >>> from sympy.physics.optics import RayTransferMatrix + >>> mat = RayTransferMatrix(1, 2, 3, 4) + >>> mat.C + 3 + """ + return self[1, 0] + + @property + def D(self): + """ + The D parameter of the Matrix. + + Examples + ======== + + >>> from sympy.physics.optics import RayTransferMatrix + >>> mat = RayTransferMatrix(1, 2, 3, 4) + >>> mat.D + 4 + """ + return self[1, 1] + + +class FreeSpace(RayTransferMatrix): + """ + Ray Transfer Matrix for free space. + + Parameters + ========== + + distance + + See Also + ======== + + RayTransferMatrix + + Examples + ======== + + >>> from sympy.physics.optics import FreeSpace + >>> from sympy import symbols + >>> d = symbols('d') + >>> FreeSpace(d) + Matrix([ + [1, d], + [0, 1]]) + """ + def __new__(cls, d): + return RayTransferMatrix.__new__(cls, 1, d, 0, 1) + + +class FlatRefraction(RayTransferMatrix): + """ + Ray Transfer Matrix for refraction. + + Parameters + ========== + + n1 : + Refractive index of one medium. + n2 : + Refractive index of other medium. + + See Also + ======== + + RayTransferMatrix + + Examples + ======== + + >>> from sympy.physics.optics import FlatRefraction + >>> from sympy import symbols + >>> n1, n2 = symbols('n1 n2') + >>> FlatRefraction(n1, n2) + Matrix([ + [1, 0], + [0, n1/n2]]) + """ + def __new__(cls, n1, n2): + n1, n2 = map(sympify, (n1, n2)) + return RayTransferMatrix.__new__(cls, 1, 0, 0, n1/n2) + + +class CurvedRefraction(RayTransferMatrix): + """ + Ray Transfer Matrix for refraction on curved interface. + + Parameters + ========== + + R : + Radius of curvature (positive for concave). + n1 : + Refractive index of one medium. + n2 : + Refractive index of other medium. + + See Also + ======== + + RayTransferMatrix + + Examples + ======== + + >>> from sympy.physics.optics import CurvedRefraction + >>> from sympy import symbols + >>> R, n1, n2 = symbols('R n1 n2') + >>> CurvedRefraction(R, n1, n2) + Matrix([ + [ 1, 0], + [(n1 - n2)/(R*n2), n1/n2]]) + """ + def __new__(cls, R, n1, n2): + R, n1, n2 = map(sympify, (R, n1, n2)) + return RayTransferMatrix.__new__(cls, 1, 0, (n1 - n2)/R/n2, n1/n2) + + +class FlatMirror(RayTransferMatrix): + """ + Ray Transfer Matrix for reflection. + + See Also + ======== + + RayTransferMatrix + + Examples + ======== + + >>> from sympy.physics.optics import FlatMirror + >>> FlatMirror() + Matrix([ + [1, 0], + [0, 1]]) + """ + def __new__(cls): + return RayTransferMatrix.__new__(cls, 1, 0, 0, 1) + + +class CurvedMirror(RayTransferMatrix): + """ + Ray Transfer Matrix for reflection from curved surface. + + Parameters + ========== + + R : radius of curvature (positive for concave) + + See Also + ======== + + RayTransferMatrix + + Examples + ======== + + >>> from sympy.physics.optics import CurvedMirror + >>> from sympy import symbols + >>> R = symbols('R') + >>> CurvedMirror(R) + Matrix([ + [ 1, 0], + [-2/R, 1]]) + """ + def __new__(cls, R): + R = sympify(R) + return RayTransferMatrix.__new__(cls, 1, 0, -2/R, 1) + + +class ThinLens(RayTransferMatrix): + """ + Ray Transfer Matrix for a thin lens. + + Parameters + ========== + + f : + The focal distance. + + See Also + ======== + + RayTransferMatrix + + Examples + ======== + + >>> from sympy.physics.optics import ThinLens + >>> from sympy import symbols + >>> f = symbols('f') + >>> ThinLens(f) + Matrix([ + [ 1, 0], + [-1/f, 1]]) + """ + def __new__(cls, f): + f = sympify(f) + return RayTransferMatrix.__new__(cls, 1, 0, -1/f, 1) + + +### +# Representation for geometric ray +### + +class GeometricRay(MutableDenseMatrix): + """ + Representation for a geometric ray in the Ray Transfer Matrix formalism. + + Parameters + ========== + + h : height, and + angle : angle, or + matrix : a 2x1 matrix (Matrix(2, 1, [height, angle])) + + Examples + ======== + + >>> from sympy.physics.optics import GeometricRay, FreeSpace + >>> from sympy import symbols, Matrix + >>> d, h, angle = symbols('d, h, angle') + + >>> GeometricRay(h, angle) + Matrix([ + [ h], + [angle]]) + + >>> FreeSpace(d)*GeometricRay(h, angle) + Matrix([ + [angle*d + h], + [ angle]]) + + >>> GeometricRay( Matrix( ((h,), (angle,)) ) ) + Matrix([ + [ h], + [angle]]) + + See Also + ======== + + RayTransferMatrix + + """ + + def __new__(cls, *args): + if len(args) == 1 and isinstance(args[0], Matrix) \ + and args[0].shape == (2, 1): + temp = args[0] + elif len(args) == 2: + temp = ((args[0],), (args[1],)) + else: + raise ValueError(filldedent(''' + Expecting 2x1 Matrix or the 2 elements of + the Matrix but got %s''' % str(args))) + return Matrix.__new__(cls, temp) + + @property + def height(self): + """ + The distance from the optical axis. + + Examples + ======== + + >>> from sympy.physics.optics import GeometricRay + >>> from sympy import symbols + >>> h, angle = symbols('h, angle') + >>> gRay = GeometricRay(h, angle) + >>> gRay.height + h + """ + return self[0] + + @property + def angle(self): + """ + The angle with the optical axis. + + Examples + ======== + + >>> from sympy.physics.optics import GeometricRay + >>> from sympy import symbols + >>> h, angle = symbols('h, angle') + >>> gRay = GeometricRay(h, angle) + >>> gRay.angle + angle + """ + return self[1] + + +### +# Representation for gauss beam +### + +class BeamParameter(Expr): + """ + Representation for a gaussian ray in the Ray Transfer Matrix formalism. + + Parameters + ========== + + wavelen : the wavelength, + z : the distance to waist, and + w : the waist, or + z_r : the rayleigh range. + n : the refractive index of medium. + + Examples + ======== + + >>> from sympy.physics.optics import BeamParameter + >>> p = BeamParameter(530e-9, 1, w=1e-3) + >>> p.q + 1 + 1.88679245283019*I*pi + + >>> p.q.n() + 1.0 + 5.92753330865999*I + >>> p.w_0.n() + 0.00100000000000000 + >>> p.z_r.n() + 5.92753330865999 + + >>> from sympy.physics.optics import FreeSpace + >>> fs = FreeSpace(10) + >>> p1 = fs*p + >>> p.w.n() + 0.00101413072159615 + >>> p1.w.n() + 0.00210803120913829 + + See Also + ======== + + RayTransferMatrix + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Complex_beam_parameter + .. [2] https://en.wikipedia.org/wiki/Gaussian_beam + """ + #TODO A class Complex may be implemented. The BeamParameter may + # subclass it. See: + # https://groups.google.com/d/topic/sympy/7XkU07NRBEs/discussion + + def __new__(cls, wavelen, z, z_r=None, w=None, n=1): + wavelen = sympify(wavelen) + z = sympify(z) + n = sympify(n) + + if z_r is not None and w is None: + z_r = sympify(z_r) + elif w is not None and z_r is None: + z_r = waist2rayleigh(sympify(w), wavelen, n) + elif z_r is None and w is None: + raise ValueError('Must specify one of w and z_r.') + + return Expr.__new__(cls, wavelen, z, z_r, n) + + @property + def wavelen(self): + return self.args[0] + + @property + def z(self): + return self.args[1] + + @property + def z_r(self): + return self.args[2] + + @property + def n(self): + return self.args[3] + + @property + def q(self): + """ + The complex parameter representing the beam. + + Examples + ======== + + >>> from sympy.physics.optics import BeamParameter + >>> p = BeamParameter(530e-9, 1, w=1e-3) + >>> p.q + 1 + 1.88679245283019*I*pi + """ + return self.z + I*self.z_r + + @property + def radius(self): + """ + The radius of curvature of the phase front. + + Examples + ======== + + >>> from sympy.physics.optics import BeamParameter + >>> p = BeamParameter(530e-9, 1, w=1e-3) + >>> p.radius + 1 + 3.55998576005696*pi**2 + """ + return self.z*(1 + (self.z_r/self.z)**2) + + @property + def w(self): + """ + The radius of the beam w(z), at any position z along the beam. + The beam radius at `1/e^2` intensity (axial value). + + See Also + ======== + + w_0 : + The minimal radius of beam. + + Examples + ======== + + >>> from sympy.physics.optics import BeamParameter + >>> p = BeamParameter(530e-9, 1, w=1e-3) + >>> p.w + 0.001*sqrt(0.2809/pi**2 + 1) + """ + return self.w_0*sqrt(1 + (self.z/self.z_r)**2) + + @property + def w_0(self): + """ + The minimal radius of beam at `1/e^2` intensity (peak value). + + See Also + ======== + + w : the beam radius at `1/e^2` intensity (axial value). + + Examples + ======== + + >>> from sympy.physics.optics import BeamParameter + >>> p = BeamParameter(530e-9, 1, w=1e-3) + >>> p.w_0 + 0.00100000000000000 + """ + return sqrt(self.z_r/(pi*self.n)*self.wavelen) + + @property + def divergence(self): + """ + Half of the total angular spread. + + Examples + ======== + + >>> from sympy.physics.optics import BeamParameter + >>> p = BeamParameter(530e-9, 1, w=1e-3) + >>> p.divergence + 0.00053/pi + """ + return self.wavelen/pi/self.w_0 + + @property + def gouy(self): + """ + The Gouy phase. + + Examples + ======== + + >>> from sympy.physics.optics import BeamParameter + >>> p = BeamParameter(530e-9, 1, w=1e-3) + >>> p.gouy + atan(0.53/pi) + """ + return atan2(self.z, self.z_r) + + @property + def waist_approximation_limit(self): + """ + The minimal waist for which the gauss beam approximation is valid. + + Explanation + =========== + + The gauss beam is a solution to the paraxial equation. For curvatures + that are too great it is not a valid approximation. + + Examples + ======== + + >>> from sympy.physics.optics import BeamParameter + >>> p = BeamParameter(530e-9, 1, w=1e-3) + >>> p.waist_approximation_limit + 1.06e-6/pi + """ + return 2*self.wavelen/pi + + +### +# Utilities +### + +def waist2rayleigh(w, wavelen, n=1): + """ + Calculate the rayleigh range from the waist of a gaussian beam. + + See Also + ======== + + rayleigh2waist, BeamParameter + + Examples + ======== + + >>> from sympy.physics.optics import waist2rayleigh + >>> from sympy import symbols + >>> w, wavelen = symbols('w wavelen') + >>> waist2rayleigh(w, wavelen) + pi*w**2/wavelen + """ + w, wavelen = map(sympify, (w, wavelen)) + return w**2*n*pi/wavelen + + +def rayleigh2waist(z_r, wavelen): + """Calculate the waist from the rayleigh range of a gaussian beam. + + See Also + ======== + + waist2rayleigh, BeamParameter + + Examples + ======== + + >>> from sympy.physics.optics import rayleigh2waist + >>> from sympy import symbols + >>> z_r, wavelen = symbols('z_r wavelen') + >>> rayleigh2waist(z_r, wavelen) + sqrt(wavelen*z_r)/sqrt(pi) + """ + z_r, wavelen = map(sympify, (z_r, wavelen)) + return sqrt(z_r/pi*wavelen) + + +def geometric_conj_ab(a, b): + """ + Conjugation relation for geometrical beams under paraxial conditions. + + Explanation + =========== + + Takes the distances to the optical element and returns the needed + focal distance. + + See Also + ======== + + geometric_conj_af, geometric_conj_bf + + Examples + ======== + + >>> from sympy.physics.optics import geometric_conj_ab + >>> from sympy import symbols + >>> a, b = symbols('a b') + >>> geometric_conj_ab(a, b) + a*b/(a + b) + """ + a, b = map(sympify, (a, b)) + if a.is_infinite or b.is_infinite: + return a if b.is_infinite else b + else: + return a*b/(a + b) + + +def geometric_conj_af(a, f): + """ + Conjugation relation for geometrical beams under paraxial conditions. + + Explanation + =========== + + Takes the object distance (for geometric_conj_af) or the image distance + (for geometric_conj_bf) to the optical element and the focal distance. + Then it returns the other distance needed for conjugation. + + See Also + ======== + + geometric_conj_ab + + Examples + ======== + + >>> from sympy.physics.optics.gaussopt import geometric_conj_af, geometric_conj_bf + >>> from sympy import symbols + >>> a, b, f = symbols('a b f') + >>> geometric_conj_af(a, f) + a*f/(a - f) + >>> geometric_conj_bf(b, f) + b*f/(b - f) + """ + a, f = map(sympify, (a, f)) + return -geometric_conj_ab(a, -f) + +geometric_conj_bf = geometric_conj_af + + +def gaussian_conj(s_in, z_r_in, f): + """ + Conjugation relation for gaussian beams. + + Parameters + ========== + + s_in : + The distance to optical element from the waist. + z_r_in : + The rayleigh range of the incident beam. + f : + The focal length of the optical element. + + Returns + ======= + + a tuple containing (s_out, z_r_out, m) + s_out : + The distance between the new waist and the optical element. + z_r_out : + The rayleigh range of the emergent beam. + m : + The ration between the new and the old waists. + + Examples + ======== + + >>> from sympy.physics.optics import gaussian_conj + >>> from sympy import symbols + >>> s_in, z_r_in, f = symbols('s_in z_r_in f') + + >>> gaussian_conj(s_in, z_r_in, f)[0] + 1/(-1/(s_in + z_r_in**2/(-f + s_in)) + 1/f) + + >>> gaussian_conj(s_in, z_r_in, f)[1] + z_r_in/(1 - s_in**2/f**2 + z_r_in**2/f**2) + + >>> gaussian_conj(s_in, z_r_in, f)[2] + 1/sqrt(1 - s_in**2/f**2 + z_r_in**2/f**2) + """ + s_in, z_r_in, f = map(sympify, (s_in, z_r_in, f)) + s_out = 1 / ( -1/(s_in + z_r_in**2/(s_in - f)) + 1/f ) + m = 1/sqrt((1 - (s_in/f)**2) + (z_r_in/f)**2) + z_r_out = z_r_in / ((1 - (s_in/f)**2) + (z_r_in/f)**2) + return (s_out, z_r_out, m) + + +def conjugate_gauss_beams(wavelen, waist_in, waist_out, **kwargs): + """ + Find the optical setup conjugating the object/image waists. + + Parameters + ========== + + wavelen : + The wavelength of the beam. + waist_in and waist_out : + The waists to be conjugated. + f : + The focal distance of the element used in the conjugation. + + Returns + ======= + + a tuple containing (s_in, s_out, f) + s_in : + The distance before the optical element. + s_out : + The distance after the optical element. + f : + The focal distance of the optical element. + + Examples + ======== + + >>> from sympy.physics.optics import conjugate_gauss_beams + >>> from sympy import symbols, factor + >>> l, w_i, w_o, f = symbols('l w_i w_o f') + + >>> conjugate_gauss_beams(l, w_i, w_o, f=f)[0] + f*(1 - sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2))) + + >>> factor(conjugate_gauss_beams(l, w_i, w_o, f=f)[1]) + f*w_o**2*(w_i**2/w_o**2 - sqrt(w_i**2/w_o**2 - + pi**2*w_i**4/(f**2*l**2)))/w_i**2 + + >>> conjugate_gauss_beams(l, w_i, w_o, f=f)[2] + f + """ + #TODO add the other possible arguments + wavelen, waist_in, waist_out = map(sympify, (wavelen, waist_in, waist_out)) + m = waist_out / waist_in + z = waist2rayleigh(waist_in, wavelen) + if len(kwargs) != 1: + raise ValueError("The function expects only one named argument") + elif 'dist' in kwargs: + raise NotImplementedError(filldedent(''' + Currently only focal length is supported as a parameter''')) + elif 'f' in kwargs: + f = sympify(kwargs['f']) + s_in = f * (1 - sqrt(1/m**2 - z**2/f**2)) + s_out = gaussian_conj(s_in, z, f)[0] + elif 's_in' in kwargs: + raise NotImplementedError(filldedent(''' + Currently only focal length is supported as a parameter''')) + else: + raise ValueError(filldedent(''' + The functions expects the focal length as a named argument''')) + return (s_in, s_out, f) + +#TODO +#def plot_beam(): +# """Plot the beam radius as it propagates in space.""" +# pass + +#TODO +#def plot_beam_conjugation(): +# """ +# Plot the intersection of two beams. +# +# Represents the conjugation relation. +# +# See Also +# ======== +# +# conjugate_gauss_beams +# """ +# pass diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/optics/tests/test_waves.py b/wemm/lib/python3.10/site-packages/sympy/physics/optics/tests/test_waves.py new file mode 100644 index 0000000000000000000000000000000000000000..3cb8f804fb5be86d6174cb7c7b15fd8979c85ff8 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/optics/tests/test_waves.py @@ -0,0 +1,82 @@ +from sympy.core.function import (Derivative, Function) +from sympy.core.numbers import (I, pi) +from sympy.core.symbol import (Symbol, symbols) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (atan2, cos, sin) +from sympy.simplify.simplify import simplify +from sympy.abc import epsilon, mu +from sympy.functions.elementary.exponential import exp +from sympy.physics.units import speed_of_light, m, s +from sympy.physics.optics import TWave + +from sympy.testing.pytest import raises + +c = speed_of_light.convert_to(m/s) + +def test_twave(): + A1, phi1, A2, phi2, f = symbols('A1, phi1, A2, phi2, f') + n = Symbol('n') # Refractive index + t = Symbol('t') # Time + x = Symbol('x') # Spatial variable + E = Function('E') + w1 = TWave(A1, f, phi1) + w2 = TWave(A2, f, phi2) + assert w1.amplitude == A1 + assert w1.frequency == f + assert w1.phase == phi1 + assert w1.wavelength == c/(f*n) + assert w1.time_period == 1/f + assert w1.angular_velocity == 2*pi*f + assert w1.wavenumber == 2*pi*f*n/c + assert w1.speed == c/n + + w3 = w1 + w2 + assert w3.amplitude == sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + A2**2) + assert w3.frequency == f + assert w3.phase == atan2(A1*sin(phi1) + A2*sin(phi2), A1*cos(phi1) + A2*cos(phi2)) + assert w3.wavelength == c/(f*n) + assert w3.time_period == 1/f + assert w3.angular_velocity == 2*pi*f + assert w3.wavenumber == 2*pi*f*n/c + assert w3.speed == c/n + assert simplify(w3.rewrite(sin) - w2.rewrite(sin) - w1.rewrite(sin)) == 0 + assert w3.rewrite('pde') == epsilon*mu*Derivative(E(x, t), t, t) + Derivative(E(x, t), x, x) + assert w3.rewrite(cos) == sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + + A2**2)*cos(pi*f*n*x*s/(149896229*m) - 2*pi*f*t + atan2(A1*sin(phi1) + + A2*sin(phi2), A1*cos(phi1) + A2*cos(phi2))) + assert w3.rewrite(exp) == sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + + A2**2)*exp(I*(-2*pi*f*t + atan2(A1*sin(phi1) + A2*sin(phi2), A1*cos(phi1) + + A2*cos(phi2)) + pi*s*f*n*x/(149896229*m))) + + w4 = TWave(A1, None, 0, 1/f) + assert w4.frequency == f + + w5 = w1 - w2 + assert w5.amplitude == sqrt(A1**2 - 2*A1*A2*cos(phi1 - phi2) + A2**2) + assert w5.frequency == f + assert w5.phase == atan2(A1*sin(phi1) - A2*sin(phi2), A1*cos(phi1) - A2*cos(phi2)) + assert w5.wavelength == c/(f*n) + assert w5.time_period == 1/f + assert w5.angular_velocity == 2*pi*f + assert w5.wavenumber == 2*pi*f*n/c + assert w5.speed == c/n + assert simplify(w5.rewrite(sin) - w1.rewrite(sin) + w2.rewrite(sin)) == 0 + assert w5.rewrite('pde') == epsilon*mu*Derivative(E(x, t), t, t) + Derivative(E(x, t), x, x) + assert w5.rewrite(cos) == sqrt(A1**2 - 2*A1*A2*cos(phi1 - phi2) + + A2**2)*cos(-2*pi*f*t + atan2(A1*sin(phi1) - A2*sin(phi2), A1*cos(phi1) + - A2*cos(phi2)) + pi*s*f*n*x/(149896229*m)) + assert w5.rewrite(exp) == sqrt(A1**2 - 2*A1*A2*cos(phi1 - phi2) + + A2**2)*exp(I*(-2*pi*f*t + atan2(A1*sin(phi1) - A2*sin(phi2), A1*cos(phi1) + - A2*cos(phi2)) + pi*s*f*n*x/(149896229*m))) + + w6 = 2*w1 + assert w6.amplitude == 2*A1 + assert w6.frequency == f + assert w6.phase == phi1 + w7 = -w6 + assert w7.amplitude == -2*A1 + assert w7.frequency == f + assert w7.phase == phi1 + + raises(ValueError, lambda:TWave(A1)) + raises(ValueError, lambda:TWave(A1, f, phi1, t)) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/optics/utils.py b/wemm/lib/python3.10/site-packages/sympy/physics/optics/utils.py new file mode 100644 index 0000000000000000000000000000000000000000..72c3b78bd4b09eb069757fb3f8d3632f09ec4b80 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/optics/utils.py @@ -0,0 +1,698 @@ +""" +**Contains** + +* refraction_angle +* fresnel_coefficients +* deviation +* brewster_angle +* critical_angle +* lens_makers_formula +* mirror_formula +* lens_formula +* hyperfocal_distance +* transverse_magnification +""" + +__all__ = ['refraction_angle', + 'deviation', + 'fresnel_coefficients', + 'brewster_angle', + 'critical_angle', + 'lens_makers_formula', + 'mirror_formula', + 'lens_formula', + 'hyperfocal_distance', + 'transverse_magnification' + ] + +from sympy.core.numbers import (Float, I, oo, pi, zoo) +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.core.sympify import sympify +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (acos, asin, atan2, cos, sin, tan) +from sympy.matrices.dense import Matrix +from sympy.polys.polytools import cancel +from sympy.series.limits import Limit +from sympy.geometry.line import Ray3D +from sympy.geometry.util import intersection +from sympy.geometry.plane import Plane +from sympy.utilities.iterables import is_sequence +from .medium import Medium + + +def refractive_index_of_medium(medium): + """ + Helper function that returns refractive index, given a medium + """ + if isinstance(medium, Medium): + n = medium.refractive_index + else: + n = sympify(medium) + return n + + +def refraction_angle(incident, medium1, medium2, normal=None, plane=None): + """ + This function calculates transmitted vector after refraction at planar + surface. ``medium1`` and ``medium2`` can be ``Medium`` or any sympifiable object. + If ``incident`` is a number then treated as angle of incidence (in radians) + in which case refraction angle is returned. + + If ``incident`` is an object of `Ray3D`, `normal` also has to be an instance + of `Ray3D` in order to get the output as a `Ray3D`. Please note that if + plane of separation is not provided and normal is an instance of `Ray3D`, + ``normal`` will be assumed to be intersecting incident ray at the plane of + separation. This will not be the case when `normal` is a `Matrix` or + any other sequence. + If ``incident`` is an instance of `Ray3D` and `plane` has not been provided + and ``normal`` is not `Ray3D`, output will be a `Matrix`. + + Parameters + ========== + + incident : Matrix, Ray3D, sequence or a number + Incident vector or angle of incidence + medium1 : sympy.physics.optics.medium.Medium or sympifiable + Medium 1 or its refractive index + medium2 : sympy.physics.optics.medium.Medium or sympifiable + Medium 2 or its refractive index + normal : Matrix, Ray3D, or sequence + Normal vector + plane : Plane + Plane of separation of the two media. + + Returns + ======= + + Returns an angle of refraction or a refracted ray depending on inputs. + + Examples + ======== + + >>> from sympy.physics.optics import refraction_angle + >>> from sympy.geometry import Point3D, Ray3D, Plane + >>> from sympy.matrices import Matrix + >>> from sympy import symbols, pi + >>> n = Matrix([0, 0, 1]) + >>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1]) + >>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0)) + >>> refraction_angle(r1, 1, 1, n) + Matrix([ + [ 1], + [ 1], + [-1]]) + >>> refraction_angle(r1, 1, 1, plane=P) + Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1)) + + With different index of refraction of the two media + + >>> n1, n2 = symbols('n1, n2') + >>> refraction_angle(r1, n1, n2, n) + Matrix([ + [ n1/n2], + [ n1/n2], + [-sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)]]) + >>> refraction_angle(r1, n1, n2, plane=P) + Ray3D(Point3D(0, 0, 0), Point3D(n1/n2, n1/n2, -sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1))) + >>> round(refraction_angle(pi/6, 1.2, 1.5), 5) + 0.41152 + """ + + n1 = refractive_index_of_medium(medium1) + n2 = refractive_index_of_medium(medium2) + + # check if an incidence angle was supplied instead of a ray + try: + angle_of_incidence = float(incident) + except TypeError: + angle_of_incidence = None + + try: + critical_angle_ = critical_angle(medium1, medium2) + except (ValueError, TypeError): + critical_angle_ = None + + if angle_of_incidence is not None: + if normal is not None or plane is not None: + raise ValueError('Normal/plane not allowed if incident is an angle') + + if not 0.0 <= angle_of_incidence < pi*0.5: + raise ValueError('Angle of incidence not in range [0:pi/2)') + + if critical_angle_ and angle_of_incidence > critical_angle_: + raise ValueError('Ray undergoes total internal reflection') + return asin(n1*sin(angle_of_incidence)/n2) + + # Treat the incident as ray below + # A flag to check whether to return Ray3D or not + return_ray = False + + if plane is not None and normal is not None: + raise ValueError("Either plane or normal is acceptable.") + + if not isinstance(incident, Matrix): + if is_sequence(incident): + _incident = Matrix(incident) + elif isinstance(incident, Ray3D): + _incident = Matrix(incident.direction_ratio) + else: + raise TypeError( + "incident should be a Matrix, Ray3D, or sequence") + else: + _incident = incident + + # If plane is provided, get direction ratios of the normal + # to the plane from the plane else go with `normal` param. + if plane is not None: + if not isinstance(plane, Plane): + raise TypeError("plane should be an instance of geometry.plane.Plane") + # If we have the plane, we can get the intersection + # point of incident ray and the plane and thus return + # an instance of Ray3D. + if isinstance(incident, Ray3D): + return_ray = True + intersection_pt = plane.intersection(incident)[0] + _normal = Matrix(plane.normal_vector) + else: + if not isinstance(normal, Matrix): + if is_sequence(normal): + _normal = Matrix(normal) + elif isinstance(normal, Ray3D): + _normal = Matrix(normal.direction_ratio) + if isinstance(incident, Ray3D): + intersection_pt = intersection(incident, normal) + if len(intersection_pt) == 0: + raise ValueError( + "Normal isn't concurrent with the incident ray.") + else: + return_ray = True + intersection_pt = intersection_pt[0] + else: + raise TypeError( + "Normal should be a Matrix, Ray3D, or sequence") + else: + _normal = normal + + eta = n1/n2 # Relative index of refraction + # Calculating magnitude of the vectors + mag_incident = sqrt(sum(i**2 for i in _incident)) + mag_normal = sqrt(sum(i**2 for i in _normal)) + # Converting vectors to unit vectors by dividing + # them with their magnitudes + _incident /= mag_incident + _normal /= mag_normal + c1 = -_incident.dot(_normal) # cos(angle_of_incidence) + cs2 = 1 - eta**2*(1 - c1**2) # cos(angle_of_refraction)**2 + if cs2.is_negative: # This is the case of total internal reflection(TIR). + return S.Zero + drs = eta*_incident + (eta*c1 - sqrt(cs2))*_normal + # Multiplying unit vector by its magnitude + drs = drs*mag_incident + if not return_ray: + return drs + else: + return Ray3D(intersection_pt, direction_ratio=drs) + + +def fresnel_coefficients(angle_of_incidence, medium1, medium2): + """ + This function uses Fresnel equations to calculate reflection and + transmission coefficients. Those are obtained for both polarisations + when the electric field vector is in the plane of incidence (labelled 'p') + and when the electric field vector is perpendicular to the plane of + incidence (labelled 's'). There are four real coefficients unless the + incident ray reflects in total internal in which case there are two complex + ones. Angle of incidence is the angle between the incident ray and the + surface normal. ``medium1`` and ``medium2`` can be ``Medium`` or any + sympifiable object. + + Parameters + ========== + + angle_of_incidence : sympifiable + + medium1 : Medium or sympifiable + Medium 1 or its refractive index + + medium2 : Medium or sympifiable + Medium 2 or its refractive index + + Returns + ======= + + Returns a list with four real Fresnel coefficients: + [reflection p (TM), reflection s (TE), + transmission p (TM), transmission s (TE)] + If the ray is undergoes total internal reflection then returns a + list of two complex Fresnel coefficients: + [reflection p (TM), reflection s (TE)] + + Examples + ======== + + >>> from sympy.physics.optics import fresnel_coefficients + >>> fresnel_coefficients(0.3, 1, 2) + [0.317843553417859, -0.348645229818821, + 0.658921776708929, 0.651354770181179] + >>> fresnel_coefficients(0.6, 2, 1) + [-0.235625382192159 - 0.971843958291041*I, + 0.816477005968898 - 0.577377951366403*I] + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Fresnel_equations + """ + if not 0 <= 2*angle_of_incidence < pi: + raise ValueError('Angle of incidence not in range [0:pi/2)') + + n1 = refractive_index_of_medium(medium1) + n2 = refractive_index_of_medium(medium2) + + angle_of_refraction = asin(n1*sin(angle_of_incidence)/n2) + try: + angle_of_total_internal_reflection_onset = critical_angle(n1, n2) + except ValueError: + angle_of_total_internal_reflection_onset = None + + if angle_of_total_internal_reflection_onset is None or\ + angle_of_total_internal_reflection_onset > angle_of_incidence: + R_s = -sin(angle_of_incidence - angle_of_refraction)\ + /sin(angle_of_incidence + angle_of_refraction) + R_p = tan(angle_of_incidence - angle_of_refraction)\ + /tan(angle_of_incidence + angle_of_refraction) + T_s = 2*sin(angle_of_refraction)*cos(angle_of_incidence)\ + /sin(angle_of_incidence + angle_of_refraction) + T_p = 2*sin(angle_of_refraction)*cos(angle_of_incidence)\ + /(sin(angle_of_incidence + angle_of_refraction)\ + *cos(angle_of_incidence - angle_of_refraction)) + return [R_p, R_s, T_p, T_s] + else: + n = n2/n1 + R_s = cancel((cos(angle_of_incidence)-\ + I*sqrt(sin(angle_of_incidence)**2 - n**2))\ + /(cos(angle_of_incidence)+\ + I*sqrt(sin(angle_of_incidence)**2 - n**2))) + R_p = cancel((n**2*cos(angle_of_incidence)-\ + I*sqrt(sin(angle_of_incidence)**2 - n**2))\ + /(n**2*cos(angle_of_incidence)+\ + I*sqrt(sin(angle_of_incidence)**2 - n**2))) + return [R_p, R_s] + + +def deviation(incident, medium1, medium2, normal=None, plane=None): + """ + This function calculates the angle of deviation of a ray + due to refraction at planar surface. + + Parameters + ========== + + incident : Matrix, Ray3D, sequence or float + Incident vector or angle of incidence + medium1 : sympy.physics.optics.medium.Medium or sympifiable + Medium 1 or its refractive index + medium2 : sympy.physics.optics.medium.Medium or sympifiable + Medium 2 or its refractive index + normal : Matrix, Ray3D, or sequence + Normal vector + plane : Plane + Plane of separation of the two media. + + Returns angular deviation between incident and refracted rays + + Examples + ======== + + >>> from sympy.physics.optics import deviation + >>> from sympy.geometry import Point3D, Ray3D, Plane + >>> from sympy.matrices import Matrix + >>> from sympy import symbols + >>> n1, n2 = symbols('n1, n2') + >>> n = Matrix([0, 0, 1]) + >>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1]) + >>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0)) + >>> deviation(r1, 1, 1, n) + 0 + >>> deviation(r1, n1, n2, plane=P) + -acos(-sqrt(-2*n1**2/(3*n2**2) + 1)) + acos(-sqrt(3)/3) + >>> round(deviation(0.1, 1.2, 1.5), 5) + -0.02005 + """ + refracted = refraction_angle(incident, + medium1, + medium2, + normal=normal, + plane=plane) + try: + angle_of_incidence = Float(incident) + except TypeError: + angle_of_incidence = None + + if angle_of_incidence is not None: + return float(refracted) - angle_of_incidence + + if refracted != 0: + if isinstance(refracted, Ray3D): + refracted = Matrix(refracted.direction_ratio) + + if not isinstance(incident, Matrix): + if is_sequence(incident): + _incident = Matrix(incident) + elif isinstance(incident, Ray3D): + _incident = Matrix(incident.direction_ratio) + else: + raise TypeError( + "incident should be a Matrix, Ray3D, or sequence") + else: + _incident = incident + + if plane is None: + if not isinstance(normal, Matrix): + if is_sequence(normal): + _normal = Matrix(normal) + elif isinstance(normal, Ray3D): + _normal = Matrix(normal.direction_ratio) + else: + raise TypeError( + "normal should be a Matrix, Ray3D, or sequence") + else: + _normal = normal + else: + _normal = Matrix(plane.normal_vector) + + mag_incident = sqrt(sum(i**2 for i in _incident)) + mag_normal = sqrt(sum(i**2 for i in _normal)) + mag_refracted = sqrt(sum(i**2 for i in refracted)) + _incident /= mag_incident + _normal /= mag_normal + refracted /= mag_refracted + i = acos(_incident.dot(_normal)) + r = acos(refracted.dot(_normal)) + return i - r + + +def brewster_angle(medium1, medium2): + """ + This function calculates the Brewster's angle of incidence to Medium 2 from + Medium 1 in radians. + + Parameters + ========== + + medium 1 : Medium or sympifiable + Refractive index of Medium 1 + medium 2 : Medium or sympifiable + Refractive index of Medium 1 + + Examples + ======== + + >>> from sympy.physics.optics import brewster_angle + >>> brewster_angle(1, 1.33) + 0.926093295503462 + + """ + + n1 = refractive_index_of_medium(medium1) + n2 = refractive_index_of_medium(medium2) + + return atan2(n2, n1) + +def critical_angle(medium1, medium2): + """ + This function calculates the critical angle of incidence (marking the onset + of total internal) to Medium 2 from Medium 1 in radians. + + Parameters + ========== + + medium 1 : Medium or sympifiable + Refractive index of Medium 1. + medium 2 : Medium or sympifiable + Refractive index of Medium 1. + + Examples + ======== + + >>> from sympy.physics.optics import critical_angle + >>> critical_angle(1.33, 1) + 0.850908514477849 + + """ + + n1 = refractive_index_of_medium(medium1) + n2 = refractive_index_of_medium(medium2) + + if n2 > n1: + raise ValueError('Total internal reflection impossible for n1 < n2') + else: + return asin(n2/n1) + + + +def lens_makers_formula(n_lens, n_surr, r1, r2, d=0): + """ + This function calculates focal length of a lens. + It follows cartesian sign convention. + + Parameters + ========== + + n_lens : Medium or sympifiable + Index of refraction of lens. + n_surr : Medium or sympifiable + Index of reflection of surrounding. + r1 : sympifiable + Radius of curvature of first surface. + r2 : sympifiable + Radius of curvature of second surface. + d : sympifiable, optional + Thickness of lens, default value is 0. + + Examples + ======== + + >>> from sympy.physics.optics import lens_makers_formula + >>> from sympy import S + >>> lens_makers_formula(1.33, 1, 10, -10) + 15.1515151515151 + >>> lens_makers_formula(1.2, 1, 10, S.Infinity) + 50.0000000000000 + >>> lens_makers_formula(1.33, 1, 10, -10, d=1) + 15.3418463277618 + + """ + + if isinstance(n_lens, Medium): + n_lens = n_lens.refractive_index + else: + n_lens = sympify(n_lens) + if isinstance(n_surr, Medium): + n_surr = n_surr.refractive_index + else: + n_surr = sympify(n_surr) + d = sympify(d) + + focal_length = 1/((n_lens - n_surr) / n_surr*(1/r1 - 1/r2 + (((n_lens - n_surr) * d) / (n_lens * r1 * r2)))) + + if focal_length == zoo: + return S.Infinity + return focal_length + + +def mirror_formula(focal_length=None, u=None, v=None): + """ + This function provides one of the three parameters + when two of them are supplied. + This is valid only for paraxial rays. + + Parameters + ========== + + focal_length : sympifiable + Focal length of the mirror. + u : sympifiable + Distance of object from the pole on + the principal axis. + v : sympifiable + Distance of the image from the pole + on the principal axis. + + Examples + ======== + + >>> from sympy.physics.optics import mirror_formula + >>> from sympy.abc import f, u, v + >>> mirror_formula(focal_length=f, u=u) + f*u/(-f + u) + >>> mirror_formula(focal_length=f, v=v) + f*v/(-f + v) + >>> mirror_formula(u=u, v=v) + u*v/(u + v) + + """ + if focal_length and u and v: + raise ValueError("Please provide only two parameters") + + focal_length = sympify(focal_length) + u = sympify(u) + v = sympify(v) + if u is oo: + _u = Symbol('u') + if v is oo: + _v = Symbol('v') + if focal_length is oo: + _f = Symbol('f') + if focal_length is None: + if u is oo and v is oo: + return Limit(Limit(_v*_u/(_v + _u), _u, oo), _v, oo).doit() + if u is oo: + return Limit(v*_u/(v + _u), _u, oo).doit() + if v is oo: + return Limit(_v*u/(_v + u), _v, oo).doit() + return v*u/(v + u) + if u is None: + if v is oo and focal_length is oo: + return Limit(Limit(_v*_f/(_v - _f), _v, oo), _f, oo).doit() + if v is oo: + return Limit(_v*focal_length/(_v - focal_length), _v, oo).doit() + if focal_length is oo: + return Limit(v*_f/(v - _f), _f, oo).doit() + return v*focal_length/(v - focal_length) + if v is None: + if u is oo and focal_length is oo: + return Limit(Limit(_u*_f/(_u - _f), _u, oo), _f, oo).doit() + if u is oo: + return Limit(_u*focal_length/(_u - focal_length), _u, oo).doit() + if focal_length is oo: + return Limit(u*_f/(u - _f), _f, oo).doit() + return u*focal_length/(u - focal_length) + + +def lens_formula(focal_length=None, u=None, v=None): + """ + This function provides one of the three parameters + when two of them are supplied. + This is valid only for paraxial rays. + + Parameters + ========== + + focal_length : sympifiable + Focal length of the mirror. + u : sympifiable + Distance of object from the optical center on + the principal axis. + v : sympifiable + Distance of the image from the optical center + on the principal axis. + + Examples + ======== + + >>> from sympy.physics.optics import lens_formula + >>> from sympy.abc import f, u, v + >>> lens_formula(focal_length=f, u=u) + f*u/(f + u) + >>> lens_formula(focal_length=f, v=v) + f*v/(f - v) + >>> lens_formula(u=u, v=v) + u*v/(u - v) + + """ + if focal_length and u and v: + raise ValueError("Please provide only two parameters") + + focal_length = sympify(focal_length) + u = sympify(u) + v = sympify(v) + if u is oo: + _u = Symbol('u') + if v is oo: + _v = Symbol('v') + if focal_length is oo: + _f = Symbol('f') + if focal_length is None: + if u is oo and v is oo: + return Limit(Limit(_v*_u/(_u - _v), _u, oo), _v, oo).doit() + if u is oo: + return Limit(v*_u/(_u - v), _u, oo).doit() + if v is oo: + return Limit(_v*u/(u - _v), _v, oo).doit() + return v*u/(u - v) + if u is None: + if v is oo and focal_length is oo: + return Limit(Limit(_v*_f/(_f - _v), _v, oo), _f, oo).doit() + if v is oo: + return Limit(_v*focal_length/(focal_length - _v), _v, oo).doit() + if focal_length is oo: + return Limit(v*_f/(_f - v), _f, oo).doit() + return v*focal_length/(focal_length - v) + if v is None: + if u is oo and focal_length is oo: + return Limit(Limit(_u*_f/(_u + _f), _u, oo), _f, oo).doit() + if u is oo: + return Limit(_u*focal_length/(_u + focal_length), _u, oo).doit() + if focal_length is oo: + return Limit(u*_f/(u + _f), _f, oo).doit() + return u*focal_length/(u + focal_length) + +def hyperfocal_distance(f, N, c): + """ + + Parameters + ========== + + f: sympifiable + Focal length of a given lens. + + N: sympifiable + F-number of a given lens. + + c: sympifiable + Circle of Confusion (CoC) of a given image format. + + Example + ======= + + >>> from sympy.physics.optics import hyperfocal_distance + >>> round(hyperfocal_distance(f = 0.5, N = 8, c = 0.0033), 2) + 9.47 + """ + + f = sympify(f) + N = sympify(N) + c = sympify(c) + + return (1/(N * c))*(f**2) + +def transverse_magnification(si, so): + """ + + Calculates the transverse magnification upon reflection in a mirror, + which is the ratio of the image size to the object size. + + Parameters + ========== + + so: sympifiable + Lens-object distance. + + si: sympifiable + Lens-image distance. + + Example + ======= + + >>> from sympy.physics.optics import transverse_magnification + >>> transverse_magnification(30, 15) + -2 + + """ + + si = sympify(si) + so = sympify(so) + + return (-(si/so)) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/optics/waves.py b/wemm/lib/python3.10/site-packages/sympy/physics/optics/waves.py new file mode 100644 index 0000000000000000000000000000000000000000..61e2ff4db578543f9f2694f239f03439bfab2c41 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/optics/waves.py @@ -0,0 +1,340 @@ +""" +This module has all the classes and functions related to waves in optics. + +**Contains** + +* TWave +""" + +__all__ = ['TWave'] + +from sympy.core.basic import Basic +from sympy.core.expr import Expr +from sympy.core.function import Derivative, Function +from sympy.core.numbers import (Number, pi, I) +from sympy.core.singleton import S +from sympy.core.symbol import (Symbol, symbols) +from sympy.core.sympify import _sympify, sympify +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (atan2, cos, sin) +from sympy.physics.units import speed_of_light, meter, second + + +c = speed_of_light.convert_to(meter/second) + + +class TWave(Expr): + + r""" + This is a simple transverse sine wave travelling in a one-dimensional space. + Basic properties are required at the time of creation of the object, + but they can be changed later with respective methods provided. + + Explanation + =========== + + It is represented as :math:`A \times cos(k*x - \omega \times t + \phi )`, + where :math:`A` is the amplitude, :math:`\omega` is the angular frequency, + :math:`k` is the wavenumber (spatial frequency), :math:`x` is a spatial variable + to represent the position on the dimension on which the wave propagates, + and :math:`\phi` is the phase angle of the wave. + + + Arguments + ========= + + amplitude : Sympifyable + Amplitude of the wave. + frequency : Sympifyable + Frequency of the wave. + phase : Sympifyable + Phase angle of the wave. + time_period : Sympifyable + Time period of the wave. + n : Sympifyable + Refractive index of the medium. + + Raises + ======= + + ValueError : When neither frequency nor time period is provided + or they are not consistent. + TypeError : When anything other than TWave objects is added. + + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.physics.optics import TWave + >>> A1, phi1, A2, phi2, f = symbols('A1, phi1, A2, phi2, f') + >>> w1 = TWave(A1, f, phi1) + >>> w2 = TWave(A2, f, phi2) + >>> w3 = w1 + w2 # Superposition of two waves + >>> w3 + TWave(sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + A2**2), f, + atan2(A1*sin(phi1) + A2*sin(phi2), A1*cos(phi1) + A2*cos(phi2)), 1/f, n) + >>> w3.amplitude + sqrt(A1**2 + 2*A1*A2*cos(phi1 - phi2) + A2**2) + >>> w3.phase + atan2(A1*sin(phi1) + A2*sin(phi2), A1*cos(phi1) + A2*cos(phi2)) + >>> w3.speed + 299792458*meter/(second*n) + >>> w3.angular_velocity + 2*pi*f + + """ + + def __new__( + cls, + amplitude, + frequency=None, + phase=S.Zero, + time_period=None, + n=Symbol('n')): + if time_period is not None: + time_period = _sympify(time_period) + _frequency = S.One/time_period + if frequency is not None: + frequency = _sympify(frequency) + _time_period = S.One/frequency + if time_period is not None: + if frequency != S.One/time_period: + raise ValueError("frequency and time_period should be consistent.") + if frequency is None and time_period is None: + raise ValueError("Either frequency or time period is needed.") + if frequency is None: + frequency = _frequency + if time_period is None: + time_period = _time_period + + amplitude = _sympify(amplitude) + phase = _sympify(phase) + n = sympify(n) + obj = Basic.__new__(cls, amplitude, frequency, phase, time_period, n) + return obj + + @property + def amplitude(self): + """ + Returns the amplitude of the wave. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.physics.optics import TWave + >>> A, phi, f = symbols('A, phi, f') + >>> w = TWave(A, f, phi) + >>> w.amplitude + A + """ + return self.args[0] + + @property + def frequency(self): + """ + Returns the frequency of the wave, + in cycles per second. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.physics.optics import TWave + >>> A, phi, f = symbols('A, phi, f') + >>> w = TWave(A, f, phi) + >>> w.frequency + f + """ + return self.args[1] + + @property + def phase(self): + """ + Returns the phase angle of the wave, + in radians. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.physics.optics import TWave + >>> A, phi, f = symbols('A, phi, f') + >>> w = TWave(A, f, phi) + >>> w.phase + phi + """ + return self.args[2] + + @property + def time_period(self): + """ + Returns the temporal period of the wave, + in seconds per cycle. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.physics.optics import TWave + >>> A, phi, f = symbols('A, phi, f') + >>> w = TWave(A, f, phi) + >>> w.time_period + 1/f + """ + return self.args[3] + + @property + def n(self): + """ + Returns the refractive index of the medium + """ + return self.args[4] + + @property + def wavelength(self): + """ + Returns the wavelength (spatial period) of the wave, + in meters per cycle. + It depends on the medium of the wave. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.physics.optics import TWave + >>> A, phi, f = symbols('A, phi, f') + >>> w = TWave(A, f, phi) + >>> w.wavelength + 299792458*meter/(second*f*n) + """ + return c/(self.frequency*self.n) + + + @property + def speed(self): + """ + Returns the propagation speed of the wave, + in meters per second. + It is dependent on the propagation medium. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.physics.optics import TWave + >>> A, phi, f = symbols('A, phi, f') + >>> w = TWave(A, f, phi) + >>> w.speed + 299792458*meter/(second*n) + """ + return self.wavelength*self.frequency + + @property + def angular_velocity(self): + """ + Returns the angular velocity of the wave, + in radians per second. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.physics.optics import TWave + >>> A, phi, f = symbols('A, phi, f') + >>> w = TWave(A, f, phi) + >>> w.angular_velocity + 2*pi*f + """ + return 2*pi*self.frequency + + @property + def wavenumber(self): + """ + Returns the wavenumber of the wave, + in radians per meter. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.physics.optics import TWave + >>> A, phi, f = symbols('A, phi, f') + >>> w = TWave(A, f, phi) + >>> w.wavenumber + pi*second*f*n/(149896229*meter) + """ + return 2*pi/self.wavelength + + def __str__(self): + """String representation of a TWave.""" + from sympy.printing import sstr + return type(self).__name__ + sstr(self.args) + + __repr__ = __str__ + + def __add__(self, other): + """ + Addition of two waves will result in their superposition. + The type of interference will depend on their phase angles. + """ + if isinstance(other, TWave): + if self.frequency == other.frequency and self.wavelength == other.wavelength: + return TWave(sqrt(self.amplitude**2 + other.amplitude**2 + 2 * + self.amplitude*other.amplitude*cos( + self.phase - other.phase)), + self.frequency, + atan2(self.amplitude*sin(self.phase) + + other.amplitude*sin(other.phase), + self.amplitude*cos(self.phase) + + other.amplitude*cos(other.phase)) + ) + else: + raise NotImplementedError("Interference of waves with different frequencies" + " has not been implemented.") + else: + raise TypeError(type(other).__name__ + " and TWave objects cannot be added.") + + def __mul__(self, other): + """ + Multiplying a wave by a scalar rescales the amplitude of the wave. + """ + other = sympify(other) + if isinstance(other, Number): + return TWave(self.amplitude*other, *self.args[1:]) + else: + raise TypeError(type(other).__name__ + " and TWave objects cannot be multiplied.") + + def __sub__(self, other): + return self.__add__(-1*other) + + def __neg__(self): + return self.__mul__(-1) + + def __radd__(self, other): + return self.__add__(other) + + def __rmul__(self, other): + return self.__mul__(other) + + def __rsub__(self, other): + return (-self).__radd__(other) + + def _eval_rewrite_as_sin(self, *args, **kwargs): + return self.amplitude*sin(self.wavenumber*Symbol('x') + - self.angular_velocity*Symbol('t') + self.phase + pi/2, evaluate=False) + + def _eval_rewrite_as_cos(self, *args, **kwargs): + return self.amplitude*cos(self.wavenumber*Symbol('x') + - self.angular_velocity*Symbol('t') + self.phase) + + def _eval_rewrite_as_pde(self, *args, **kwargs): + mu, epsilon, x, t = symbols('mu, epsilon, x, t') + E = Function('E') + return Derivative(E(x, t), x, 2) + mu*epsilon*Derivative(E(x, t), t, 2) + + def _eval_rewrite_as_exp(self, *args, **kwargs): + return self.amplitude*exp(I*(self.wavenumber*Symbol('x') + - self.angular_velocity*Symbol('t') + self.phase)) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/boson.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/boson.py new file mode 100644 index 0000000000000000000000000000000000000000..3be2ebc45c392e8733de7e58528e9a0567273e73 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/boson.py @@ -0,0 +1,259 @@ +"""Bosonic quantum operators.""" + +from sympy.core.mul import Mul +from sympy.core.numbers import Integer +from sympy.core.singleton import S +from sympy.functions.elementary.complexes import conjugate +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.physics.quantum import Operator +from sympy.physics.quantum import HilbertSpace, FockSpace, Ket, Bra, IdentityOperator +from sympy.functions.special.tensor_functions import KroneckerDelta + + +__all__ = [ + 'BosonOp', + 'BosonFockKet', + 'BosonFockBra', + 'BosonCoherentKet', + 'BosonCoherentBra' +] + + +class BosonOp(Operator): + """A bosonic operator that satisfies [a, Dagger(a)] == 1. + + Parameters + ========== + + name : str + A string that labels the bosonic mode. + + annihilation : bool + A bool that indicates if the bosonic operator is an annihilation (True, + default value) or creation operator (False) + + Examples + ======== + + >>> from sympy.physics.quantum import Dagger, Commutator + >>> from sympy.physics.quantum.boson import BosonOp + >>> a = BosonOp("a") + >>> Commutator(a, Dagger(a)).doit() + 1 + """ + + @property + def name(self): + return self.args[0] + + @property + def is_annihilation(self): + return bool(self.args[1]) + + @classmethod + def default_args(self): + return ("a", True) + + def __new__(cls, *args, **hints): + if not len(args) in [1, 2]: + raise ValueError('1 or 2 parameters expected, got %s' % args) + + if len(args) == 1: + args = (args[0], S.One) + + if len(args) == 2: + args = (args[0], Integer(args[1])) + + return Operator.__new__(cls, *args) + + def _eval_commutator_BosonOp(self, other, **hints): + if self.name == other.name: + # [a^\dagger, a] = -1 + if not self.is_annihilation and other.is_annihilation: + return S.NegativeOne + + elif 'independent' in hints and hints['independent']: + # [a, b] = 0 + return S.Zero + + return None + + def _eval_commutator_FermionOp(self, other, **hints): + return S.Zero + + def _eval_anticommutator_BosonOp(self, other, **hints): + if 'independent' in hints and hints['independent']: + # {a, b} = 2 * a * b, because [a, b] = 0 + return 2 * self * other + + return None + + def _eval_adjoint(self): + return BosonOp(str(self.name), not self.is_annihilation) + + def __mul__(self, other): + + if other == IdentityOperator(2): + return self + + if isinstance(other, Mul): + args1 = tuple(arg for arg in other.args if arg.is_commutative) + args2 = tuple(arg for arg in other.args if not arg.is_commutative) + x = self + for y in args2: + x = x * y + return Mul(*args1) * x + + return Mul(self, other) + + def _print_contents_latex(self, printer, *args): + if self.is_annihilation: + return r'{%s}' % str(self.name) + else: + return r'{{%s}^\dagger}' % str(self.name) + + def _print_contents(self, printer, *args): + if self.is_annihilation: + return r'%s' % str(self.name) + else: + return r'Dagger(%s)' % str(self.name) + + def _print_contents_pretty(self, printer, *args): + from sympy.printing.pretty.stringpict import prettyForm + pform = printer._print(self.args[0], *args) + if self.is_annihilation: + return pform + else: + return pform**prettyForm('\N{DAGGER}') + + +class BosonFockKet(Ket): + """Fock state ket for a bosonic mode. + + Parameters + ========== + + n : Number + The Fock state number. + + """ + + def __new__(cls, n): + return Ket.__new__(cls, n) + + @property + def n(self): + return self.label[0] + + @classmethod + def dual_class(self): + return BosonFockBra + + @classmethod + def _eval_hilbert_space(cls, label): + return FockSpace() + + def _eval_innerproduct_BosonFockBra(self, bra, **hints): + return KroneckerDelta(self.n, bra.n) + + def _apply_from_right_to_BosonOp(self, op, **options): + if op.is_annihilation: + return sqrt(self.n) * BosonFockKet(self.n - 1) + else: + return sqrt(self.n + 1) * BosonFockKet(self.n + 1) + + +class BosonFockBra(Bra): + """Fock state bra for a bosonic mode. + + Parameters + ========== + + n : Number + The Fock state number. + + """ + + def __new__(cls, n): + return Bra.__new__(cls, n) + + @property + def n(self): + return self.label[0] + + @classmethod + def dual_class(self): + return BosonFockKet + + @classmethod + def _eval_hilbert_space(cls, label): + return FockSpace() + + +class BosonCoherentKet(Ket): + """Coherent state ket for a bosonic mode. + + Parameters + ========== + + alpha : Number, Symbol + The complex amplitude of the coherent state. + + """ + + def __new__(cls, alpha): + return Ket.__new__(cls, alpha) + + @property + def alpha(self): + return self.label[0] + + @classmethod + def dual_class(self): + return BosonCoherentBra + + @classmethod + def _eval_hilbert_space(cls, label): + return HilbertSpace() + + def _eval_innerproduct_BosonCoherentBra(self, bra, **hints): + if self.alpha == bra.alpha: + return S.One + else: + return exp(-(abs(self.alpha)**2 + abs(bra.alpha)**2 - 2 * conjugate(bra.alpha) * self.alpha)/2) + + def _apply_from_right_to_BosonOp(self, op, **options): + if op.is_annihilation: + return self.alpha * self + else: + return None + + +class BosonCoherentBra(Bra): + """Coherent state bra for a bosonic mode. + + Parameters + ========== + + alpha : Number, Symbol + The complex amplitude of the coherent state. + + """ + + def __new__(cls, alpha): + return Bra.__new__(cls, alpha) + + @property + def alpha(self): + return self.label[0] + + @classmethod + def dual_class(self): + return BosonCoherentKet + + def _apply_operator_BosonOp(self, op, **options): + if not op.is_annihilation: + return self.alpha * self + else: + return None diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/commutator.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/commutator.py new file mode 100644 index 0000000000000000000000000000000000000000..627158657481a4b66875e1d23107c1ca3bdb6969 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/commutator.py @@ -0,0 +1,239 @@ +"""The commutator: [A,B] = A*B - B*A.""" + +from sympy.core.add import Add +from sympy.core.expr import Expr +from sympy.core.mul import Mul +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.printing.pretty.stringpict import prettyForm + +from sympy.physics.quantum.dagger import Dagger +from sympy.physics.quantum.operator import Operator + + +__all__ = [ + 'Commutator' +] + +#----------------------------------------------------------------------------- +# Commutator +#----------------------------------------------------------------------------- + + +class Commutator(Expr): + """The standard commutator, in an unevaluated state. + + Explanation + =========== + + Evaluating a commutator is defined [1]_ as: ``[A, B] = A*B - B*A``. This + class returns the commutator in an unevaluated form. To evaluate the + commutator, use the ``.doit()`` method. + + Canonical ordering of a commutator is ``[A, B]`` for ``A < B``. The + arguments of the commutator are put into canonical order using ``__cmp__``. + If ``B < A``, then ``[B, A]`` is returned as ``-[A, B]``. + + Parameters + ========== + + A : Expr + The first argument of the commutator [A,B]. + B : Expr + The second argument of the commutator [A,B]. + + Examples + ======== + + >>> from sympy.physics.quantum import Commutator, Dagger, Operator + >>> from sympy.abc import x, y + >>> A = Operator('A') + >>> B = Operator('B') + >>> C = Operator('C') + + Create a commutator and use ``.doit()`` to evaluate it: + + >>> comm = Commutator(A, B) + >>> comm + [A,B] + >>> comm.doit() + A*B - B*A + + The commutator orders it arguments in canonical order: + + >>> comm = Commutator(B, A); comm + -[A,B] + + Commutative constants are factored out: + + >>> Commutator(3*x*A, x*y*B) + 3*x**2*y*[A,B] + + Using ``.expand(commutator=True)``, the standard commutator expansion rules + can be applied: + + >>> Commutator(A+B, C).expand(commutator=True) + [A,C] + [B,C] + >>> Commutator(A, B+C).expand(commutator=True) + [A,B] + [A,C] + >>> Commutator(A*B, C).expand(commutator=True) + [A,C]*B + A*[B,C] + >>> Commutator(A, B*C).expand(commutator=True) + [A,B]*C + B*[A,C] + + Adjoint operations applied to the commutator are properly applied to the + arguments: + + >>> Dagger(Commutator(A, B)) + -[Dagger(A),Dagger(B)] + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Commutator + """ + is_commutative = False + + def __new__(cls, A, B): + r = cls.eval(A, B) + if r is not None: + return r + obj = Expr.__new__(cls, A, B) + return obj + + @classmethod + def eval(cls, a, b): + if not (a and b): + return S.Zero + if a == b: + return S.Zero + if a.is_commutative or b.is_commutative: + return S.Zero + + # [xA,yB] -> xy*[A,B] + ca, nca = a.args_cnc() + cb, ncb = b.args_cnc() + c_part = ca + cb + if c_part: + return Mul(Mul(*c_part), cls(Mul._from_args(nca), Mul._from_args(ncb))) + + # Canonical ordering of arguments + # The Commutator [A, B] is in canonical form if A < B. + if a.compare(b) == 1: + return S.NegativeOne*cls(b, a) + + def _expand_pow(self, A, B, sign): + exp = A.exp + if not exp.is_integer or not exp.is_constant() or abs(exp) <= 1: + # nothing to do + return self + base = A.base + if exp.is_negative: + base = A.base**-1 + exp = -exp + comm = Commutator(base, B).expand(commutator=True) + + result = base**(exp - 1) * comm + for i in range(1, exp): + result += base**(exp - 1 - i) * comm * base**i + return sign*result.expand() + + def _eval_expand_commutator(self, **hints): + A = self.args[0] + B = self.args[1] + + if isinstance(A, Add): + # [A + B, C] -> [A, C] + [B, C] + sargs = [] + for term in A.args: + comm = Commutator(term, B) + if isinstance(comm, Commutator): + comm = comm._eval_expand_commutator() + sargs.append(comm) + return Add(*sargs) + elif isinstance(B, Add): + # [A, B + C] -> [A, B] + [A, C] + sargs = [] + for term in B.args: + comm = Commutator(A, term) + if isinstance(comm, Commutator): + comm = comm._eval_expand_commutator() + sargs.append(comm) + return Add(*sargs) + elif isinstance(A, Mul): + # [A*B, C] -> A*[B, C] + [A, C]*B + a = A.args[0] + b = Mul(*A.args[1:]) + c = B + comm1 = Commutator(b, c) + comm2 = Commutator(a, c) + if isinstance(comm1, Commutator): + comm1 = comm1._eval_expand_commutator() + if isinstance(comm2, Commutator): + comm2 = comm2._eval_expand_commutator() + first = Mul(a, comm1) + second = Mul(comm2, b) + return Add(first, second) + elif isinstance(B, Mul): + # [A, B*C] -> [A, B]*C + B*[A, C] + a = A + b = B.args[0] + c = Mul(*B.args[1:]) + comm1 = Commutator(a, b) + comm2 = Commutator(a, c) + if isinstance(comm1, Commutator): + comm1 = comm1._eval_expand_commutator() + if isinstance(comm2, Commutator): + comm2 = comm2._eval_expand_commutator() + first = Mul(comm1, c) + second = Mul(b, comm2) + return Add(first, second) + elif isinstance(A, Pow): + # [A**n, C] -> A**(n - 1)*[A, C] + A**(n - 2)*[A, C]*A + ... + [A, C]*A**(n-1) + return self._expand_pow(A, B, 1) + elif isinstance(B, Pow): + # [A, C**n] -> C**(n - 1)*[C, A] + C**(n - 2)*[C, A]*C + ... + [C, A]*C**(n-1) + return self._expand_pow(B, A, -1) + + # No changes, so return self + return self + + def doit(self, **hints): + """ Evaluate commutator """ + A = self.args[0] + B = self.args[1] + if isinstance(A, Operator) and isinstance(B, Operator): + try: + comm = A._eval_commutator(B, **hints) + except NotImplementedError: + try: + comm = -1*B._eval_commutator(A, **hints) + except NotImplementedError: + comm = None + if comm is not None: + return comm.doit(**hints) + return (A*B - B*A).doit(**hints) + + def _eval_adjoint(self): + return Commutator(Dagger(self.args[1]), Dagger(self.args[0])) + + def _sympyrepr(self, printer, *args): + return "%s(%s,%s)" % ( + self.__class__.__name__, printer._print( + self.args[0]), printer._print(self.args[1]) + ) + + def _sympystr(self, printer, *args): + return "[%s,%s]" % ( + printer._print(self.args[0]), printer._print(self.args[1])) + + def _pretty(self, printer, *args): + pform = printer._print(self.args[0], *args) + pform = prettyForm(*pform.right(prettyForm(','))) + pform = prettyForm(*pform.right(printer._print(self.args[1], *args))) + pform = prettyForm(*pform.parens(left='[', right=']')) + return pform + + def _latex(self, printer, *args): + return "\\left[%s,%s\\right]" % tuple([ + printer._print(arg, *args) for arg in self.args]) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/gate.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/gate.py new file mode 100644 index 0000000000000000000000000000000000000000..f8bcf5cd3611173cd9ebd6308dbbc896f5257f20 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/gate.py @@ -0,0 +1,1309 @@ +"""An implementation of gates that act on qubits. + +Gates are unitary operators that act on the space of qubits. + +Medium Term Todo: + +* Optimize Gate._apply_operators_Qubit to remove the creation of many + intermediate Qubit objects. +* Add commutation relationships to all operators and use this in gate_sort. +* Fix gate_sort and gate_simp. +* Get multi-target UGates plotting properly. +* Get UGate to work with either sympy/numpy matrices and output either + format. This should also use the matrix slots. +""" + +from itertools import chain +import random + +from sympy.core.add import Add +from sympy.core.containers import Tuple +from sympy.core.mul import Mul +from sympy.core.numbers import (I, Integer) +from sympy.core.power import Pow +from sympy.core.numbers import Number +from sympy.core.singleton import S as _S +from sympy.core.sorting import default_sort_key +from sympy.core.sympify import _sympify +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.printing.pretty.stringpict import prettyForm, stringPict + +from sympy.physics.quantum.anticommutator import AntiCommutator +from sympy.physics.quantum.commutator import Commutator +from sympy.physics.quantum.qexpr import QuantumError +from sympy.physics.quantum.hilbert import ComplexSpace +from sympy.physics.quantum.operator import (UnitaryOperator, Operator, + HermitianOperator) +from sympy.physics.quantum.matrixutils import matrix_tensor_product, matrix_eye +from sympy.physics.quantum.matrixcache import matrix_cache + +from sympy.matrices.matrixbase import MatrixBase + +from sympy.utilities.iterables import is_sequence + +__all__ = [ + 'Gate', + 'CGate', + 'UGate', + 'OneQubitGate', + 'TwoQubitGate', + 'IdentityGate', + 'HadamardGate', + 'XGate', + 'YGate', + 'ZGate', + 'TGate', + 'PhaseGate', + 'SwapGate', + 'CNotGate', + # Aliased gate names + 'CNOT', + 'SWAP', + 'H', + 'X', + 'Y', + 'Z', + 'T', + 'S', + 'Phase', + 'normalized', + 'gate_sort', + 'gate_simp', + 'random_circuit', + 'CPHASE', + 'CGateS', +] + +#----------------------------------------------------------------------------- +# Gate Super-Classes +#----------------------------------------------------------------------------- + +_normalized = True + + +def _max(*args, **kwargs): + if "key" not in kwargs: + kwargs["key"] = default_sort_key + return max(*args, **kwargs) + + +def _min(*args, **kwargs): + if "key" not in kwargs: + kwargs["key"] = default_sort_key + return min(*args, **kwargs) + + +def normalized(normalize): + r"""Set flag controlling normalization of Hadamard gates by `1/\sqrt{2}`. + + This is a global setting that can be used to simplify the look of various + expressions, by leaving off the leading `1/\sqrt{2}` of the Hadamard gate. + + Parameters + ---------- + normalize : bool + Should the Hadamard gate include the `1/\sqrt{2}` normalization factor? + When True, the Hadamard gate will have the `1/\sqrt{2}`. When False, the + Hadamard gate will not have this factor. + """ + global _normalized + _normalized = normalize + + +def _validate_targets_controls(tandc): + tandc = list(tandc) + # Check for integers + for bit in tandc: + if not bit.is_Integer and not bit.is_Symbol: + raise TypeError('Integer expected, got: %r' % tandc[bit]) + # Detect duplicates + if len(set(tandc)) != len(tandc): + raise QuantumError( + 'Target/control qubits in a gate cannot be duplicated' + ) + + +class Gate(UnitaryOperator): + """Non-controlled unitary gate operator that acts on qubits. + + This is a general abstract gate that needs to be subclassed to do anything + useful. + + Parameters + ---------- + label : tuple, int + A list of the target qubits (as ints) that the gate will apply to. + + Examples + ======== + + + """ + + _label_separator = ',' + + gate_name = 'G' + gate_name_latex = 'G' + + #------------------------------------------------------------------------- + # Initialization/creation + #------------------------------------------------------------------------- + + @classmethod + def _eval_args(cls, args): + args = Tuple(*UnitaryOperator._eval_args(args)) + _validate_targets_controls(args) + return args + + @classmethod + def _eval_hilbert_space(cls, args): + """This returns the smallest possible Hilbert space.""" + return ComplexSpace(2)**(_max(args) + 1) + + #------------------------------------------------------------------------- + # Properties + #------------------------------------------------------------------------- + + @property + def nqubits(self): + """The total number of qubits this gate acts on. + + For controlled gate subclasses this includes both target and control + qubits, so that, for examples the CNOT gate acts on 2 qubits. + """ + return len(self.targets) + + @property + def min_qubits(self): + """The minimum number of qubits this gate needs to act on.""" + return _max(self.targets) + 1 + + @property + def targets(self): + """A tuple of target qubits.""" + return self.label + + @property + def gate_name_plot(self): + return r'$%s$' % self.gate_name_latex + + #------------------------------------------------------------------------- + # Gate methods + #------------------------------------------------------------------------- + + def get_target_matrix(self, format='sympy'): + """The matrix representation of the target part of the gate. + + Parameters + ---------- + format : str + The format string ('sympy','numpy', etc.) + """ + raise NotImplementedError( + 'get_target_matrix is not implemented in Gate.') + + #------------------------------------------------------------------------- + # Apply + #------------------------------------------------------------------------- + + def _apply_operator_IntQubit(self, qubits, **options): + """Redirect an apply from IntQubit to Qubit""" + return self._apply_operator_Qubit(qubits, **options) + + def _apply_operator_Qubit(self, qubits, **options): + """Apply this gate to a Qubit.""" + + # Check number of qubits this gate acts on. + if qubits.nqubits < self.min_qubits: + raise QuantumError( + 'Gate needs a minimum of %r qubits to act on, got: %r' % + (self.min_qubits, qubits.nqubits) + ) + + # If the controls are not met, just return + if isinstance(self, CGate): + if not self.eval_controls(qubits): + return qubits + + targets = self.targets + target_matrix = self.get_target_matrix(format='sympy') + + # Find which column of the target matrix this applies to. + column_index = 0 + n = 1 + for target in targets: + column_index += n*qubits[target] + n = n << 1 + column = target_matrix[:, int(column_index)] + + # Now apply each column element to the qubit. + result = 0 + for index in range(column.rows): + # TODO: This can be optimized to reduce the number of Qubit + # creations. We should simply manipulate the raw list of qubit + # values and then build the new Qubit object once. + # Make a copy of the incoming qubits. + new_qubit = qubits.__class__(*qubits.args) + # Flip the bits that need to be flipped. + for bit, target in enumerate(targets): + if new_qubit[target] != (index >> bit) & 1: + new_qubit = new_qubit.flip(target) + # The value in that row and column times the flipped-bit qubit + # is the result for that part. + result += column[index]*new_qubit + return result + + #------------------------------------------------------------------------- + # Represent + #------------------------------------------------------------------------- + + def _represent_default_basis(self, **options): + return self._represent_ZGate(None, **options) + + def _represent_ZGate(self, basis, **options): + format = options.get('format', 'sympy') + nqubits = options.get('nqubits', 0) + if nqubits == 0: + raise QuantumError( + 'The number of qubits must be given as nqubits.') + + # Make sure we have enough qubits for the gate. + if nqubits < self.min_qubits: + raise QuantumError( + 'The number of qubits %r is too small for the gate.' % nqubits + ) + + target_matrix = self.get_target_matrix(format) + targets = self.targets + if isinstance(self, CGate): + controls = self.controls + else: + controls = [] + m = represent_zbasis( + controls, targets, target_matrix, nqubits, format + ) + return m + + #------------------------------------------------------------------------- + # Print methods + #------------------------------------------------------------------------- + + def _sympystr(self, printer, *args): + label = self._print_label(printer, *args) + return '%s(%s)' % (self.gate_name, label) + + def _pretty(self, printer, *args): + a = stringPict(self.gate_name) + b = self._print_label_pretty(printer, *args) + return self._print_subscript_pretty(a, b) + + def _latex(self, printer, *args): + label = self._print_label(printer, *args) + return '%s_{%s}' % (self.gate_name_latex, label) + + def plot_gate(self, axes, gate_idx, gate_grid, wire_grid): + raise NotImplementedError('plot_gate is not implemented.') + + +class CGate(Gate): + """A general unitary gate with control qubits. + + A general control gate applies a target gate to a set of targets if all + of the control qubits have a particular values (set by + ``CGate.control_value``). + + Parameters + ---------- + label : tuple + The label in this case has the form (controls, gate), where controls + is a tuple/list of control qubits (as ints) and gate is a ``Gate`` + instance that is the target operator. + + Examples + ======== + + """ + + gate_name = 'C' + gate_name_latex = 'C' + + # The values this class controls for. + control_value = _S.One + + simplify_cgate = False + + #------------------------------------------------------------------------- + # Initialization + #------------------------------------------------------------------------- + + @classmethod + def _eval_args(cls, args): + # _eval_args has the right logic for the controls argument. + controls = args[0] + gate = args[1] + if not is_sequence(controls): + controls = (controls,) + controls = UnitaryOperator._eval_args(controls) + _validate_targets_controls(chain(controls, gate.targets)) + return (Tuple(*controls), gate) + + @classmethod + def _eval_hilbert_space(cls, args): + """This returns the smallest possible Hilbert space.""" + return ComplexSpace(2)**_max(_max(args[0]) + 1, args[1].min_qubits) + + #------------------------------------------------------------------------- + # Properties + #------------------------------------------------------------------------- + + @property + def nqubits(self): + """The total number of qubits this gate acts on. + + For controlled gate subclasses this includes both target and control + qubits, so that, for examples the CNOT gate acts on 2 qubits. + """ + return len(self.targets) + len(self.controls) + + @property + def min_qubits(self): + """The minimum number of qubits this gate needs to act on.""" + return _max(_max(self.controls), _max(self.targets)) + 1 + + @property + def targets(self): + """A tuple of target qubits.""" + return self.gate.targets + + @property + def controls(self): + """A tuple of control qubits.""" + return tuple(self.label[0]) + + @property + def gate(self): + """The non-controlled gate that will be applied to the targets.""" + return self.label[1] + + #------------------------------------------------------------------------- + # Gate methods + #------------------------------------------------------------------------- + + def get_target_matrix(self, format='sympy'): + return self.gate.get_target_matrix(format) + + def eval_controls(self, qubit): + """Return True/False to indicate if the controls are satisfied.""" + return all(qubit[bit] == self.control_value for bit in self.controls) + + def decompose(self, **options): + """Decompose the controlled gate into CNOT and single qubits gates.""" + if len(self.controls) == 1: + c = self.controls[0] + t = self.gate.targets[0] + if isinstance(self.gate, YGate): + g1 = PhaseGate(t) + g2 = CNotGate(c, t) + g3 = PhaseGate(t) + g4 = ZGate(t) + return g1*g2*g3*g4 + if isinstance(self.gate, ZGate): + g1 = HadamardGate(t) + g2 = CNotGate(c, t) + g3 = HadamardGate(t) + return g1*g2*g3 + else: + return self + + #------------------------------------------------------------------------- + # Print methods + #------------------------------------------------------------------------- + + def _print_label(self, printer, *args): + controls = self._print_sequence(self.controls, ',', printer, *args) + gate = printer._print(self.gate, *args) + return '(%s),%s' % (controls, gate) + + def _pretty(self, printer, *args): + controls = self._print_sequence_pretty( + self.controls, ',', printer, *args) + gate = printer._print(self.gate) + gate_name = stringPict(self.gate_name) + first = self._print_subscript_pretty(gate_name, controls) + gate = self._print_parens_pretty(gate) + final = prettyForm(*first.right(gate)) + return final + + def _latex(self, printer, *args): + controls = self._print_sequence(self.controls, ',', printer, *args) + gate = printer._print(self.gate, *args) + return r'%s_{%s}{\left(%s\right)}' % \ + (self.gate_name_latex, controls, gate) + + def plot_gate(self, circ_plot, gate_idx): + """ + Plot the controlled gate. If *simplify_cgate* is true, simplify + C-X and C-Z gates into their more familiar forms. + """ + min_wire = int(_min(chain(self.controls, self.targets))) + max_wire = int(_max(chain(self.controls, self.targets))) + circ_plot.control_line(gate_idx, min_wire, max_wire) + for c in self.controls: + circ_plot.control_point(gate_idx, int(c)) + if self.simplify_cgate: + if self.gate.gate_name == 'X': + self.gate.plot_gate_plus(circ_plot, gate_idx) + elif self.gate.gate_name == 'Z': + circ_plot.control_point(gate_idx, self.targets[0]) + else: + self.gate.plot_gate(circ_plot, gate_idx) + else: + self.gate.plot_gate(circ_plot, gate_idx) + + #------------------------------------------------------------------------- + # Miscellaneous + #------------------------------------------------------------------------- + + def _eval_dagger(self): + if isinstance(self.gate, HermitianOperator): + return self + else: + return Gate._eval_dagger(self) + + def _eval_inverse(self): + if isinstance(self.gate, HermitianOperator): + return self + else: + return Gate._eval_inverse(self) + + def _eval_power(self, exp): + if isinstance(self.gate, HermitianOperator): + if exp == -1: + return Gate._eval_power(self, exp) + elif abs(exp) % 2 == 0: + return self*(Gate._eval_inverse(self)) + else: + return self + else: + return Gate._eval_power(self, exp) + +class CGateS(CGate): + """Version of CGate that allows gate simplifications. + I.e. cnot looks like an oplus, cphase has dots, etc. + """ + simplify_cgate=True + + +class UGate(Gate): + """General gate specified by a set of targets and a target matrix. + + Parameters + ---------- + label : tuple + A tuple of the form (targets, U), where targets is a tuple of the + target qubits and U is a unitary matrix with dimension of + len(targets). + """ + gate_name = 'U' + gate_name_latex = 'U' + + #------------------------------------------------------------------------- + # Initialization + #------------------------------------------------------------------------- + + @classmethod + def _eval_args(cls, args): + targets = args[0] + if not is_sequence(targets): + targets = (targets,) + targets = Gate._eval_args(targets) + _validate_targets_controls(targets) + mat = args[1] + if not isinstance(mat, MatrixBase): + raise TypeError('Matrix expected, got: %r' % mat) + #make sure this matrix is of a Basic type + mat = _sympify(mat) + dim = 2**len(targets) + if not all(dim == shape for shape in mat.shape): + raise IndexError( + 'Number of targets must match the matrix size: %r %r' % + (targets, mat) + ) + return (targets, mat) + + @classmethod + def _eval_hilbert_space(cls, args): + """This returns the smallest possible Hilbert space.""" + return ComplexSpace(2)**(_max(args[0]) + 1) + + #------------------------------------------------------------------------- + # Properties + #------------------------------------------------------------------------- + + @property + def targets(self): + """A tuple of target qubits.""" + return tuple(self.label[0]) + + #------------------------------------------------------------------------- + # Gate methods + #------------------------------------------------------------------------- + + def get_target_matrix(self, format='sympy'): + """The matrix rep. of the target part of the gate. + + Parameters + ---------- + format : str + The format string ('sympy','numpy', etc.) + """ + return self.label[1] + + #------------------------------------------------------------------------- + # Print methods + #------------------------------------------------------------------------- + def _pretty(self, printer, *args): + targets = self._print_sequence_pretty( + self.targets, ',', printer, *args) + gate_name = stringPict(self.gate_name) + return self._print_subscript_pretty(gate_name, targets) + + def _latex(self, printer, *args): + targets = self._print_sequence(self.targets, ',', printer, *args) + return r'%s_{%s}' % (self.gate_name_latex, targets) + + def plot_gate(self, circ_plot, gate_idx): + circ_plot.one_qubit_box( + self.gate_name_plot, + gate_idx, int(self.targets[0]) + ) + + +class OneQubitGate(Gate): + """A single qubit unitary gate base class.""" + + nqubits = _S.One + + def plot_gate(self, circ_plot, gate_idx): + circ_plot.one_qubit_box( + self.gate_name_plot, + gate_idx, int(self.targets[0]) + ) + + def _eval_commutator(self, other, **hints): + if isinstance(other, OneQubitGate): + if self.targets != other.targets or self.__class__ == other.__class__: + return _S.Zero + return Operator._eval_commutator(self, other, **hints) + + def _eval_anticommutator(self, other, **hints): + if isinstance(other, OneQubitGate): + if self.targets != other.targets or self.__class__ == other.__class__: + return Integer(2)*self*other + return Operator._eval_anticommutator(self, other, **hints) + + +class TwoQubitGate(Gate): + """A two qubit unitary gate base class.""" + + nqubits = Integer(2) + +#----------------------------------------------------------------------------- +# Single Qubit Gates +#----------------------------------------------------------------------------- + + +class IdentityGate(OneQubitGate): + """The single qubit identity gate. + + Parameters + ---------- + target : int + The target qubit this gate will apply to. + + Examples + ======== + + """ + is_hermitian = True + gate_name = '1' + gate_name_latex = '1' + + # Short cut version of gate._apply_operator_Qubit + def _apply_operator_Qubit(self, qubits, **options): + # Check number of qubits this gate acts on (see gate._apply_operator_Qubit) + if qubits.nqubits < self.min_qubits: + raise QuantumError( + 'Gate needs a minimum of %r qubits to act on, got: %r' % + (self.min_qubits, qubits.nqubits) + ) + return qubits # no computation required for IdentityGate + + def get_target_matrix(self, format='sympy'): + return matrix_cache.get_matrix('eye2', format) + + def _eval_commutator(self, other, **hints): + return _S.Zero + + def _eval_anticommutator(self, other, **hints): + return Integer(2)*other + + +class HadamardGate(HermitianOperator, OneQubitGate): + """The single qubit Hadamard gate. + + Parameters + ---------- + target : int + The target qubit this gate will apply to. + + Examples + ======== + + >>> from sympy import sqrt + >>> from sympy.physics.quantum.qubit import Qubit + >>> from sympy.physics.quantum.gate import HadamardGate + >>> from sympy.physics.quantum.qapply import qapply + >>> qapply(HadamardGate(0)*Qubit('1')) + sqrt(2)*|0>/2 - sqrt(2)*|1>/2 + >>> # Hadamard on bell state, applied on 2 qubits. + >>> psi = 1/sqrt(2)*(Qubit('00')+Qubit('11')) + >>> qapply(HadamardGate(0)*HadamardGate(1)*psi) + sqrt(2)*|00>/2 + sqrt(2)*|11>/2 + + """ + gate_name = 'H' + gate_name_latex = 'H' + + def get_target_matrix(self, format='sympy'): + if _normalized: + return matrix_cache.get_matrix('H', format) + else: + return matrix_cache.get_matrix('Hsqrt2', format) + + def _eval_commutator_XGate(self, other, **hints): + return I*sqrt(2)*YGate(self.targets[0]) + + def _eval_commutator_YGate(self, other, **hints): + return I*sqrt(2)*(ZGate(self.targets[0]) - XGate(self.targets[0])) + + def _eval_commutator_ZGate(self, other, **hints): + return -I*sqrt(2)*YGate(self.targets[0]) + + def _eval_anticommutator_XGate(self, other, **hints): + return sqrt(2)*IdentityGate(self.targets[0]) + + def _eval_anticommutator_YGate(self, other, **hints): + return _S.Zero + + def _eval_anticommutator_ZGate(self, other, **hints): + return sqrt(2)*IdentityGate(self.targets[0]) + + +class XGate(HermitianOperator, OneQubitGate): + """The single qubit X, or NOT, gate. + + Parameters + ---------- + target : int + The target qubit this gate will apply to. + + Examples + ======== + + """ + gate_name = 'X' + gate_name_latex = 'X' + + def get_target_matrix(self, format='sympy'): + return matrix_cache.get_matrix('X', format) + + def plot_gate(self, circ_plot, gate_idx): + OneQubitGate.plot_gate(self,circ_plot,gate_idx) + + def plot_gate_plus(self, circ_plot, gate_idx): + circ_plot.not_point( + gate_idx, int(self.label[0]) + ) + + def _eval_commutator_YGate(self, other, **hints): + return Integer(2)*I*ZGate(self.targets[0]) + + def _eval_anticommutator_XGate(self, other, **hints): + return Integer(2)*IdentityGate(self.targets[0]) + + def _eval_anticommutator_YGate(self, other, **hints): + return _S.Zero + + def _eval_anticommutator_ZGate(self, other, **hints): + return _S.Zero + + +class YGate(HermitianOperator, OneQubitGate): + """The single qubit Y gate. + + Parameters + ---------- + target : int + The target qubit this gate will apply to. + + Examples + ======== + + """ + gate_name = 'Y' + gate_name_latex = 'Y' + + def get_target_matrix(self, format='sympy'): + return matrix_cache.get_matrix('Y', format) + + def _eval_commutator_ZGate(self, other, **hints): + return Integer(2)*I*XGate(self.targets[0]) + + def _eval_anticommutator_YGate(self, other, **hints): + return Integer(2)*IdentityGate(self.targets[0]) + + def _eval_anticommutator_ZGate(self, other, **hints): + return _S.Zero + + +class ZGate(HermitianOperator, OneQubitGate): + """The single qubit Z gate. + + Parameters + ---------- + target : int + The target qubit this gate will apply to. + + Examples + ======== + + """ + gate_name = 'Z' + gate_name_latex = 'Z' + + def get_target_matrix(self, format='sympy'): + return matrix_cache.get_matrix('Z', format) + + def _eval_commutator_XGate(self, other, **hints): + return Integer(2)*I*YGate(self.targets[0]) + + def _eval_anticommutator_YGate(self, other, **hints): + return _S.Zero + + +class PhaseGate(OneQubitGate): + """The single qubit phase, or S, gate. + + This gate rotates the phase of the state by pi/2 if the state is ``|1>`` and + does nothing if the state is ``|0>``. + + Parameters + ---------- + target : int + The target qubit this gate will apply to. + + Examples + ======== + + """ + is_hermitian = False + gate_name = 'S' + gate_name_latex = 'S' + + def get_target_matrix(self, format='sympy'): + return matrix_cache.get_matrix('S', format) + + def _eval_commutator_ZGate(self, other, **hints): + return _S.Zero + + def _eval_commutator_TGate(self, other, **hints): + return _S.Zero + + +class TGate(OneQubitGate): + """The single qubit pi/8 gate. + + This gate rotates the phase of the state by pi/4 if the state is ``|1>`` and + does nothing if the state is ``|0>``. + + Parameters + ---------- + target : int + The target qubit this gate will apply to. + + Examples + ======== + + """ + is_hermitian = False + gate_name = 'T' + gate_name_latex = 'T' + + def get_target_matrix(self, format='sympy'): + return matrix_cache.get_matrix('T', format) + + def _eval_commutator_ZGate(self, other, **hints): + return _S.Zero + + def _eval_commutator_PhaseGate(self, other, **hints): + return _S.Zero + + +# Aliases for gate names. +H = HadamardGate +X = XGate +Y = YGate +Z = ZGate +T = TGate +Phase = S = PhaseGate + + +#----------------------------------------------------------------------------- +# 2 Qubit Gates +#----------------------------------------------------------------------------- + + +class CNotGate(HermitianOperator, CGate, TwoQubitGate): + """Two qubit controlled-NOT. + + This gate performs the NOT or X gate on the target qubit if the control + qubits all have the value 1. + + Parameters + ---------- + label : tuple + A tuple of the form (control, target). + + Examples + ======== + + >>> from sympy.physics.quantum.gate import CNOT + >>> from sympy.physics.quantum.qapply import qapply + >>> from sympy.physics.quantum.qubit import Qubit + >>> c = CNOT(1,0) + >>> qapply(c*Qubit('10')) # note that qubits are indexed from right to left + |11> + + """ + gate_name = 'CNOT' + gate_name_latex = r'\text{CNOT}' + simplify_cgate = True + + #------------------------------------------------------------------------- + # Initialization + #------------------------------------------------------------------------- + + @classmethod + def _eval_args(cls, args): + args = Gate._eval_args(args) + return args + + @classmethod + def _eval_hilbert_space(cls, args): + """This returns the smallest possible Hilbert space.""" + return ComplexSpace(2)**(_max(args) + 1) + + #------------------------------------------------------------------------- + # Properties + #------------------------------------------------------------------------- + + @property + def min_qubits(self): + """The minimum number of qubits this gate needs to act on.""" + return _max(self.label) + 1 + + @property + def targets(self): + """A tuple of target qubits.""" + return (self.label[1],) + + @property + def controls(self): + """A tuple of control qubits.""" + return (self.label[0],) + + @property + def gate(self): + """The non-controlled gate that will be applied to the targets.""" + return XGate(self.label[1]) + + #------------------------------------------------------------------------- + # Properties + #------------------------------------------------------------------------- + + # The default printing of Gate works better than those of CGate, so we + # go around the overridden methods in CGate. + + def _print_label(self, printer, *args): + return Gate._print_label(self, printer, *args) + + def _pretty(self, printer, *args): + return Gate._pretty(self, printer, *args) + + def _latex(self, printer, *args): + return Gate._latex(self, printer, *args) + + #------------------------------------------------------------------------- + # Commutator/AntiCommutator + #------------------------------------------------------------------------- + + def _eval_commutator_ZGate(self, other, **hints): + """[CNOT(i, j), Z(i)] == 0.""" + if self.controls[0] == other.targets[0]: + return _S.Zero + else: + raise NotImplementedError('Commutator not implemented: %r' % other) + + def _eval_commutator_TGate(self, other, **hints): + """[CNOT(i, j), T(i)] == 0.""" + return self._eval_commutator_ZGate(other, **hints) + + def _eval_commutator_PhaseGate(self, other, **hints): + """[CNOT(i, j), S(i)] == 0.""" + return self._eval_commutator_ZGate(other, **hints) + + def _eval_commutator_XGate(self, other, **hints): + """[CNOT(i, j), X(j)] == 0.""" + if self.targets[0] == other.targets[0]: + return _S.Zero + else: + raise NotImplementedError('Commutator not implemented: %r' % other) + + def _eval_commutator_CNotGate(self, other, **hints): + """[CNOT(i, j), CNOT(i,k)] == 0.""" + if self.controls[0] == other.controls[0]: + return _S.Zero + else: + raise NotImplementedError('Commutator not implemented: %r' % other) + + +class SwapGate(TwoQubitGate): + """Two qubit SWAP gate. + + This gate swap the values of the two qubits. + + Parameters + ---------- + label : tuple + A tuple of the form (target1, target2). + + Examples + ======== + + """ + is_hermitian = True + gate_name = 'SWAP' + gate_name_latex = r'\text{SWAP}' + + def get_target_matrix(self, format='sympy'): + return matrix_cache.get_matrix('SWAP', format) + + def decompose(self, **options): + """Decompose the SWAP gate into CNOT gates.""" + i, j = self.targets[0], self.targets[1] + g1 = CNotGate(i, j) + g2 = CNotGate(j, i) + return g1*g2*g1 + + def plot_gate(self, circ_plot, gate_idx): + min_wire = int(_min(self.targets)) + max_wire = int(_max(self.targets)) + circ_plot.control_line(gate_idx, min_wire, max_wire) + circ_plot.swap_point(gate_idx, min_wire) + circ_plot.swap_point(gate_idx, max_wire) + + def _represent_ZGate(self, basis, **options): + """Represent the SWAP gate in the computational basis. + + The following representation is used to compute this: + + SWAP = |1><1|x|1><1| + |0><0|x|0><0| + |1><0|x|0><1| + |0><1|x|1><0| + """ + format = options.get('format', 'sympy') + targets = [int(t) for t in self.targets] + min_target = _min(targets) + max_target = _max(targets) + nqubits = options.get('nqubits', self.min_qubits) + + op01 = matrix_cache.get_matrix('op01', format) + op10 = matrix_cache.get_matrix('op10', format) + op11 = matrix_cache.get_matrix('op11', format) + op00 = matrix_cache.get_matrix('op00', format) + eye2 = matrix_cache.get_matrix('eye2', format) + + result = None + for i, j in ((op01, op10), (op10, op01), (op00, op00), (op11, op11)): + product = nqubits*[eye2] + product[nqubits - min_target - 1] = i + product[nqubits - max_target - 1] = j + new_result = matrix_tensor_product(*product) + if result is None: + result = new_result + else: + result = result + new_result + + return result + + +# Aliases for gate names. +CNOT = CNotGate +SWAP = SwapGate +def CPHASE(a,b): return CGateS((a,),Z(b)) + + +#----------------------------------------------------------------------------- +# Represent +#----------------------------------------------------------------------------- + + +def represent_zbasis(controls, targets, target_matrix, nqubits, format='sympy'): + """Represent a gate with controls, targets and target_matrix. + + This function does the low-level work of representing gates as matrices + in the standard computational basis (ZGate). Currently, we support two + main cases: + + 1. One target qubit and no control qubits. + 2. One target qubits and multiple control qubits. + + For the base of multiple controls, we use the following expression [1]: + + 1_{2**n} + (|1><1|)^{(n-1)} x (target-matrix - 1_{2}) + + Parameters + ---------- + controls : list, tuple + A sequence of control qubits. + targets : list, tuple + A sequence of target qubits. + target_matrix : sympy.Matrix, numpy.matrix, scipy.sparse + The matrix form of the transformation to be performed on the target + qubits. The format of this matrix must match that passed into + the `format` argument. + nqubits : int + The total number of qubits used for the representation. + format : str + The format of the final matrix ('sympy', 'numpy', 'scipy.sparse'). + + Examples + ======== + + References + ---------- + [1] http://www.johnlapeyre.com/qinf/qinf_html/node6.html. + """ + controls = [int(x) for x in controls] + targets = [int(x) for x in targets] + nqubits = int(nqubits) + + # This checks for the format as well. + op11 = matrix_cache.get_matrix('op11', format) + eye2 = matrix_cache.get_matrix('eye2', format) + + # Plain single qubit case + if len(controls) == 0 and len(targets) == 1: + product = [] + bit = targets[0] + # Fill product with [I1,Gate,I2] such that the unitaries, + # I, cause the gate to be applied to the correct Qubit + if bit != nqubits - 1: + product.append(matrix_eye(2**(nqubits - bit - 1), format=format)) + product.append(target_matrix) + if bit != 0: + product.append(matrix_eye(2**bit, format=format)) + return matrix_tensor_product(*product) + + # Single target, multiple controls. + elif len(targets) == 1 and len(controls) >= 1: + target = targets[0] + + # Build the non-trivial part. + product2 = [] + for i in range(nqubits): + product2.append(matrix_eye(2, format=format)) + for control in controls: + product2[nqubits - 1 - control] = op11 + product2[nqubits - 1 - target] = target_matrix - eye2 + + return matrix_eye(2**nqubits, format=format) + \ + matrix_tensor_product(*product2) + + # Multi-target, multi-control is not yet implemented. + else: + raise NotImplementedError( + 'The representation of multi-target, multi-control gates ' + 'is not implemented.' + ) + + +#----------------------------------------------------------------------------- +# Gate manipulation functions. +#----------------------------------------------------------------------------- + + +def gate_simp(circuit): + """Simplifies gates symbolically + + It first sorts gates using gate_sort. It then applies basic + simplification rules to the circuit, e.g., XGate**2 = Identity + """ + + # Bubble sort out gates that commute. + circuit = gate_sort(circuit) + + # Do simplifications by subing a simplification into the first element + # which can be simplified. We recursively call gate_simp with new circuit + # as input more simplifications exist. + if isinstance(circuit, Add): + return sum(gate_simp(t) for t in circuit.args) + elif isinstance(circuit, Mul): + circuit_args = circuit.args + elif isinstance(circuit, Pow): + b, e = circuit.as_base_exp() + circuit_args = (gate_simp(b)**e,) + else: + return circuit + + # Iterate through each element in circuit, simplify if possible. + for i in range(len(circuit_args)): + # H,X,Y or Z squared is 1. + # T**2 = S, S**2 = Z + if isinstance(circuit_args[i], Pow): + if isinstance(circuit_args[i].base, + (HadamardGate, XGate, YGate, ZGate)) \ + and isinstance(circuit_args[i].exp, Number): + # Build a new circuit taking replacing the + # H,X,Y,Z squared with one. + newargs = (circuit_args[:i] + + (circuit_args[i].base**(circuit_args[i].exp % 2),) + + circuit_args[i + 1:]) + # Recursively simplify the new circuit. + circuit = gate_simp(Mul(*newargs)) + break + elif isinstance(circuit_args[i].base, PhaseGate): + # Build a new circuit taking old circuit but splicing + # in simplification. + newargs = circuit_args[:i] + # Replace PhaseGate**2 with ZGate. + newargs = newargs + (ZGate(circuit_args[i].base.args[0])** + (Integer(circuit_args[i].exp/2)), circuit_args[i].base** + (circuit_args[i].exp % 2)) + # Append the last elements. + newargs = newargs + circuit_args[i + 1:] + # Recursively simplify the new circuit. + circuit = gate_simp(Mul(*newargs)) + break + elif isinstance(circuit_args[i].base, TGate): + # Build a new circuit taking all the old elements. + newargs = circuit_args[:i] + + # Put an Phasegate in place of any TGate**2. + newargs = newargs + (PhaseGate(circuit_args[i].base.args[0])** + Integer(circuit_args[i].exp/2), circuit_args[i].base** + (circuit_args[i].exp % 2)) + + # Append the last elements. + newargs = newargs + circuit_args[i + 1:] + # Recursively simplify the new circuit. + circuit = gate_simp(Mul(*newargs)) + break + return circuit + + +def gate_sort(circuit): + """Sorts the gates while keeping track of commutation relations + + This function uses a bubble sort to rearrange the order of gate + application. Keeps track of Quantum computations special commutation + relations (e.g. things that apply to the same Qubit do not commute with + each other) + + circuit is the Mul of gates that are to be sorted. + """ + # Make sure we have an Add or Mul. + if isinstance(circuit, Add): + return sum(gate_sort(t) for t in circuit.args) + if isinstance(circuit, Pow): + return gate_sort(circuit.base)**circuit.exp + elif isinstance(circuit, Gate): + return circuit + if not isinstance(circuit, Mul): + return circuit + + changes = True + while changes: + changes = False + circ_array = circuit.args + for i in range(len(circ_array) - 1): + # Go through each element and switch ones that are in wrong order + if isinstance(circ_array[i], (Gate, Pow)) and \ + isinstance(circ_array[i + 1], (Gate, Pow)): + # If we have a Pow object, look at only the base + first_base, first_exp = circ_array[i].as_base_exp() + second_base, second_exp = circ_array[i + 1].as_base_exp() + + # Use SymPy's hash based sorting. This is not mathematical + # sorting, but is rather based on comparing hashes of objects. + # See Basic.compare for details. + if first_base.compare(second_base) > 0: + if Commutator(first_base, second_base).doit() == 0: + new_args = (circuit.args[:i] + (circuit.args[i + 1],) + + (circuit.args[i],) + circuit.args[i + 2:]) + circuit = Mul(*new_args) + changes = True + break + if AntiCommutator(first_base, second_base).doit() == 0: + new_args = (circuit.args[:i] + (circuit.args[i + 1],) + + (circuit.args[i],) + circuit.args[i + 2:]) + sign = _S.NegativeOne**(first_exp*second_exp) + circuit = sign*Mul(*new_args) + changes = True + break + return circuit + + +#----------------------------------------------------------------------------- +# Utility functions +#----------------------------------------------------------------------------- + + +def random_circuit(ngates, nqubits, gate_space=(X, Y, Z, S, T, H, CNOT, SWAP)): + """Return a random circuit of ngates and nqubits. + + This uses an equally weighted sample of (X, Y, Z, S, T, H, CNOT, SWAP) + gates. + + Parameters + ---------- + ngates : int + The number of gates in the circuit. + nqubits : int + The number of qubits in the circuit. + gate_space : tuple + A tuple of the gate classes that will be used in the circuit. + Repeating gate classes multiple times in this tuple will increase + the frequency they appear in the random circuit. + """ + qubit_space = range(nqubits) + result = [] + for i in range(ngates): + g = random.choice(gate_space) + if g == CNotGate or g == SwapGate: + qubits = random.sample(qubit_space, 2) + g = g(*qubits) + else: + qubit = random.choice(qubit_space) + g = g(qubit) + result.append(g) + return Mul(*result) + + +def zx_basis_transform(self, format='sympy'): + """Transformation matrix from Z to X basis.""" + return matrix_cache.get_matrix('ZX', format) + + +def zy_basis_transform(self, format='sympy'): + """Transformation matrix from Z to Y basis.""" + return matrix_cache.get_matrix('ZY', format) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/grover.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/grover.py new file mode 100644 index 0000000000000000000000000000000000000000..a03bd3a61a6e0960ab66d55bcc0fc7f25936199e --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/grover.py @@ -0,0 +1,345 @@ +"""Grover's algorithm and helper functions. + +Todo: + +* W gate construction (or perhaps -W gate based on Mermin's book) +* Generalize the algorithm for an unknown function that returns 1 on multiple + qubit states, not just one. +* Implement _represent_ZGate in OracleGate +""" + +from sympy.core.numbers import pi +from sympy.core.sympify import sympify +from sympy.core.basic import Atom +from sympy.functions.elementary.integers import floor +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.matrices.dense import eye +from sympy.core.numbers import NegativeOne +from sympy.physics.quantum.qapply import qapply +from sympy.physics.quantum.qexpr import QuantumError +from sympy.physics.quantum.hilbert import ComplexSpace +from sympy.physics.quantum.operator import UnitaryOperator +from sympy.physics.quantum.gate import Gate +from sympy.physics.quantum.qubit import IntQubit + +__all__ = [ + 'OracleGate', + 'WGate', + 'superposition_basis', + 'grover_iteration', + 'apply_grover' +] + + +def superposition_basis(nqubits): + """Creates an equal superposition of the computational basis. + + Parameters + ========== + + nqubits : int + The number of qubits. + + Returns + ======= + + state : Qubit + An equal superposition of the computational basis with nqubits. + + Examples + ======== + + Create an equal superposition of 2 qubits:: + + >>> from sympy.physics.quantum.grover import superposition_basis + >>> superposition_basis(2) + |0>/2 + |1>/2 + |2>/2 + |3>/2 + """ + + amp = 1/sqrt(2**nqubits) + return sum(amp*IntQubit(n, nqubits=nqubits) for n in range(2**nqubits)) + +class OracleGateFunction(Atom): + """Wrapper for python functions used in `OracleGate`s""" + + def __new__(cls, function): + if not callable(function): + raise TypeError('Callable expected, got: %r' % function) + obj = Atom.__new__(cls) + obj.function = function + return obj + + def _hashable_content(self): + return type(self), self.function + + def __call__(self, *args): + return self.function(*args) + + +class OracleGate(Gate): + """A black box gate. + + The gate marks the desired qubits of an unknown function by flipping + the sign of the qubits. The unknown function returns true when it + finds its desired qubits and false otherwise. + + Parameters + ========== + + qubits : int + Number of qubits. + + oracle : callable + A callable function that returns a boolean on a computational basis. + + Examples + ======== + + Apply an Oracle gate that flips the sign of ``|2>`` on different qubits:: + + >>> from sympy.physics.quantum.qubit import IntQubit + >>> from sympy.physics.quantum.qapply import qapply + >>> from sympy.physics.quantum.grover import OracleGate + >>> f = lambda qubits: qubits == IntQubit(2) + >>> v = OracleGate(2, f) + >>> qapply(v*IntQubit(2)) + -|2> + >>> qapply(v*IntQubit(3)) + |3> + """ + + gate_name = 'V' + gate_name_latex = 'V' + + #------------------------------------------------------------------------- + # Initialization/creation + #------------------------------------------------------------------------- + + @classmethod + def _eval_args(cls, args): + if len(args) != 2: + raise QuantumError( + 'Insufficient/excessive arguments to Oracle. Please ' + + 'supply the number of qubits and an unknown function.' + ) + sub_args = (args[0],) + sub_args = UnitaryOperator._eval_args(sub_args) + if not sub_args[0].is_Integer: + raise TypeError('Integer expected, got: %r' % sub_args[0]) + + function = args[1] + if not isinstance(function, OracleGateFunction): + function = OracleGateFunction(function) + + return (sub_args[0], function) + + @classmethod + def _eval_hilbert_space(cls, args): + """This returns the smallest possible Hilbert space.""" + return ComplexSpace(2)**args[0] + + #------------------------------------------------------------------------- + # Properties + #------------------------------------------------------------------------- + + @property + def search_function(self): + """The unknown function that helps find the sought after qubits.""" + return self.label[1] + + @property + def targets(self): + """A tuple of target qubits.""" + return sympify(tuple(range(self.args[0]))) + + #------------------------------------------------------------------------- + # Apply + #------------------------------------------------------------------------- + + def _apply_operator_Qubit(self, qubits, **options): + """Apply this operator to a Qubit subclass. + + Parameters + ========== + + qubits : Qubit + The qubit subclass to apply this operator to. + + Returns + ======= + + state : Expr + The resulting quantum state. + """ + if qubits.nqubits != self.nqubits: + raise QuantumError( + 'OracleGate operates on %r qubits, got: %r' + % (self.nqubits, qubits.nqubits) + ) + # If function returns 1 on qubits + # return the negative of the qubits (flip the sign) + if self.search_function(qubits): + return -qubits + else: + return qubits + + #------------------------------------------------------------------------- + # Represent + #------------------------------------------------------------------------- + + def _represent_ZGate(self, basis, **options): + """ + Represent the OracleGate in the computational basis. + """ + nbasis = 2**self.nqubits # compute it only once + matrixOracle = eye(nbasis) + # Flip the sign given the output of the oracle function + for i in range(nbasis): + if self.search_function(IntQubit(i, nqubits=self.nqubits)): + matrixOracle[i, i] = NegativeOne() + return matrixOracle + + +class WGate(Gate): + """General n qubit W Gate in Grover's algorithm. + + The gate performs the operation ``2|phi> = (tensor product of n Hadamards)*(|0> with n qubits)`` + + Parameters + ========== + + nqubits : int + The number of qubits to operate on + + """ + + gate_name = 'W' + gate_name_latex = 'W' + + @classmethod + def _eval_args(cls, args): + if len(args) != 1: + raise QuantumError( + 'Insufficient/excessive arguments to W gate. Please ' + + 'supply the number of qubits to operate on.' + ) + args = UnitaryOperator._eval_args(args) + if not args[0].is_Integer: + raise TypeError('Integer expected, got: %r' % args[0]) + return args + + #------------------------------------------------------------------------- + # Properties + #------------------------------------------------------------------------- + + @property + def targets(self): + return sympify(tuple(reversed(range(self.args[0])))) + + #------------------------------------------------------------------------- + # Apply + #------------------------------------------------------------------------- + + def _apply_operator_Qubit(self, qubits, **options): + """ + qubits: a set of qubits (Qubit) + Returns: quantum object (quantum expression - QExpr) + """ + if qubits.nqubits != self.nqubits: + raise QuantumError( + 'WGate operates on %r qubits, got: %r' + % (self.nqubits, qubits.nqubits) + ) + + # See 'Quantum Computer Science' by David Mermin p.92 -> W|a> result + # Return (2/(sqrt(2^n)))|phi> - |a> where |a> is the current basis + # state and phi is the superposition of basis states (see function + # create_computational_basis above) + basis_states = superposition_basis(self.nqubits) + change_to_basis = (2/sqrt(2**self.nqubits))*basis_states + return change_to_basis - qubits + + +def grover_iteration(qstate, oracle): + """Applies one application of the Oracle and W Gate, WV. + + Parameters + ========== + + qstate : Qubit + A superposition of qubits. + oracle : OracleGate + The black box operator that flips the sign of the desired basis qubits. + + Returns + ======= + + Qubit : The qubits after applying the Oracle and W gate. + + Examples + ======== + + Perform one iteration of grover's algorithm to see a phase change:: + + >>> from sympy.physics.quantum.qapply import qapply + >>> from sympy.physics.quantum.qubit import IntQubit + >>> from sympy.physics.quantum.grover import OracleGate + >>> from sympy.physics.quantum.grover import superposition_basis + >>> from sympy.physics.quantum.grover import grover_iteration + >>> numqubits = 2 + >>> basis_states = superposition_basis(numqubits) + >>> f = lambda qubits: qubits == IntQubit(2) + >>> v = OracleGate(numqubits, f) + >>> qapply(grover_iteration(basis_states, v)) + |2> + + """ + wgate = WGate(oracle.nqubits) + return wgate*oracle*qstate + + +def apply_grover(oracle, nqubits, iterations=None): + """Applies grover's algorithm. + + Parameters + ========== + + oracle : callable + The unknown callable function that returns true when applied to the + desired qubits and false otherwise. + + Returns + ======= + + state : Expr + The resulting state after Grover's algorithm has been iterated. + + Examples + ======== + + Apply grover's algorithm to an even superposition of 2 qubits:: + + >>> from sympy.physics.quantum.qapply import qapply + >>> from sympy.physics.quantum.qubit import IntQubit + >>> from sympy.physics.quantum.grover import apply_grover + >>> f = lambda qubits: qubits == IntQubit(2) + >>> qapply(apply_grover(f, 2)) + |2> + + """ + if nqubits <= 0: + raise QuantumError( + 'Grover\'s algorithm needs nqubits > 0, received %r qubits' + % nqubits + ) + if iterations is None: + iterations = floor(sqrt(2**nqubits)*(pi/4)) + + v = OracleGate(nqubits, oracle) + iterated = superposition_basis(nqubits) + for iter in range(iterations): + iterated = grover_iteration(iterated, v) + iterated = qapply(iterated) + + return iterated diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/hilbert.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/hilbert.py new file mode 100644 index 0000000000000000000000000000000000000000..f475a9e83a6ccc93e9e2dbb9873ad111c1d05f93 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/hilbert.py @@ -0,0 +1,653 @@ +"""Hilbert spaces for quantum mechanics. + +Authors: +* Brian Granger +* Matt Curry +""" + +from functools import reduce + +from sympy.core.basic import Basic +from sympy.core.singleton import S +from sympy.core.sympify import sympify +from sympy.sets.sets import Interval +from sympy.printing.pretty.stringpict import prettyForm +from sympy.physics.quantum.qexpr import QuantumError + + +__all__ = [ + 'HilbertSpaceError', + 'HilbertSpace', + 'TensorProductHilbertSpace', + 'TensorPowerHilbertSpace', + 'DirectSumHilbertSpace', + 'ComplexSpace', + 'L2', + 'FockSpace' +] + +#----------------------------------------------------------------------------- +# Main objects +#----------------------------------------------------------------------------- + + +class HilbertSpaceError(QuantumError): + pass + +#----------------------------------------------------------------------------- +# Main objects +#----------------------------------------------------------------------------- + + +class HilbertSpace(Basic): + """An abstract Hilbert space for quantum mechanics. + + In short, a Hilbert space is an abstract vector space that is complete + with inner products defined [1]_. + + Examples + ======== + + >>> from sympy.physics.quantum.hilbert import HilbertSpace + >>> hs = HilbertSpace() + >>> hs + H + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Hilbert_space + """ + + def __new__(cls): + obj = Basic.__new__(cls) + return obj + + @property + def dimension(self): + """Return the Hilbert dimension of the space.""" + raise NotImplementedError('This Hilbert space has no dimension.') + + def __add__(self, other): + return DirectSumHilbertSpace(self, other) + + def __radd__(self, other): + return DirectSumHilbertSpace(other, self) + + def __mul__(self, other): + return TensorProductHilbertSpace(self, other) + + def __rmul__(self, other): + return TensorProductHilbertSpace(other, self) + + def __pow__(self, other, mod=None): + if mod is not None: + raise ValueError('The third argument to __pow__ is not supported \ + for Hilbert spaces.') + return TensorPowerHilbertSpace(self, other) + + def __contains__(self, other): + """Is the operator or state in this Hilbert space. + + This is checked by comparing the classes of the Hilbert spaces, not + the instances. This is to allow Hilbert Spaces with symbolic + dimensions. + """ + if other.hilbert_space.__class__ == self.__class__: + return True + else: + return False + + def _sympystr(self, printer, *args): + return 'H' + + def _pretty(self, printer, *args): + ustr = '\N{LATIN CAPITAL LETTER H}' + return prettyForm(ustr) + + def _latex(self, printer, *args): + return r'\mathcal{H}' + + +class ComplexSpace(HilbertSpace): + """Finite dimensional Hilbert space of complex vectors. + + The elements of this Hilbert space are n-dimensional complex valued + vectors with the usual inner product that takes the complex conjugate + of the vector on the right. + + A classic example of this type of Hilbert space is spin-1/2, which is + ``ComplexSpace(2)``. Generalizing to spin-s, the space is + ``ComplexSpace(2*s+1)``. Quantum computing with N qubits is done with the + direct product space ``ComplexSpace(2)**N``. + + Examples + ======== + + >>> from sympy import symbols + >>> from sympy.physics.quantum.hilbert import ComplexSpace + >>> c1 = ComplexSpace(2) + >>> c1 + C(2) + >>> c1.dimension + 2 + + >>> n = symbols('n') + >>> c2 = ComplexSpace(n) + >>> c2 + C(n) + >>> c2.dimension + n + + """ + + def __new__(cls, dimension): + dimension = sympify(dimension) + r = cls.eval(dimension) + if isinstance(r, Basic): + return r + obj = Basic.__new__(cls, dimension) + return obj + + @classmethod + def eval(cls, dimension): + if len(dimension.atoms()) == 1: + if not (dimension.is_Integer and dimension > 0 or dimension is S.Infinity + or dimension.is_Symbol): + raise TypeError('The dimension of a ComplexSpace can only' + 'be a positive integer, oo, or a Symbol: %r' + % dimension) + else: + for dim in dimension.atoms(): + if not (dim.is_Integer or dim is S.Infinity or dim.is_Symbol): + raise TypeError('The dimension of a ComplexSpace can only' + ' contain integers, oo, or a Symbol: %r' + % dim) + + @property + def dimension(self): + return self.args[0] + + def _sympyrepr(self, printer, *args): + return "%s(%s)" % (self.__class__.__name__, + printer._print(self.dimension, *args)) + + def _sympystr(self, printer, *args): + return "C(%s)" % printer._print(self.dimension, *args) + + def _pretty(self, printer, *args): + ustr = '\N{LATIN CAPITAL LETTER C}' + pform_exp = printer._print(self.dimension, *args) + pform_base = prettyForm(ustr) + return pform_base**pform_exp + + def _latex(self, printer, *args): + return r'\mathcal{C}^{%s}' % printer._print(self.dimension, *args) + + +class L2(HilbertSpace): + """The Hilbert space of square integrable functions on an interval. + + An L2 object takes in a single SymPy Interval argument which represents + the interval its functions (vectors) are defined on. + + Examples + ======== + + >>> from sympy import Interval, oo + >>> from sympy.physics.quantum.hilbert import L2 + >>> hs = L2(Interval(0,oo)) + >>> hs + L2(Interval(0, oo)) + >>> hs.dimension + oo + >>> hs.interval + Interval(0, oo) + + """ + + def __new__(cls, interval): + if not isinstance(interval, Interval): + raise TypeError('L2 interval must be an Interval instance: %r' + % interval) + obj = Basic.__new__(cls, interval) + return obj + + @property + def dimension(self): + return S.Infinity + + @property + def interval(self): + return self.args[0] + + def _sympyrepr(self, printer, *args): + return "L2(%s)" % printer._print(self.interval, *args) + + def _sympystr(self, printer, *args): + return "L2(%s)" % printer._print(self.interval, *args) + + def _pretty(self, printer, *args): + pform_exp = prettyForm('2') + pform_base = prettyForm('L') + return pform_base**pform_exp + + def _latex(self, printer, *args): + interval = printer._print(self.interval, *args) + return r'{\mathcal{L}^2}\left( %s \right)' % interval + + +class FockSpace(HilbertSpace): + """The Hilbert space for second quantization. + + Technically, this Hilbert space is a infinite direct sum of direct + products of single particle Hilbert spaces [1]_. This is a mess, so we have + a class to represent it directly. + + Examples + ======== + + >>> from sympy.physics.quantum.hilbert import FockSpace + >>> hs = FockSpace() + >>> hs + F + >>> hs.dimension + oo + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Fock_space + """ + + def __new__(cls): + obj = Basic.__new__(cls) + return obj + + @property + def dimension(self): + return S.Infinity + + def _sympyrepr(self, printer, *args): + return "FockSpace()" + + def _sympystr(self, printer, *args): + return "F" + + def _pretty(self, printer, *args): + ustr = '\N{LATIN CAPITAL LETTER F}' + return prettyForm(ustr) + + def _latex(self, printer, *args): + return r'\mathcal{F}' + + +class TensorProductHilbertSpace(HilbertSpace): + """A tensor product of Hilbert spaces [1]_. + + The tensor product between Hilbert spaces is represented by the + operator ``*`` Products of the same Hilbert space will be combined into + tensor powers. + + A ``TensorProductHilbertSpace`` object takes in an arbitrary number of + ``HilbertSpace`` objects as its arguments. In addition, multiplication of + ``HilbertSpace`` objects will automatically return this tensor product + object. + + Examples + ======== + + >>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace + >>> from sympy import symbols + + >>> c = ComplexSpace(2) + >>> f = FockSpace() + >>> hs = c*f + >>> hs + C(2)*F + >>> hs.dimension + oo + >>> hs.spaces + (C(2), F) + + >>> c1 = ComplexSpace(2) + >>> n = symbols('n') + >>> c2 = ComplexSpace(n) + >>> hs = c1*c2 + >>> hs + C(2)*C(n) + >>> hs.dimension + 2*n + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Hilbert_space#Tensor_products + """ + + def __new__(cls, *args): + r = cls.eval(args) + if isinstance(r, Basic): + return r + obj = Basic.__new__(cls, *args) + return obj + + @classmethod + def eval(cls, args): + """Evaluates the direct product.""" + new_args = [] + recall = False + #flatten arguments + for arg in args: + if isinstance(arg, TensorProductHilbertSpace): + new_args.extend(arg.args) + recall = True + elif isinstance(arg, (HilbertSpace, TensorPowerHilbertSpace)): + new_args.append(arg) + else: + raise TypeError('Hilbert spaces can only be multiplied by \ + other Hilbert spaces: %r' % arg) + #combine like arguments into direct powers + comb_args = [] + prev_arg = None + for new_arg in new_args: + if prev_arg is not None: + if isinstance(new_arg, TensorPowerHilbertSpace) and \ + isinstance(prev_arg, TensorPowerHilbertSpace) and \ + new_arg.base == prev_arg.base: + prev_arg = new_arg.base**(new_arg.exp + prev_arg.exp) + elif isinstance(new_arg, TensorPowerHilbertSpace) and \ + new_arg.base == prev_arg: + prev_arg = prev_arg**(new_arg.exp + 1) + elif isinstance(prev_arg, TensorPowerHilbertSpace) and \ + new_arg == prev_arg.base: + prev_arg = new_arg**(prev_arg.exp + 1) + elif new_arg == prev_arg: + prev_arg = new_arg**2 + else: + comb_args.append(prev_arg) + prev_arg = new_arg + elif prev_arg is None: + prev_arg = new_arg + comb_args.append(prev_arg) + if recall: + return TensorProductHilbertSpace(*comb_args) + elif len(comb_args) == 1: + return TensorPowerHilbertSpace(comb_args[0].base, comb_args[0].exp) + else: + return None + + @property + def dimension(self): + arg_list = [arg.dimension for arg in self.args] + if S.Infinity in arg_list: + return S.Infinity + else: + return reduce(lambda x, y: x*y, arg_list) + + @property + def spaces(self): + """A tuple of the Hilbert spaces in this tensor product.""" + return self.args + + def _spaces_printer(self, printer, *args): + spaces_strs = [] + for arg in self.args: + s = printer._print(arg, *args) + if isinstance(arg, DirectSumHilbertSpace): + s = '(%s)' % s + spaces_strs.append(s) + return spaces_strs + + def _sympyrepr(self, printer, *args): + spaces_reprs = self._spaces_printer(printer, *args) + return "TensorProductHilbertSpace(%s)" % ','.join(spaces_reprs) + + def _sympystr(self, printer, *args): + spaces_strs = self._spaces_printer(printer, *args) + return '*'.join(spaces_strs) + + def _pretty(self, printer, *args): + length = len(self.args) + pform = printer._print('', *args) + for i in range(length): + next_pform = printer._print(self.args[i], *args) + if isinstance(self.args[i], (DirectSumHilbertSpace, + TensorProductHilbertSpace)): + next_pform = prettyForm( + *next_pform.parens(left='(', right=')') + ) + pform = prettyForm(*pform.right(next_pform)) + if i != length - 1: + if printer._use_unicode: + pform = prettyForm(*pform.right(' ' + '\N{N-ARY CIRCLED TIMES OPERATOR}' + ' ')) + else: + pform = prettyForm(*pform.right(' x ')) + return pform + + def _latex(self, printer, *args): + length = len(self.args) + s = '' + for i in range(length): + arg_s = printer._print(self.args[i], *args) + if isinstance(self.args[i], (DirectSumHilbertSpace, + TensorProductHilbertSpace)): + arg_s = r'\left(%s\right)' % arg_s + s = s + arg_s + if i != length - 1: + s = s + r'\otimes ' + return s + + +class DirectSumHilbertSpace(HilbertSpace): + """A direct sum of Hilbert spaces [1]_. + + This class uses the ``+`` operator to represent direct sums between + different Hilbert spaces. + + A ``DirectSumHilbertSpace`` object takes in an arbitrary number of + ``HilbertSpace`` objects as its arguments. Also, addition of + ``HilbertSpace`` objects will automatically return a direct sum object. + + Examples + ======== + + >>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace + + >>> c = ComplexSpace(2) + >>> f = FockSpace() + >>> hs = c+f + >>> hs + C(2)+F + >>> hs.dimension + oo + >>> list(hs.spaces) + [C(2), F] + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Hilbert_space#Direct_sums + """ + def __new__(cls, *args): + r = cls.eval(args) + if isinstance(r, Basic): + return r + obj = Basic.__new__(cls, *args) + return obj + + @classmethod + def eval(cls, args): + """Evaluates the direct product.""" + new_args = [] + recall = False + #flatten arguments + for arg in args: + if isinstance(arg, DirectSumHilbertSpace): + new_args.extend(arg.args) + recall = True + elif isinstance(arg, HilbertSpace): + new_args.append(arg) + else: + raise TypeError('Hilbert spaces can only be summed with other \ + Hilbert spaces: %r' % arg) + if recall: + return DirectSumHilbertSpace(*new_args) + else: + return None + + @property + def dimension(self): + arg_list = [arg.dimension for arg in self.args] + if S.Infinity in arg_list: + return S.Infinity + else: + return reduce(lambda x, y: x + y, arg_list) + + @property + def spaces(self): + """A tuple of the Hilbert spaces in this direct sum.""" + return self.args + + def _sympyrepr(self, printer, *args): + spaces_reprs = [printer._print(arg, *args) for arg in self.args] + return "DirectSumHilbertSpace(%s)" % ','.join(spaces_reprs) + + def _sympystr(self, printer, *args): + spaces_strs = [printer._print(arg, *args) for arg in self.args] + return '+'.join(spaces_strs) + + def _pretty(self, printer, *args): + length = len(self.args) + pform = printer._print('', *args) + for i in range(length): + next_pform = printer._print(self.args[i], *args) + if isinstance(self.args[i], (DirectSumHilbertSpace, + TensorProductHilbertSpace)): + next_pform = prettyForm( + *next_pform.parens(left='(', right=')') + ) + pform = prettyForm(*pform.right(next_pform)) + if i != length - 1: + if printer._use_unicode: + pform = prettyForm(*pform.right(' \N{CIRCLED PLUS} ')) + else: + pform = prettyForm(*pform.right(' + ')) + return pform + + def _latex(self, printer, *args): + length = len(self.args) + s = '' + for i in range(length): + arg_s = printer._print(self.args[i], *args) + if isinstance(self.args[i], (DirectSumHilbertSpace, + TensorProductHilbertSpace)): + arg_s = r'\left(%s\right)' % arg_s + s = s + arg_s + if i != length - 1: + s = s + r'\oplus ' + return s + + +class TensorPowerHilbertSpace(HilbertSpace): + """An exponentiated Hilbert space [1]_. + + Tensor powers (repeated tensor products) are represented by the + operator ``**`` Identical Hilbert spaces that are multiplied together + will be automatically combined into a single tensor power object. + + Any Hilbert space, product, or sum may be raised to a tensor power. The + ``TensorPowerHilbertSpace`` takes two arguments: the Hilbert space; and the + tensor power (number). + + Examples + ======== + + >>> from sympy.physics.quantum.hilbert import ComplexSpace, FockSpace + >>> from sympy import symbols + + >>> n = symbols('n') + >>> c = ComplexSpace(2) + >>> hs = c**n + >>> hs + C(2)**n + >>> hs.dimension + 2**n + + >>> c = ComplexSpace(2) + >>> c*c + C(2)**2 + >>> f = FockSpace() + >>> c*f*f + C(2)*F**2 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Hilbert_space#Tensor_products + """ + + def __new__(cls, *args): + r = cls.eval(args) + if isinstance(r, Basic): + return r + return Basic.__new__(cls, *r) + + @classmethod + def eval(cls, args): + new_args = args[0], sympify(args[1]) + exp = new_args[1] + #simplify hs**1 -> hs + if exp is S.One: + return args[0] + #simplify hs**0 -> 1 + if exp is S.Zero: + return S.One + #check (and allow) for hs**(x+42+y...) case + if len(exp.atoms()) == 1: + if not (exp.is_Integer and exp >= 0 or exp.is_Symbol): + raise ValueError('Hilbert spaces can only be raised to \ + positive integers or Symbols: %r' % exp) + else: + for power in exp.atoms(): + if not (power.is_Integer or power.is_Symbol): + raise ValueError('Tensor powers can only contain integers \ + or Symbols: %r' % power) + return new_args + + @property + def base(self): + return self.args[0] + + @property + def exp(self): + return self.args[1] + + @property + def dimension(self): + if self.base.dimension is S.Infinity: + return S.Infinity + else: + return self.base.dimension**self.exp + + def _sympyrepr(self, printer, *args): + return "TensorPowerHilbertSpace(%s,%s)" % (printer._print(self.base, + *args), printer._print(self.exp, *args)) + + def _sympystr(self, printer, *args): + return "%s**%s" % (printer._print(self.base, *args), + printer._print(self.exp, *args)) + + def _pretty(self, printer, *args): + pform_exp = printer._print(self.exp, *args) + if printer._use_unicode: + pform_exp = prettyForm(*pform_exp.left(prettyForm('\N{N-ARY CIRCLED TIMES OPERATOR}'))) + else: + pform_exp = prettyForm(*pform_exp.left(prettyForm('x'))) + pform_base = printer._print(self.base, *args) + return pform_base**pform_exp + + def _latex(self, printer, *args): + base = printer._print(self.base, *args) + exp = printer._print(self.exp, *args) + return r'{%s}^{\otimes %s}' % (base, exp) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/identitysearch.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/identitysearch.py new file mode 100644 index 0000000000000000000000000000000000000000..9a178e9b808450b7ce91175600d6b393fc9797d6 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/identitysearch.py @@ -0,0 +1,853 @@ +from collections import deque +from sympy.core.random import randint + +from sympy.external import import_module +from sympy.core.basic import Basic +from sympy.core.mul import Mul +from sympy.core.numbers import Number, equal_valued +from sympy.core.power import Pow +from sympy.core.singleton import S +from sympy.physics.quantum.represent import represent +from sympy.physics.quantum.dagger import Dagger + +__all__ = [ + # Public interfaces + 'generate_gate_rules', + 'generate_equivalent_ids', + 'GateIdentity', + 'bfs_identity_search', + 'random_identity_search', + + # "Private" functions + 'is_scalar_sparse_matrix', + 'is_scalar_nonsparse_matrix', + 'is_degenerate', + 'is_reducible', +] + +np = import_module('numpy') +scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']}) + + +def is_scalar_sparse_matrix(circuit, nqubits, identity_only, eps=1e-11): + """Checks if a given scipy.sparse matrix is a scalar matrix. + + A scalar matrix is such that B = bI, where B is the scalar + matrix, b is some scalar multiple, and I is the identity + matrix. A scalar matrix would have only the element b along + it's main diagonal and zeroes elsewhere. + + Parameters + ========== + + circuit : Gate tuple + Sequence of quantum gates representing a quantum circuit + nqubits : int + Number of qubits in the circuit + identity_only : bool + Check for only identity matrices + eps : number + The tolerance value for zeroing out elements in the matrix. + Values in the range [-eps, +eps] will be changed to a zero. + """ + + if not np or not scipy: + pass + + matrix = represent(Mul(*circuit), nqubits=nqubits, + format='scipy.sparse') + + # In some cases, represent returns a 1D scalar value in place + # of a multi-dimensional scalar matrix + if (isinstance(matrix, int)): + return matrix == 1 if identity_only else True + + # If represent returns a matrix, check if the matrix is diagonal + # and if every item along the diagonal is the same + else: + # Due to floating pointing operations, must zero out + # elements that are "very" small in the dense matrix + # See parameter for default value. + + # Get the ndarray version of the dense matrix + dense_matrix = matrix.todense().getA() + # Since complex values can't be compared, must split + # the matrix into real and imaginary components + # Find the real values in between -eps and eps + bool_real = np.logical_and(dense_matrix.real > -eps, + dense_matrix.real < eps) + # Find the imaginary values between -eps and eps + bool_imag = np.logical_and(dense_matrix.imag > -eps, + dense_matrix.imag < eps) + # Replaces values between -eps and eps with 0 + corrected_real = np.where(bool_real, 0.0, dense_matrix.real) + corrected_imag = np.where(bool_imag, 0.0, dense_matrix.imag) + # Convert the matrix with real values into imaginary values + corrected_imag = corrected_imag * complex(1j) + # Recombine the real and imaginary components + corrected_dense = corrected_real + corrected_imag + + # Check if it's diagonal + row_indices = corrected_dense.nonzero()[0] + col_indices = corrected_dense.nonzero()[1] + # Check if the rows indices and columns indices are the same + # If they match, then matrix only contains elements along diagonal + bool_indices = row_indices == col_indices + is_diagonal = bool_indices.all() + + first_element = corrected_dense[0][0] + # If the first element is a zero, then can't rescale matrix + # and definitely not diagonal + if (first_element == 0.0 + 0.0j): + return False + + # The dimensions of the dense matrix should still + # be 2^nqubits if there are elements all along the + # the main diagonal + trace_of_corrected = (corrected_dense/first_element).trace() + expected_trace = pow(2, nqubits) + has_correct_trace = trace_of_corrected == expected_trace + + # If only looking for identity matrices + # first element must be a 1 + real_is_one = abs(first_element.real - 1.0) < eps + imag_is_zero = abs(first_element.imag) < eps + is_one = real_is_one and imag_is_zero + is_identity = is_one if identity_only else True + return bool(is_diagonal and has_correct_trace and is_identity) + + +def is_scalar_nonsparse_matrix(circuit, nqubits, identity_only, eps=None): + """Checks if a given circuit, in matrix form, is equivalent to + a scalar value. + + Parameters + ========== + + circuit : Gate tuple + Sequence of quantum gates representing a quantum circuit + nqubits : int + Number of qubits in the circuit + identity_only : bool + Check for only identity matrices + eps : number + This argument is ignored. It is just for signature compatibility with + is_scalar_sparse_matrix. + + Note: Used in situations when is_scalar_sparse_matrix has bugs + """ + + matrix = represent(Mul(*circuit), nqubits=nqubits) + + # In some cases, represent returns a 1D scalar value in place + # of a multi-dimensional scalar matrix + if (isinstance(matrix, Number)): + return matrix == 1 if identity_only else True + + # If represent returns a matrix, check if the matrix is diagonal + # and if every item along the diagonal is the same + else: + # Added up the diagonal elements + matrix_trace = matrix.trace() + # Divide the trace by the first element in the matrix + # if matrix is not required to be the identity matrix + adjusted_matrix_trace = (matrix_trace/matrix[0] + if not identity_only + else matrix_trace) + + is_identity = equal_valued(matrix[0], 1) if identity_only else True + + has_correct_trace = adjusted_matrix_trace == pow(2, nqubits) + + # The matrix is scalar if it's diagonal and the adjusted trace + # value is equal to 2^nqubits + return bool( + matrix.is_diagonal() and has_correct_trace and is_identity) + +if np and scipy: + is_scalar_matrix = is_scalar_sparse_matrix +else: + is_scalar_matrix = is_scalar_nonsparse_matrix + + +def _get_min_qubits(a_gate): + if isinstance(a_gate, Pow): + return a_gate.base.min_qubits + else: + return a_gate.min_qubits + + +def ll_op(left, right): + """Perform a LL operation. + + A LL operation multiplies both left and right circuits + with the dagger of the left circuit's leftmost gate, and + the dagger is multiplied on the left side of both circuits. + + If a LL is possible, it returns the new gate rule as a + 2-tuple (LHS, RHS), where LHS is the left circuit and + and RHS is the right circuit of the new rule. + If a LL is not possible, None is returned. + + Parameters + ========== + + left : Gate tuple + The left circuit of a gate rule expression. + right : Gate tuple + The right circuit of a gate rule expression. + + Examples + ======== + + Generate a new gate rule using a LL operation: + + >>> from sympy.physics.quantum.identitysearch import ll_op + >>> from sympy.physics.quantum.gate import X, Y, Z + >>> x = X(0); y = Y(0); z = Z(0) + >>> ll_op((x, y, z), ()) + ((Y(0), Z(0)), (X(0),)) + + >>> ll_op((y, z), (x,)) + ((Z(0),), (Y(0), X(0))) + """ + + if (len(left) > 0): + ll_gate = left[0] + ll_gate_is_unitary = is_scalar_matrix( + (Dagger(ll_gate), ll_gate), _get_min_qubits(ll_gate), True) + + if (len(left) > 0 and ll_gate_is_unitary): + # Get the new left side w/o the leftmost gate + new_left = left[1:len(left)] + # Add the leftmost gate to the left position on the right side + new_right = (Dagger(ll_gate),) + right + # Return the new gate rule + return (new_left, new_right) + + return None + + +def lr_op(left, right): + """Perform a LR operation. + + A LR operation multiplies both left and right circuits + with the dagger of the left circuit's rightmost gate, and + the dagger is multiplied on the right side of both circuits. + + If a LR is possible, it returns the new gate rule as a + 2-tuple (LHS, RHS), where LHS is the left circuit and + and RHS is the right circuit of the new rule. + If a LR is not possible, None is returned. + + Parameters + ========== + + left : Gate tuple + The left circuit of a gate rule expression. + right : Gate tuple + The right circuit of a gate rule expression. + + Examples + ======== + + Generate a new gate rule using a LR operation: + + >>> from sympy.physics.quantum.identitysearch import lr_op + >>> from sympy.physics.quantum.gate import X, Y, Z + >>> x = X(0); y = Y(0); z = Z(0) + >>> lr_op((x, y, z), ()) + ((X(0), Y(0)), (Z(0),)) + + >>> lr_op((x, y), (z,)) + ((X(0),), (Z(0), Y(0))) + """ + + if (len(left) > 0): + lr_gate = left[len(left) - 1] + lr_gate_is_unitary = is_scalar_matrix( + (Dagger(lr_gate), lr_gate), _get_min_qubits(lr_gate), True) + + if (len(left) > 0 and lr_gate_is_unitary): + # Get the new left side w/o the rightmost gate + new_left = left[0:len(left) - 1] + # Add the rightmost gate to the right position on the right side + new_right = right + (Dagger(lr_gate),) + # Return the new gate rule + return (new_left, new_right) + + return None + + +def rl_op(left, right): + """Perform a RL operation. + + A RL operation multiplies both left and right circuits + with the dagger of the right circuit's leftmost gate, and + the dagger is multiplied on the left side of both circuits. + + If a RL is possible, it returns the new gate rule as a + 2-tuple (LHS, RHS), where LHS is the left circuit and + and RHS is the right circuit of the new rule. + If a RL is not possible, None is returned. + + Parameters + ========== + + left : Gate tuple + The left circuit of a gate rule expression. + right : Gate tuple + The right circuit of a gate rule expression. + + Examples + ======== + + Generate a new gate rule using a RL operation: + + >>> from sympy.physics.quantum.identitysearch import rl_op + >>> from sympy.physics.quantum.gate import X, Y, Z + >>> x = X(0); y = Y(0); z = Z(0) + >>> rl_op((x,), (y, z)) + ((Y(0), X(0)), (Z(0),)) + + >>> rl_op((x, y), (z,)) + ((Z(0), X(0), Y(0)), ()) + """ + + if (len(right) > 0): + rl_gate = right[0] + rl_gate_is_unitary = is_scalar_matrix( + (Dagger(rl_gate), rl_gate), _get_min_qubits(rl_gate), True) + + if (len(right) > 0 and rl_gate_is_unitary): + # Get the new right side w/o the leftmost gate + new_right = right[1:len(right)] + # Add the leftmost gate to the left position on the left side + new_left = (Dagger(rl_gate),) + left + # Return the new gate rule + return (new_left, new_right) + + return None + + +def rr_op(left, right): + """Perform a RR operation. + + A RR operation multiplies both left and right circuits + with the dagger of the right circuit's rightmost gate, and + the dagger is multiplied on the right side of both circuits. + + If a RR is possible, it returns the new gate rule as a + 2-tuple (LHS, RHS), where LHS is the left circuit and + and RHS is the right circuit of the new rule. + If a RR is not possible, None is returned. + + Parameters + ========== + + left : Gate tuple + The left circuit of a gate rule expression. + right : Gate tuple + The right circuit of a gate rule expression. + + Examples + ======== + + Generate a new gate rule using a RR operation: + + >>> from sympy.physics.quantum.identitysearch import rr_op + >>> from sympy.physics.quantum.gate import X, Y, Z + >>> x = X(0); y = Y(0); z = Z(0) + >>> rr_op((x, y), (z,)) + ((X(0), Y(0), Z(0)), ()) + + >>> rr_op((x,), (y, z)) + ((X(0), Z(0)), (Y(0),)) + """ + + if (len(right) > 0): + rr_gate = right[len(right) - 1] + rr_gate_is_unitary = is_scalar_matrix( + (Dagger(rr_gate), rr_gate), _get_min_qubits(rr_gate), True) + + if (len(right) > 0 and rr_gate_is_unitary): + # Get the new right side w/o the rightmost gate + new_right = right[0:len(right) - 1] + # Add the rightmost gate to the right position on the right side + new_left = left + (Dagger(rr_gate),) + # Return the new gate rule + return (new_left, new_right) + + return None + + +def generate_gate_rules(gate_seq, return_as_muls=False): + """Returns a set of gate rules. Each gate rules is represented + as a 2-tuple of tuples or Muls. An empty tuple represents an arbitrary + scalar value. + + This function uses the four operations (LL, LR, RL, RR) + to generate the gate rules. + + A gate rule is an expression such as ABC = D or AB = CD, where + A, B, C, and D are gates. Each value on either side of the + equal sign represents a circuit. The four operations allow + one to find a set of equivalent circuits from a gate identity. + The letters denoting the operation tell the user what + activities to perform on each expression. The first letter + indicates which side of the equal sign to focus on. The + second letter indicates which gate to focus on given the + side. Once this information is determined, the inverse + of the gate is multiplied on both circuits to create a new + gate rule. + + For example, given the identity, ABCD = 1, a LL operation + means look at the left value and multiply both left sides by the + inverse of the leftmost gate A. If A is Hermitian, the inverse + of A is still A. The resulting new rule is BCD = A. + + The following is a summary of the four operations. Assume + that in the examples, all gates are Hermitian. + + LL : left circuit, left multiply + ABCD = E -> AABCD = AE -> BCD = AE + LR : left circuit, right multiply + ABCD = E -> ABCDD = ED -> ABC = ED + RL : right circuit, left multiply + ABC = ED -> EABC = EED -> EABC = D + RR : right circuit, right multiply + AB = CD -> ABD = CDD -> ABD = C + + The number of gate rules generated is n*(n+1), where n + is the number of gates in the sequence (unproven). + + Parameters + ========== + + gate_seq : Gate tuple, Mul, or Number + A variable length tuple or Mul of Gates whose product is equal to + a scalar matrix + return_as_muls : bool + True to return a set of Muls; False to return a set of tuples + + Examples + ======== + + Find the gate rules of the current circuit using tuples: + + >>> from sympy.physics.quantum.identitysearch import generate_gate_rules + >>> from sympy.physics.quantum.gate import X, Y, Z + >>> x = X(0); y = Y(0); z = Z(0) + >>> generate_gate_rules((x, x)) + {((X(0),), (X(0),)), ((X(0), X(0)), ())} + + >>> generate_gate_rules((x, y, z)) + {((), (X(0), Z(0), Y(0))), ((), (Y(0), X(0), Z(0))), + ((), (Z(0), Y(0), X(0))), ((X(0),), (Z(0), Y(0))), + ((Y(0),), (X(0), Z(0))), ((Z(0),), (Y(0), X(0))), + ((X(0), Y(0)), (Z(0),)), ((Y(0), Z(0)), (X(0),)), + ((Z(0), X(0)), (Y(0),)), ((X(0), Y(0), Z(0)), ()), + ((Y(0), Z(0), X(0)), ()), ((Z(0), X(0), Y(0)), ())} + + Find the gate rules of the current circuit using Muls: + + >>> generate_gate_rules(x*x, return_as_muls=True) + {(1, 1)} + + >>> generate_gate_rules(x*y*z, return_as_muls=True) + {(1, X(0)*Z(0)*Y(0)), (1, Y(0)*X(0)*Z(0)), + (1, Z(0)*Y(0)*X(0)), (X(0)*Y(0), Z(0)), + (Y(0)*Z(0), X(0)), (Z(0)*X(0), Y(0)), + (X(0)*Y(0)*Z(0), 1), (Y(0)*Z(0)*X(0), 1), + (Z(0)*X(0)*Y(0), 1), (X(0), Z(0)*Y(0)), + (Y(0), X(0)*Z(0)), (Z(0), Y(0)*X(0))} + """ + + if isinstance(gate_seq, Number): + if return_as_muls: + return {(S.One, S.One)} + else: + return {((), ())} + + elif isinstance(gate_seq, Mul): + gate_seq = gate_seq.args + + # Each item in queue is a 3-tuple: + # i) first item is the left side of an equality + # ii) second item is the right side of an equality + # iii) third item is the number of operations performed + # The argument, gate_seq, will start on the left side, and + # the right side will be empty, implying the presence of an + # identity. + queue = deque() + # A set of gate rules + rules = set() + # Maximum number of operations to perform + max_ops = len(gate_seq) + + def process_new_rule(new_rule, ops): + if new_rule is not None: + new_left, new_right = new_rule + + if new_rule not in rules and (new_right, new_left) not in rules: + rules.add(new_rule) + # If haven't reached the max limit on operations + if ops + 1 < max_ops: + queue.append(new_rule + (ops + 1,)) + + queue.append((gate_seq, (), 0)) + rules.add((gate_seq, ())) + + while len(queue) > 0: + left, right, ops = queue.popleft() + + # Do a LL + new_rule = ll_op(left, right) + process_new_rule(new_rule, ops) + # Do a LR + new_rule = lr_op(left, right) + process_new_rule(new_rule, ops) + # Do a RL + new_rule = rl_op(left, right) + process_new_rule(new_rule, ops) + # Do a RR + new_rule = rr_op(left, right) + process_new_rule(new_rule, ops) + + if return_as_muls: + # Convert each rule as tuples into a rule as muls + mul_rules = set() + for rule in rules: + left, right = rule + mul_rules.add((Mul(*left), Mul(*right))) + + rules = mul_rules + + return rules + + +def generate_equivalent_ids(gate_seq, return_as_muls=False): + """Returns a set of equivalent gate identities. + + A gate identity is a quantum circuit such that the product + of the gates in the circuit is equal to a scalar value. + For example, XYZ = i, where X, Y, Z are the Pauli gates and + i is the imaginary value, is considered a gate identity. + + This function uses the four operations (LL, LR, RL, RR) + to generate the gate rules and, subsequently, to locate equivalent + gate identities. + + Note that all equivalent identities are reachable in n operations + from the starting gate identity, where n is the number of gates + in the sequence. + + The max number of gate identities is 2n, where n is the number + of gates in the sequence (unproven). + + Parameters + ========== + + gate_seq : Gate tuple, Mul, or Number + A variable length tuple or Mul of Gates whose product is equal to + a scalar matrix. + return_as_muls: bool + True to return as Muls; False to return as tuples + + Examples + ======== + + Find equivalent gate identities from the current circuit with tuples: + + >>> from sympy.physics.quantum.identitysearch import generate_equivalent_ids + >>> from sympy.physics.quantum.gate import X, Y, Z + >>> x = X(0); y = Y(0); z = Z(0) + >>> generate_equivalent_ids((x, x)) + {(X(0), X(0))} + + >>> generate_equivalent_ids((x, y, z)) + {(X(0), Y(0), Z(0)), (X(0), Z(0), Y(0)), (Y(0), X(0), Z(0)), + (Y(0), Z(0), X(0)), (Z(0), X(0), Y(0)), (Z(0), Y(0), X(0))} + + Find equivalent gate identities from the current circuit with Muls: + + >>> generate_equivalent_ids(x*x, return_as_muls=True) + {1} + + >>> generate_equivalent_ids(x*y*z, return_as_muls=True) + {X(0)*Y(0)*Z(0), X(0)*Z(0)*Y(0), Y(0)*X(0)*Z(0), + Y(0)*Z(0)*X(0), Z(0)*X(0)*Y(0), Z(0)*Y(0)*X(0)} + """ + + if isinstance(gate_seq, Number): + return {S.One} + elif isinstance(gate_seq, Mul): + gate_seq = gate_seq.args + + # Filter through the gate rules and keep the rules + # with an empty tuple either on the left or right side + + # A set of equivalent gate identities + eq_ids = set() + + gate_rules = generate_gate_rules(gate_seq) + for rule in gate_rules: + l, r = rule + if l == (): + eq_ids.add(r) + elif r == (): + eq_ids.add(l) + + if return_as_muls: + convert_to_mul = lambda id_seq: Mul(*id_seq) + eq_ids = set(map(convert_to_mul, eq_ids)) + + return eq_ids + + +class GateIdentity(Basic): + """Wrapper class for circuits that reduce to a scalar value. + + A gate identity is a quantum circuit such that the product + of the gates in the circuit is equal to a scalar value. + For example, XYZ = i, where X, Y, Z are the Pauli gates and + i is the imaginary value, is considered a gate identity. + + Parameters + ========== + + args : Gate tuple + A variable length tuple of Gates that form an identity. + + Examples + ======== + + Create a GateIdentity and look at its attributes: + + >>> from sympy.physics.quantum.identitysearch import GateIdentity + >>> from sympy.physics.quantum.gate import X, Y, Z + >>> x = X(0); y = Y(0); z = Z(0) + >>> an_identity = GateIdentity(x, y, z) + >>> an_identity.circuit + X(0)*Y(0)*Z(0) + + >>> an_identity.equivalent_ids + {(X(0), Y(0), Z(0)), (X(0), Z(0), Y(0)), (Y(0), X(0), Z(0)), + (Y(0), Z(0), X(0)), (Z(0), X(0), Y(0)), (Z(0), Y(0), X(0))} + """ + + def __new__(cls, *args): + # args should be a tuple - a variable length argument list + obj = Basic.__new__(cls, *args) + obj._circuit = Mul(*args) + obj._rules = generate_gate_rules(args) + obj._eq_ids = generate_equivalent_ids(args) + + return obj + + @property + def circuit(self): + return self._circuit + + @property + def gate_rules(self): + return self._rules + + @property + def equivalent_ids(self): + return self._eq_ids + + @property + def sequence(self): + return self.args + + def __str__(self): + """Returns the string of gates in a tuple.""" + return str(self.circuit) + + +def is_degenerate(identity_set, gate_identity): + """Checks if a gate identity is a permutation of another identity. + + Parameters + ========== + + identity_set : set + A Python set with GateIdentity objects. + gate_identity : GateIdentity + The GateIdentity to check for existence in the set. + + Examples + ======== + + Check if the identity is a permutation of another identity: + + >>> from sympy.physics.quantum.identitysearch import ( + ... GateIdentity, is_degenerate) + >>> from sympy.physics.quantum.gate import X, Y, Z + >>> x = X(0); y = Y(0); z = Z(0) + >>> an_identity = GateIdentity(x, y, z) + >>> id_set = {an_identity} + >>> another_id = (y, z, x) + >>> is_degenerate(id_set, another_id) + True + + >>> another_id = (x, x) + >>> is_degenerate(id_set, another_id) + False + """ + + # For now, just iteratively go through the set and check if the current + # gate_identity is a permutation of an identity in the set + for an_id in identity_set: + if (gate_identity in an_id.equivalent_ids): + return True + return False + + +def is_reducible(circuit, nqubits, begin, end): + """Determines if a circuit is reducible by checking + if its subcircuits are scalar values. + + Parameters + ========== + + circuit : Gate tuple + A tuple of Gates representing a circuit. The circuit to check + if a gate identity is contained in a subcircuit. + nqubits : int + The number of qubits the circuit operates on. + begin : int + The leftmost gate in the circuit to include in a subcircuit. + end : int + The rightmost gate in the circuit to include in a subcircuit. + + Examples + ======== + + Check if the circuit can be reduced: + + >>> from sympy.physics.quantum.identitysearch import is_reducible + >>> from sympy.physics.quantum.gate import X, Y, Z + >>> x = X(0); y = Y(0); z = Z(0) + >>> is_reducible((x, y, z), 1, 0, 3) + True + + Check if an interval in the circuit can be reduced: + + >>> is_reducible((x, y, z), 1, 1, 3) + False + + >>> is_reducible((x, y, y), 1, 1, 3) + True + """ + + current_circuit = () + # Start from the gate at "end" and go down to almost the gate at "begin" + for ndx in reversed(range(begin, end)): + next_gate = circuit[ndx] + current_circuit = (next_gate,) + current_circuit + + # If a circuit as a matrix is equivalent to a scalar value + if (is_scalar_matrix(current_circuit, nqubits, False)): + return True + + return False + + +def bfs_identity_search(gate_list, nqubits, max_depth=None, + identity_only=False): + """Constructs a set of gate identities from the list of possible gates. + + Performs a breadth first search over the space of gate identities. + This allows the finding of the shortest gate identities first. + + Parameters + ========== + + gate_list : list, Gate + A list of Gates from which to search for gate identities. + nqubits : int + The number of qubits the quantum circuit operates on. + max_depth : int + The longest quantum circuit to construct from gate_list. + identity_only : bool + True to search for gate identities that reduce to identity; + False to search for gate identities that reduce to a scalar. + + Examples + ======== + + Find a list of gate identities: + + >>> from sympy.physics.quantum.identitysearch import bfs_identity_search + >>> from sympy.physics.quantum.gate import X, Y, Z + >>> x = X(0); y = Y(0); z = Z(0) + >>> bfs_identity_search([x], 1, max_depth=2) + {GateIdentity(X(0), X(0))} + + >>> bfs_identity_search([x, y, z], 1) + {GateIdentity(X(0), X(0)), GateIdentity(Y(0), Y(0)), + GateIdentity(Z(0), Z(0)), GateIdentity(X(0), Y(0), Z(0))} + + Find a list of identities that only equal to 1: + + >>> bfs_identity_search([x, y, z], 1, identity_only=True) + {GateIdentity(X(0), X(0)), GateIdentity(Y(0), Y(0)), + GateIdentity(Z(0), Z(0))} + """ + + if max_depth is None or max_depth <= 0: + max_depth = len(gate_list) + + id_only = identity_only + + # Start with an empty sequence (implicitly contains an IdentityGate) + queue = deque([()]) + + # Create an empty set of gate identities + ids = set() + + # Begin searching for gate identities in given space. + while (len(queue) > 0): + current_circuit = queue.popleft() + + for next_gate in gate_list: + new_circuit = current_circuit + (next_gate,) + + # Determines if a (strict) subcircuit is a scalar matrix + circuit_reducible = is_reducible(new_circuit, nqubits, + 1, len(new_circuit)) + + # In many cases when the matrix is a scalar value, + # the evaluated matrix will actually be an integer + if (is_scalar_matrix(new_circuit, nqubits, id_only) and + not is_degenerate(ids, new_circuit) and + not circuit_reducible): + ids.add(GateIdentity(*new_circuit)) + + elif (len(new_circuit) < max_depth and + not circuit_reducible): + queue.append(new_circuit) + + return ids + + +def random_identity_search(gate_list, numgates, nqubits): + """Randomly selects numgates from gate_list and checks if it is + a gate identity. + + If the circuit is a gate identity, the circuit is returned; + Otherwise, None is returned. + """ + + gate_size = len(gate_list) + circuit = () + + for i in range(numgates): + next_gate = gate_list[randint(0, gate_size - 1)] + circuit = circuit + (next_gate,) + + is_scalar = is_scalar_matrix(circuit, nqubits, False) + + return circuit if is_scalar else None diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/operator.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/operator.py new file mode 100644 index 0000000000000000000000000000000000000000..8c540dc016fc1a1043f3c25acf71ae0e1996e1c6 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/operator.py @@ -0,0 +1,657 @@ +"""Quantum mechanical operators. + +TODO: + +* Fix early 0 in apply_operators. +* Debug and test apply_operators. +* Get cse working with classes in this file. +* Doctests and documentation of special methods for InnerProduct, Commutator, + AntiCommutator, represent, apply_operators. +""" +from typing import Optional + +from sympy.core.add import Add +from sympy.core.expr import Expr +from sympy.core.function import (Derivative, expand) +from sympy.core.mul import Mul +from sympy.core.numbers import oo +from sympy.core.singleton import S +from sympy.printing.pretty.stringpict import prettyForm +from sympy.physics.quantum.dagger import Dagger +from sympy.physics.quantum.qexpr import QExpr, dispatch_method +from sympy.matrices import eye + +__all__ = [ + 'Operator', + 'HermitianOperator', + 'UnitaryOperator', + 'IdentityOperator', + 'OuterProduct', + 'DifferentialOperator' +] + +#----------------------------------------------------------------------------- +# Operators and outer products +#----------------------------------------------------------------------------- + + +class Operator(QExpr): + """Base class for non-commuting quantum operators. + + An operator maps between quantum states [1]_. In quantum mechanics, + observables (including, but not limited to, measured physical values) are + represented as Hermitian operators [2]_. + + Parameters + ========== + + args : tuple + The list of numbers or parameters that uniquely specify the + operator. For time-dependent operators, this will include the time. + + Examples + ======== + + Create an operator and examine its attributes:: + + >>> from sympy.physics.quantum import Operator + >>> from sympy import I + >>> A = Operator('A') + >>> A + A + >>> A.hilbert_space + H + >>> A.label + (A,) + >>> A.is_commutative + False + + Create another operator and do some arithmetic operations:: + + >>> B = Operator('B') + >>> C = 2*A*A + I*B + >>> C + 2*A**2 + I*B + + Operators do not commute:: + + >>> A.is_commutative + False + >>> B.is_commutative + False + >>> A*B == B*A + False + + Polymonials of operators respect the commutation properties:: + + >>> e = (A+B)**3 + >>> e.expand() + A*B*A + A*B**2 + A**2*B + A**3 + B*A*B + B*A**2 + B**2*A + B**3 + + Operator inverses are handle symbolically:: + + >>> A.inv() + A**(-1) + >>> A*A.inv() + 1 + + References + ========== + + .. [1] https://en.wikipedia.org/wiki/Operator_%28physics%29 + .. [2] https://en.wikipedia.org/wiki/Observable + """ + is_hermitian: Optional[bool] = None + is_unitary: Optional[bool] = None + @classmethod + def default_args(self): + return ("O",) + + #------------------------------------------------------------------------- + # Printing + #------------------------------------------------------------------------- + + _label_separator = ',' + + def _print_operator_name(self, printer, *args): + return self.__class__.__name__ + + _print_operator_name_latex = _print_operator_name + + def _print_operator_name_pretty(self, printer, *args): + return prettyForm(self.__class__.__name__) + + def _print_contents(self, printer, *args): + if len(self.label) == 1: + return self._print_label(printer, *args) + else: + return '%s(%s)' % ( + self._print_operator_name(printer, *args), + self._print_label(printer, *args) + ) + + def _print_contents_pretty(self, printer, *args): + if len(self.label) == 1: + return self._print_label_pretty(printer, *args) + else: + pform = self._print_operator_name_pretty(printer, *args) + label_pform = self._print_label_pretty(printer, *args) + label_pform = prettyForm( + *label_pform.parens(left='(', right=')') + ) + pform = prettyForm(*pform.right(label_pform)) + return pform + + def _print_contents_latex(self, printer, *args): + if len(self.label) == 1: + return self._print_label_latex(printer, *args) + else: + return r'%s\left(%s\right)' % ( + self._print_operator_name_latex(printer, *args), + self._print_label_latex(printer, *args) + ) + + #------------------------------------------------------------------------- + # _eval_* methods + #------------------------------------------------------------------------- + + def _eval_commutator(self, other, **options): + """Evaluate [self, other] if known, return None if not known.""" + return dispatch_method(self, '_eval_commutator', other, **options) + + def _eval_anticommutator(self, other, **options): + """Evaluate [self, other] if known.""" + return dispatch_method(self, '_eval_anticommutator', other, **options) + + #------------------------------------------------------------------------- + # Operator application + #------------------------------------------------------------------------- + + def _apply_operator(self, ket, **options): + return dispatch_method(self, '_apply_operator', ket, **options) + + def _apply_from_right_to(self, bra, **options): + return None + + def matrix_element(self, *args): + raise NotImplementedError('matrix_elements is not defined') + + def inverse(self): + return self._eval_inverse() + + inv = inverse + + def _eval_inverse(self): + return self**(-1) + + def __mul__(self, other): + + if isinstance(other, IdentityOperator): + return self + + return Mul(self, other) + + +class HermitianOperator(Operator): + """A Hermitian operator that satisfies H == Dagger(H). + + Parameters + ========== + + args : tuple + The list of numbers or parameters that uniquely specify the + operator. For time-dependent operators, this will include the time. + + Examples + ======== + + >>> from sympy.physics.quantum import Dagger, HermitianOperator + >>> H = HermitianOperator('H') + >>> Dagger(H) + H + """ + + is_hermitian = True + + def _eval_inverse(self): + if isinstance(self, UnitaryOperator): + return self + else: + return Operator._eval_inverse(self) + + def _eval_power(self, exp): + if isinstance(self, UnitaryOperator): + # so all eigenvalues of self are 1 or -1 + if exp.is_even: + from sympy.core.singleton import S + return S.One # is identity, see Issue 24153. + elif exp.is_odd: + return self + # No simplification in all other cases + return Operator._eval_power(self, exp) + + +class UnitaryOperator(Operator): + """A unitary operator that satisfies U*Dagger(U) == 1. + + Parameters + ========== + + args : tuple + The list of numbers or parameters that uniquely specify the + operator. For time-dependent operators, this will include the time. + + Examples + ======== + + >>> from sympy.physics.quantum import Dagger, UnitaryOperator + >>> U = UnitaryOperator('U') + >>> U*Dagger(U) + 1 + """ + is_unitary = True + def _eval_adjoint(self): + return self._eval_inverse() + + +class IdentityOperator(Operator): + """An identity operator I that satisfies op * I == I * op == op for any + operator op. + + Parameters + ========== + + N : Integer + Optional parameter that specifies the dimension of the Hilbert space + of operator. This is used when generating a matrix representation. + + Examples + ======== + + >>> from sympy.physics.quantum import IdentityOperator + >>> IdentityOperator() + I + """ + is_hermitian = True + is_unitary = True + @property + def dimension(self): + return self.N + + @classmethod + def default_args(self): + return (oo,) + + def __init__(self, *args, **hints): + if not len(args) in (0, 1): + raise ValueError('0 or 1 parameters expected, got %s' % args) + + self.N = args[0] if (len(args) == 1 and args[0]) else oo + + def _eval_commutator(self, other, **hints): + return S.Zero + + def _eval_anticommutator(self, other, **hints): + return 2 * other + + def _eval_inverse(self): + return self + + def _eval_adjoint(self): + return self + + def _apply_operator(self, ket, **options): + return ket + + def _apply_from_right_to(self, bra, **options): + return bra + + def _eval_power(self, exp): + return self + + def _print_contents(self, printer, *args): + return 'I' + + def _print_contents_pretty(self, printer, *args): + return prettyForm('I') + + def _print_contents_latex(self, printer, *args): + return r'{\mathcal{I}}' + + def __mul__(self, other): + + if isinstance(other, (Operator, Dagger)): + return other + + return Mul(self, other) + + def _represent_default_basis(self, **options): + if not self.N or self.N == oo: + raise NotImplementedError('Cannot represent infinite dimensional' + + ' identity operator as a matrix') + + format = options.get('format', 'sympy') + if format != 'sympy': + raise NotImplementedError('Representation in format ' + + '%s not implemented.' % format) + + return eye(self.N) + + +class OuterProduct(Operator): + """An unevaluated outer product between a ket and bra. + + This constructs an outer product between any subclass of ``KetBase`` and + ``BraBase`` as ``|a>>> from sympy.physics.quantum import Ket, Bra, OuterProduct, Dagger + >>> from sympy.physics.quantum import Operator + + >>> k = Ket('k') + >>> b = Bra('b') + >>> op = OuterProduct(k, b) + >>> op + |k>>> op.hilbert_space + H + >>> op.ket + |k> + >>> op.bra + >> Dagger(op) + |b>>> k*b + |k>>> A = Operator('A') + >>> A*k*b + A*|k>*>> A*(k*b) + A*|k>>> from sympy import Derivative, Function, Symbol + >>> from sympy.physics.quantum.operator import DifferentialOperator + >>> from sympy.physics.quantum.state import Wavefunction + >>> from sympy.physics.quantum.qapply import qapply + >>> f = Function('f') + >>> x = Symbol('x') + >>> d = DifferentialOperator(1/x*Derivative(f(x), x), f(x)) + >>> w = Wavefunction(x**2, x) + >>> d.function + f(x) + >>> d.variables + (x,) + >>> qapply(d*w) + Wavefunction(2, x) + + """ + + @property + def variables(self): + """ + Returns the variables with which the function in the specified + arbitrary expression is evaluated + + Examples + ======== + + >>> from sympy.physics.quantum.operator import DifferentialOperator + >>> from sympy import Symbol, Function, Derivative + >>> x = Symbol('x') + >>> f = Function('f') + >>> d = DifferentialOperator(1/x*Derivative(f(x), x), f(x)) + >>> d.variables + (x,) + >>> y = Symbol('y') + >>> d = DifferentialOperator(Derivative(f(x, y), x) + + ... Derivative(f(x, y), y), f(x, y)) + >>> d.variables + (x, y) + """ + + return self.args[-1].args + + @property + def function(self): + """ + Returns the function which is to be replaced with the Wavefunction + + Examples + ======== + + >>> from sympy.physics.quantum.operator import DifferentialOperator + >>> from sympy import Function, Symbol, Derivative + >>> x = Symbol('x') + >>> f = Function('f') + >>> d = DifferentialOperator(Derivative(f(x), x), f(x)) + >>> d.function + f(x) + >>> y = Symbol('y') + >>> d = DifferentialOperator(Derivative(f(x, y), x) + + ... Derivative(f(x, y), y), f(x, y)) + >>> d.function + f(x, y) + """ + + return self.args[-1] + + @property + def expr(self): + """ + Returns the arbitrary expression which is to have the Wavefunction + substituted into it + + Examples + ======== + + >>> from sympy.physics.quantum.operator import DifferentialOperator + >>> from sympy import Function, Symbol, Derivative + >>> x = Symbol('x') + >>> f = Function('f') + >>> d = DifferentialOperator(Derivative(f(x), x), f(x)) + >>> d.expr + Derivative(f(x), x) + >>> y = Symbol('y') + >>> d = DifferentialOperator(Derivative(f(x, y), x) + + ... Derivative(f(x, y), y), f(x, y)) + >>> d.expr + Derivative(f(x, y), x) + Derivative(f(x, y), y) + """ + + return self.args[0] + + @property + def free_symbols(self): + """ + Return the free symbols of the expression. + """ + + return self.expr.free_symbols + + def _apply_operator_Wavefunction(self, func, **options): + from sympy.physics.quantum.state import Wavefunction + var = self.variables + wf_vars = func.args[1:] + + f = self.function + new_expr = self.expr.subs(f, func(*var)) + new_expr = new_expr.doit() + + return Wavefunction(new_expr, *wf_vars) + + def _eval_derivative(self, symbol): + new_expr = Derivative(self.expr, symbol) + return DifferentialOperator(new_expr, self.args[-1]) + + #------------------------------------------------------------------------- + # Printing + #------------------------------------------------------------------------- + + def _print(self, printer, *args): + return '%s(%s)' % ( + self._print_operator_name(printer, *args), + self._print_label(printer, *args) + ) + + def _print_pretty(self, printer, *args): + pform = self._print_operator_name_pretty(printer, *args) + label_pform = self._print_label_pretty(printer, *args) + label_pform = prettyForm( + *label_pform.parens(left='(', right=')') + ) + pform = prettyForm(*pform.right(label_pform)) + return pform diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/shor.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/shor.py new file mode 100644 index 0000000000000000000000000000000000000000..fc9e55229d74634bdb82efc03c2d1649e088efb3 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/shor.py @@ -0,0 +1,173 @@ +"""Shor's algorithm and helper functions. + +Todo: + +* Get the CMod gate working again using the new Gate API. +* Fix everything. +* Update docstrings and reformat. +""" + +import math +import random + +from sympy.core.mul import Mul +from sympy.core.singleton import S +from sympy.functions.elementary.exponential import log +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.core.intfunc import igcd +from sympy.ntheory import continued_fraction_periodic as continued_fraction +from sympy.utilities.iterables import variations + +from sympy.physics.quantum.gate import Gate +from sympy.physics.quantum.qubit import Qubit, measure_partial_oneshot +from sympy.physics.quantum.qapply import qapply +from sympy.physics.quantum.qft import QFT +from sympy.physics.quantum.qexpr import QuantumError + + +class OrderFindingException(QuantumError): + pass + + +class CMod(Gate): + """A controlled mod gate. + + This is black box controlled Mod function for use by shor's algorithm. + TODO: implement a decompose property that returns how to do this in terms + of elementary gates + """ + + @classmethod + def _eval_args(cls, args): + # t = args[0] + # a = args[1] + # N = args[2] + raise NotImplementedError('The CMod gate has not been completed.') + + @property + def t(self): + """Size of 1/2 input register. First 1/2 holds output.""" + return self.label[0] + + @property + def a(self): + """Base of the controlled mod function.""" + return self.label[1] + + @property + def N(self): + """N is the type of modular arithmetic we are doing.""" + return self.label[2] + + def _apply_operator_Qubit(self, qubits, **options): + """ + This directly calculates the controlled mod of the second half of + the register and puts it in the second + This will look pretty when we get Tensor Symbolically working + """ + n = 1 + k = 0 + # Determine the value stored in high memory. + for i in range(self.t): + k += n*qubits[self.t + i] + n *= 2 + + # The value to go in low memory will be out. + out = int(self.a**k % self.N) + + # Create array for new qbit-ket which will have high memory unaffected + outarray = list(qubits.args[0][:self.t]) + + # Place out in low memory + for i in reversed(range(self.t)): + outarray.append((out >> i) & 1) + + return Qubit(*outarray) + + +def shor(N): + """This function implements Shor's factoring algorithm on the Integer N + + The algorithm starts by picking a random number (a) and seeing if it is + coprime with N. If it is not, then the gcd of the two numbers is a factor + and we are done. Otherwise, it begins the period_finding subroutine which + finds the period of a in modulo N arithmetic. This period, if even, can + be used to calculate factors by taking a**(r/2)-1 and a**(r/2)+1. + These values are returned. + """ + a = random.randrange(N - 2) + 2 + if igcd(N, a) != 1: + return igcd(N, a) + r = period_find(a, N) + if r % 2 == 1: + shor(N) + answer = (igcd(a**(r/2) - 1, N), igcd(a**(r/2) + 1, N)) + return answer + + +def getr(x, y, N): + fraction = continued_fraction(x, y) + # Now convert into r + total = ratioize(fraction, N) + return total + + +def ratioize(list, N): + if list[0] > N: + return S.Zero + if len(list) == 1: + return list[0] + return list[0] + ratioize(list[1:], N) + + +def period_find(a, N): + """Finds the period of a in modulo N arithmetic + + This is quantum part of Shor's algorithm. It takes two registers, + puts first in superposition of states with Hadamards so: ``|k>|0>`` + with k being all possible choices. It then does a controlled mod and + a QFT to determine the order of a. + """ + epsilon = .5 + # picks out t's such that maintains accuracy within epsilon + t = int(2*math.ceil(log(N, 2))) + # make the first half of register be 0's |000...000> + start = [0 for x in range(t)] + # Put second half into superposition of states so we have |1>x|0> + |2>x|0> + ... |k>x>|0> + ... + |2**n-1>x|0> + factor = 1/sqrt(2**t) + qubits = 0 + for arr in variations(range(2), t, repetition=True): + qbitArray = list(arr) + start + qubits = qubits + Qubit(*qbitArray) + circuit = (factor*qubits).expand() + # Controlled second half of register so that we have: + # |1>x|a**1 %N> + |2>x|a**2 %N> + ... + |k>x|a**k %N >+ ... + |2**n-1=k>x|a**k % n> + circuit = CMod(t, a, N)*circuit + # will measure first half of register giving one of the a**k%N's + + circuit = qapply(circuit) + for i in range(t): + circuit = measure_partial_oneshot(circuit, i) + # Now apply Inverse Quantum Fourier Transform on the second half of the register + + circuit = qapply(QFT(t, t*2).decompose()*circuit, floatingPoint=True) + for i in range(t): + circuit = measure_partial_oneshot(circuit, i + t) + if isinstance(circuit, Qubit): + register = circuit + elif isinstance(circuit, Mul): + register = circuit.args[-1] + else: + register = circuit.args[-1].args[-1] + + n = 1 + answer = 0 + for i in range(len(register)/2): + answer += n*register[i + t] + n = n << 1 + if answer == 0: + raise OrderFindingException( + "Order finder returned 0. Happens with chance %f" % epsilon) + #turn answer into r using continued fractions + g = getr(answer, 2**t, N) + return g diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/spin.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/spin.py new file mode 100644 index 0000000000000000000000000000000000000000..6c568d36c57be38702b770f6fa95f4dc6a00ed15 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/spin.py @@ -0,0 +1,2150 @@ +"""Quantum mechanical angular momemtum.""" + +from sympy.concrete.summations import Sum +from sympy.core.add import Add +from sympy.core.containers import Tuple +from sympy.core.expr import Expr +from sympy.core.numbers import int_valued +from sympy.core.mul import Mul +from sympy.core.numbers import (I, Integer, Rational, pi) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, symbols) +from sympy.core.sympify import sympify +from sympy.functions.combinatorial.factorials import (binomial, factorial) +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.simplify.simplify import simplify +from sympy.matrices import zeros +from sympy.printing.pretty.stringpict import prettyForm, stringPict +from sympy.printing.pretty.pretty_symbology import pretty_symbol + +from sympy.physics.quantum.qexpr import QExpr +from sympy.physics.quantum.operator import (HermitianOperator, Operator, + UnitaryOperator) +from sympy.physics.quantum.state import Bra, Ket, State +from sympy.functions.special.tensor_functions import KroneckerDelta +from sympy.physics.quantum.constants import hbar +from sympy.physics.quantum.hilbert import ComplexSpace, DirectSumHilbertSpace +from sympy.physics.quantum.tensorproduct import TensorProduct +from sympy.physics.quantum.cg import CG +from sympy.physics.quantum.qapply import qapply + + +__all__ = [ + 'm_values', + 'Jplus', + 'Jminus', + 'Jx', + 'Jy', + 'Jz', + 'J2', + 'Rotation', + 'WignerD', + 'JxKet', + 'JxBra', + 'JyKet', + 'JyBra', + 'JzKet', + 'JzBra', + 'JzOp', + 'J2Op', + 'JxKetCoupled', + 'JxBraCoupled', + 'JyKetCoupled', + 'JyBraCoupled', + 'JzKetCoupled', + 'JzBraCoupled', + 'couple', + 'uncouple' +] + + +def m_values(j): + j = sympify(j) + size = 2*j + 1 + if not size.is_Integer or not size > 0: + raise ValueError( + 'Only integer or half-integer values allowed for j, got: : %r' % j + ) + return size, [j - i for i in range(int(2*j + 1))] + + +#----------------------------------------------------------------------------- +# Spin Operators +#----------------------------------------------------------------------------- + + +class SpinOpBase: + """Base class for spin operators.""" + + @classmethod + def _eval_hilbert_space(cls, label): + # We consider all j values so our space is infinite. + return ComplexSpace(S.Infinity) + + @property + def name(self): + return self.args[0] + + def _print_contents(self, printer, *args): + return '%s%s' % (self.name, self._coord) + + def _print_contents_pretty(self, printer, *args): + a = stringPict(str(self.name)) + b = stringPict(self._coord) + return self._print_subscript_pretty(a, b) + + def _print_contents_latex(self, printer, *args): + return r'%s_%s' % ((self.name, self._coord)) + + def _represent_base(self, basis, **options): + j = options.get('j', S.Half) + size, mvals = m_values(j) + result = zeros(size, size) + for p in range(size): + for q in range(size): + me = self.matrix_element(j, mvals[p], j, mvals[q]) + result[p, q] = me + return result + + def _apply_op(self, ket, orig_basis, **options): + state = ket.rewrite(self.basis) + # If the state has only one term + if isinstance(state, State): + ret = (hbar*state.m)*state + # state is a linear combination of states + elif isinstance(state, Sum): + ret = self._apply_operator_Sum(state, **options) + else: + ret = qapply(self*state) + if ret == self*state: + raise NotImplementedError + return ret.rewrite(orig_basis) + + def _apply_operator_JxKet(self, ket, **options): + return self._apply_op(ket, 'Jx', **options) + + def _apply_operator_JxKetCoupled(self, ket, **options): + return self._apply_op(ket, 'Jx', **options) + + def _apply_operator_JyKet(self, ket, **options): + return self._apply_op(ket, 'Jy', **options) + + def _apply_operator_JyKetCoupled(self, ket, **options): + return self._apply_op(ket, 'Jy', **options) + + def _apply_operator_JzKet(self, ket, **options): + return self._apply_op(ket, 'Jz', **options) + + def _apply_operator_JzKetCoupled(self, ket, **options): + return self._apply_op(ket, 'Jz', **options) + + def _apply_operator_TensorProduct(self, tp, **options): + # Uncoupling operator is only easily found for coordinate basis spin operators + # TODO: add methods for uncoupling operators + if not isinstance(self, (JxOp, JyOp, JzOp)): + raise NotImplementedError + result = [] + for n in range(len(tp.args)): + arg = [] + arg.extend(tp.args[:n]) + arg.append(self._apply_operator(tp.args[n])) + arg.extend(tp.args[n + 1:]) + result.append(tp.__class__(*arg)) + return Add(*result).expand() + + # TODO: move this to qapply_Mul + def _apply_operator_Sum(self, s, **options): + new_func = qapply(self*s.function) + if new_func == self*s.function: + raise NotImplementedError + return Sum(new_func, *s.limits) + + def _eval_trace(self, **options): + #TODO: use options to use different j values + #For now eval at default basis + + # is it efficient to represent each time + # to do a trace? + return self._represent_default_basis().trace() + + +class JplusOp(SpinOpBase, Operator): + """The J+ operator.""" + + _coord = '+' + + basis = 'Jz' + + def _eval_commutator_JminusOp(self, other): + return 2*hbar*JzOp(self.name) + + def _apply_operator_JzKet(self, ket, **options): + j = ket.j + m = ket.m + if m.is_Number and j.is_Number: + if m >= j: + return S.Zero + return hbar*sqrt(j*(j + S.One) - m*(m + S.One))*JzKet(j, m + S.One) + + def _apply_operator_JzKetCoupled(self, ket, **options): + j = ket.j + m = ket.m + jn = ket.jn + coupling = ket.coupling + if m.is_Number and j.is_Number: + if m >= j: + return S.Zero + return hbar*sqrt(j*(j + S.One) - m*(m + S.One))*JzKetCoupled(j, m + S.One, jn, coupling) + + def matrix_element(self, j, m, jp, mp): + result = hbar*sqrt(j*(j + S.One) - mp*(mp + S.One)) + result *= KroneckerDelta(m, mp + 1) + result *= KroneckerDelta(j, jp) + return result + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JzOp(self, basis, **options): + return self._represent_base(basis, **options) + + def _eval_rewrite_as_xyz(self, *args, **kwargs): + return JxOp(args[0]) + I*JyOp(args[0]) + + +class JminusOp(SpinOpBase, Operator): + """The J- operator.""" + + _coord = '-' + + basis = 'Jz' + + def _apply_operator_JzKet(self, ket, **options): + j = ket.j + m = ket.m + if m.is_Number and j.is_Number: + if m <= -j: + return S.Zero + return hbar*sqrt(j*(j + S.One) - m*(m - S.One))*JzKet(j, m - S.One) + + def _apply_operator_JzKetCoupled(self, ket, **options): + j = ket.j + m = ket.m + jn = ket.jn + coupling = ket.coupling + if m.is_Number and j.is_Number: + if m <= -j: + return S.Zero + return hbar*sqrt(j*(j + S.One) - m*(m - S.One))*JzKetCoupled(j, m - S.One, jn, coupling) + + def matrix_element(self, j, m, jp, mp): + result = hbar*sqrt(j*(j + S.One) - mp*(mp - S.One)) + result *= KroneckerDelta(m, mp - 1) + result *= KroneckerDelta(j, jp) + return result + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JzOp(self, basis, **options): + return self._represent_base(basis, **options) + + def _eval_rewrite_as_xyz(self, *args, **kwargs): + return JxOp(args[0]) - I*JyOp(args[0]) + + +class JxOp(SpinOpBase, HermitianOperator): + """The Jx operator.""" + + _coord = 'x' + + basis = 'Jx' + + def _eval_commutator_JyOp(self, other): + return I*hbar*JzOp(self.name) + + def _eval_commutator_JzOp(self, other): + return -I*hbar*JyOp(self.name) + + def _apply_operator_JzKet(self, ket, **options): + jp = JplusOp(self.name)._apply_operator_JzKet(ket, **options) + jm = JminusOp(self.name)._apply_operator_JzKet(ket, **options) + return (jp + jm)/Integer(2) + + def _apply_operator_JzKetCoupled(self, ket, **options): + jp = JplusOp(self.name)._apply_operator_JzKetCoupled(ket, **options) + jm = JminusOp(self.name)._apply_operator_JzKetCoupled(ket, **options) + return (jp + jm)/Integer(2) + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JzOp(self, basis, **options): + jp = JplusOp(self.name)._represent_JzOp(basis, **options) + jm = JminusOp(self.name)._represent_JzOp(basis, **options) + return (jp + jm)/Integer(2) + + def _eval_rewrite_as_plusminus(self, *args, **kwargs): + return (JplusOp(args[0]) + JminusOp(args[0]))/2 + + +class JyOp(SpinOpBase, HermitianOperator): + """The Jy operator.""" + + _coord = 'y' + + basis = 'Jy' + + def _eval_commutator_JzOp(self, other): + return I*hbar*JxOp(self.name) + + def _eval_commutator_JxOp(self, other): + return -I*hbar*J2Op(self.name) + + def _apply_operator_JzKet(self, ket, **options): + jp = JplusOp(self.name)._apply_operator_JzKet(ket, **options) + jm = JminusOp(self.name)._apply_operator_JzKet(ket, **options) + return (jp - jm)/(Integer(2)*I) + + def _apply_operator_JzKetCoupled(self, ket, **options): + jp = JplusOp(self.name)._apply_operator_JzKetCoupled(ket, **options) + jm = JminusOp(self.name)._apply_operator_JzKetCoupled(ket, **options) + return (jp - jm)/(Integer(2)*I) + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JzOp(self, basis, **options): + jp = JplusOp(self.name)._represent_JzOp(basis, **options) + jm = JminusOp(self.name)._represent_JzOp(basis, **options) + return (jp - jm)/(Integer(2)*I) + + def _eval_rewrite_as_plusminus(self, *args, **kwargs): + return (JplusOp(args[0]) - JminusOp(args[0]))/(2*I) + + +class JzOp(SpinOpBase, HermitianOperator): + """The Jz operator.""" + + _coord = 'z' + + basis = 'Jz' + + def _eval_commutator_JxOp(self, other): + return I*hbar*JyOp(self.name) + + def _eval_commutator_JyOp(self, other): + return -I*hbar*JxOp(self.name) + + def _eval_commutator_JplusOp(self, other): + return hbar*JplusOp(self.name) + + def _eval_commutator_JminusOp(self, other): + return -hbar*JminusOp(self.name) + + def matrix_element(self, j, m, jp, mp): + result = hbar*mp + result *= KroneckerDelta(m, mp) + result *= KroneckerDelta(j, jp) + return result + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JzOp(self, basis, **options): + return self._represent_base(basis, **options) + + +class J2Op(SpinOpBase, HermitianOperator): + """The J^2 operator.""" + + _coord = '2' + + def _eval_commutator_JxOp(self, other): + return S.Zero + + def _eval_commutator_JyOp(self, other): + return S.Zero + + def _eval_commutator_JzOp(self, other): + return S.Zero + + def _eval_commutator_JplusOp(self, other): + return S.Zero + + def _eval_commutator_JminusOp(self, other): + return S.Zero + + def _apply_operator_JxKet(self, ket, **options): + j = ket.j + return hbar**2*j*(j + 1)*ket + + def _apply_operator_JxKetCoupled(self, ket, **options): + j = ket.j + return hbar**2*j*(j + 1)*ket + + def _apply_operator_JyKet(self, ket, **options): + j = ket.j + return hbar**2*j*(j + 1)*ket + + def _apply_operator_JyKetCoupled(self, ket, **options): + j = ket.j + return hbar**2*j*(j + 1)*ket + + def _apply_operator_JzKet(self, ket, **options): + j = ket.j + return hbar**2*j*(j + 1)*ket + + def _apply_operator_JzKetCoupled(self, ket, **options): + j = ket.j + return hbar**2*j*(j + 1)*ket + + def matrix_element(self, j, m, jp, mp): + result = (hbar**2)*j*(j + 1) + result *= KroneckerDelta(m, mp) + result *= KroneckerDelta(j, jp) + return result + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JzOp(self, basis, **options): + return self._represent_base(basis, **options) + + def _print_contents_pretty(self, printer, *args): + a = prettyForm(str(self.name)) + b = prettyForm('2') + return a**b + + def _print_contents_latex(self, printer, *args): + return r'%s^2' % str(self.name) + + def _eval_rewrite_as_xyz(self, *args, **kwargs): + return JxOp(args[0])**2 + JyOp(args[0])**2 + JzOp(args[0])**2 + + def _eval_rewrite_as_plusminus(self, *args, **kwargs): + a = args[0] + return JzOp(a)**2 + \ + S.Half*(JplusOp(a)*JminusOp(a) + JminusOp(a)*JplusOp(a)) + + +class Rotation(UnitaryOperator): + """Wigner D operator in terms of Euler angles. + + Defines the rotation operator in terms of the Euler angles defined by + the z-y-z convention for a passive transformation. That is the coordinate + axes are rotated first about the z-axis, giving the new x'-y'-z' axes. Then + this new coordinate system is rotated about the new y'-axis, giving new + x''-y''-z'' axes. Then this new coordinate system is rotated about the + z''-axis. Conventions follow those laid out in [1]_. + + Parameters + ========== + + alpha : Number, Symbol + First Euler Angle + beta : Number, Symbol + Second Euler angle + gamma : Number, Symbol + Third Euler angle + + Examples + ======== + + A simple example rotation operator: + + >>> from sympy import pi + >>> from sympy.physics.quantum.spin import Rotation + >>> Rotation(pi, 0, pi/2) + R(pi,0,pi/2) + + With symbolic Euler angles and calculating the inverse rotation operator: + + >>> from sympy import symbols + >>> a, b, c = symbols('a b c') + >>> Rotation(a, b, c) + R(a,b,c) + >>> Rotation(a, b, c).inverse() + R(-c,-b,-a) + + See Also + ======== + + WignerD: Symbolic Wigner-D function + D: Wigner-D function + d: Wigner small-d function + + References + ========== + + .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. + """ + + @classmethod + def _eval_args(cls, args): + args = QExpr._eval_args(args) + if len(args) != 3: + raise ValueError('3 Euler angles required, got: %r' % args) + return args + + @classmethod + def _eval_hilbert_space(cls, label): + # We consider all j values so our space is infinite. + return ComplexSpace(S.Infinity) + + @property + def alpha(self): + return self.label[0] + + @property + def beta(self): + return self.label[1] + + @property + def gamma(self): + return self.label[2] + + def _print_operator_name(self, printer, *args): + return 'R' + + def _print_operator_name_pretty(self, printer, *args): + if printer._use_unicode: + return prettyForm('\N{SCRIPT CAPITAL R}' + ' ') + else: + return prettyForm("R ") + + def _print_operator_name_latex(self, printer, *args): + return r'\mathcal{R}' + + def _eval_inverse(self): + return Rotation(-self.gamma, -self.beta, -self.alpha) + + @classmethod + def D(cls, j, m, mp, alpha, beta, gamma): + """Wigner D-function. + + Returns an instance of the WignerD class corresponding to the Wigner-D + function specified by the parameters. + + Parameters + =========== + + j : Number + Total angular momentum + m : Number + Eigenvalue of angular momentum along axis after rotation + mp : Number + Eigenvalue of angular momentum along rotated axis + alpha : Number, Symbol + First Euler angle of rotation + beta : Number, Symbol + Second Euler angle of rotation + gamma : Number, Symbol + Third Euler angle of rotation + + Examples + ======== + + Return the Wigner-D matrix element for a defined rotation, both + numerical and symbolic: + + >>> from sympy.physics.quantum.spin import Rotation + >>> from sympy import pi, symbols + >>> alpha, beta, gamma = symbols('alpha beta gamma') + >>> Rotation.D(1, 1, 0,pi, pi/2,-pi) + WignerD(1, 1, 0, pi, pi/2, -pi) + + See Also + ======== + + WignerD: Symbolic Wigner-D function + + """ + return WignerD(j, m, mp, alpha, beta, gamma) + + @classmethod + def d(cls, j, m, mp, beta): + """Wigner small-d function. + + Returns an instance of the WignerD class corresponding to the Wigner-D + function specified by the parameters with the alpha and gamma angles + given as 0. + + Parameters + =========== + + j : Number + Total angular momentum + m : Number + Eigenvalue of angular momentum along axis after rotation + mp : Number + Eigenvalue of angular momentum along rotated axis + beta : Number, Symbol + Second Euler angle of rotation + + Examples + ======== + + Return the Wigner-D matrix element for a defined rotation, both + numerical and symbolic: + + >>> from sympy.physics.quantum.spin import Rotation + >>> from sympy import pi, symbols + >>> beta = symbols('beta') + >>> Rotation.d(1, 1, 0, pi/2) + WignerD(1, 1, 0, 0, pi/2, 0) + + See Also + ======== + + WignerD: Symbolic Wigner-D function + + """ + return WignerD(j, m, mp, 0, beta, 0) + + def matrix_element(self, j, m, jp, mp): + result = self.__class__.D( + jp, m, mp, self.alpha, self.beta, self.gamma + ) + result *= KroneckerDelta(j, jp) + return result + + def _represent_base(self, basis, **options): + j = sympify(options.get('j', S.Half)) + # TODO: move evaluation up to represent function/implement elsewhere + evaluate = sympify(options.get('doit')) + size, mvals = m_values(j) + result = zeros(size, size) + for p in range(size): + for q in range(size): + me = self.matrix_element(j, mvals[p], j, mvals[q]) + if evaluate: + result[p, q] = me.doit() + else: + result[p, q] = me + return result + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JzOp(self, basis, **options): + return self._represent_base(basis, **options) + + def _apply_operator_uncoupled(self, state, ket, *, dummy=True, **options): + a = self.alpha + b = self.beta + g = self.gamma + j = ket.j + m = ket.m + if j.is_number: + s = [] + size = m_values(j) + sz = size[1] + for mp in sz: + r = Rotation.D(j, m, mp, a, b, g) + z = r.doit() + s.append(z*state(j, mp)) + return Add(*s) + else: + if dummy: + mp = Dummy('mp') + else: + mp = symbols('mp') + return Sum(Rotation.D(j, m, mp, a, b, g)*state(j, mp), (mp, -j, j)) + + def _apply_operator_JxKet(self, ket, **options): + return self._apply_operator_uncoupled(JxKet, ket, **options) + + def _apply_operator_JyKet(self, ket, **options): + return self._apply_operator_uncoupled(JyKet, ket, **options) + + def _apply_operator_JzKet(self, ket, **options): + return self._apply_operator_uncoupled(JzKet, ket, **options) + + def _apply_operator_coupled(self, state, ket, *, dummy=True, **options): + a = self.alpha + b = self.beta + g = self.gamma + j = ket.j + m = ket.m + jn = ket.jn + coupling = ket.coupling + if j.is_number: + s = [] + size = m_values(j) + sz = size[1] + for mp in sz: + r = Rotation.D(j, m, mp, a, b, g) + z = r.doit() + s.append(z*state(j, mp, jn, coupling)) + return Add(*s) + else: + if dummy: + mp = Dummy('mp') + else: + mp = symbols('mp') + return Sum(Rotation.D(j, m, mp, a, b, g)*state( + j, mp, jn, coupling), (mp, -j, j)) + + def _apply_operator_JxKetCoupled(self, ket, **options): + return self._apply_operator_coupled(JxKetCoupled, ket, **options) + + def _apply_operator_JyKetCoupled(self, ket, **options): + return self._apply_operator_coupled(JyKetCoupled, ket, **options) + + def _apply_operator_JzKetCoupled(self, ket, **options): + return self._apply_operator_coupled(JzKetCoupled, ket, **options) + +class WignerD(Expr): + r"""Wigner-D function + + The Wigner D-function gives the matrix elements of the rotation + operator in the jm-representation. For the Euler angles `\alpha`, + `\beta`, `\gamma`, the D-function is defined such that: + + .. math :: + = \delta_{jj'} D(j, m, m', \alpha, \beta, \gamma) + + Where the rotation operator is as defined by the Rotation class [1]_. + + The Wigner D-function defined in this way gives: + + .. math :: + D(j, m, m', \alpha, \beta, \gamma) = e^{-i m \alpha} d(j, m, m', \beta) e^{-i m' \gamma} + + Where d is the Wigner small-d function, which is given by Rotation.d. + + The Wigner small-d function gives the component of the Wigner + D-function that is determined by the second Euler angle. That is the + Wigner D-function is: + + .. math :: + D(j, m, m', \alpha, \beta, \gamma) = e^{-i m \alpha} d(j, m, m', \beta) e^{-i m' \gamma} + + Where d is the small-d function. The Wigner D-function is given by + Rotation.D. + + Note that to evaluate the D-function, the j, m and mp parameters must + be integer or half integer numbers. + + Parameters + ========== + + j : Number + Total angular momentum + m : Number + Eigenvalue of angular momentum along axis after rotation + mp : Number + Eigenvalue of angular momentum along rotated axis + alpha : Number, Symbol + First Euler angle of rotation + beta : Number, Symbol + Second Euler angle of rotation + gamma : Number, Symbol + Third Euler angle of rotation + + Examples + ======== + + Evaluate the Wigner-D matrix elements of a simple rotation: + + >>> from sympy.physics.quantum.spin import Rotation + >>> from sympy import pi + >>> rot = Rotation.D(1, 1, 0, pi, pi/2, 0) + >>> rot + WignerD(1, 1, 0, pi, pi/2, 0) + >>> rot.doit() + sqrt(2)/2 + + Evaluate the Wigner-d matrix elements of a simple rotation + + >>> rot = Rotation.d(1, 1, 0, pi/2) + >>> rot + WignerD(1, 1, 0, 0, pi/2, 0) + >>> rot.doit() + -sqrt(2)/2 + + See Also + ======== + + Rotation: Rotation operator + + References + ========== + + .. [1] Varshalovich, D A, Quantum Theory of Angular Momentum. 1988. + """ + + is_commutative = True + + def __new__(cls, *args, **hints): + if not len(args) == 6: + raise ValueError('6 parameters expected, got %s' % args) + args = sympify(args) + evaluate = hints.get('evaluate', False) + if evaluate: + return Expr.__new__(cls, *args)._eval_wignerd() + return Expr.__new__(cls, *args) + + @property + def j(self): + return self.args[0] + + @property + def m(self): + return self.args[1] + + @property + def mp(self): + return self.args[2] + + @property + def alpha(self): + return self.args[3] + + @property + def beta(self): + return self.args[4] + + @property + def gamma(self): + return self.args[5] + + def _latex(self, printer, *args): + if self.alpha == 0 and self.gamma == 0: + return r'd^{%s}_{%s,%s}\left(%s\right)' % \ + ( + printer._print(self.j), printer._print( + self.m), printer._print(self.mp), + printer._print(self.beta) ) + return r'D^{%s}_{%s,%s}\left(%s,%s,%s\right)' % \ + ( + printer._print( + self.j), printer._print(self.m), printer._print(self.mp), + printer._print(self.alpha), printer._print(self.beta), printer._print(self.gamma) ) + + def _pretty(self, printer, *args): + top = printer._print(self.j) + + bot = printer._print(self.m) + bot = prettyForm(*bot.right(',')) + bot = prettyForm(*bot.right(printer._print(self.mp))) + + pad = max(top.width(), bot.width()) + top = prettyForm(*top.left(' ')) + bot = prettyForm(*bot.left(' ')) + if pad > top.width(): + top = prettyForm(*top.right(' '*(pad - top.width()))) + if pad > bot.width(): + bot = prettyForm(*bot.right(' '*(pad - bot.width()))) + if self.alpha == 0 and self.gamma == 0: + args = printer._print(self.beta) + s = stringPict('d' + ' '*pad) + else: + args = printer._print(self.alpha) + args = prettyForm(*args.right(',')) + args = prettyForm(*args.right(printer._print(self.beta))) + args = prettyForm(*args.right(',')) + args = prettyForm(*args.right(printer._print(self.gamma))) + + s = stringPict('D' + ' '*pad) + + args = prettyForm(*args.parens()) + s = prettyForm(*s.above(top)) + s = prettyForm(*s.below(bot)) + s = prettyForm(*s.right(args)) + return s + + def doit(self, **hints): + hints['evaluate'] = True + return WignerD(*self.args, **hints) + + def _eval_wignerd(self): + j = self.j + m = self.m + mp = self.mp + alpha = self.alpha + beta = self.beta + gamma = self.gamma + if alpha == 0 and beta == 0 and gamma == 0: + return KroneckerDelta(m, mp) + if not j.is_number: + raise ValueError( + 'j parameter must be numerical to evaluate, got %s' % j) + r = 0 + if beta == pi/2: + # Varshalovich Equation (5), Section 4.16, page 113, setting + # alpha=gamma=0. + for k in range(2*j + 1): + if k > j + mp or k > j - m or k < mp - m: + continue + r += (S.NegativeOne)**k*binomial(j + mp, k)*binomial(j - mp, k + m - mp) + r *= (S.NegativeOne)**(m - mp) / 2**j*sqrt(factorial(j + m) * + factorial(j - m) / (factorial(j + mp)*factorial(j - mp))) + else: + # Varshalovich Equation(5), Section 4.7.2, page 87, where we set + # beta1=beta2=pi/2, and we get alpha=gamma=pi/2 and beta=phi+pi, + # then we use the Eq. (1), Section 4.4. page 79, to simplify: + # d(j, m, mp, beta+pi) = (-1)**(j-mp)*d(j, m, -mp, beta) + # This happens to be almost the same as in Eq.(10), Section 4.16, + # except that we need to substitute -mp for mp. + size, mvals = m_values(j) + for mpp in mvals: + r += Rotation.d(j, m, mpp, pi/2).doit()*(cos(-mpp*beta) + I*sin(-mpp*beta))*\ + Rotation.d(j, mpp, -mp, pi/2).doit() + # Empirical normalization factor so results match Varshalovich + # Tables 4.3-4.12 + # Note that this exact normalization does not follow from the + # above equations + r = r*I**(2*j - m - mp)*(-1)**(2*m) + # Finally, simplify the whole expression + r = simplify(r) + r *= exp(-I*m*alpha)*exp(-I*mp*gamma) + return r + + +Jx = JxOp('J') +Jy = JyOp('J') +Jz = JzOp('J') +J2 = J2Op('J') +Jplus = JplusOp('J') +Jminus = JminusOp('J') + + +#----------------------------------------------------------------------------- +# Spin States +#----------------------------------------------------------------------------- + + +class SpinState(State): + """Base class for angular momentum states.""" + + _label_separator = ',' + + def __new__(cls, j, m): + j = sympify(j) + m = sympify(m) + if j.is_number: + if 2*j != int(2*j): + raise ValueError( + 'j must be integer or half-integer, got: %s' % j) + if j < 0: + raise ValueError('j must be >= 0, got: %s' % j) + if m.is_number: + if 2*m != int(2*m): + raise ValueError( + 'm must be integer or half-integer, got: %s' % m) + if j.is_number and m.is_number: + if abs(m) > j: + raise ValueError('Allowed values for m are -j <= m <= j, got j, m: %s, %s' % (j, m)) + if int(j - m) != j - m: + raise ValueError('Both j and m must be integer or half-integer, got j, m: %s, %s' % (j, m)) + return State.__new__(cls, j, m) + + @property + def j(self): + return self.label[0] + + @property + def m(self): + return self.label[1] + + @classmethod + def _eval_hilbert_space(cls, label): + return ComplexSpace(2*label[0] + 1) + + def _represent_base(self, **options): + j = self.j + m = self.m + alpha = sympify(options.get('alpha', 0)) + beta = sympify(options.get('beta', 0)) + gamma = sympify(options.get('gamma', 0)) + size, mvals = m_values(j) + result = zeros(size, 1) + # breaks finding angles on L930 + for p, mval in enumerate(mvals): + if m.is_number: + result[p, 0] = Rotation.D( + self.j, mval, self.m, alpha, beta, gamma).doit() + else: + result[p, 0] = Rotation.D(self.j, mval, + self.m, alpha, beta, gamma) + return result + + def _eval_rewrite_as_Jx(self, *args, **options): + if isinstance(self, Bra): + return self._rewrite_basis(Jx, JxBra, **options) + return self._rewrite_basis(Jx, JxKet, **options) + + def _eval_rewrite_as_Jy(self, *args, **options): + if isinstance(self, Bra): + return self._rewrite_basis(Jy, JyBra, **options) + return self._rewrite_basis(Jy, JyKet, **options) + + def _eval_rewrite_as_Jz(self, *args, **options): + if isinstance(self, Bra): + return self._rewrite_basis(Jz, JzBra, **options) + return self._rewrite_basis(Jz, JzKet, **options) + + def _rewrite_basis(self, basis, evect, **options): + from sympy.physics.quantum.represent import represent + j = self.j + args = self.args[2:] + if j.is_number: + if isinstance(self, CoupledSpinState): + if j == int(j): + start = j**2 + else: + start = (2*j - 1)*(2*j + 1)/4 + else: + start = 0 + vect = represent(self, basis=basis, **options) + result = Add( + *[vect[start + i]*evect(j, j - i, *args) for i in range(2*j + 1)]) + if isinstance(self, CoupledSpinState) and options.get('coupled') is False: + return uncouple(result) + return result + else: + i = 0 + mi = symbols('mi') + # make sure not to introduce a symbol already in the state + while self.subs(mi, 0) != self: + i += 1 + mi = symbols('mi%d' % i) + break + # TODO: better way to get angles of rotation + if isinstance(self, CoupledSpinState): + test_args = (0, mi, (0, 0)) + else: + test_args = (0, mi) + if isinstance(self, Ket): + angles = represent( + self.__class__(*test_args), basis=basis)[0].args[3:6] + else: + angles = represent(self.__class__( + *test_args), basis=basis)[0].args[0].args[3:6] + if angles == (0, 0, 0): + return self + else: + state = evect(j, mi, *args) + lt = Rotation.D(j, mi, self.m, *angles) + return Sum(lt*state, (mi, -j, j)) + + def _eval_innerproduct_JxBra(self, bra, **hints): + result = KroneckerDelta(self.j, bra.j) + if bra.dual_class() is not self.__class__: + result *= self._represent_JxOp(None)[bra.j - bra.m] + else: + result *= KroneckerDelta( + self.j, bra.j)*KroneckerDelta(self.m, bra.m) + return result + + def _eval_innerproduct_JyBra(self, bra, **hints): + result = KroneckerDelta(self.j, bra.j) + if bra.dual_class() is not self.__class__: + result *= self._represent_JyOp(None)[bra.j - bra.m] + else: + result *= KroneckerDelta( + self.j, bra.j)*KroneckerDelta(self.m, bra.m) + return result + + def _eval_innerproduct_JzBra(self, bra, **hints): + result = KroneckerDelta(self.j, bra.j) + if bra.dual_class() is not self.__class__: + result *= self._represent_JzOp(None)[bra.j - bra.m] + else: + result *= KroneckerDelta( + self.j, bra.j)*KroneckerDelta(self.m, bra.m) + return result + + def _eval_trace(self, bra, **hints): + + # One way to implement this method is to assume the basis set k is + # passed. + # Then we can apply the discrete form of Trace formula here + # Tr(|i> + #then we do qapply() on each each inner product and sum over them. + + # OR + + # Inner product of |i>>> from sympy.physics.quantum.spin import JzKet, JxKet + >>> from sympy import symbols + >>> JzKet(1, 0) + |1,0> + >>> j, m = symbols('j m') + >>> JzKet(j, m) + |j,m> + + Rewriting the JzKet in terms of eigenkets of the Jx operator: + Note: that the resulting eigenstates are JxKet's + + >>> JzKet(1,1).rewrite("Jx") + |1,-1>/2 - sqrt(2)*|1,0>/2 + |1,1>/2 + + Get the vector representation of a state in terms of the basis elements + of the Jx operator: + + >>> from sympy.physics.quantum.represent import represent + >>> from sympy.physics.quantum.spin import Jx, Jz + >>> represent(JzKet(1,-1), basis=Jx) + Matrix([ + [ 1/2], + [sqrt(2)/2], + [ 1/2]]) + + Apply innerproducts between states: + + >>> from sympy.physics.quantum.innerproduct import InnerProduct + >>> from sympy.physics.quantum.spin import JxBra + >>> i = InnerProduct(JxBra(1,1), JzKet(1,1)) + >>> i + <1,1|1,1> + >>> i.doit() + 1/2 + + *Uncoupled States:* + + Define an uncoupled state as a TensorProduct between two Jz eigenkets: + + >>> from sympy.physics.quantum.tensorproduct import TensorProduct + >>> j1,m1,j2,m2 = symbols('j1 m1 j2 m2') + >>> TensorProduct(JzKet(1,0), JzKet(1,1)) + |1,0>x|1,1> + >>> TensorProduct(JzKet(j1,m1), JzKet(j2,m2)) + |j1,m1>x|j2,m2> + + A TensorProduct can be rewritten, in which case the eigenstates that make + up the tensor product is rewritten to the new basis: + + >>> TensorProduct(JzKet(1,1),JxKet(1,1)).rewrite('Jz') + |1,1>x|1,-1>/2 + sqrt(2)*|1,1>x|1,0>/2 + |1,1>x|1,1>/2 + + The represent method for TensorProduct's gives the vector representation of + the state. Note that the state in the product basis is the equivalent of the + tensor product of the vector representation of the component eigenstates: + + >>> represent(TensorProduct(JzKet(1,0),JzKet(1,1))) + Matrix([ + [0], + [0], + [0], + [1], + [0], + [0], + [0], + [0], + [0]]) + >>> represent(TensorProduct(JzKet(1,1),JxKet(1,1)), basis=Jz) + Matrix([ + [ 1/2], + [sqrt(2)/2], + [ 1/2], + [ 0], + [ 0], + [ 0], + [ 0], + [ 0], + [ 0]]) + + See Also + ======== + + JzKetCoupled: Coupled eigenstates + sympy.physics.quantum.tensorproduct.TensorProduct: Used to specify uncoupled states + uncouple: Uncouples states given coupling parameters + couple: Couples uncoupled states + + """ + + @classmethod + def dual_class(self): + return JzBra + + @classmethod + def coupled_class(self): + return JzKetCoupled + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JxOp(self, basis, **options): + return self._represent_base(beta=pi*Rational(3, 2), **options) + + def _represent_JyOp(self, basis, **options): + return self._represent_base(alpha=pi*Rational(3, 2), beta=pi/2, gamma=pi/2, **options) + + def _represent_JzOp(self, basis, **options): + return self._represent_base(**options) + + +class JzBra(SpinState, Bra): + """Eigenbra of Jz. + + See the JzKet for the usage of spin eigenstates. + + See Also + ======== + + JzKet: Usage of spin states + + """ + + @classmethod + def dual_class(self): + return JzKet + + @classmethod + def coupled_class(self): + return JzBraCoupled + + +# Method used primarily to create coupled_n and coupled_jn by __new__ in +# CoupledSpinState +# This same method is also used by the uncouple method, and is separated from +# the CoupledSpinState class to maintain consistency in defining coupling +def _build_coupled(jcoupling, length): + n_list = [ [n + 1] for n in range(length) ] + coupled_jn = [] + coupled_n = [] + for n1, n2, j_new in jcoupling: + coupled_jn.append(j_new) + coupled_n.append( (n_list[n1 - 1], n_list[n2 - 1]) ) + n_sort = sorted(n_list[n1 - 1] + n_list[n2 - 1]) + n_list[n_sort[0] - 1] = n_sort + return coupled_n, coupled_jn + + +class CoupledSpinState(SpinState): + """Base class for coupled angular momentum states.""" + + def __new__(cls, j, m, jn, *jcoupling): + # Check j and m values using SpinState + SpinState(j, m) + # Build and check coupling scheme from arguments + if len(jcoupling) == 0: + # Use default coupling scheme + jcoupling = [] + for n in range(2, len(jn)): + jcoupling.append( (1, n, Add(*[jn[i] for i in range(n)])) ) + jcoupling.append( (1, len(jn), j) ) + elif len(jcoupling) == 1: + # Use specified coupling scheme + jcoupling = jcoupling[0] + else: + raise TypeError("CoupledSpinState only takes 3 or 4 arguments, got: %s" % (len(jcoupling) + 3) ) + # Check arguments have correct form + if not isinstance(jn, (list, tuple, Tuple)): + raise TypeError('jn must be Tuple, list or tuple, got %s' % + jn.__class__.__name__) + if not isinstance(jcoupling, (list, tuple, Tuple)): + raise TypeError('jcoupling must be Tuple, list or tuple, got %s' % + jcoupling.__class__.__name__) + if not all(isinstance(term, (list, tuple, Tuple)) for term in jcoupling): + raise TypeError( + 'All elements of jcoupling must be list, tuple or Tuple') + if not len(jn) - 1 == len(jcoupling): + raise ValueError('jcoupling must have length of %d, got %d' % + (len(jn) - 1, len(jcoupling))) + if not all(len(x) == 3 for x in jcoupling): + raise ValueError('All elements of jcoupling must have length 3') + # Build sympified args + j = sympify(j) + m = sympify(m) + jn = Tuple( *[sympify(ji) for ji in jn] ) + jcoupling = Tuple( *[Tuple(sympify( + n1), sympify(n2), sympify(ji)) for (n1, n2, ji) in jcoupling] ) + # Check values in coupling scheme give physical state + if any(2*ji != int(2*ji) for ji in jn if ji.is_number): + raise ValueError('All elements of jn must be integer or half-integer, got: %s' % jn) + if any(n1 != int(n1) or n2 != int(n2) for (n1, n2, _) in jcoupling): + raise ValueError('Indices in jcoupling must be integers') + if any(n1 < 1 or n2 < 1 or n1 > len(jn) or n2 > len(jn) for (n1, n2, _) in jcoupling): + raise ValueError('Indices must be between 1 and the number of coupled spin spaces') + if any(2*ji != int(2*ji) for (_, _, ji) in jcoupling if ji.is_number): + raise ValueError('All coupled j values in coupling scheme must be integer or half-integer') + coupled_n, coupled_jn = _build_coupled(jcoupling, len(jn)) + jvals = list(jn) + for n, (n1, n2) in enumerate(coupled_n): + j1 = jvals[min(n1) - 1] + j2 = jvals[min(n2) - 1] + j3 = coupled_jn[n] + if sympify(j1).is_number and sympify(j2).is_number and sympify(j3).is_number: + if j1 + j2 < j3: + raise ValueError('All couplings must have j1+j2 >= j3, ' + 'in coupling number %d got j1,j2,j3: %d,%d,%d' % (n + 1, j1, j2, j3)) + if abs(j1 - j2) > j3: + raise ValueError("All couplings must have |j1+j2| <= j3, " + "in coupling number %d got j1,j2,j3: %d,%d,%d" % (n + 1, j1, j2, j3)) + if int_valued(j1 + j2): + pass + jvals[min(n1 + n2) - 1] = j3 + if len(jcoupling) > 0 and jcoupling[-1][2] != j: + raise ValueError('Last j value coupled together must be the final j of the state') + # Return state + return State.__new__(cls, j, m, jn, jcoupling) + + def _print_label(self, printer, *args): + label = [printer._print(self.j), printer._print(self.m)] + for i, ji in enumerate(self.jn, start=1): + label.append('j%d=%s' % ( + i, printer._print(ji) + )) + for jn, (n1, n2) in zip(self.coupled_jn[:-1], self.coupled_n[:-1]): + label.append('j(%s)=%s' % ( + ','.join(str(i) for i in sorted(n1 + n2)), printer._print(jn) + )) + return ','.join(label) + + def _print_label_pretty(self, printer, *args): + label = [self.j, self.m] + for i, ji in enumerate(self.jn, start=1): + symb = 'j%d' % i + symb = pretty_symbol(symb) + symb = prettyForm(symb + '=') + item = prettyForm(*symb.right(printer._print(ji))) + label.append(item) + for jn, (n1, n2) in zip(self.coupled_jn[:-1], self.coupled_n[:-1]): + n = ','.join(pretty_symbol("j%d" % i)[-1] for i in sorted(n1 + n2)) + symb = prettyForm('j' + n + '=') + item = prettyForm(*symb.right(printer._print(jn))) + label.append(item) + return self._print_sequence_pretty( + label, self._label_separator, printer, *args + ) + + def _print_label_latex(self, printer, *args): + label = [ + printer._print(self.j, *args), + printer._print(self.m, *args) + ] + for i, ji in enumerate(self.jn, start=1): + label.append('j_{%d}=%s' % (i, printer._print(ji, *args)) ) + for jn, (n1, n2) in zip(self.coupled_jn[:-1], self.coupled_n[:-1]): + n = ','.join(str(i) for i in sorted(n1 + n2)) + label.append('j_{%s}=%s' % (n, printer._print(jn, *args)) ) + return self._label_separator.join(label) + + @property + def jn(self): + return self.label[2] + + @property + def coupling(self): + return self.label[3] + + @property + def coupled_jn(self): + return _build_coupled(self.label[3], len(self.label[2]))[1] + + @property + def coupled_n(self): + return _build_coupled(self.label[3], len(self.label[2]))[0] + + @classmethod + def _eval_hilbert_space(cls, label): + j = Add(*label[2]) + if j.is_number: + return DirectSumHilbertSpace(*[ ComplexSpace(x) for x in range(int(2*j + 1), 0, -2) ]) + else: + # TODO: Need hilbert space fix, see issue 5732 + # Desired behavior: + #ji = symbols('ji') + #ret = Sum(ComplexSpace(2*ji + 1), (ji, 0, j)) + # Temporary fix: + return ComplexSpace(2*j + 1) + + def _represent_coupled_base(self, **options): + evect = self.uncoupled_class() + if not self.j.is_number: + raise ValueError( + 'State must not have symbolic j value to represent') + if not self.hilbert_space.dimension.is_number: + raise ValueError( + 'State must not have symbolic j values to represent') + result = zeros(self.hilbert_space.dimension, 1) + if self.j == int(self.j): + start = self.j**2 + else: + start = (2*self.j - 1)*(1 + 2*self.j)/4 + result[start:start + 2*self.j + 1, 0] = evect( + self.j, self.m)._represent_base(**options) + return result + + def _eval_rewrite_as_Jx(self, *args, **options): + if isinstance(self, Bra): + return self._rewrite_basis(Jx, JxBraCoupled, **options) + return self._rewrite_basis(Jx, JxKetCoupled, **options) + + def _eval_rewrite_as_Jy(self, *args, **options): + if isinstance(self, Bra): + return self._rewrite_basis(Jy, JyBraCoupled, **options) + return self._rewrite_basis(Jy, JyKetCoupled, **options) + + def _eval_rewrite_as_Jz(self, *args, **options): + if isinstance(self, Bra): + return self._rewrite_basis(Jz, JzBraCoupled, **options) + return self._rewrite_basis(Jz, JzKetCoupled, **options) + + +class JxKetCoupled(CoupledSpinState, Ket): + """Coupled eigenket of Jx. + + See JzKetCoupled for the usage of coupled spin eigenstates. + + See Also + ======== + + JzKetCoupled: Usage of coupled spin states + + """ + + @classmethod + def dual_class(self): + return JxBraCoupled + + @classmethod + def uncoupled_class(self): + return JxKet + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JxOp(self, basis, **options): + return self._represent_coupled_base(**options) + + def _represent_JyOp(self, basis, **options): + return self._represent_coupled_base(alpha=pi*Rational(3, 2), **options) + + def _represent_JzOp(self, basis, **options): + return self._represent_coupled_base(beta=pi/2, **options) + + +class JxBraCoupled(CoupledSpinState, Bra): + """Coupled eigenbra of Jx. + + See JzKetCoupled for the usage of coupled spin eigenstates. + + See Also + ======== + + JzKetCoupled: Usage of coupled spin states + + """ + + @classmethod + def dual_class(self): + return JxKetCoupled + + @classmethod + def uncoupled_class(self): + return JxBra + + +class JyKetCoupled(CoupledSpinState, Ket): + """Coupled eigenket of Jy. + + See JzKetCoupled for the usage of coupled spin eigenstates. + + See Also + ======== + + JzKetCoupled: Usage of coupled spin states + + """ + + @classmethod + def dual_class(self): + return JyBraCoupled + + @classmethod + def uncoupled_class(self): + return JyKet + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JxOp(self, basis, **options): + return self._represent_coupled_base(gamma=pi/2, **options) + + def _represent_JyOp(self, basis, **options): + return self._represent_coupled_base(**options) + + def _represent_JzOp(self, basis, **options): + return self._represent_coupled_base(alpha=pi*Rational(3, 2), beta=-pi/2, gamma=pi/2, **options) + + +class JyBraCoupled(CoupledSpinState, Bra): + """Coupled eigenbra of Jy. + + See JzKetCoupled for the usage of coupled spin eigenstates. + + See Also + ======== + + JzKetCoupled: Usage of coupled spin states + + """ + + @classmethod + def dual_class(self): + return JyKetCoupled + + @classmethod + def uncoupled_class(self): + return JyBra + + +class JzKetCoupled(CoupledSpinState, Ket): + r"""Coupled eigenket of Jz + + Spin state that is an eigenket of Jz which represents the coupling of + separate spin spaces. + + The arguments for creating instances of JzKetCoupled are ``j``, ``m``, + ``jn`` and an optional ``jcoupling`` argument. The ``j`` and ``m`` options + are the total angular momentum quantum numbers, as used for normal states + (e.g. JzKet). + + The other required parameter in ``jn``, which is a tuple defining the `j_n` + angular momentum quantum numbers of the product spaces. So for example, if + a state represented the coupling of the product basis state + `\left|j_1,m_1\right\rangle\times\left|j_2,m_2\right\rangle`, the ``jn`` + for this state would be ``(j1,j2)``. + + The final option is ``jcoupling``, which is used to define how the spaces + specified by ``jn`` are coupled, which includes both the order these spaces + are coupled together and the quantum numbers that arise from these + couplings. The ``jcoupling`` parameter itself is a list of lists, such that + each of the sublists defines a single coupling between the spin spaces. If + there are N coupled angular momentum spaces, that is ``jn`` has N elements, + then there must be N-1 sublists. Each of these sublists making up the + ``jcoupling`` parameter have length 3. The first two elements are the + indices of the product spaces that are considered to be coupled together. + For example, if we want to couple `j_1` and `j_4`, the indices would be 1 + and 4. If a state has already been coupled, it is referenced by the + smallest index that is coupled, so if `j_2` and `j_4` has already been + coupled to some `j_{24}`, then this value can be coupled by referencing it + with index 2. The final element of the sublist is the quantum number of the + coupled state. So putting everything together, into a valid sublist for + ``jcoupling``, if `j_1` and `j_2` are coupled to an angular momentum space + with quantum number `j_{12}` with the value ``j12``, the sublist would be + ``(1,2,j12)``, N-1 of these sublists are used in the list for + ``jcoupling``. + + Note the ``jcoupling`` parameter is optional, if it is not specified, the + default coupling is taken. This default value is to coupled the spaces in + order and take the quantum number of the coupling to be the maximum value. + For example, if the spin spaces are `j_1`, `j_2`, `j_3`, `j_4`, then the + default coupling couples `j_1` and `j_2` to `j_{12}=j_1+j_2`, then, + `j_{12}` and `j_3` are coupled to `j_{123}=j_{12}+j_3`, and finally + `j_{123}` and `j_4` to `j=j_{123}+j_4`. The jcoupling value that would + correspond to this is: + + ``((1,2,j1+j2),(1,3,j1+j2+j3))`` + + Parameters + ========== + + args : tuple + The arguments that must be passed are ``j``, ``m``, ``jn``, and + ``jcoupling``. The ``j`` value is the total angular momentum. The ``m`` + value is the eigenvalue of the Jz spin operator. The ``jn`` list are + the j values of argular momentum spaces coupled together. The + ``jcoupling`` parameter is an optional parameter defining how the spaces + are coupled together. See the above description for how these coupling + parameters are defined. + + Examples + ======== + + Defining simple spin states, both numerical and symbolic: + + >>> from sympy.physics.quantum.spin import JzKetCoupled + >>> from sympy import symbols + >>> JzKetCoupled(1, 0, (1, 1)) + |1,0,j1=1,j2=1> + >>> j, m, j1, j2 = symbols('j m j1 j2') + >>> JzKetCoupled(j, m, (j1, j2)) + |j,m,j1=j1,j2=j2> + + Defining coupled spin states for more than 2 coupled spaces with various + coupling parameters: + + >>> JzKetCoupled(2, 1, (1, 1, 1)) + |2,1,j1=1,j2=1,j3=1,j(1,2)=2> + >>> JzKetCoupled(2, 1, (1, 1, 1), ((1,2,2),(1,3,2)) ) + |2,1,j1=1,j2=1,j3=1,j(1,2)=2> + >>> JzKetCoupled(2, 1, (1, 1, 1), ((2,3,1),(1,2,2)) ) + |2,1,j1=1,j2=1,j3=1,j(2,3)=1> + + Rewriting the JzKetCoupled in terms of eigenkets of the Jx operator: + Note: that the resulting eigenstates are JxKetCoupled + + >>> JzKetCoupled(1,1,(1,1)).rewrite("Jx") + |1,-1,j1=1,j2=1>/2 - sqrt(2)*|1,0,j1=1,j2=1>/2 + |1,1,j1=1,j2=1>/2 + + The rewrite method can be used to convert a coupled state to an uncoupled + state. This is done by passing coupled=False to the rewrite function: + + >>> JzKetCoupled(1, 0, (1, 1)).rewrite('Jz', coupled=False) + -sqrt(2)*|1,-1>x|1,1>/2 + sqrt(2)*|1,1>x|1,-1>/2 + + Get the vector representation of a state in terms of the basis elements + of the Jx operator: + + >>> from sympy.physics.quantum.represent import represent + >>> from sympy.physics.quantum.spin import Jx + >>> from sympy import S + >>> represent(JzKetCoupled(1,-1,(S(1)/2,S(1)/2)), basis=Jx) + Matrix([ + [ 0], + [ 1/2], + [sqrt(2)/2], + [ 1/2]]) + + See Also + ======== + + JzKet: Normal spin eigenstates + uncouple: Uncoupling of coupling spin states + couple: Coupling of uncoupled spin states + + """ + + @classmethod + def dual_class(self): + return JzBraCoupled + + @classmethod + def uncoupled_class(self): + return JzKet + + def _represent_default_basis(self, **options): + return self._represent_JzOp(None, **options) + + def _represent_JxOp(self, basis, **options): + return self._represent_coupled_base(beta=pi*Rational(3, 2), **options) + + def _represent_JyOp(self, basis, **options): + return self._represent_coupled_base(alpha=pi*Rational(3, 2), beta=pi/2, gamma=pi/2, **options) + + def _represent_JzOp(self, basis, **options): + return self._represent_coupled_base(**options) + + +class JzBraCoupled(CoupledSpinState, Bra): + """Coupled eigenbra of Jz. + + See the JzKetCoupled for the usage of coupled spin eigenstates. + + See Also + ======== + + JzKetCoupled: Usage of coupled spin states + + """ + + @classmethod + def dual_class(self): + return JzKetCoupled + + @classmethod + def uncoupled_class(self): + return JzBra + +#----------------------------------------------------------------------------- +# Coupling/uncoupling +#----------------------------------------------------------------------------- + + +def couple(expr, jcoupling_list=None): + """ Couple a tensor product of spin states + + This function can be used to couple an uncoupled tensor product of spin + states. All of the eigenstates to be coupled must be of the same class. It + will return a linear combination of eigenstates that are subclasses of + CoupledSpinState determined by Clebsch-Gordan angular momentum coupling + coefficients. + + Parameters + ========== + + expr : Expr + An expression involving TensorProducts of spin states to be coupled. + Each state must be a subclass of SpinState and they all must be the + same class. + + jcoupling_list : list or tuple + Elements of this list are sub-lists of length 2 specifying the order of + the coupling of the spin spaces. The length of this must be N-1, where N + is the number of states in the tensor product to be coupled. The + elements of this sublist are the same as the first two elements of each + sublist in the ``jcoupling`` parameter defined for JzKetCoupled. If this + parameter is not specified, the default value is taken, which couples + the first and second product basis spaces, then couples this new coupled + space to the third product space, etc + + Examples + ======== + + Couple a tensor product of numerical states for two spaces: + + >>> from sympy.physics.quantum.spin import JzKet, couple + >>> from sympy.physics.quantum.tensorproduct import TensorProduct + >>> couple(TensorProduct(JzKet(1,0), JzKet(1,1))) + -sqrt(2)*|1,1,j1=1,j2=1>/2 + sqrt(2)*|2,1,j1=1,j2=1>/2 + + + Numerical coupling of three spaces using the default coupling method, i.e. + first and second spaces couple, then this couples to the third space: + + >>> couple(TensorProduct(JzKet(1,1), JzKet(1,1), JzKet(1,0))) + sqrt(6)*|2,2,j1=1,j2=1,j3=1,j(1,2)=2>/3 + sqrt(3)*|3,2,j1=1,j2=1,j3=1,j(1,2)=2>/3 + + Perform this same coupling, but we define the coupling to first couple + the first and third spaces: + + >>> couple(TensorProduct(JzKet(1,1), JzKet(1,1), JzKet(1,0)), ((1,3),(1,2)) ) + sqrt(2)*|2,2,j1=1,j2=1,j3=1,j(1,3)=1>/2 - sqrt(6)*|2,2,j1=1,j2=1,j3=1,j(1,3)=2>/6 + sqrt(3)*|3,2,j1=1,j2=1,j3=1,j(1,3)=2>/3 + + Couple a tensor product of symbolic states: + + >>> from sympy import symbols + >>> j1,m1,j2,m2 = symbols('j1 m1 j2 m2') + >>> couple(TensorProduct(JzKet(j1,m1), JzKet(j2,m2))) + Sum(CG(j1, m1, j2, m2, j, m1 + m2)*|j,m1 + m2,j1=j1,j2=j2>, (j, m1 + m2, j1 + j2)) + + """ + a = expr.atoms(TensorProduct) + for tp in a: + # Allow other tensor products to be in expression + if not all(isinstance(state, SpinState) for state in tp.args): + continue + # If tensor product has all spin states, raise error for invalid tensor product state + if not all(state.__class__ is tp.args[0].__class__ for state in tp.args): + raise TypeError('All states must be the same basis') + expr = expr.subs(tp, _couple(tp, jcoupling_list)) + return expr + + +def _couple(tp, jcoupling_list): + states = tp.args + coupled_evect = states[0].coupled_class() + + # Define default coupling if none is specified + if jcoupling_list is None: + jcoupling_list = [] + for n in range(1, len(states)): + jcoupling_list.append( (1, n + 1) ) + + # Check jcoupling_list valid + if not len(jcoupling_list) == len(states) - 1: + raise TypeError('jcoupling_list must be length %d, got %d' % + (len(states) - 1, len(jcoupling_list))) + if not all( len(coupling) == 2 for coupling in jcoupling_list): + raise ValueError('Each coupling must define 2 spaces') + if any(n1 == n2 for n1, n2 in jcoupling_list): + raise ValueError('Spin spaces cannot couple to themselves') + if all(sympify(n1).is_number and sympify(n2).is_number for n1, n2 in jcoupling_list): + j_test = [0]*len(states) + for n1, n2 in jcoupling_list: + if j_test[n1 - 1] == -1 or j_test[n2 - 1] == -1: + raise ValueError('Spaces coupling j_n\'s are referenced by smallest n value') + j_test[max(n1, n2) - 1] = -1 + + # j values of states to be coupled together + jn = [state.j for state in states] + mn = [state.m for state in states] + + # Create coupling_list, which defines all the couplings between all + # the spaces from jcoupling_list + coupling_list = [] + n_list = [ [i + 1] for i in range(len(states)) ] + for j_coupling in jcoupling_list: + # Least n for all j_n which is coupled as first and second spaces + n1, n2 = j_coupling + # List of all n's coupled in first and second spaces + j1_n = list(n_list[n1 - 1]) + j2_n = list(n_list[n2 - 1]) + coupling_list.append( (j1_n, j2_n) ) + # Set new j_n to be coupling of all j_n in both first and second spaces + n_list[ min(n1, n2) - 1 ] = sorted(j1_n + j2_n) + + if all(state.j.is_number and state.m.is_number for state in states): + # Numerical coupling + # Iterate over difference between maximum possible j value of each coupling and the actual value + diff_max = [ Add( *[ jn[n - 1] - mn[n - 1] for n in coupling[0] + + coupling[1] ] ) for coupling in coupling_list ] + result = [] + for diff in range(diff_max[-1] + 1): + # Determine available configurations + n = len(coupling_list) + tot = binomial(diff + n - 1, diff) + + for config_num in range(tot): + diff_list = _confignum_to_difflist(config_num, diff, n) + + # Skip the configuration if non-physical + # This is a lazy check for physical states given the loose restrictions of diff_max + if any(d > m for d, m in zip(diff_list, diff_max)): + continue + + # Determine term + cg_terms = [] + coupled_j = list(jn) + jcoupling = [] + for (j1_n, j2_n), coupling_diff in zip(coupling_list, diff_list): + j1 = coupled_j[ min(j1_n) - 1 ] + j2 = coupled_j[ min(j2_n) - 1 ] + j3 = j1 + j2 - coupling_diff + coupled_j[ min(j1_n + j2_n) - 1 ] = j3 + m1 = Add( *[ mn[x - 1] for x in j1_n] ) + m2 = Add( *[ mn[x - 1] for x in j2_n] ) + m3 = m1 + m2 + cg_terms.append( (j1, m1, j2, m2, j3, m3) ) + jcoupling.append( (min(j1_n), min(j2_n), j3) ) + # Better checks that state is physical + if any(abs(term[5]) > term[4] for term in cg_terms): + continue + if any(term[0] + term[2] < term[4] for term in cg_terms): + continue + if any(abs(term[0] - term[2]) > term[4] for term in cg_terms): + continue + coeff = Mul( *[ CG(*term).doit() for term in cg_terms] ) + state = coupled_evect(j3, m3, jn, jcoupling) + result.append(coeff*state) + return Add(*result) + else: + # Symbolic coupling + cg_terms = [] + jcoupling = [] + sum_terms = [] + coupled_j = list(jn) + for j1_n, j2_n in coupling_list: + j1 = coupled_j[ min(j1_n) - 1 ] + j2 = coupled_j[ min(j2_n) - 1 ] + if len(j1_n + j2_n) == len(states): + j3 = symbols('j') + else: + j3_name = 'j' + ''.join(["%s" % n for n in j1_n + j2_n]) + j3 = symbols(j3_name) + coupled_j[ min(j1_n + j2_n) - 1 ] = j3 + m1 = Add( *[ mn[x - 1] for x in j1_n] ) + m2 = Add( *[ mn[x - 1] for x in j2_n] ) + m3 = m1 + m2 + cg_terms.append( (j1, m1, j2, m2, j3, m3) ) + jcoupling.append( (min(j1_n), min(j2_n), j3) ) + sum_terms.append((j3, m3, j1 + j2)) + coeff = Mul( *[ CG(*term) for term in cg_terms] ) + state = coupled_evect(j3, m3, jn, jcoupling) + return Sum(coeff*state, *sum_terms) + + +def uncouple(expr, jn=None, jcoupling_list=None): + """ Uncouple a coupled spin state + + Gives the uncoupled representation of a coupled spin state. Arguments must + be either a spin state that is a subclass of CoupledSpinState or a spin + state that is a subclass of SpinState and an array giving the j values + of the spaces that are to be coupled + + Parameters + ========== + + expr : Expr + The expression containing states that are to be coupled. If the states + are a subclass of SpinState, the ``jn`` and ``jcoupling`` parameters + must be defined. If the states are a subclass of CoupledSpinState, + ``jn`` and ``jcoupling`` will be taken from the state. + + jn : list or tuple + The list of the j-values that are coupled. If state is a + CoupledSpinState, this parameter is ignored. This must be defined if + state is not a subclass of CoupledSpinState. The syntax of this + parameter is the same as the ``jn`` parameter of JzKetCoupled. + + jcoupling_list : list or tuple + The list defining how the j-values are coupled together. If state is a + CoupledSpinState, this parameter is ignored. This must be defined if + state is not a subclass of CoupledSpinState. The syntax of this + parameter is the same as the ``jcoupling`` parameter of JzKetCoupled. + + Examples + ======== + + Uncouple a numerical state using a CoupledSpinState state: + + >>> from sympy.physics.quantum.spin import JzKetCoupled, uncouple + >>> from sympy import S + >>> uncouple(JzKetCoupled(1, 0, (S(1)/2, S(1)/2))) + sqrt(2)*|1/2,-1/2>x|1/2,1/2>/2 + sqrt(2)*|1/2,1/2>x|1/2,-1/2>/2 + + Perform the same calculation using a SpinState state: + + >>> from sympy.physics.quantum.spin import JzKet + >>> uncouple(JzKet(1, 0), (S(1)/2, S(1)/2)) + sqrt(2)*|1/2,-1/2>x|1/2,1/2>/2 + sqrt(2)*|1/2,1/2>x|1/2,-1/2>/2 + + Uncouple a numerical state of three coupled spaces using a CoupledSpinState state: + + >>> uncouple(JzKetCoupled(1, 1, (1, 1, 1), ((1,3,1),(1,2,1)) )) + |1,-1>x|1,1>x|1,1>/2 - |1,0>x|1,0>x|1,1>/2 + |1,1>x|1,0>x|1,0>/2 - |1,1>x|1,1>x|1,-1>/2 + + Perform the same calculation using a SpinState state: + + >>> uncouple(JzKet(1, 1), (1, 1, 1), ((1,3,1),(1,2,1)) ) + |1,-1>x|1,1>x|1,1>/2 - |1,0>x|1,0>x|1,1>/2 + |1,1>x|1,0>x|1,0>/2 - |1,1>x|1,1>x|1,-1>/2 + + Uncouple a symbolic state using a CoupledSpinState state: + + >>> from sympy import symbols + >>> j,m,j1,j2 = symbols('j m j1 j2') + >>> uncouple(JzKetCoupled(j, m, (j1, j2))) + Sum(CG(j1, m1, j2, m2, j, m)*|j1,m1>x|j2,m2>, (m1, -j1, j1), (m2, -j2, j2)) + + Perform the same calculation using a SpinState state + + >>> uncouple(JzKet(j, m), (j1, j2)) + Sum(CG(j1, m1, j2, m2, j, m)*|j1,m1>x|j2,m2>, (m1, -j1, j1), (m2, -j2, j2)) + + """ + a = expr.atoms(SpinState) + for state in a: + expr = expr.subs(state, _uncouple(state, jn, jcoupling_list)) + return expr + + +def _uncouple(state, jn, jcoupling_list): + if isinstance(state, CoupledSpinState): + jn = state.jn + coupled_n = state.coupled_n + coupled_jn = state.coupled_jn + evect = state.uncoupled_class() + elif isinstance(state, SpinState): + if jn is None: + raise ValueError("Must specify j-values for coupled state") + if not isinstance(jn, (list, tuple)): + raise TypeError("jn must be list or tuple") + if jcoupling_list is None: + # Use default + jcoupling_list = [] + for i in range(1, len(jn)): + jcoupling_list.append( + (1, 1 + i, Add(*[jn[j] for j in range(i + 1)])) ) + if not isinstance(jcoupling_list, (list, tuple)): + raise TypeError("jcoupling must be a list or tuple") + if not len(jcoupling_list) == len(jn) - 1: + raise ValueError("Must specify 2 fewer coupling terms than the number of j values") + coupled_n, coupled_jn = _build_coupled(jcoupling_list, len(jn)) + evect = state.__class__ + else: + raise TypeError("state must be a spin state") + j = state.j + m = state.m + coupling_list = [] + j_list = list(jn) + + # Create coupling, which defines all the couplings between all the spaces + for j3, (n1, n2) in zip(coupled_jn, coupled_n): + # j's which are coupled as first and second spaces + j1 = j_list[n1[0] - 1] + j2 = j_list[n2[0] - 1] + # Build coupling list + coupling_list.append( (n1, n2, j1, j2, j3) ) + # Set new value in j_list + j_list[min(n1 + n2) - 1] = j3 + + if j.is_number and m.is_number: + diff_max = [ 2*x for x in jn ] + diff = Add(*jn) - m + + n = len(jn) + tot = binomial(diff + n - 1, diff) + + result = [] + for config_num in range(tot): + diff_list = _confignum_to_difflist(config_num, diff, n) + if any(d > p for d, p in zip(diff_list, diff_max)): + continue + + cg_terms = [] + for coupling in coupling_list: + j1_n, j2_n, j1, j2, j3 = coupling + m1 = Add( *[ jn[x - 1] - diff_list[x - 1] for x in j1_n ] ) + m2 = Add( *[ jn[x - 1] - diff_list[x - 1] for x in j2_n ] ) + m3 = m1 + m2 + cg_terms.append( (j1, m1, j2, m2, j3, m3) ) + coeff = Mul( *[ CG(*term).doit() for term in cg_terms ] ) + state = TensorProduct( + *[ evect(j, j - d) for j, d in zip(jn, diff_list) ] ) + result.append(coeff*state) + return Add(*result) + else: + # Symbolic coupling + m_str = "m1:%d" % (len(jn) + 1) + mvals = symbols(m_str) + cg_terms = [(j1, Add(*[mvals[n - 1] for n in j1_n]), + j2, Add(*[mvals[n - 1] for n in j2_n]), + j3, Add(*[mvals[n - 1] for n in j1_n + j2_n])) for j1_n, j2_n, j1, j2, j3 in coupling_list[:-1] ] + cg_terms.append(*[(j1, Add(*[mvals[n - 1] for n in j1_n]), + j2, Add(*[mvals[n - 1] for n in j2_n]), + j, m) for j1_n, j2_n, j1, j2, j3 in [coupling_list[-1]] ]) + cg_coeff = Mul(*[CG(*cg_term) for cg_term in cg_terms]) + sum_terms = [ (m, -j, j) for j, m in zip(jn, mvals) ] + state = TensorProduct( *[ evect(j, m) for j, m in zip(jn, mvals) ] ) + return Sum(cg_coeff*state, *sum_terms) + + +def _confignum_to_difflist(config_num, diff, list_len): + # Determines configuration of diffs into list_len number of slots + diff_list = [] + for n in range(list_len): + prev_diff = diff + # Number of spots after current one + rem_spots = list_len - n - 1 + # Number of configurations of distributing diff among the remaining spots + rem_configs = binomial(diff + rem_spots - 1, diff) + while config_num >= rem_configs: + config_num -= rem_configs + diff -= 1 + rem_configs = binomial(diff + rem_spots - 1, diff) + diff_list.append(prev_diff - diff) + return diff_list diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/tests/test_identitysearch.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/tests/test_identitysearch.py new file mode 100644 index 0000000000000000000000000000000000000000..8747b1f9d9630e699695f67734333f9d61581fb8 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/tests/test_identitysearch.py @@ -0,0 +1,492 @@ +from sympy.external import import_module +from sympy.core.mul import Mul +from sympy.core.numbers import Integer +from sympy.physics.quantum.dagger import Dagger +from sympy.physics.quantum.gate import (X, Y, Z, H, CNOT, + IdentityGate, CGate, PhaseGate, TGate) +from sympy.physics.quantum.identitysearch import (generate_gate_rules, + generate_equivalent_ids, GateIdentity, bfs_identity_search, + is_scalar_sparse_matrix, + is_scalar_nonsparse_matrix, is_degenerate, is_reducible) +from sympy.testing.pytest import skip + + +def create_gate_sequence(qubit=0): + gates = (X(qubit), Y(qubit), Z(qubit), H(qubit)) + return gates + + +def test_generate_gate_rules_1(): + # Test with tuples + (x, y, z, h) = create_gate_sequence() + ph = PhaseGate(0) + cgate_t = CGate(0, TGate(1)) + + assert generate_gate_rules((x,)) == {((x,), ())} + + gate_rules = {((x, x), ()), + ((x,), (x,))} + assert generate_gate_rules((x, x)) == gate_rules + + gate_rules = {((x, y, x), ()), + ((y, x, x), ()), + ((x, x, y), ()), + ((y, x), (x,)), + ((x, y), (x,)), + ((y,), (x, x))} + assert generate_gate_rules((x, y, x)) == gate_rules + + gate_rules = {((x, y, z), ()), ((y, z, x), ()), ((z, x, y), ()), + ((), (x, z, y)), ((), (y, x, z)), ((), (z, y, x)), + ((x,), (z, y)), ((y, z), (x,)), ((y,), (x, z)), + ((z, x), (y,)), ((z,), (y, x)), ((x, y), (z,))} + actual = generate_gate_rules((x, y, z)) + assert actual == gate_rules + + gate_rules = { + ((), (h, z, y, x)), ((), (x, h, z, y)), ((), (y, x, h, z)), + ((), (z, y, x, h)), ((h,), (z, y, x)), ((x,), (h, z, y)), + ((y,), (x, h, z)), ((z,), (y, x, h)), ((h, x), (z, y)), + ((x, y), (h, z)), ((y, z), (x, h)), ((z, h), (y, x)), + ((h, x, y), (z,)), ((x, y, z), (h,)), ((y, z, h), (x,)), + ((z, h, x), (y,)), ((h, x, y, z), ()), ((x, y, z, h), ()), + ((y, z, h, x), ()), ((z, h, x, y), ())} + actual = generate_gate_rules((x, y, z, h)) + assert actual == gate_rules + + gate_rules = {((), (cgate_t**(-1), ph**(-1), x)), + ((), (ph**(-1), x, cgate_t**(-1))), + ((), (x, cgate_t**(-1), ph**(-1))), + ((cgate_t,), (ph**(-1), x)), + ((ph,), (x, cgate_t**(-1))), + ((x,), (cgate_t**(-1), ph**(-1))), + ((cgate_t, x), (ph**(-1),)), + ((ph, cgate_t), (x,)), + ((x, ph), (cgate_t**(-1),)), + ((cgate_t, x, ph), ()), + ((ph, cgate_t, x), ()), + ((x, ph, cgate_t), ())} + actual = generate_gate_rules((x, ph, cgate_t)) + assert actual == gate_rules + + gate_rules = {(Integer(1), cgate_t**(-1)*ph**(-1)*x), + (Integer(1), ph**(-1)*x*cgate_t**(-1)), + (Integer(1), x*cgate_t**(-1)*ph**(-1)), + (cgate_t, ph**(-1)*x), + (ph, x*cgate_t**(-1)), + (x, cgate_t**(-1)*ph**(-1)), + (cgate_t*x, ph**(-1)), + (ph*cgate_t, x), + (x*ph, cgate_t**(-1)), + (cgate_t*x*ph, Integer(1)), + (ph*cgate_t*x, Integer(1)), + (x*ph*cgate_t, Integer(1))} + actual = generate_gate_rules((x, ph, cgate_t), return_as_muls=True) + assert actual == gate_rules + + +def test_generate_gate_rules_2(): + # Test with Muls + (x, y, z, h) = create_gate_sequence() + ph = PhaseGate(0) + cgate_t = CGate(0, TGate(1)) + + # Note: 1 (type int) is not the same as 1 (type One) + expected = {(x, Integer(1))} + assert generate_gate_rules((x,), return_as_muls=True) == expected + + expected = {(Integer(1), Integer(1))} + assert generate_gate_rules(x*x, return_as_muls=True) == expected + + expected = {((), ())} + assert generate_gate_rules(x*x, return_as_muls=False) == expected + + gate_rules = {(x*y*x, Integer(1)), + (y, Integer(1)), + (y*x, x), + (x*y, x)} + assert generate_gate_rules(x*y*x, return_as_muls=True) == gate_rules + + gate_rules = {(x*y*z, Integer(1)), + (y*z*x, Integer(1)), + (z*x*y, Integer(1)), + (Integer(1), x*z*y), + (Integer(1), y*x*z), + (Integer(1), z*y*x), + (x, z*y), + (y*z, x), + (y, x*z), + (z*x, y), + (z, y*x), + (x*y, z)} + actual = generate_gate_rules(x*y*z, return_as_muls=True) + assert actual == gate_rules + + gate_rules = {(Integer(1), h*z*y*x), + (Integer(1), x*h*z*y), + (Integer(1), y*x*h*z), + (Integer(1), z*y*x*h), + (h, z*y*x), (x, h*z*y), + (y, x*h*z), (z, y*x*h), + (h*x, z*y), (z*h, y*x), + (x*y, h*z), (y*z, x*h), + (h*x*y, z), (x*y*z, h), + (y*z*h, x), (z*h*x, y), + (h*x*y*z, Integer(1)), + (x*y*z*h, Integer(1)), + (y*z*h*x, Integer(1)), + (z*h*x*y, Integer(1))} + actual = generate_gate_rules(x*y*z*h, return_as_muls=True) + assert actual == gate_rules + + gate_rules = {(Integer(1), cgate_t**(-1)*ph**(-1)*x), + (Integer(1), ph**(-1)*x*cgate_t**(-1)), + (Integer(1), x*cgate_t**(-1)*ph**(-1)), + (cgate_t, ph**(-1)*x), + (ph, x*cgate_t**(-1)), + (x, cgate_t**(-1)*ph**(-1)), + (cgate_t*x, ph**(-1)), + (ph*cgate_t, x), + (x*ph, cgate_t**(-1)), + (cgate_t*x*ph, Integer(1)), + (ph*cgate_t*x, Integer(1)), + (x*ph*cgate_t, Integer(1))} + actual = generate_gate_rules(x*ph*cgate_t, return_as_muls=True) + assert actual == gate_rules + + gate_rules = {((), (cgate_t**(-1), ph**(-1), x)), + ((), (ph**(-1), x, cgate_t**(-1))), + ((), (x, cgate_t**(-1), ph**(-1))), + ((cgate_t,), (ph**(-1), x)), + ((ph,), (x, cgate_t**(-1))), + ((x,), (cgate_t**(-1), ph**(-1))), + ((cgate_t, x), (ph**(-1),)), + ((ph, cgate_t), (x,)), + ((x, ph), (cgate_t**(-1),)), + ((cgate_t, x, ph), ()), + ((ph, cgate_t, x), ()), + ((x, ph, cgate_t), ())} + actual = generate_gate_rules(x*ph*cgate_t) + assert actual == gate_rules + + +def test_generate_equivalent_ids_1(): + # Test with tuples + (x, y, z, h) = create_gate_sequence() + + assert generate_equivalent_ids((x,)) == {(x,)} + assert generate_equivalent_ids((x, x)) == {(x, x)} + assert generate_equivalent_ids((x, y)) == {(x, y), (y, x)} + + gate_seq = (x, y, z) + gate_ids = {(x, y, z), (y, z, x), (z, x, y), (z, y, x), + (y, x, z), (x, z, y)} + assert generate_equivalent_ids(gate_seq) == gate_ids + + gate_ids = {Mul(x, y, z), Mul(y, z, x), Mul(z, x, y), + Mul(z, y, x), Mul(y, x, z), Mul(x, z, y)} + assert generate_equivalent_ids(gate_seq, return_as_muls=True) == gate_ids + + gate_seq = (x, y, z, h) + gate_ids = {(x, y, z, h), (y, z, h, x), + (h, x, y, z), (h, z, y, x), + (z, y, x, h), (y, x, h, z), + (z, h, x, y), (x, h, z, y)} + assert generate_equivalent_ids(gate_seq) == gate_ids + + gate_seq = (x, y, x, y) + gate_ids = {(x, y, x, y), (y, x, y, x)} + assert generate_equivalent_ids(gate_seq) == gate_ids + + cgate_y = CGate((1,), y) + gate_seq = (y, cgate_y, y, cgate_y) + gate_ids = {(y, cgate_y, y, cgate_y), (cgate_y, y, cgate_y, y)} + assert generate_equivalent_ids(gate_seq) == gate_ids + + cnot = CNOT(1, 0) + cgate_z = CGate((0,), Z(1)) + gate_seq = (cnot, h, cgate_z, h) + gate_ids = {(cnot, h, cgate_z, h), (h, cgate_z, h, cnot), + (h, cnot, h, cgate_z), (cgate_z, h, cnot, h)} + assert generate_equivalent_ids(gate_seq) == gate_ids + + +def test_generate_equivalent_ids_2(): + # Test with Muls + (x, y, z, h) = create_gate_sequence() + + assert generate_equivalent_ids((x,), return_as_muls=True) == {x} + + gate_ids = {Integer(1)} + assert generate_equivalent_ids(x*x, return_as_muls=True) == gate_ids + + gate_ids = {x*y, y*x} + assert generate_equivalent_ids(x*y, return_as_muls=True) == gate_ids + + gate_ids = {(x, y), (y, x)} + assert generate_equivalent_ids(x*y) == gate_ids + + circuit = Mul(*(x, y, z)) + gate_ids = {x*y*z, y*z*x, z*x*y, z*y*x, + y*x*z, x*z*y} + assert generate_equivalent_ids(circuit, return_as_muls=True) == gate_ids + + circuit = Mul(*(x, y, z, h)) + gate_ids = {x*y*z*h, y*z*h*x, + h*x*y*z, h*z*y*x, + z*y*x*h, y*x*h*z, + z*h*x*y, x*h*z*y} + assert generate_equivalent_ids(circuit, return_as_muls=True) == gate_ids + + circuit = Mul(*(x, y, x, y)) + gate_ids = {x*y*x*y, y*x*y*x} + assert generate_equivalent_ids(circuit, return_as_muls=True) == gate_ids + + cgate_y = CGate((1,), y) + circuit = Mul(*(y, cgate_y, y, cgate_y)) + gate_ids = {y*cgate_y*y*cgate_y, cgate_y*y*cgate_y*y} + assert generate_equivalent_ids(circuit, return_as_muls=True) == gate_ids + + cnot = CNOT(1, 0) + cgate_z = CGate((0,), Z(1)) + circuit = Mul(*(cnot, h, cgate_z, h)) + gate_ids = {cnot*h*cgate_z*h, h*cgate_z*h*cnot, + h*cnot*h*cgate_z, cgate_z*h*cnot*h} + assert generate_equivalent_ids(circuit, return_as_muls=True) == gate_ids + + +def test_is_scalar_nonsparse_matrix(): + numqubits = 2 + id_only = False + + id_gate = (IdentityGate(1),) + actual = is_scalar_nonsparse_matrix(id_gate, numqubits, id_only) + assert actual is True + + x0 = X(0) + xx_circuit = (x0, x0) + actual = is_scalar_nonsparse_matrix(xx_circuit, numqubits, id_only) + assert actual is True + + x1 = X(1) + y1 = Y(1) + xy_circuit = (x1, y1) + actual = is_scalar_nonsparse_matrix(xy_circuit, numqubits, id_only) + assert actual is False + + z1 = Z(1) + xyz_circuit = (x1, y1, z1) + actual = is_scalar_nonsparse_matrix(xyz_circuit, numqubits, id_only) + assert actual is True + + cnot = CNOT(1, 0) + cnot_circuit = (cnot, cnot) + actual = is_scalar_nonsparse_matrix(cnot_circuit, numqubits, id_only) + assert actual is True + + h = H(0) + hh_circuit = (h, h) + actual = is_scalar_nonsparse_matrix(hh_circuit, numqubits, id_only) + assert actual is True + + h1 = H(1) + xhzh_circuit = (x1, h1, z1, h1) + actual = is_scalar_nonsparse_matrix(xhzh_circuit, numqubits, id_only) + assert actual is True + + id_only = True + actual = is_scalar_nonsparse_matrix(xhzh_circuit, numqubits, id_only) + assert actual is True + actual = is_scalar_nonsparse_matrix(xyz_circuit, numqubits, id_only) + assert actual is False + actual = is_scalar_nonsparse_matrix(cnot_circuit, numqubits, id_only) + assert actual is True + actual = is_scalar_nonsparse_matrix(hh_circuit, numqubits, id_only) + assert actual is True + + +def test_is_scalar_sparse_matrix(): + np = import_module('numpy') + if not np: + skip("numpy not installed.") + + scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']}) + if not scipy: + skip("scipy not installed.") + + numqubits = 2 + id_only = False + + id_gate = (IdentityGate(1),) + assert is_scalar_sparse_matrix(id_gate, numqubits, id_only) is True + + x0 = X(0) + xx_circuit = (x0, x0) + assert is_scalar_sparse_matrix(xx_circuit, numqubits, id_only) is True + + x1 = X(1) + y1 = Y(1) + xy_circuit = (x1, y1) + assert is_scalar_sparse_matrix(xy_circuit, numqubits, id_only) is False + + z1 = Z(1) + xyz_circuit = (x1, y1, z1) + assert is_scalar_sparse_matrix(xyz_circuit, numqubits, id_only) is True + + cnot = CNOT(1, 0) + cnot_circuit = (cnot, cnot) + assert is_scalar_sparse_matrix(cnot_circuit, numqubits, id_only) is True + + h = H(0) + hh_circuit = (h, h) + assert is_scalar_sparse_matrix(hh_circuit, numqubits, id_only) is True + + # NOTE: + # The elements of the sparse matrix for the following circuit + # is actually 1.0000000000000002+0.0j. + h1 = H(1) + xhzh_circuit = (x1, h1, z1, h1) + assert is_scalar_sparse_matrix(xhzh_circuit, numqubits, id_only) is True + + id_only = True + assert is_scalar_sparse_matrix(xhzh_circuit, numqubits, id_only) is True + assert is_scalar_sparse_matrix(xyz_circuit, numqubits, id_only) is False + assert is_scalar_sparse_matrix(cnot_circuit, numqubits, id_only) is True + assert is_scalar_sparse_matrix(hh_circuit, numqubits, id_only) is True + + +def test_is_degenerate(): + (x, y, z, h) = create_gate_sequence() + + gate_id = GateIdentity(x, y, z) + ids = {gate_id} + + another_id = (z, y, x) + assert is_degenerate(ids, another_id) is True + + +def test_is_reducible(): + nqubits = 2 + (x, y, z, h) = create_gate_sequence() + + circuit = (x, y, y) + assert is_reducible(circuit, nqubits, 1, 3) is True + + circuit = (x, y, x) + assert is_reducible(circuit, nqubits, 1, 3) is False + + circuit = (x, y, y, x) + assert is_reducible(circuit, nqubits, 0, 4) is True + + circuit = (x, y, y, x) + assert is_reducible(circuit, nqubits, 1, 3) is True + + circuit = (x, y, z, y, y) + assert is_reducible(circuit, nqubits, 1, 5) is True + + +def test_bfs_identity_search(): + assert bfs_identity_search([], 1) == set() + + (x, y, z, h) = create_gate_sequence() + + gate_list = [x] + id_set = {GateIdentity(x, x)} + assert bfs_identity_search(gate_list, 1, max_depth=2) == id_set + + # Set should not contain degenerate quantum circuits + gate_list = [x, y, z] + id_set = {GateIdentity(x, x), + GateIdentity(y, y), + GateIdentity(z, z), + GateIdentity(x, y, z)} + assert bfs_identity_search(gate_list, 1) == id_set + + id_set = {GateIdentity(x, x), + GateIdentity(y, y), + GateIdentity(z, z), + GateIdentity(x, y, z), + GateIdentity(x, y, x, y), + GateIdentity(x, z, x, z), + GateIdentity(y, z, y, z)} + assert bfs_identity_search(gate_list, 1, max_depth=4) == id_set + assert bfs_identity_search(gate_list, 1, max_depth=5) == id_set + + gate_list = [x, y, z, h] + id_set = {GateIdentity(x, x), + GateIdentity(y, y), + GateIdentity(z, z), + GateIdentity(h, h), + GateIdentity(x, y, z), + GateIdentity(x, y, x, y), + GateIdentity(x, z, x, z), + GateIdentity(x, h, z, h), + GateIdentity(y, z, y, z), + GateIdentity(y, h, y, h)} + assert bfs_identity_search(gate_list, 1) == id_set + + id_set = {GateIdentity(x, x), + GateIdentity(y, y), + GateIdentity(z, z), + GateIdentity(h, h)} + assert id_set == bfs_identity_search(gate_list, 1, max_depth=3, + identity_only=True) + + id_set = {GateIdentity(x, x), + GateIdentity(y, y), + GateIdentity(z, z), + GateIdentity(h, h), + GateIdentity(x, y, z), + GateIdentity(x, y, x, y), + GateIdentity(x, z, x, z), + GateIdentity(x, h, z, h), + GateIdentity(y, z, y, z), + GateIdentity(y, h, y, h), + GateIdentity(x, y, h, x, h), + GateIdentity(x, z, h, y, h), + GateIdentity(y, z, h, z, h)} + assert bfs_identity_search(gate_list, 1, max_depth=5) == id_set + + id_set = {GateIdentity(x, x), + GateIdentity(y, y), + GateIdentity(z, z), + GateIdentity(h, h), + GateIdentity(x, h, z, h)} + assert id_set == bfs_identity_search(gate_list, 1, max_depth=4, + identity_only=True) + + cnot = CNOT(1, 0) + gate_list = [x, cnot] + id_set = {GateIdentity(x, x), + GateIdentity(cnot, cnot), + GateIdentity(x, cnot, x, cnot)} + assert bfs_identity_search(gate_list, 2, max_depth=4) == id_set + + cgate_x = CGate((1,), x) + gate_list = [x, cgate_x] + id_set = {GateIdentity(x, x), + GateIdentity(cgate_x, cgate_x), + GateIdentity(x, cgate_x, x, cgate_x)} + assert bfs_identity_search(gate_list, 2, max_depth=4) == id_set + + cgate_z = CGate((0,), Z(1)) + gate_list = [cnot, cgate_z, h] + id_set = {GateIdentity(h, h), + GateIdentity(cgate_z, cgate_z), + GateIdentity(cnot, cnot), + GateIdentity(cnot, h, cgate_z, h)} + assert bfs_identity_search(gate_list, 2, max_depth=4) == id_set + + s = PhaseGate(0) + t = TGate(0) + gate_list = [s, t] + id_set = {GateIdentity(s, s, s, s)} + assert bfs_identity_search(gate_list, 1, max_depth=4) == id_set + + +def test_bfs_identity_search_xfail(): + s = PhaseGate(0) + t = TGate(0) + gate_list = [Dagger(s), t] + id_set = {GateIdentity(Dagger(s), t, t)} + assert bfs_identity_search(gate_list, 1, max_depth=3) == id_set diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/quantum/tests/test_represent.py b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/tests/test_represent.py new file mode 100644 index 0000000000000000000000000000000000000000..c49dcbd7e7876f30cbe8e5426c91419903add5ff --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/quantum/tests/test_represent.py @@ -0,0 +1,186 @@ +from sympy.core.numbers import (Float, I, Integer) +from sympy.matrices.dense import Matrix +from sympy.external import import_module +from sympy.testing.pytest import skip + +from sympy.physics.quantum.dagger import Dagger +from sympy.physics.quantum.represent import (represent, rep_innerproduct, + rep_expectation, enumerate_states) +from sympy.physics.quantum.state import Bra, Ket +from sympy.physics.quantum.operator import Operator, OuterProduct +from sympy.physics.quantum.tensorproduct import TensorProduct +from sympy.physics.quantum.tensorproduct import matrix_tensor_product +from sympy.physics.quantum.commutator import Commutator +from sympy.physics.quantum.anticommutator import AntiCommutator +from sympy.physics.quantum.innerproduct import InnerProduct +from sympy.physics.quantum.matrixutils import (numpy_ndarray, + scipy_sparse_matrix, to_numpy, + to_scipy_sparse, to_sympy) +from sympy.physics.quantum.cartesian import XKet, XOp, XBra +from sympy.physics.quantum.qapply import qapply +from sympy.physics.quantum.operatorset import operators_to_state +from sympy.testing.pytest import raises + +Amat = Matrix([[1, I], [-I, 1]]) +Bmat = Matrix([[1, 2], [3, 4]]) +Avec = Matrix([[1], [I]]) + + +class AKet(Ket): + + @classmethod + def dual_class(self): + return ABra + + def _represent_default_basis(self, **options): + return self._represent_AOp(None, **options) + + def _represent_AOp(self, basis, **options): + return Avec + + +class ABra(Bra): + + @classmethod + def dual_class(self): + return AKet + + +class AOp(Operator): + + def _represent_default_basis(self, **options): + return self._represent_AOp(None, **options) + + def _represent_AOp(self, basis, **options): + return Amat + + +class BOp(Operator): + + def _represent_default_basis(self, **options): + return self._represent_AOp(None, **options) + + def _represent_AOp(self, basis, **options): + return Bmat + + +k = AKet('a') +b = ABra('a') +A = AOp('A') +B = BOp('B') + +_tests = [ + # Bra + (b, Dagger(Avec)), + (Dagger(b), Avec), + # Ket + (k, Avec), + (Dagger(k), Dagger(Avec)), + # Operator + (A, Amat), + (Dagger(A), Dagger(Amat)), + # OuterProduct + (OuterProduct(k, b), Avec*Avec.H), + # TensorProduct + (TensorProduct(A, B), matrix_tensor_product(Amat, Bmat)), + # Pow + (A**2, Amat**2), + # Add/Mul + (A*B + 2*A, Amat*Bmat + 2*Amat), + # Commutator + (Commutator(A, B), Amat*Bmat - Bmat*Amat), + # AntiCommutator + (AntiCommutator(A, B), Amat*Bmat + Bmat*Amat), + # InnerProduct + (InnerProduct(b, k), (Avec.H*Avec)[0]) +] + + +def test_format_sympy(): + for test in _tests: + lhs = represent(test[0], basis=A, format='sympy') + rhs = to_sympy(test[1]) + assert lhs == rhs + + +def test_scalar_sympy(): + assert represent(Integer(1)) == Integer(1) + assert represent(Float(1.0)) == Float(1.0) + assert represent(1.0 + I) == 1.0 + I + + +np = import_module('numpy') + + +def test_format_numpy(): + if not np: + skip("numpy not installed.") + + for test in _tests: + lhs = represent(test[0], basis=A, format='numpy') + rhs = to_numpy(test[1]) + if isinstance(lhs, numpy_ndarray): + assert (lhs == rhs).all() + else: + assert lhs == rhs + + +def test_scalar_numpy(): + if not np: + skip("numpy not installed.") + + assert represent(Integer(1), format='numpy') == 1 + assert represent(Float(1.0), format='numpy') == 1.0 + assert represent(1.0 + I, format='numpy') == 1.0 + 1.0j + + +scipy = import_module('scipy', import_kwargs={'fromlist': ['sparse']}) + + +def test_format_scipy_sparse(): + if not np: + skip("numpy not installed.") + if not scipy: + skip("scipy not installed.") + + for test in _tests: + lhs = represent(test[0], basis=A, format='scipy.sparse') + rhs = to_scipy_sparse(test[1]) + if isinstance(lhs, scipy_sparse_matrix): + assert np.linalg.norm((lhs - rhs).todense()) == 0.0 + else: + assert lhs == rhs + + +def test_scalar_scipy_sparse(): + if not np: + skip("numpy not installed.") + if not scipy: + skip("scipy not installed.") + + assert represent(Integer(1), format='scipy.sparse') == 1 + assert represent(Float(1.0), format='scipy.sparse') == 1.0 + assert represent(1.0 + I, format='scipy.sparse') == 1.0 + 1.0j + +x_ket = XKet('x') +x_bra = XBra('x') +x_op = XOp('X') + + +def test_innerprod_represent(): + assert rep_innerproduct(x_ket) == InnerProduct(XBra("x_1"), x_ket).doit() + assert rep_innerproduct(x_bra) == InnerProduct(x_bra, XKet("x_1")).doit() + raises(TypeError, lambda: rep_innerproduct(x_op)) + + +def test_operator_represent(): + basis_kets = enumerate_states(operators_to_state(x_op), 1, 2) + assert rep_expectation( + x_op) == qapply(basis_kets[1].dual*x_op*basis_kets[0]) + + +def test_enumerate_states(): + test = XKet("foo") + assert enumerate_states(test, 1, 1) == [XKet("foo_1")] + assert enumerate_states( + test, [1, 2, 4]) == [XKet("foo_1"), XKet("foo_2"), XKet("foo_4")] diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/tests/__init__.py b/wemm/lib/python3.10/site-packages/sympy/physics/tests/__init__.py new file mode 100644 index 0000000000000000000000000000000000000000..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/tests/__pycache__/test_clebsch_gordan.cpython-310.pyc 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Binary files /dev/null and b/wemm/lib/python3.10/site-packages/sympy/physics/tests/__pycache__/test_sho.cpython-310.pyc differ diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_hydrogen.py b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_hydrogen.py new file mode 100644 index 0000000000000000000000000000000000000000..eb11744dd8e731f24fcd6f6be2a92ada4fffc554 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_hydrogen.py @@ -0,0 +1,126 @@ +from sympy.core.numbers import (I, Rational, oo, pi) +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.integrals.integrals import integrate +from sympy.simplify.simplify import simplify +from sympy.physics.hydrogen import R_nl, E_nl, E_nl_dirac, Psi_nlm +from sympy.testing.pytest import raises + +n, r, Z = symbols('n r Z') + + +def feq(a, b, max_relative_error=1e-12, max_absolute_error=1e-12): + a = float(a) + b = float(b) + # if the numbers are close enough (absolutely), then they are equal + if abs(a - b) < max_absolute_error: + return True + # if not, they can still be equal if their relative error is small + if abs(b) > abs(a): + relative_error = abs((a - b)/b) + else: + relative_error = abs((a - b)/a) + return relative_error <= max_relative_error + + +def test_wavefunction(): + a = 1/Z + R = { + (1, 0): 2*sqrt(1/a**3) * exp(-r/a), + (2, 0): sqrt(1/(2*a**3)) * exp(-r/(2*a)) * (1 - r/(2*a)), + (2, 1): S.Half * sqrt(1/(6*a**3)) * exp(-r/(2*a)) * r/a, + (3, 0): Rational(2, 3) * sqrt(1/(3*a**3)) * exp(-r/(3*a)) * + (1 - 2*r/(3*a) + Rational(2, 27) * (r/a)**2), + (3, 1): Rational(4, 27) * sqrt(2/(3*a**3)) * exp(-r/(3*a)) * + (1 - r/(6*a)) * r/a, + (3, 2): Rational(2, 81) * sqrt(2/(15*a**3)) * exp(-r/(3*a)) * (r/a)**2, + (4, 0): Rational(1, 4) * sqrt(1/a**3) * exp(-r/(4*a)) * + (1 - 3*r/(4*a) + Rational(1, 8) * (r/a)**2 - Rational(1, 192) * (r/a)**3), + (4, 1): Rational(1, 16) * sqrt(5/(3*a**3)) * exp(-r/(4*a)) * + (1 - r/(4*a) + Rational(1, 80) * (r/a)**2) * (r/a), + (4, 2): Rational(1, 64) * sqrt(1/(5*a**3)) * exp(-r/(4*a)) * + (1 - r/(12*a)) * (r/a)**2, + (4, 3): Rational(1, 768) * sqrt(1/(35*a**3)) * exp(-r/(4*a)) * (r/a)**3, + } + for n, l in R: + assert simplify(R_nl(n, l, r, Z) - R[(n, l)]) == 0 + + +def test_norm(): + # Maximum "n" which is tested: + n_max = 2 # it works, but is slow, for n_max > 2 + for n in range(n_max + 1): + for l in range(n): + assert integrate(R_nl(n, l, r)**2 * r**2, (r, 0, oo)) == 1 + +def test_psi_nlm(): + r=S('r') + phi=S('phi') + theta=S('theta') + assert (Psi_nlm(1, 0, 0, r, phi, theta) == exp(-r) / sqrt(pi)) + assert (Psi_nlm(2, 1, -1, r, phi, theta)) == S.Half * exp(-r / (2)) * r \ + * (sin(theta) * exp(-I * phi) / (4 * sqrt(pi))) + assert (Psi_nlm(3, 2, 1, r, phi, theta, 2) == -sqrt(2) * sin(theta) \ + * exp(I * phi) * cos(theta) / (4 * sqrt(pi)) * S(2) / 81 \ + * sqrt(2 * 2 ** 3) * exp(-2 * r / (3)) * (r * 2) ** 2) + +def test_hydrogen_energies(): + assert E_nl(n, Z) == -Z**2/(2*n**2) + assert E_nl(n) == -1/(2*n**2) + + assert E_nl(1, 47) == -S(47)**2/(2*1**2) + assert E_nl(2, 47) == -S(47)**2/(2*2**2) + + assert E_nl(1) == -S.One/(2*1**2) + assert E_nl(2) == -S.One/(2*2**2) + assert E_nl(3) == -S.One/(2*3**2) + assert E_nl(4) == -S.One/(2*4**2) + assert E_nl(100) == -S.One/(2*100**2) + + raises(ValueError, lambda: E_nl(0)) + + +def test_hydrogen_energies_relat(): + # First test exact formulas for small "c" so that we get nice expressions: + assert E_nl_dirac(2, 0, Z=1, c=1) == 1/sqrt(2) - 1 + assert simplify(E_nl_dirac(2, 0, Z=1, c=2) - ( (8*sqrt(3) + 16) + / sqrt(16*sqrt(3) + 32) - 4)) == 0 + assert simplify(E_nl_dirac(2, 0, Z=1, c=3) - ( (54*sqrt(2) + 81) + / sqrt(108*sqrt(2) + 162) - 9)) == 0 + + # Now test for almost the correct speed of light, without floating point + # numbers: + assert simplify(E_nl_dirac(2, 0, Z=1, c=137) - ( (352275361 + 10285412 * + sqrt(1173)) / sqrt(704550722 + 20570824 * sqrt(1173)) - 18769)) == 0 + assert simplify(E_nl_dirac(2, 0, Z=82, c=137) - ( (352275361 + 2571353 * + sqrt(12045)) / sqrt(704550722 + 5142706*sqrt(12045)) - 18769)) == 0 + + # Test using exact speed of light, and compare against the nonrelativistic + # energies: + for n in range(1, 5): + for l in range(n): + assert feq(E_nl_dirac(n, l), E_nl(n), 1e-5, 1e-5) + if l > 0: + assert feq(E_nl_dirac(n, l, False), E_nl(n), 1e-5, 1e-5) + + Z = 2 + for n in range(1, 5): + for l in range(n): + assert feq(E_nl_dirac(n, l, Z=Z), E_nl(n, Z), 1e-4, 1e-4) + if l > 0: + assert feq(E_nl_dirac(n, l, False, Z), E_nl(n, Z), 1e-4, 1e-4) + + Z = 3 + for n in range(1, 5): + for l in range(n): + assert feq(E_nl_dirac(n, l, Z=Z), E_nl(n, Z), 1e-3, 1e-3) + if l > 0: + assert feq(E_nl_dirac(n, l, False, Z), E_nl(n, Z), 1e-3, 1e-3) + + # Test the exceptions: + raises(ValueError, lambda: E_nl_dirac(0, 0)) + raises(ValueError, lambda: E_nl_dirac(1, -1)) + raises(ValueError, lambda: E_nl_dirac(1, 0, False)) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_paulialgebra.py b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_paulialgebra.py new file mode 100644 index 0000000000000000000000000000000000000000..f773470a1802f2864b79f56d38be1de030ff86dc --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_paulialgebra.py @@ -0,0 +1,57 @@ +from sympy.core.numbers import I +from sympy.core.symbol import symbols +from sympy.physics.paulialgebra import Pauli +from sympy.testing.pytest import XFAIL +from sympy.physics.quantum import TensorProduct + +sigma1 = Pauli(1) +sigma2 = Pauli(2) +sigma3 = Pauli(3) + +tau1 = symbols("tau1", commutative = False) + + +def test_Pauli(): + + assert sigma1 == sigma1 + assert sigma1 != sigma2 + + assert sigma1*sigma2 == I*sigma3 + assert sigma3*sigma1 == I*sigma2 + assert sigma2*sigma3 == I*sigma1 + + assert sigma1*sigma1 == 1 + assert sigma2*sigma2 == 1 + assert sigma3*sigma3 == 1 + + assert sigma1**0 == 1 + assert sigma1**1 == sigma1 + assert sigma1**2 == 1 + assert sigma1**3 == sigma1 + assert sigma1**4 == 1 + + assert sigma3**2 == 1 + + assert sigma1*2*sigma1 == 2 + + +def test_evaluate_pauli_product(): + from sympy.physics.paulialgebra import evaluate_pauli_product + + assert evaluate_pauli_product(I*sigma2*sigma3) == -sigma1 + + # Check issue 6471 + assert evaluate_pauli_product(-I*4*sigma1*sigma2) == 4*sigma3 + + assert evaluate_pauli_product( + 1 + I*sigma1*sigma2*sigma1*sigma2 + \ + I*sigma1*sigma2*tau1*sigma1*sigma3 + \ + ((tau1**2).subs(tau1, I*sigma1)) + \ + sigma3*((tau1**2).subs(tau1, I*sigma1)) + \ + TensorProduct(I*sigma1*sigma2*sigma1*sigma2, 1) + ) == 1 -I + I*sigma3*tau1*sigma2 - 1 - sigma3 - I*TensorProduct(1,1) + + +@XFAIL +def test_Pauli_should_work(): + assert sigma1*sigma3*sigma1 == -sigma3 diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_physics_matrices.py b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_physics_matrices.py new file mode 100644 index 0000000000000000000000000000000000000000..14fa47668d0760826e0354c8cafae787a24256eb --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_physics_matrices.py @@ -0,0 +1,84 @@ +from sympy.physics.matrices import msigma, mgamma, minkowski_tensor, pat_matrix, mdft +from sympy.core.numbers import (I, Rational) +from sympy.core.singleton import S +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.matrices.dense import (Matrix, eye, zeros) +from sympy.testing.pytest import warns_deprecated_sympy + + +def test_parallel_axis_theorem(): + # This tests the parallel axis theorem matrix by comparing to test + # matrices. + + # First case, 1 in all directions. + mat1 = Matrix(((2, -1, -1), (-1, 2, -1), (-1, -1, 2))) + assert pat_matrix(1, 1, 1, 1) == mat1 + assert pat_matrix(2, 1, 1, 1) == 2*mat1 + + # Second case, 1 in x, 0 in all others + mat2 = Matrix(((0, 0, 0), (0, 1, 0), (0, 0, 1))) + assert pat_matrix(1, 1, 0, 0) == mat2 + assert pat_matrix(2, 1, 0, 0) == 2*mat2 + + # Third case, 1 in y, 0 in all others + mat3 = Matrix(((1, 0, 0), (0, 0, 0), (0, 0, 1))) + assert pat_matrix(1, 0, 1, 0) == mat3 + assert pat_matrix(2, 0, 1, 0) == 2*mat3 + + # Fourth case, 1 in z, 0 in all others + mat4 = Matrix(((1, 0, 0), (0, 1, 0), (0, 0, 0))) + assert pat_matrix(1, 0, 0, 1) == mat4 + assert pat_matrix(2, 0, 0, 1) == 2*mat4 + + +def test_Pauli(): + #this and the following test are testing both Pauli and Dirac matrices + #and also that the general Matrix class works correctly in a real world + #situation + sigma1 = msigma(1) + sigma2 = msigma(2) + sigma3 = msigma(3) + + assert sigma1 == sigma1 + assert sigma1 != sigma2 + + # sigma*I -> I*sigma (see #354) + assert sigma1*sigma2 == sigma3*I + assert sigma3*sigma1 == sigma2*I + assert sigma2*sigma3 == sigma1*I + + assert sigma1*sigma1 == eye(2) + assert sigma2*sigma2 == eye(2) + assert sigma3*sigma3 == eye(2) + + assert sigma1*2*sigma1 == 2*eye(2) + assert sigma1*sigma3*sigma1 == -sigma3 + + +def test_Dirac(): + gamma0 = mgamma(0) + gamma1 = mgamma(1) + gamma2 = mgamma(2) + gamma3 = mgamma(3) + gamma5 = mgamma(5) + + # gamma*I -> I*gamma (see #354) + assert gamma5 == gamma0 * gamma1 * gamma2 * gamma3 * I + assert gamma1 * gamma2 + gamma2 * gamma1 == zeros(4) + assert gamma0 * gamma0 == eye(4) * minkowski_tensor[0, 0] + assert gamma2 * gamma2 != eye(4) * minkowski_tensor[0, 0] + assert gamma2 * gamma2 == eye(4) * minkowski_tensor[2, 2] + + assert mgamma(5, True) == \ + mgamma(0, True)*mgamma(1, True)*mgamma(2, True)*mgamma(3, True)*I + +def test_mdft(): + with warns_deprecated_sympy(): + assert mdft(1) == Matrix([[1]]) + with warns_deprecated_sympy(): + assert mdft(2) == 1/sqrt(2)*Matrix([[1,1],[1,-1]]) + with warns_deprecated_sympy(): + assert mdft(4) == Matrix([[S.Half, S.Half, S.Half, S.Half], + [S.Half, -I/2, Rational(-1,2), I/2], + [S.Half, Rational(-1,2), S.Half, Rational(-1,2)], + [S.Half, I/2, Rational(-1,2), -I/2]]) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_pring.py b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_pring.py new file mode 100644 index 0000000000000000000000000000000000000000..ed7398eac4a8bb1cd4af810825caf3fcefb5f18f --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_pring.py @@ -0,0 +1,41 @@ +from sympy.physics.pring import wavefunction, energy +from sympy.core.numbers import (I, pi) +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.integrals.integrals import integrate +from sympy.simplify.simplify import simplify +from sympy.abc import m, x, r +from sympy.physics.quantum.constants import hbar + + +def test_wavefunction(): + Psi = { + 0: (1/sqrt(2 * pi)), + 1: (1/sqrt(2 * pi)) * exp(I * x), + 2: (1/sqrt(2 * pi)) * exp(2 * I * x), + 3: (1/sqrt(2 * pi)) * exp(3 * I * x) + } + for n in Psi: + assert simplify(wavefunction(n, x) - Psi[n]) == 0 + + +def test_norm(n=1): + # Maximum "n" which is tested: + for i in range(n + 1): + assert integrate( + wavefunction(i, x) * wavefunction(-i, x), (x, 0, 2 * pi)) == 1 + + +def test_orthogonality(n=1): + # Maximum "n" which is tested: + for i in range(n + 1): + for j in range(i+1, n+1): + assert integrate( + wavefunction(i, x) * wavefunction(j, x), (x, 0, 2 * pi)) == 0 + + +def test_energy(n=1): + # Maximum "n" which is tested: + for i in range(n+1): + assert simplify( + energy(i, m, r) - ((i**2 * hbar**2) / (2 * m * r**2))) == 0 diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_qho_1d.py b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_qho_1d.py new file mode 100644 index 0000000000000000000000000000000000000000..34e52c9e3a721496fc61f7d2b31414db15caa7a8 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_qho_1d.py @@ -0,0 +1,50 @@ +from sympy.core.numbers import (Rational, oo, pi) +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.functions.elementary.exponential import exp +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.integrals.integrals import integrate +from sympy.simplify.simplify import simplify +from sympy.abc import omega, m, x +from sympy.physics.qho_1d import psi_n, E_n, coherent_state +from sympy.physics.quantum.constants import hbar + +nu = m * omega / hbar + + +def test_wavefunction(): + Psi = { + 0: (nu/pi)**Rational(1, 4) * exp(-nu * x**2 /2), + 1: (nu/pi)**Rational(1, 4) * sqrt(2*nu) * x * exp(-nu * x**2 /2), + 2: (nu/pi)**Rational(1, 4) * (2 * nu * x**2 - 1)/sqrt(2) * exp(-nu * x**2 /2), + 3: (nu/pi)**Rational(1, 4) * sqrt(nu/3) * (2 * nu * x**3 - 3 * x) * exp(-nu * x**2 /2) + } + for n in Psi: + assert simplify(psi_n(n, x, m, omega) - Psi[n]) == 0 + + +def test_norm(n=1): + # Maximum "n" which is tested: + for i in range(n + 1): + assert integrate(psi_n(i, x, 1, 1)**2, (x, -oo, oo)) == 1 + + +def test_orthogonality(n=1): + # Maximum "n" which is tested: + for i in range(n + 1): + for j in range(i + 1, n + 1): + assert integrate( + psi_n(i, x, 1, 1)*psi_n(j, x, 1, 1), (x, -oo, oo)) == 0 + + +def test_energies(n=1): + # Maximum "n" which is tested: + for i in range(n + 1): + assert E_n(i, omega) == hbar * omega * (i + S.Half) + +def test_coherent_state(n=10): + # Maximum "n" which is tested: + # test whether coherent state is the eigenstate of annihilation operator + alpha = Symbol("alpha") + for i in range(n + 1): + assert simplify(sqrt(n + 1) * coherent_state(n + 1, alpha)) == simplify(alpha * coherent_state(n, alpha)) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_secondquant.py b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_secondquant.py new file mode 100644 index 0000000000000000000000000000000000000000..dc9f4a499a7bee96d5fb5c76e83d84a72db5db8a --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_secondquant.py @@ -0,0 +1,1280 @@ +from sympy.physics.secondquant import ( + Dagger, Bd, VarBosonicBasis, BBra, B, BKet, FixedBosonicBasis, + matrix_rep, apply_operators, InnerProduct, Commutator, KroneckerDelta, + AnnihilateBoson, CreateBoson, BosonicOperator, + F, Fd, FKet, BosonState, CreateFermion, AnnihilateFermion, + evaluate_deltas, AntiSymmetricTensor, contraction, NO, wicks, + PermutationOperator, simplify_index_permutations, + _sort_anticommuting_fermions, _get_ordered_dummies, + substitute_dummies, FockStateBosonKet, + ContractionAppliesOnlyToFermions +) + +from sympy.concrete.summations import Sum +from sympy.core.function import (Function, expand) +from sympy.core.numbers import (I, Rational) +from sympy.core.singleton import S +from sympy.core.symbol import (Dummy, Symbol, symbols) +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.printing.repr import srepr +from sympy.simplify.simplify import simplify + +from sympy.testing.pytest import slow, raises +from sympy.printing.latex import latex + + +def test_PermutationOperator(): + p, q, r, s = symbols('p,q,r,s') + f, g, h, i = map(Function, 'fghi') + P = PermutationOperator + assert P(p, q).get_permuted(f(p)*g(q)) == -f(q)*g(p) + assert P(p, q).get_permuted(f(p, q)) == -f(q, p) + assert P(p, q).get_permuted(f(p)) == f(p) + expr = (f(p)*g(q)*h(r)*i(s) + - f(q)*g(p)*h(r)*i(s) + - f(p)*g(q)*h(s)*i(r) + + f(q)*g(p)*h(s)*i(r)) + perms = [P(p, q), P(r, s)] + assert (simplify_index_permutations(expr, perms) == + P(p, q)*P(r, s)*f(p)*g(q)*h(r)*i(s)) + assert latex(P(p, q)) == 'P(pq)' + + +def test_index_permutations_with_dummies(): + a, b, c, d = symbols('a b c d') + p, q, r, s = symbols('p q r s', cls=Dummy) + f, g = map(Function, 'fg') + P = PermutationOperator + + # No dummy substitution necessary + expr = f(a, b, p, q) - f(b, a, p, q) + assert simplify_index_permutations( + expr, [P(a, b)]) == P(a, b)*f(a, b, p, q) + + # Cases where dummy substitution is needed + expected = P(a, b)*substitute_dummies(f(a, b, p, q)) + + expr = f(a, b, p, q) - f(b, a, q, p) + result = simplify_index_permutations(expr, [P(a, b)]) + assert expected == substitute_dummies(result) + + expr = f(a, b, q, p) - f(b, a, p, q) + result = simplify_index_permutations(expr, [P(a, b)]) + assert expected == substitute_dummies(result) + + # A case where nothing can be done + expr = f(a, b, q, p) - g(b, a, p, q) + result = simplify_index_permutations(expr, [P(a, b)]) + assert expr == result + + +def test_dagger(): + i, j, n, m = symbols('i,j,n,m') + assert Dagger(1) == 1 + assert Dagger(1.0) == 1.0 + assert Dagger(2*I) == -2*I + assert Dagger(S.Half*I/3.0) == I*Rational(-1, 2)/3.0 + assert Dagger(BKet([n])) == BBra([n]) + assert Dagger(B(0)) == Bd(0) + assert Dagger(Bd(0)) == B(0) + assert Dagger(B(n)) == Bd(n) + assert Dagger(Bd(n)) == B(n) + assert Dagger(B(0) + B(1)) == Bd(0) + Bd(1) + assert Dagger(n*m) == Dagger(n)*Dagger(m) # n, m commute + assert Dagger(B(n)*B(m)) == Bd(m)*Bd(n) + assert Dagger(B(n)**10) == Dagger(B(n))**10 + assert Dagger('a') == Dagger(Symbol('a')) + assert Dagger(Dagger('a')) == Symbol('a') + + +def test_operator(): + i, j = symbols('i,j') + o = BosonicOperator(i) + assert o.state == i + assert o.is_symbolic + o = BosonicOperator(1) + assert o.state == 1 + assert not o.is_symbolic + + +def test_create(): + i, j, n, m = symbols('i,j,n,m') + o = Bd(i) + assert latex(o) == "{b^\\dagger_{i}}" + assert isinstance(o, CreateBoson) + o = o.subs(i, j) + assert o.atoms(Symbol) == {j} + o = Bd(0) + assert o.apply_operator(BKet([n])) == sqrt(n + 1)*BKet([n + 1]) + o = Bd(n) + assert o.apply_operator(BKet([n])) == o*BKet([n]) + + +def test_annihilate(): + i, j, n, m = symbols('i,j,n,m') + o = B(i) + assert latex(o) == "b_{i}" + assert isinstance(o, AnnihilateBoson) + o = o.subs(i, j) + assert o.atoms(Symbol) == {j} + o = B(0) + assert o.apply_operator(BKet([n])) == sqrt(n)*BKet([n - 1]) + o = B(n) + assert o.apply_operator(BKet([n])) == o*BKet([n]) + + +def test_basic_state(): + i, j, n, m = symbols('i,j,n,m') + s = BosonState([0, 1, 2, 3, 4]) + assert len(s) == 5 + assert s.args[0] == tuple(range(5)) + assert s.up(0) == BosonState([1, 1, 2, 3, 4]) + assert s.down(4) == BosonState([0, 1, 2, 3, 3]) + for i in range(5): + assert s.up(i).down(i) == s + assert s.down(0) == 0 + for i in range(5): + assert s[i] == i + s = BosonState([n, m]) + assert s.down(0) == BosonState([n - 1, m]) + assert s.up(0) == BosonState([n + 1, m]) + + +def test_basic_apply(): + n = symbols("n") + e = B(0)*BKet([n]) + assert apply_operators(e) == sqrt(n)*BKet([n - 1]) + e = Bd(0)*BKet([n]) + assert apply_operators(e) == sqrt(n + 1)*BKet([n + 1]) + + +def test_complex_apply(): + n, m = symbols("n,m") + o = Bd(0)*B(0)*Bd(1)*B(0) + e = apply_operators(o*BKet([n, m])) + answer = sqrt(n)*sqrt(m + 1)*(-1 + n)*BKet([-1 + n, 1 + m]) + assert expand(e) == expand(answer) + + +def test_number_operator(): + n = symbols("n") + o = Bd(0)*B(0) + e = apply_operators(o*BKet([n])) + assert e == n*BKet([n]) + + +def test_inner_product(): + i, j, k, l = symbols('i,j,k,l') + s1 = BBra([0]) + s2 = BKet([1]) + assert InnerProduct(s1, Dagger(s1)) == 1 + assert InnerProduct(s1, s2) == 0 + s1 = BBra([i, j]) + s2 = BKet([k, l]) + r = InnerProduct(s1, s2) + assert r == KroneckerDelta(i, k)*KroneckerDelta(j, l) + + +def test_symbolic_matrix_elements(): + n, m = symbols('n,m') + s1 = BBra([n]) + s2 = BKet([m]) + o = B(0) + e = apply_operators(s1*o*s2) + assert e == sqrt(m)*KroneckerDelta(n, m - 1) + + +def test_matrix_elements(): + b = VarBosonicBasis(5) + o = B(0) + m = matrix_rep(o, b) + for i in range(4): + assert m[i, i + 1] == sqrt(i + 1) + o = Bd(0) + m = matrix_rep(o, b) + for i in range(4): + assert m[i + 1, i] == sqrt(i + 1) + + +def test_fixed_bosonic_basis(): + b = FixedBosonicBasis(2, 2) + # assert b == [FockState((2, 0)), FockState((1, 1)), FockState((0, 2))] + state = b.state(1) + assert state == FockStateBosonKet((1, 1)) + assert b.index(state) == 1 + assert b.state(1) == b[1] + assert len(b) == 3 + assert str(b) == '[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]' + assert repr(b) == '[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]' + assert srepr(b) == '[FockState((2, 0)), FockState((1, 1)), FockState((0, 2))]' + + +@slow +def test_sho(): + n, m = symbols('n,m') + h_n = Bd(n)*B(n)*(n + S.Half) + H = Sum(h_n, (n, 0, 5)) + o = H.doit(deep=False) + b = FixedBosonicBasis(2, 6) + m = matrix_rep(o, b) + # We need to double check these energy values to make sure that they + # are correct and have the proper degeneracies! + diag = [1, 2, 3, 3, 4, 5, 4, 5, 6, 7, 5, 6, 7, 8, 9, 6, 7, 8, 9, 10, 11] + for i in range(len(diag)): + assert diag[i] == m[i, i] + + +def test_commutation(): + n, m = symbols("n,m", above_fermi=True) + c = Commutator(B(0), Bd(0)) + assert c == 1 + c = Commutator(Bd(0), B(0)) + assert c == -1 + c = Commutator(B(n), Bd(0)) + assert c == KroneckerDelta(n, 0) + c = Commutator(B(0), B(0)) + assert c == 0 + c = Commutator(B(0), Bd(0)) + e = simplify(apply_operators(c*BKet([n]))) + assert e == BKet([n]) + c = Commutator(B(0), B(1)) + e = simplify(apply_operators(c*BKet([n, m]))) + assert e == 0 + + c = Commutator(F(m), Fd(m)) + assert c == +1 - 2*NO(Fd(m)*F(m)) + c = Commutator(Fd(m), F(m)) + assert c.expand() == -1 + 2*NO(Fd(m)*F(m)) + + C = Commutator + X, Y, Z = symbols('X,Y,Z', commutative=False) + assert C(C(X, Y), Z) != 0 + assert C(C(X, Z), Y) != 0 + assert C(Y, C(X, Z)) != 0 + + i, j, k, l = symbols('i,j,k,l', below_fermi=True) + a, b, c, d = symbols('a,b,c,d', above_fermi=True) + p, q, r, s = symbols('p,q,r,s') + D = KroneckerDelta + + assert C(Fd(a), F(i)) == -2*NO(F(i)*Fd(a)) + assert C(Fd(j), NO(Fd(a)*F(i))).doit(wicks=True) == -D(j, i)*Fd(a) + assert C(Fd(a)*F(i), Fd(b)*F(j)).doit(wicks=True) == 0 + + c1 = Commutator(F(a), Fd(a)) + assert Commutator.eval(c1, c1) == 0 + c = Commutator(Fd(a)*F(i),Fd(b)*F(j)) + assert latex(c) == r'\left[{a^\dagger_{a}} a_{i},{a^\dagger_{b}} a_{j}\right]' + assert repr(c) == 'Commutator(CreateFermion(a)*AnnihilateFermion(i),CreateFermion(b)*AnnihilateFermion(j))' + assert str(c) == '[CreateFermion(a)*AnnihilateFermion(i),CreateFermion(b)*AnnihilateFermion(j)]' + + +def test_create_f(): + i, j, n, m = symbols('i,j,n,m') + o = Fd(i) + assert isinstance(o, CreateFermion) + o = o.subs(i, j) + assert o.atoms(Symbol) == {j} + o = Fd(1) + assert o.apply_operator(FKet([n])) == FKet([1, n]) + assert o.apply_operator(FKet([n])) == -FKet([n, 1]) + o = Fd(n) + assert o.apply_operator(FKet([])) == FKet([n]) + + vacuum = FKet([], fermi_level=4) + assert vacuum == FKet([], fermi_level=4) + + i, j, k, l = symbols('i,j,k,l', below_fermi=True) + a, b, c, d = symbols('a,b,c,d', above_fermi=True) + p, q, r, s = symbols('p,q,r,s') + + assert Fd(i).apply_operator(FKet([i, j, k], 4)) == FKet([j, k], 4) + assert Fd(a).apply_operator(FKet([i, b, k], 4)) == FKet([a, i, b, k], 4) + + assert Dagger(B(p)).apply_operator(q) == q*CreateBoson(p) + assert repr(Fd(p)) == 'CreateFermion(p)' + assert srepr(Fd(p)) == "CreateFermion(Symbol('p'))" + assert latex(Fd(p)) == r'{a^\dagger_{p}}' + + +def test_annihilate_f(): + i, j, n, m = symbols('i,j,n,m') + o = F(i) + assert isinstance(o, AnnihilateFermion) + o = o.subs(i, j) + assert o.atoms(Symbol) == {j} + o = F(1) + assert o.apply_operator(FKet([1, n])) == FKet([n]) + assert o.apply_operator(FKet([n, 1])) == -FKet([n]) + o = F(n) + assert o.apply_operator(FKet([n])) == FKet([]) + + i, j, k, l = symbols('i,j,k,l', below_fermi=True) + a, b, c, d = symbols('a,b,c,d', above_fermi=True) + p, q, r, s = symbols('p,q,r,s') + assert F(i).apply_operator(FKet([i, j, k], 4)) == 0 + assert F(a).apply_operator(FKet([i, b, k], 4)) == 0 + assert F(l).apply_operator(FKet([i, j, k], 3)) == 0 + assert F(l).apply_operator(FKet([i, j, k], 4)) == FKet([l, i, j, k], 4) + assert str(F(p)) == 'f(p)' + assert repr(F(p)) == 'AnnihilateFermion(p)' + assert srepr(F(p)) == "AnnihilateFermion(Symbol('p'))" + assert latex(F(p)) == 'a_{p}' + + +def test_create_b(): + i, j, n, m = symbols('i,j,n,m') + o = Bd(i) + assert isinstance(o, CreateBoson) + o = o.subs(i, j) + assert o.atoms(Symbol) == {j} + o = Bd(0) + assert o.apply_operator(BKet([n])) == sqrt(n + 1)*BKet([n + 1]) + o = Bd(n) + assert o.apply_operator(BKet([n])) == o*BKet([n]) + + +def test_annihilate_b(): + i, j, n, m = symbols('i,j,n,m') + o = B(i) + assert isinstance(o, AnnihilateBoson) + o = o.subs(i, j) + assert o.atoms(Symbol) == {j} + o = B(0) + + +def test_wicks(): + p, q, r, s = symbols('p,q,r,s', above_fermi=True) + + # Testing for particles only + + str = F(p)*Fd(q) + assert wicks(str) == NO(F(p)*Fd(q)) + KroneckerDelta(p, q) + str = Fd(p)*F(q) + assert wicks(str) == NO(Fd(p)*F(q)) + + str = F(p)*Fd(q)*F(r)*Fd(s) + nstr = wicks(str) + fasit = NO( + KroneckerDelta(p, q)*KroneckerDelta(r, s) + + KroneckerDelta(p, q)*AnnihilateFermion(r)*CreateFermion(s) + + KroneckerDelta(r, s)*AnnihilateFermion(p)*CreateFermion(q) + - KroneckerDelta(p, s)*AnnihilateFermion(r)*CreateFermion(q) + - AnnihilateFermion(p)*AnnihilateFermion(r)*CreateFermion(q)*CreateFermion(s)) + assert nstr == fasit + + assert (p*q*nstr).expand() == wicks(p*q*str) + assert (nstr*p*q*2).expand() == wicks(str*p*q*2) + + # Testing CC equations particles and holes + i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy) + a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy) + p, q, r, s = symbols('p q r s', cls=Dummy) + + assert (wicks(F(a)*NO(F(i)*F(j))*Fd(b)) == + NO(F(a)*F(i)*F(j)*Fd(b)) + + KroneckerDelta(a, b)*NO(F(i)*F(j))) + assert (wicks(F(a)*NO(F(i)*F(j)*F(k))*Fd(b)) == + NO(F(a)*F(i)*F(j)*F(k)*Fd(b)) - + KroneckerDelta(a, b)*NO(F(i)*F(j)*F(k))) + + expr = wicks(Fd(i)*NO(Fd(j)*F(k))*F(l)) + assert (expr == + -KroneckerDelta(i, k)*NO(Fd(j)*F(l)) - + KroneckerDelta(j, l)*NO(Fd(i)*F(k)) - + KroneckerDelta(i, k)*KroneckerDelta(j, l) + + KroneckerDelta(i, l)*NO(Fd(j)*F(k)) + + NO(Fd(i)*Fd(j)*F(k)*F(l))) + expr = wicks(F(a)*NO(F(b)*Fd(c))*Fd(d)) + assert (expr == + -KroneckerDelta(a, c)*NO(F(b)*Fd(d)) - + KroneckerDelta(b, d)*NO(F(a)*Fd(c)) - + KroneckerDelta(a, c)*KroneckerDelta(b, d) + + KroneckerDelta(a, d)*NO(F(b)*Fd(c)) + + NO(F(a)*F(b)*Fd(c)*Fd(d))) + + +def test_NO(): + i, j, k, l = symbols('i j k l', below_fermi=True) + a, b, c, d = symbols('a b c d', above_fermi=True) + p, q, r, s = symbols('p q r s', cls=Dummy) + + assert (NO(Fd(p)*F(q) + Fd(a)*F(b)) == + NO(Fd(p)*F(q)) + NO(Fd(a)*F(b))) + assert (NO(Fd(i)*NO(F(j)*Fd(a))) == + NO(Fd(i)*F(j)*Fd(a))) + assert NO(1) == 1 + assert NO(i) == i + assert (NO(Fd(a)*Fd(b)*(F(c) + F(d))) == + NO(Fd(a)*Fd(b)*F(c)) + + NO(Fd(a)*Fd(b)*F(d))) + + assert NO(Fd(a)*F(b))._remove_brackets() == Fd(a)*F(b) + assert NO(F(j)*Fd(i))._remove_brackets() == F(j)*Fd(i) + + assert (NO(Fd(p)*F(q)).subs(Fd(p), Fd(a) + Fd(i)) == + NO(Fd(a)*F(q)) + NO(Fd(i)*F(q))) + assert (NO(Fd(p)*F(q)).subs(F(q), F(a) + F(i)) == + NO(Fd(p)*F(a)) + NO(Fd(p)*F(i))) + + expr = NO(Fd(p)*F(q))._remove_brackets() + assert wicks(expr) == NO(expr) + + assert NO(Fd(a)*F(b)) == - NO(F(b)*Fd(a)) + + no = NO(Fd(a)*F(i)*F(b)*Fd(j)) + l1 = list(no.iter_q_creators()) + assert l1 == [0, 1] + l2 = list(no.iter_q_annihilators()) + assert l2 == [3, 2] + no = NO(Fd(a)*Fd(i)) + assert no.has_q_creators == 1 + assert no.has_q_annihilators == -1 + assert str(no) == ':CreateFermion(a)*CreateFermion(i):' + assert repr(no) == 'NO(CreateFermion(a)*CreateFermion(i))' + assert latex(no) == r'\left\{{a^\dagger_{a}} {a^\dagger_{i}}\right\}' + raises(NotImplementedError, lambda: NO(Bd(p)*F(q))) + + +def test_sorting(): + i, j = symbols('i,j', below_fermi=True) + a, b = symbols('a,b', above_fermi=True) + p, q = symbols('p,q') + + # p, q + assert _sort_anticommuting_fermions([Fd(p), F(q)]) == ([Fd(p), F(q)], 0) + assert _sort_anticommuting_fermions([F(p), Fd(q)]) == ([Fd(q), F(p)], 1) + + # i, p + assert _sort_anticommuting_fermions([F(p), Fd(i)]) == ([F(p), Fd(i)], 0) + assert _sort_anticommuting_fermions([Fd(i), F(p)]) == ([F(p), Fd(i)], 1) + assert _sort_anticommuting_fermions([Fd(p), Fd(i)]) == ([Fd(p), Fd(i)], 0) + assert _sort_anticommuting_fermions([Fd(i), Fd(p)]) == ([Fd(p), Fd(i)], 1) + assert _sort_anticommuting_fermions([F(p), F(i)]) == ([F(i), F(p)], 1) + assert _sort_anticommuting_fermions([F(i), F(p)]) == ([F(i), F(p)], 0) + assert _sort_anticommuting_fermions([Fd(p), F(i)]) == ([F(i), Fd(p)], 1) + assert _sort_anticommuting_fermions([F(i), Fd(p)]) == ([F(i), Fd(p)], 0) + + # a, p + assert _sort_anticommuting_fermions([F(p), Fd(a)]) == ([Fd(a), F(p)], 1) + assert _sort_anticommuting_fermions([Fd(a), F(p)]) == ([Fd(a), F(p)], 0) + assert _sort_anticommuting_fermions([Fd(p), Fd(a)]) == ([Fd(a), Fd(p)], 1) + assert _sort_anticommuting_fermions([Fd(a), Fd(p)]) == ([Fd(a), Fd(p)], 0) + assert _sort_anticommuting_fermions([F(p), F(a)]) == ([F(p), F(a)], 0) + assert _sort_anticommuting_fermions([F(a), F(p)]) == ([F(p), F(a)], 1) + assert _sort_anticommuting_fermions([Fd(p), F(a)]) == ([Fd(p), F(a)], 0) + assert _sort_anticommuting_fermions([F(a), Fd(p)]) == ([Fd(p), F(a)], 1) + + # i, a + assert _sort_anticommuting_fermions([F(i), Fd(j)]) == ([F(i), Fd(j)], 0) + assert _sort_anticommuting_fermions([Fd(j), F(i)]) == ([F(i), Fd(j)], 1) + assert _sort_anticommuting_fermions([Fd(a), Fd(i)]) == ([Fd(a), Fd(i)], 0) + assert _sort_anticommuting_fermions([Fd(i), Fd(a)]) == ([Fd(a), Fd(i)], 1) + assert _sort_anticommuting_fermions([F(a), F(i)]) == ([F(i), F(a)], 1) + assert _sort_anticommuting_fermions([F(i), F(a)]) == ([F(i), F(a)], 0) + + +def test_contraction(): + i, j, k, l = symbols('i,j,k,l', below_fermi=True) + a, b, c, d = symbols('a,b,c,d', above_fermi=True) + p, q, r, s = symbols('p,q,r,s') + assert contraction(Fd(i), F(j)) == KroneckerDelta(i, j) + assert contraction(F(a), Fd(b)) == KroneckerDelta(a, b) + assert contraction(F(a), Fd(i)) == 0 + assert contraction(Fd(a), F(i)) == 0 + assert contraction(F(i), Fd(a)) == 0 + assert contraction(Fd(i), F(a)) == 0 + assert contraction(Fd(i), F(p)) == KroneckerDelta(i, p) + restr = evaluate_deltas(contraction(Fd(p), F(q))) + assert restr.is_only_below_fermi + restr = evaluate_deltas(contraction(F(p), Fd(q))) + assert restr.is_only_above_fermi + raises(ContractionAppliesOnlyToFermions, lambda: contraction(B(a), Fd(b))) + + +def test_evaluate_deltas(): + i, j, k = symbols('i,j,k') + + r = KroneckerDelta(i, j) * KroneckerDelta(j, k) + assert evaluate_deltas(r) == KroneckerDelta(i, k) + + r = KroneckerDelta(i, 0) * KroneckerDelta(j, k) + assert evaluate_deltas(r) == KroneckerDelta(i, 0) * KroneckerDelta(j, k) + + r = KroneckerDelta(1, j) * KroneckerDelta(j, k) + assert evaluate_deltas(r) == KroneckerDelta(1, k) + + r = KroneckerDelta(j, 2) * KroneckerDelta(k, j) + assert evaluate_deltas(r) == KroneckerDelta(2, k) + + r = KroneckerDelta(i, 0) * KroneckerDelta(i, j) * KroneckerDelta(j, 1) + assert evaluate_deltas(r) == 0 + + r = (KroneckerDelta(0, i) * KroneckerDelta(0, j) + * KroneckerDelta(1, j) * KroneckerDelta(1, j)) + assert evaluate_deltas(r) == 0 + + +def test_Tensors(): + i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy) + a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy) + p, q, r, s = symbols('p q r s') + + AT = AntiSymmetricTensor + assert AT('t', (a, b), (i, j)) == -AT('t', (b, a), (i, j)) + assert AT('t', (a, b), (i, j)) == AT('t', (b, a), (j, i)) + assert AT('t', (a, b), (i, j)) == -AT('t', (a, b), (j, i)) + assert AT('t', (a, a), (i, j)) == 0 + assert AT('t', (a, b), (i, i)) == 0 + assert AT('t', (a, b, c), (i, j)) == -AT('t', (b, a, c), (i, j)) + assert AT('t', (a, b, c), (i, j, k)) == AT('t', (b, a, c), (i, k, j)) + + tabij = AT('t', (a, b), (i, j)) + assert tabij.has(a) + assert tabij.has(b) + assert tabij.has(i) + assert tabij.has(j) + assert tabij.subs(b, c) == AT('t', (a, c), (i, j)) + assert (2*tabij).subs(i, c) == 2*AT('t', (a, b), (c, j)) + assert tabij.symbol == Symbol('t') + assert latex(tabij) == '{t^{ab}_{ij}}' + assert str(tabij) == 't((_a, _b),(_i, _j))' + + assert AT('t', (a, a), (i, j)).subs(a, b) == AT('t', (b, b), (i, j)) + assert AT('t', (a, i), (a, j)).subs(a, b) == AT('t', (b, i), (b, j)) + + +def test_fully_contracted(): + i, j, k, l = symbols('i j k l', below_fermi=True) + a, b, c, d = symbols('a b c d', above_fermi=True) + p, q, r, s = symbols('p q r s', cls=Dummy) + + Fock = (AntiSymmetricTensor('f', (p,), (q,))* + NO(Fd(p)*F(q))) + V = (AntiSymmetricTensor('v', (p, q), (r, s))* + NO(Fd(p)*Fd(q)*F(s)*F(r)))/4 + + Fai = wicks(NO(Fd(i)*F(a))*Fock, + keep_only_fully_contracted=True, + simplify_kronecker_deltas=True) + assert Fai == AntiSymmetricTensor('f', (a,), (i,)) + Vabij = wicks(NO(Fd(i)*Fd(j)*F(b)*F(a))*V, + keep_only_fully_contracted=True, + simplify_kronecker_deltas=True) + assert Vabij == AntiSymmetricTensor('v', (a, b), (i, j)) + + +def test_substitute_dummies_without_dummies(): + i, j = symbols('i,j') + assert substitute_dummies(att(i, j) + 2) == att(i, j) + 2 + assert substitute_dummies(att(i, j) + 1) == att(i, j) + 1 + + +def test_substitute_dummies_NO_operator(): + i, j = symbols('i j', cls=Dummy) + assert substitute_dummies(att(i, j)*NO(Fd(i)*F(j)) + - att(j, i)*NO(Fd(j)*F(i))) == 0 + + +def test_substitute_dummies_SQ_operator(): + i, j = symbols('i j', cls=Dummy) + assert substitute_dummies(att(i, j)*Fd(i)*F(j) + - att(j, i)*Fd(j)*F(i)) == 0 + + +def test_substitute_dummies_new_indices(): + i, j = symbols('i j', below_fermi=True, cls=Dummy) + a, b = symbols('a b', above_fermi=True, cls=Dummy) + p, q = symbols('p q', cls=Dummy) + f = Function('f') + assert substitute_dummies(f(i, a, p) - f(j, b, q), new_indices=True) == 0 + + +def test_substitute_dummies_substitution_order(): + i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy) + f = Function('f') + from sympy.utilities.iterables import variations + for permut in variations([i, j, k, l], 4): + assert substitute_dummies(f(*permut) - f(i, j, k, l)) == 0 + + +def test_dummy_order_inner_outer_lines_VT1T1T1(): + ii = symbols('i', below_fermi=True) + aa = symbols('a', above_fermi=True) + k, l = symbols('k l', below_fermi=True, cls=Dummy) + c, d = symbols('c d', above_fermi=True, cls=Dummy) + + v = Function('v') + t = Function('t') + dums = _get_ordered_dummies + + # Coupled-Cluster T1 terms with V*T1*T1*T1 + # t^{a}_{k} t^{c}_{i} t^{d}_{l} v^{lk}_{dc} + exprs = [ + # permut v and t <=> swapping internal lines, equivalent + # irrespective of symmetries in v + v(k, l, c, d)*t(c, ii)*t(d, l)*t(aa, k), + v(l, k, c, d)*t(c, ii)*t(d, k)*t(aa, l), + v(k, l, d, c)*t(d, ii)*t(c, l)*t(aa, k), + v(l, k, d, c)*t(d, ii)*t(c, k)*t(aa, l), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_dummy_order_inner_outer_lines_VT1T1T1T1(): + ii, jj = symbols('i j', below_fermi=True) + aa, bb = symbols('a b', above_fermi=True) + k, l = symbols('k l', below_fermi=True, cls=Dummy) + c, d = symbols('c d', above_fermi=True, cls=Dummy) + + v = Function('v') + t = Function('t') + dums = _get_ordered_dummies + + # Coupled-Cluster T2 terms with V*T1*T1*T1*T1 + exprs = [ + # permut t <=> swapping external lines, not equivalent + # except if v has certain symmetries. + v(k, l, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l), + v(k, l, c, d)*t(c, jj)*t(d, ii)*t(aa, k)*t(bb, l), + v(k, l, c, d)*t(c, ii)*t(d, jj)*t(bb, k)*t(aa, l), + v(k, l, c, d)*t(c, jj)*t(d, ii)*t(bb, k)*t(aa, l), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) != substitute_dummies(permut) + exprs = [ + # permut v <=> swapping external lines, not equivalent + # except if v has certain symmetries. + # + # Note that in contrast to above, these permutations have identical + # dummy order. That is because the proximity to external indices + # has higher influence on the canonical dummy ordering than the + # position of a dummy on the factors. In fact, the terms here are + # similar in structure as the result of the dummy substitutions above. + v(k, l, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l), + v(l, k, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l), + v(k, l, d, c)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l), + v(l, k, d, c)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) == dums(permut) + assert substitute_dummies(exprs[0]) != substitute_dummies(permut) + exprs = [ + # permut t and v <=> swapping internal lines, equivalent. + # Canonical dummy order is different, and a consistent + # substitution reveals the equivalence. + v(k, l, c, d)*t(c, ii)*t(d, jj)*t(aa, k)*t(bb, l), + v(k, l, d, c)*t(c, jj)*t(d, ii)*t(aa, k)*t(bb, l), + v(l, k, c, d)*t(c, ii)*t(d, jj)*t(bb, k)*t(aa, l), + v(l, k, d, c)*t(c, jj)*t(d, ii)*t(bb, k)*t(aa, l), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_get_subNO(): + p, q, r = symbols('p,q,r') + assert NO(F(p)*F(q)*F(r)).get_subNO(1) == NO(F(p)*F(r)) + assert NO(F(p)*F(q)*F(r)).get_subNO(0) == NO(F(q)*F(r)) + assert NO(F(p)*F(q)*F(r)).get_subNO(2) == NO(F(p)*F(q)) + + +def test_equivalent_internal_lines_VT1T1(): + i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy) + a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy) + + v = Function('v') + t = Function('t') + dums = _get_ordered_dummies + + exprs = [ # permute v. Different dummy order. Not equivalent. + v(i, j, a, b)*t(a, i)*t(b, j), + v(j, i, a, b)*t(a, i)*t(b, j), + v(i, j, b, a)*t(a, i)*t(b, j), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) != substitute_dummies(permut) + + exprs = [ # permute v. Different dummy order. Equivalent + v(i, j, a, b)*t(a, i)*t(b, j), + v(j, i, b, a)*t(a, i)*t(b, j), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + exprs = [ # permute t. Same dummy order, not equivalent. + v(i, j, a, b)*t(a, i)*t(b, j), + v(i, j, a, b)*t(b, i)*t(a, j), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) == dums(permut) + assert substitute_dummies(exprs[0]) != substitute_dummies(permut) + + exprs = [ # permute v and t. Different dummy order, equivalent + v(i, j, a, b)*t(a, i)*t(b, j), + v(j, i, a, b)*t(a, j)*t(b, i), + v(i, j, b, a)*t(b, i)*t(a, j), + v(j, i, b, a)*t(b, j)*t(a, i), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_equivalent_internal_lines_VT2conjT2(): + # this diagram requires special handling in TCE + i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy) + a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy) + p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy) + h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy) + + from sympy.utilities.iterables import variations + + v = Function('v') + t = Function('t') + dums = _get_ordered_dummies + + # v(abcd)t(abij)t(ijcd) + template = v(p1, p2, p3, p4)*t(p1, p2, i, j)*t(i, j, p3, p4) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert dums(base) != dums(expr) + assert substitute_dummies(expr) == substitute_dummies(base) + template = v(p1, p2, p3, p4)*t(p1, p2, j, i)*t(j, i, p3, p4) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert dums(base) != dums(expr) + assert substitute_dummies(expr) == substitute_dummies(base) + + # v(abcd)t(abij)t(jicd) + template = v(p1, p2, p3, p4)*t(p1, p2, i, j)*t(j, i, p3, p4) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert dums(base) != dums(expr) + assert substitute_dummies(expr) == substitute_dummies(base) + template = v(p1, p2, p3, p4)*t(p1, p2, j, i)*t(i, j, p3, p4) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert dums(base) != dums(expr) + assert substitute_dummies(expr) == substitute_dummies(base) + + +def test_equivalent_internal_lines_VT2conjT2_ambiguous_order(): + # These diagrams invokes _determine_ambiguous() because the + # dummies can not be ordered unambiguously by the key alone + i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy) + a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy) + p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy) + h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy) + + from sympy.utilities.iterables import variations + + v = Function('v') + t = Function('t') + dums = _get_ordered_dummies + + # v(abcd)t(abij)t(cdij) + template = v(p1, p2, p3, p4)*t(p1, p2, i, j)*t(p3, p4, i, j) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert dums(base) != dums(expr) + assert substitute_dummies(expr) == substitute_dummies(base) + template = v(p1, p2, p3, p4)*t(p1, p2, j, i)*t(p3, p4, i, j) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert dums(base) != dums(expr) + assert substitute_dummies(expr) == substitute_dummies(base) + + +def test_equivalent_internal_lines_VT2(): + i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy) + a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy) + + v = Function('v') + t = Function('t') + dums = _get_ordered_dummies + exprs = [ + # permute v. Same dummy order, not equivalent. + # + # This test show that the dummy order may not be sensitive to all + # index permutations. The following expressions have identical + # structure as the resulting terms from of the dummy substitutions + # in the test above. Here, all expressions have the same dummy + # order, so they cannot be simplified by means of dummy + # substitution. In order to simplify further, it is necessary to + # exploit symmetries in the objects, for instance if t or v is + # antisymmetric. + v(i, j, a, b)*t(a, b, i, j), + v(j, i, a, b)*t(a, b, i, j), + v(i, j, b, a)*t(a, b, i, j), + v(j, i, b, a)*t(a, b, i, j), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) == dums(permut) + assert substitute_dummies(exprs[0]) != substitute_dummies(permut) + + exprs = [ + # permute t. + v(i, j, a, b)*t(a, b, i, j), + v(i, j, a, b)*t(b, a, i, j), + v(i, j, a, b)*t(a, b, j, i), + v(i, j, a, b)*t(b, a, j, i), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) != substitute_dummies(permut) + + exprs = [ # permute v and t. Relabelling of dummies should be equivalent. + v(i, j, a, b)*t(a, b, i, j), + v(j, i, a, b)*t(a, b, j, i), + v(i, j, b, a)*t(b, a, i, j), + v(j, i, b, a)*t(b, a, j, i), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_internal_external_VT2T2(): + ii, jj = symbols('i j', below_fermi=True) + aa, bb = symbols('a b', above_fermi=True) + k, l = symbols('k l', below_fermi=True, cls=Dummy) + c, d = symbols('c d', above_fermi=True, cls=Dummy) + + v = Function('v') + t = Function('t') + dums = _get_ordered_dummies + + exprs = [ + v(k, l, c, d)*t(aa, c, ii, k)*t(bb, d, jj, l), + v(l, k, c, d)*t(aa, c, ii, l)*t(bb, d, jj, k), + v(k, l, d, c)*t(aa, d, ii, k)*t(bb, c, jj, l), + v(l, k, d, c)*t(aa, d, ii, l)*t(bb, c, jj, k), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + exprs = [ + v(k, l, c, d)*t(aa, c, ii, k)*t(d, bb, jj, l), + v(l, k, c, d)*t(aa, c, ii, l)*t(d, bb, jj, k), + v(k, l, d, c)*t(aa, d, ii, k)*t(c, bb, jj, l), + v(l, k, d, c)*t(aa, d, ii, l)*t(c, bb, jj, k), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + exprs = [ + v(k, l, c, d)*t(c, aa, ii, k)*t(bb, d, jj, l), + v(l, k, c, d)*t(c, aa, ii, l)*t(bb, d, jj, k), + v(k, l, d, c)*t(d, aa, ii, k)*t(bb, c, jj, l), + v(l, k, d, c)*t(d, aa, ii, l)*t(bb, c, jj, k), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_internal_external_pqrs(): + ii, jj = symbols('i j') + aa, bb = symbols('a b') + k, l = symbols('k l', cls=Dummy) + c, d = symbols('c d', cls=Dummy) + + v = Function('v') + t = Function('t') + dums = _get_ordered_dummies + + exprs = [ + v(k, l, c, d)*t(aa, c, ii, k)*t(bb, d, jj, l), + v(l, k, c, d)*t(aa, c, ii, l)*t(bb, d, jj, k), + v(k, l, d, c)*t(aa, d, ii, k)*t(bb, c, jj, l), + v(l, k, d, c)*t(aa, d, ii, l)*t(bb, c, jj, k), + ] + for permut in exprs[1:]: + assert dums(exprs[0]) != dums(permut) + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_dummy_order_well_defined(): + aa, bb = symbols('a b', above_fermi=True) + k, l, m = symbols('k l m', below_fermi=True, cls=Dummy) + c, d = symbols('c d', above_fermi=True, cls=Dummy) + p, q = symbols('p q', cls=Dummy) + + A = Function('A') + B = Function('B') + C = Function('C') + dums = _get_ordered_dummies + + # We go through all key components in the order of increasing priority, + # and consider only fully orderable expressions. Non-orderable expressions + # are tested elsewhere. + + # pos in first factor determines sort order + assert dums(A(k, l)*B(l, k)) == [k, l] + assert dums(A(l, k)*B(l, k)) == [l, k] + assert dums(A(k, l)*B(k, l)) == [k, l] + assert dums(A(l, k)*B(k, l)) == [l, k] + + # factors involving the index + assert dums(A(k, l)*B(l, m)*C(k, m)) == [l, k, m] + assert dums(A(k, l)*B(l, m)*C(m, k)) == [l, k, m] + assert dums(A(l, k)*B(l, m)*C(k, m)) == [l, k, m] + assert dums(A(l, k)*B(l, m)*C(m, k)) == [l, k, m] + assert dums(A(k, l)*B(m, l)*C(k, m)) == [l, k, m] + assert dums(A(k, l)*B(m, l)*C(m, k)) == [l, k, m] + assert dums(A(l, k)*B(m, l)*C(k, m)) == [l, k, m] + assert dums(A(l, k)*B(m, l)*C(m, k)) == [l, k, m] + + # same, but with factor order determined by non-dummies + assert dums(A(k, aa, l)*A(l, bb, m)*A(bb, k, m)) == [l, k, m] + assert dums(A(k, aa, l)*A(l, bb, m)*A(bb, m, k)) == [l, k, m] + assert dums(A(k, aa, l)*A(m, bb, l)*A(bb, k, m)) == [l, k, m] + assert dums(A(k, aa, l)*A(m, bb, l)*A(bb, m, k)) == [l, k, m] + assert dums(A(l, aa, k)*A(l, bb, m)*A(bb, k, m)) == [l, k, m] + assert dums(A(l, aa, k)*A(l, bb, m)*A(bb, m, k)) == [l, k, m] + assert dums(A(l, aa, k)*A(m, bb, l)*A(bb, k, m)) == [l, k, m] + assert dums(A(l, aa, k)*A(m, bb, l)*A(bb, m, k)) == [l, k, m] + + # index range + assert dums(A(p, c, k)*B(p, c, k)) == [k, c, p] + assert dums(A(p, k, c)*B(p, c, k)) == [k, c, p] + assert dums(A(c, k, p)*B(p, c, k)) == [k, c, p] + assert dums(A(c, p, k)*B(p, c, k)) == [k, c, p] + assert dums(A(k, c, p)*B(p, c, k)) == [k, c, p] + assert dums(A(k, p, c)*B(p, c, k)) == [k, c, p] + assert dums(B(p, c, k)*A(p, c, k)) == [k, c, p] + assert dums(B(p, k, c)*A(p, c, k)) == [k, c, p] + assert dums(B(c, k, p)*A(p, c, k)) == [k, c, p] + assert dums(B(c, p, k)*A(p, c, k)) == [k, c, p] + assert dums(B(k, c, p)*A(p, c, k)) == [k, c, p] + assert dums(B(k, p, c)*A(p, c, k)) == [k, c, p] + + +def test_dummy_order_ambiguous(): + aa, bb = symbols('a b', above_fermi=True) + i, j, k, l, m = symbols('i j k l m', below_fermi=True, cls=Dummy) + a, b, c, d, e = symbols('a b c d e', above_fermi=True, cls=Dummy) + p, q = symbols('p q', cls=Dummy) + p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy) + p5, p6, p7, p8 = symbols('p5 p6 p7 p8', above_fermi=True, cls=Dummy) + h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy) + h5, h6, h7, h8 = symbols('h5 h6 h7 h8', below_fermi=True, cls=Dummy) + + A = Function('A') + B = Function('B') + + from sympy.utilities.iterables import variations + + # A*A*A*A*B -- ordering of p5 and p4 is used to figure out the rest + template = A(p1, p2)*A(p4, p1)*A(p2, p3)*A(p3, p5)*B(p5, p4) + permutator = variations([a, b, c, d, e], 5) + base = template.subs(zip([p1, p2, p3, p4, p5], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4, p5], permut) + expr = template.subs(subslist) + assert substitute_dummies(expr) == substitute_dummies(base) + + # A*A*A*A*A -- an arbitrary index is assigned and the rest are figured out + template = A(p1, p2)*A(p4, p1)*A(p2, p3)*A(p3, p5)*A(p5, p4) + permutator = variations([a, b, c, d, e], 5) + base = template.subs(zip([p1, p2, p3, p4, p5], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4, p5], permut) + expr = template.subs(subslist) + assert substitute_dummies(expr) == substitute_dummies(base) + + # A*A*A -- ordering of p5 and p4 is used to figure out the rest + template = A(p1, p2, p4, p1)*A(p2, p3, p3, p5)*A(p5, p4) + permutator = variations([a, b, c, d, e], 5) + base = template.subs(zip([p1, p2, p3, p4, p5], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4, p5], permut) + expr = template.subs(subslist) + assert substitute_dummies(expr) == substitute_dummies(base) + + +def atv(*args): + return AntiSymmetricTensor('v', args[:2], args[2:] ) + + +def att(*args): + if len(args) == 4: + return AntiSymmetricTensor('t', args[:2], args[2:] ) + elif len(args) == 2: + return AntiSymmetricTensor('t', (args[0],), (args[1],)) + + +def test_dummy_order_inner_outer_lines_VT1T1T1_AT(): + ii = symbols('i', below_fermi=True) + aa = symbols('a', above_fermi=True) + k, l = symbols('k l', below_fermi=True, cls=Dummy) + c, d = symbols('c d', above_fermi=True, cls=Dummy) + + # Coupled-Cluster T1 terms with V*T1*T1*T1 + # t^{a}_{k} t^{c}_{i} t^{d}_{l} v^{lk}_{dc} + exprs = [ + # permut v and t <=> swapping internal lines, equivalent + # irrespective of symmetries in v + atv(k, l, c, d)*att(c, ii)*att(d, l)*att(aa, k), + atv(l, k, c, d)*att(c, ii)*att(d, k)*att(aa, l), + atv(k, l, d, c)*att(d, ii)*att(c, l)*att(aa, k), + atv(l, k, d, c)*att(d, ii)*att(c, k)*att(aa, l), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_dummy_order_inner_outer_lines_VT1T1T1T1_AT(): + ii, jj = symbols('i j', below_fermi=True) + aa, bb = symbols('a b', above_fermi=True) + k, l = symbols('k l', below_fermi=True, cls=Dummy) + c, d = symbols('c d', above_fermi=True, cls=Dummy) + + # Coupled-Cluster T2 terms with V*T1*T1*T1*T1 + # non-equivalent substitutions (change of sign) + exprs = [ + # permut t <=> swapping external lines + atv(k, l, c, d)*att(c, ii)*att(d, jj)*att(aa, k)*att(bb, l), + atv(k, l, c, d)*att(c, jj)*att(d, ii)*att(aa, k)*att(bb, l), + atv(k, l, c, d)*att(c, ii)*att(d, jj)*att(bb, k)*att(aa, l), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) == -substitute_dummies(permut) + + # equivalent substitutions + exprs = [ + atv(k, l, c, d)*att(c, ii)*att(d, jj)*att(aa, k)*att(bb, l), + # permut t <=> swapping external lines + atv(k, l, c, d)*att(c, jj)*att(d, ii)*att(bb, k)*att(aa, l), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_equivalent_internal_lines_VT1T1_AT(): + i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy) + a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy) + + exprs = [ # permute v. Different dummy order. Not equivalent. + atv(i, j, a, b)*att(a, i)*att(b, j), + atv(j, i, a, b)*att(a, i)*att(b, j), + atv(i, j, b, a)*att(a, i)*att(b, j), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) != substitute_dummies(permut) + + exprs = [ # permute v. Different dummy order. Equivalent + atv(i, j, a, b)*att(a, i)*att(b, j), + atv(j, i, b, a)*att(a, i)*att(b, j), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + exprs = [ # permute t. Same dummy order, not equivalent. + atv(i, j, a, b)*att(a, i)*att(b, j), + atv(i, j, a, b)*att(b, i)*att(a, j), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) != substitute_dummies(permut) + + exprs = [ # permute v and t. Different dummy order, equivalent + atv(i, j, a, b)*att(a, i)*att(b, j), + atv(j, i, a, b)*att(a, j)*att(b, i), + atv(i, j, b, a)*att(b, i)*att(a, j), + atv(j, i, b, a)*att(b, j)*att(a, i), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_equivalent_internal_lines_VT2conjT2_AT(): + # this diagram requires special handling in TCE + i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy) + a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy) + p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy) + h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy) + + from sympy.utilities.iterables import variations + + # atv(abcd)att(abij)att(ijcd) + template = atv(p1, p2, p3, p4)*att(p1, p2, i, j)*att(i, j, p3, p4) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert substitute_dummies(expr) == substitute_dummies(base) + template = atv(p1, p2, p3, p4)*att(p1, p2, j, i)*att(j, i, p3, p4) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert substitute_dummies(expr) == substitute_dummies(base) + + # atv(abcd)att(abij)att(jicd) + template = atv(p1, p2, p3, p4)*att(p1, p2, i, j)*att(j, i, p3, p4) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert substitute_dummies(expr) == substitute_dummies(base) + template = atv(p1, p2, p3, p4)*att(p1, p2, j, i)*att(i, j, p3, p4) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert substitute_dummies(expr) == substitute_dummies(base) + + +def test_equivalent_internal_lines_VT2conjT2_ambiguous_order_AT(): + # These diagrams invokes _determine_ambiguous() because the + # dummies can not be ordered unambiguously by the key alone + i, j, k, l, m, n = symbols('i j k l m n', below_fermi=True, cls=Dummy) + a, b, c, d, e, f = symbols('a b c d e f', above_fermi=True, cls=Dummy) + p1, p2, p3, p4 = symbols('p1 p2 p3 p4', above_fermi=True, cls=Dummy) + h1, h2, h3, h4 = symbols('h1 h2 h3 h4', below_fermi=True, cls=Dummy) + + from sympy.utilities.iterables import variations + + # atv(abcd)att(abij)att(cdij) + template = atv(p1, p2, p3, p4)*att(p1, p2, i, j)*att(p3, p4, i, j) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert substitute_dummies(expr) == substitute_dummies(base) + template = atv(p1, p2, p3, p4)*att(p1, p2, j, i)*att(p3, p4, i, j) + permutator = variations([a, b, c, d], 4) + base = template.subs(zip([p1, p2, p3, p4], next(permutator))) + for permut in permutator: + subslist = zip([p1, p2, p3, p4], permut) + expr = template.subs(subslist) + assert substitute_dummies(expr) == substitute_dummies(base) + + +def test_equivalent_internal_lines_VT2_AT(): + i, j, k, l = symbols('i j k l', below_fermi=True, cls=Dummy) + a, b, c, d = symbols('a b c d', above_fermi=True, cls=Dummy) + + exprs = [ + # permute v. Same dummy order, not equivalent. + atv(i, j, a, b)*att(a, b, i, j), + atv(j, i, a, b)*att(a, b, i, j), + atv(i, j, b, a)*att(a, b, i, j), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) != substitute_dummies(permut) + + exprs = [ + # permute t. + atv(i, j, a, b)*att(a, b, i, j), + atv(i, j, a, b)*att(b, a, i, j), + atv(i, j, a, b)*att(a, b, j, i), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) != substitute_dummies(permut) + + exprs = [ # permute v and t. Relabelling of dummies should be equivalent. + atv(i, j, a, b)*att(a, b, i, j), + atv(j, i, a, b)*att(a, b, j, i), + atv(i, j, b, a)*att(b, a, i, j), + atv(j, i, b, a)*att(b, a, j, i), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_internal_external_VT2T2_AT(): + ii, jj = symbols('i j', below_fermi=True) + aa, bb = symbols('a b', above_fermi=True) + k, l = symbols('k l', below_fermi=True, cls=Dummy) + c, d = symbols('c d', above_fermi=True, cls=Dummy) + + exprs = [ + atv(k, l, c, d)*att(aa, c, ii, k)*att(bb, d, jj, l), + atv(l, k, c, d)*att(aa, c, ii, l)*att(bb, d, jj, k), + atv(k, l, d, c)*att(aa, d, ii, k)*att(bb, c, jj, l), + atv(l, k, d, c)*att(aa, d, ii, l)*att(bb, c, jj, k), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + exprs = [ + atv(k, l, c, d)*att(aa, c, ii, k)*att(d, bb, jj, l), + atv(l, k, c, d)*att(aa, c, ii, l)*att(d, bb, jj, k), + atv(k, l, d, c)*att(aa, d, ii, k)*att(c, bb, jj, l), + atv(l, k, d, c)*att(aa, d, ii, l)*att(c, bb, jj, k), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + exprs = [ + atv(k, l, c, d)*att(c, aa, ii, k)*att(bb, d, jj, l), + atv(l, k, c, d)*att(c, aa, ii, l)*att(bb, d, jj, k), + atv(k, l, d, c)*att(d, aa, ii, k)*att(bb, c, jj, l), + atv(l, k, d, c)*att(d, aa, ii, l)*att(bb, c, jj, k), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_internal_external_pqrs_AT(): + ii, jj = symbols('i j') + aa, bb = symbols('a b') + k, l = symbols('k l', cls=Dummy) + c, d = symbols('c d', cls=Dummy) + + exprs = [ + atv(k, l, c, d)*att(aa, c, ii, k)*att(bb, d, jj, l), + atv(l, k, c, d)*att(aa, c, ii, l)*att(bb, d, jj, k), + atv(k, l, d, c)*att(aa, d, ii, k)*att(bb, c, jj, l), + atv(l, k, d, c)*att(aa, d, ii, l)*att(bb, c, jj, k), + ] + for permut in exprs[1:]: + assert substitute_dummies(exprs[0]) == substitute_dummies(permut) + + +def test_issue_19661(): + a = Symbol('0') + assert latex(Commutator(Bd(a)**2, B(a)) + ) == '- \\left[b_{0},{b^\\dagger_{0}}^{2}\\right]' + + +def test_canonical_ordering_AntiSymmetricTensor(): + v = symbols("v") + + c, d = symbols(('c','d'), above_fermi=True, + cls=Dummy) + k, l = symbols(('k','l'), below_fermi=True, + cls=Dummy) + + # formerly, the left gave either the left or the right + assert AntiSymmetricTensor(v, (k, l), (d, c) + ) == -AntiSymmetricTensor(v, (l, k), (d, c)) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_sho.py b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_sho.py new file mode 100644 index 0000000000000000000000000000000000000000..7248838b4bb9ad280fd4211bbe208063b65adcf5 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/tests/test_sho.py @@ -0,0 +1,21 @@ +from sympy.core import symbols, Rational, Function, diff +from sympy.physics.sho import R_nl, E_nl +from sympy.simplify.simplify import simplify + + +def test_sho_R_nl(): + omega, r = symbols('omega r') + l = symbols('l', integer=True) + u = Function('u') + + # check that it obeys the Schrodinger equation + for n in range(5): + schreq = ( -diff(u(r), r, 2)/2 + ((l*(l + 1))/(2*r**2) + + omega**2*r**2/2 - E_nl(n, l, omega))*u(r) ) + result = schreq.subs(u(r), r*R_nl(n, l, omega/2, r)) + assert simplify(result.doit()) == 0 + + +def test_energy(): + n, l, hw = symbols('n l hw') + assert simplify(E_nl(n, l, hw) - (2*n + l + Rational(3, 2))*hw) == 0 diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/vector/functions.py b/wemm/lib/python3.10/site-packages/sympy/physics/vector/functions.py new file mode 100644 index 0000000000000000000000000000000000000000..6775b4b23bb376992d6a9e7651ba73a951c84287 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/vector/functions.py @@ -0,0 +1,650 @@ +from functools import reduce + +from sympy import (sympify, diff, sin, cos, Matrix, symbols, + Function, S, Symbol, linear_eq_to_matrix) +from sympy.integrals.integrals import integrate +from sympy.simplify.trigsimp import trigsimp +from .vector import Vector, _check_vector +from .frame import CoordinateSym, _check_frame +from .dyadic import Dyadic +from .printing import vprint, vsprint, vpprint, vlatex, init_vprinting +from sympy.utilities.iterables import iterable +from sympy.utilities.misc import translate + +__all__ = ['cross', 'dot', 'express', 'time_derivative', 'outer', + 'kinematic_equations', 'get_motion_params', 'partial_velocity', + 'dynamicsymbols', 'vprint', 'vsprint', 'vpprint', 'vlatex', + 'init_vprinting'] + + +def cross(vec1, vec2): + """Cross product convenience wrapper for Vector.cross(): \n""" + if not isinstance(vec1, (Vector, Dyadic)): + raise TypeError('Cross product is between two vectors') + return vec1 ^ vec2 + + +cross.__doc__ += Vector.cross.__doc__ # type: ignore + + +def dot(vec1, vec2): + """Dot product convenience wrapper for Vector.dot(): \n""" + if not isinstance(vec1, (Vector, Dyadic)): + raise TypeError('Dot product is between two vectors') + return vec1 & vec2 + + +dot.__doc__ += Vector.dot.__doc__ # type: ignore + + +def express(expr, frame, frame2=None, variables=False): + """ + Global function for 'express' functionality. + + Re-expresses a Vector, scalar(sympyfiable) or Dyadic in given frame. + + Refer to the local methods of Vector and Dyadic for details. + If 'variables' is True, then the coordinate variables (CoordinateSym + instances) of other frames present in the vector/scalar field or + dyadic expression are also substituted in terms of the base scalars of + this frame. + + Parameters + ========== + + expr : Vector/Dyadic/scalar(sympyfiable) + The expression to re-express in ReferenceFrame 'frame' + + frame: ReferenceFrame + The reference frame to express expr in + + frame2 : ReferenceFrame + The other frame required for re-expression(only for Dyadic expr) + + variables : boolean + Specifies whether to substitute the coordinate variables present + in expr, in terms of those of frame + + Examples + ======== + + >>> from sympy.physics.vector import ReferenceFrame, outer, dynamicsymbols + >>> from sympy.physics.vector import init_vprinting + >>> init_vprinting(pretty_print=False) + >>> N = ReferenceFrame('N') + >>> q = dynamicsymbols('q') + >>> B = N.orientnew('B', 'Axis', [q, N.z]) + >>> d = outer(N.x, N.x) + >>> from sympy.physics.vector import express + >>> express(d, B, N) + cos(q)*(B.x|N.x) - sin(q)*(B.y|N.x) + >>> express(B.x, N) + cos(q)*N.x + sin(q)*N.y + >>> express(N[0], B, variables=True) + B_x*cos(q) - B_y*sin(q) + + """ + + _check_frame(frame) + + if expr == 0: + return expr + + if isinstance(expr, Vector): + # Given expr is a Vector + if variables: + # If variables attribute is True, substitute the coordinate + # variables in the Vector + frame_list = [x[-1] for x in expr.args] + subs_dict = {} + for f in frame_list: + subs_dict.update(f.variable_map(frame)) + expr = expr.subs(subs_dict) + # Re-express in this frame + outvec = Vector([]) + for v in expr.args: + if v[1] != frame: + temp = frame.dcm(v[1]) * v[0] + if Vector.simp: + temp = temp.applyfunc(lambda x: + trigsimp(x, method='fu')) + outvec += Vector([(temp, frame)]) + else: + outvec += Vector([v]) + return outvec + + if isinstance(expr, Dyadic): + if frame2 is None: + frame2 = frame + _check_frame(frame2) + ol = Dyadic(0) + for v in expr.args: + ol += express(v[0], frame, variables=variables) * \ + (express(v[1], frame, variables=variables) | + express(v[2], frame2, variables=variables)) + return ol + + else: + if variables: + # Given expr is a scalar field + frame_set = set() + expr = sympify(expr) + # Substitute all the coordinate variables + for x in expr.free_symbols: + if isinstance(x, CoordinateSym) and x.frame != frame: + frame_set.add(x.frame) + subs_dict = {} + for f in frame_set: + subs_dict.update(f.variable_map(frame)) + return expr.subs(subs_dict) + return expr + + +def time_derivative(expr, frame, order=1): + """ + Calculate the time derivative of a vector/scalar field function + or dyadic expression in given frame. + + References + ========== + + https://en.wikipedia.org/wiki/Rotating_reference_frame#Time_derivatives_in_the_two_frames + + Parameters + ========== + + expr : Vector/Dyadic/sympifyable + The expression whose time derivative is to be calculated + + frame : ReferenceFrame + The reference frame to calculate the time derivative in + + order : integer + The order of the derivative to be calculated + + Examples + ======== + + >>> from sympy.physics.vector import ReferenceFrame, dynamicsymbols + >>> from sympy.physics.vector import init_vprinting + >>> init_vprinting(pretty_print=False) + >>> from sympy import Symbol + >>> q1 = Symbol('q1') + >>> u1 = dynamicsymbols('u1') + >>> N = ReferenceFrame('N') + >>> A = N.orientnew('A', 'Axis', [q1, N.x]) + >>> v = u1 * N.x + >>> A.set_ang_vel(N, 10*A.x) + >>> from sympy.physics.vector import time_derivative + >>> time_derivative(v, N) + u1'*N.x + >>> time_derivative(u1*A[0], N) + N_x*u1' + >>> B = N.orientnew('B', 'Axis', [u1, N.z]) + >>> from sympy.physics.vector import outer + >>> d = outer(N.x, N.x) + >>> time_derivative(d, B) + - u1'*(N.y|N.x) - u1'*(N.x|N.y) + + """ + + t = dynamicsymbols._t + _check_frame(frame) + + if order == 0: + return expr + if order % 1 != 0 or order < 0: + raise ValueError("Unsupported value of order entered") + + if isinstance(expr, Vector): + outlist = [] + for v in expr.args: + if v[1] == frame: + outlist += [(express(v[0], frame, variables=True).diff(t), + frame)] + else: + outlist += (time_derivative(Vector([v]), v[1]) + + (v[1].ang_vel_in(frame) ^ Vector([v]))).args + outvec = Vector(outlist) + return time_derivative(outvec, frame, order - 1) + + if isinstance(expr, Dyadic): + ol = Dyadic(0) + for v in expr.args: + ol += (v[0].diff(t) * (v[1] | v[2])) + ol += (v[0] * (time_derivative(v[1], frame) | v[2])) + ol += (v[0] * (v[1] | time_derivative(v[2], frame))) + return time_derivative(ol, frame, order - 1) + + else: + return diff(express(expr, frame, variables=True), t, order) + + +def outer(vec1, vec2): + """Outer product convenience wrapper for Vector.outer():\n""" + if not isinstance(vec1, Vector): + raise TypeError('Outer product is between two Vectors') + return vec1.outer(vec2) + + +outer.__doc__ += Vector.outer.__doc__ # type: ignore + + +def kinematic_equations(speeds, coords, rot_type, rot_order=''): + """Gives equations relating the qdot's to u's for a rotation type. + + Supply rotation type and order as in orient. Speeds are assumed to be + body-fixed; if we are defining the orientation of B in A using by rot_type, + the angular velocity of B in A is assumed to be in the form: speed[0]*B.x + + speed[1]*B.y + speed[2]*B.z + + Parameters + ========== + + speeds : list of length 3 + The body fixed angular velocity measure numbers. + coords : list of length 3 or 4 + The coordinates used to define the orientation of the two frames. + rot_type : str + The type of rotation used to create the equations. Body, Space, or + Quaternion only + rot_order : str or int + If applicable, the order of a series of rotations. + + Examples + ======== + + >>> from sympy.physics.vector import dynamicsymbols + >>> from sympy.physics.vector import kinematic_equations, vprint + >>> u1, u2, u3 = dynamicsymbols('u1 u2 u3') + >>> q1, q2, q3 = dynamicsymbols('q1 q2 q3') + >>> vprint(kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313'), + ... order=None) + [-(u1*sin(q3) + u2*cos(q3))/sin(q2) + q1', -u1*cos(q3) + u2*sin(q3) + q2', (u1*sin(q3) + u2*cos(q3))*cos(q2)/sin(q2) - u3 + q3'] + + """ + + # Code below is checking and sanitizing input + approved_orders = ('123', '231', '312', '132', '213', '321', '121', '131', + '212', '232', '313', '323', '1', '2', '3', '') + # make sure XYZ => 123 and rot_type is in lower case + rot_order = translate(str(rot_order), 'XYZxyz', '123123') + rot_type = rot_type.lower() + + if not isinstance(speeds, (list, tuple)): + raise TypeError('Need to supply speeds in a list') + if len(speeds) != 3: + raise TypeError('Need to supply 3 body-fixed speeds') + if not isinstance(coords, (list, tuple)): + raise TypeError('Need to supply coordinates in a list') + if rot_type in ['body', 'space']: + if rot_order not in approved_orders: + raise ValueError('Not an acceptable rotation order') + if len(coords) != 3: + raise ValueError('Need 3 coordinates for body or space') + # Actual hard-coded kinematic differential equations + w1, w2, w3 = speeds + if w1 == w2 == w3 == 0: + return [S.Zero]*3 + q1, q2, q3 = coords + q1d, q2d, q3d = [diff(i, dynamicsymbols._t) for i in coords] + s1, s2, s3 = [sin(q1), sin(q2), sin(q3)] + c1, c2, c3 = [cos(q1), cos(q2), cos(q3)] + if rot_type == 'body': + if rot_order == '123': + return [q1d - (w1 * c3 - w2 * s3) / c2, q2d - w1 * s3 - w2 * + c3, q3d - (-w1 * c3 + w2 * s3) * s2 / c2 - w3] + if rot_order == '231': + return [q1d - (w2 * c3 - w3 * s3) / c2, q2d - w2 * s3 - w3 * + c3, q3d - w1 - (- w2 * c3 + w3 * s3) * s2 / c2] + if rot_order == '312': + return [q1d - (-w1 * s3 + w3 * c3) / c2, q2d - w1 * c3 - w3 * + s3, q3d - (w1 * s3 - w3 * c3) * s2 / c2 - w2] + if rot_order == '132': + return [q1d - (w1 * c3 + w3 * s3) / c2, q2d + w1 * s3 - w3 * + c3, q3d - (w1 * c3 + w3 * s3) * s2 / c2 - w2] + if rot_order == '213': + return [q1d - (w1 * s3 + w2 * c3) / c2, q2d - w1 * c3 + w2 * + s3, q3d - (w1 * s3 + w2 * c3) * s2 / c2 - w3] + if rot_order == '321': + return [q1d - (w2 * s3 + w3 * c3) / c2, q2d - w2 * c3 + w3 * + s3, q3d - w1 - (w2 * s3 + w3 * c3) * s2 / c2] + if rot_order == '121': + return [q1d - (w2 * s3 + w3 * c3) / s2, q2d - w2 * c3 + w3 * + s3, q3d - w1 + (w2 * s3 + w3 * c3) * c2 / s2] + if rot_order == '131': + return [q1d - (-w2 * c3 + w3 * s3) / s2, q2d - w2 * s3 - w3 * + c3, q3d - w1 - (w2 * c3 - w3 * s3) * c2 / s2] + if rot_order == '212': + return [q1d - (w1 * s3 - w3 * c3) / s2, q2d - w1 * c3 - w3 * + s3, q3d - (-w1 * s3 + w3 * c3) * c2 / s2 - w2] + if rot_order == '232': + return [q1d - (w1 * c3 + w3 * s3) / s2, q2d + w1 * s3 - w3 * + c3, q3d + (w1 * c3 + w3 * s3) * c2 / s2 - w2] + if rot_order == '313': + return [q1d - (w1 * s3 + w2 * c3) / s2, q2d - w1 * c3 + w2 * + s3, q3d + (w1 * s3 + w2 * c3) * c2 / s2 - w3] + if rot_order == '323': + return [q1d - (-w1 * c3 + w2 * s3) / s2, q2d - w1 * s3 - w2 * + c3, q3d - (w1 * c3 - w2 * s3) * c2 / s2 - w3] + if rot_type == 'space': + if rot_order == '123': + return [q1d - w1 - (w2 * s1 + w3 * c1) * s2 / c2, q2d - w2 * + c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / c2] + if rot_order == '231': + return [q1d - (w1 * c1 + w3 * s1) * s2 / c2 - w2, q2d + w1 * + s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / c2] + if rot_order == '312': + return [q1d - (w1 * s1 + w2 * c1) * s2 / c2 - w3, q2d - w1 * + c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / c2] + if rot_order == '132': + return [q1d - w1 - (-w2 * c1 + w3 * s1) * s2 / c2, q2d - w2 * + s1 - w3 * c1, q3d - (w2 * c1 - w3 * s1) / c2] + if rot_order == '213': + return [q1d - (w1 * s1 - w3 * c1) * s2 / c2 - w2, q2d - w1 * + c1 - w3 * s1, q3d - (-w1 * s1 + w3 * c1) / c2] + if rot_order == '321': + return [q1d - (-w1 * c1 + w2 * s1) * s2 / c2 - w3, q2d - w1 * + s1 - w2 * c1, q3d - (w1 * c1 - w2 * s1) / c2] + if rot_order == '121': + return [q1d - w1 + (w2 * s1 + w3 * c1) * c2 / s2, q2d - w2 * + c1 + w3 * s1, q3d - (w2 * s1 + w3 * c1) / s2] + if rot_order == '131': + return [q1d - w1 - (w2 * c1 - w3 * s1) * c2 / s2, q2d - w2 * + s1 - w3 * c1, q3d - (-w2 * c1 + w3 * s1) / s2] + if rot_order == '212': + return [q1d - (-w1 * s1 + w3 * c1) * c2 / s2 - w2, q2d - w1 * + c1 - w3 * s1, q3d - (w1 * s1 - w3 * c1) / s2] + if rot_order == '232': + return [q1d + (w1 * c1 + w3 * s1) * c2 / s2 - w2, q2d + w1 * + s1 - w3 * c1, q3d - (w1 * c1 + w3 * s1) / s2] + if rot_order == '313': + return [q1d + (w1 * s1 + w2 * c1) * c2 / s2 - w3, q2d - w1 * + c1 + w2 * s1, q3d - (w1 * s1 + w2 * c1) / s2] + if rot_order == '323': + return [q1d - (w1 * c1 - w2 * s1) * c2 / s2 - w3, q2d - w1 * + s1 - w2 * c1, q3d - (-w1 * c1 + w2 * s1) / s2] + elif rot_type == 'quaternion': + if rot_order != '': + raise ValueError('Cannot have rotation order for quaternion') + if len(coords) != 4: + raise ValueError('Need 4 coordinates for quaternion') + # Actual hard-coded kinematic differential equations + e0, e1, e2, e3 = coords + w = Matrix(speeds + [0]) + E = Matrix([[e0, -e3, e2, e1], + [e3, e0, -e1, e2], + [-e2, e1, e0, e3], + [-e1, -e2, -e3, e0]]) + edots = Matrix([diff(i, dynamicsymbols._t) for i in [e1, e2, e3, e0]]) + return list(edots.T - 0.5 * w.T * E.T) + else: + raise ValueError('Not an approved rotation type for this function') + + +def get_motion_params(frame, **kwargs): + """ + Returns the three motion parameters - (acceleration, velocity, and + position) as vectorial functions of time in the given frame. + + If a higher order differential function is provided, the lower order + functions are used as boundary conditions. For example, given the + acceleration, the velocity and position parameters are taken as + boundary conditions. + + The values of time at which the boundary conditions are specified + are taken from timevalue1(for position boundary condition) and + timevalue2(for velocity boundary condition). + + If any of the boundary conditions are not provided, they are taken + to be zero by default (zero vectors, in case of vectorial inputs). If + the boundary conditions are also functions of time, they are converted + to constants by substituting the time values in the dynamicsymbols._t + time Symbol. + + This function can also be used for calculating rotational motion + parameters. Have a look at the Parameters and Examples for more clarity. + + Parameters + ========== + + frame : ReferenceFrame + The frame to express the motion parameters in + + acceleration : Vector + Acceleration of the object/frame as a function of time + + velocity : Vector + Velocity as function of time or as boundary condition + of velocity at time = timevalue1 + + position : Vector + Velocity as function of time or as boundary condition + of velocity at time = timevalue1 + + timevalue1 : sympyfiable + Value of time for position boundary condition + + timevalue2 : sympyfiable + Value of time for velocity boundary condition + + Examples + ======== + + >>> from sympy.physics.vector import ReferenceFrame, get_motion_params, dynamicsymbols + >>> from sympy.physics.vector import init_vprinting + >>> init_vprinting(pretty_print=False) + >>> from sympy import symbols + >>> R = ReferenceFrame('R') + >>> v1, v2, v3 = dynamicsymbols('v1 v2 v3') + >>> v = v1*R.x + v2*R.y + v3*R.z + >>> get_motion_params(R, position = v) + (v1''*R.x + v2''*R.y + v3''*R.z, v1'*R.x + v2'*R.y + v3'*R.z, v1*R.x + v2*R.y + v3*R.z) + >>> a, b, c = symbols('a b c') + >>> v = a*R.x + b*R.y + c*R.z + >>> get_motion_params(R, velocity = v) + (0, a*R.x + b*R.y + c*R.z, a*t*R.x + b*t*R.y + c*t*R.z) + >>> parameters = get_motion_params(R, acceleration = v) + >>> parameters[1] + a*t*R.x + b*t*R.y + c*t*R.z + >>> parameters[2] + a*t**2/2*R.x + b*t**2/2*R.y + c*t**2/2*R.z + + """ + + def _process_vector_differential(vectdiff, condition, variable, ordinate, + frame): + """ + Helper function for get_motion methods. Finds derivative of vectdiff + wrt variable, and its integral using the specified boundary condition + at value of variable = ordinate. + Returns a tuple of - (derivative, function and integral) wrt vectdiff + + """ + + # Make sure boundary condition is independent of 'variable' + if condition != 0: + condition = express(condition, frame, variables=True) + # Special case of vectdiff == 0 + if vectdiff == Vector(0): + return (0, 0, condition) + # Express vectdiff completely in condition's frame to give vectdiff1 + vectdiff1 = express(vectdiff, frame) + # Find derivative of vectdiff + vectdiff2 = time_derivative(vectdiff, frame) + # Integrate and use boundary condition + vectdiff0 = Vector(0) + lims = (variable, ordinate, variable) + for dim in frame: + function1 = vectdiff1.dot(dim) + abscissa = dim.dot(condition).subs({variable: ordinate}) + # Indefinite integral of 'function1' wrt 'variable', using + # the given initial condition (ordinate, abscissa). + vectdiff0 += (integrate(function1, lims) + abscissa) * dim + # Return tuple + return (vectdiff2, vectdiff, vectdiff0) + + _check_frame(frame) + # Decide mode of operation based on user's input + if 'acceleration' in kwargs: + mode = 2 + elif 'velocity' in kwargs: + mode = 1 + else: + mode = 0 + # All the possible parameters in kwargs + # Not all are required for every case + # If not specified, set to default values(may or may not be used in + # calculations) + conditions = ['acceleration', 'velocity', 'position', + 'timevalue', 'timevalue1', 'timevalue2'] + for i, x in enumerate(conditions): + if x not in kwargs: + if i < 3: + kwargs[x] = Vector(0) + else: + kwargs[x] = S.Zero + elif i < 3: + _check_vector(kwargs[x]) + else: + kwargs[x] = sympify(kwargs[x]) + if mode == 2: + vel = _process_vector_differential(kwargs['acceleration'], + kwargs['velocity'], + dynamicsymbols._t, + kwargs['timevalue2'], frame)[2] + pos = _process_vector_differential(vel, kwargs['position'], + dynamicsymbols._t, + kwargs['timevalue1'], frame)[2] + return (kwargs['acceleration'], vel, pos) + elif mode == 1: + return _process_vector_differential(kwargs['velocity'], + kwargs['position'], + dynamicsymbols._t, + kwargs['timevalue1'], frame) + else: + vel = time_derivative(kwargs['position'], frame) + acc = time_derivative(vel, frame) + return (acc, vel, kwargs['position']) + + +def partial_velocity(vel_vecs, gen_speeds, frame): + """Returns a list of partial velocities with respect to the provided + generalized speeds in the given reference frame for each of the supplied + velocity vectors. + + The output is a list of lists. The outer list has a number of elements + equal to the number of supplied velocity vectors. The inner lists are, for + each velocity vector, the partial derivatives of that velocity vector with + respect to the generalized speeds supplied. + + Parameters + ========== + + vel_vecs : iterable + An iterable of velocity vectors (angular or linear). + gen_speeds : iterable + An iterable of generalized speeds. + frame : ReferenceFrame + The reference frame that the partial derivatives are going to be taken + in. + + Examples + ======== + + >>> from sympy.physics.vector import Point, ReferenceFrame + >>> from sympy.physics.vector import dynamicsymbols + >>> from sympy.physics.vector import partial_velocity + >>> u = dynamicsymbols('u') + >>> N = ReferenceFrame('N') + >>> P = Point('P') + >>> P.set_vel(N, u * N.x) + >>> vel_vecs = [P.vel(N)] + >>> gen_speeds = [u] + >>> partial_velocity(vel_vecs, gen_speeds, N) + [[N.x]] + + """ + + if not iterable(vel_vecs): + raise TypeError('Velocity vectors must be contained in an iterable.') + + if not iterable(gen_speeds): + raise TypeError('Generalized speeds must be contained in an iterable') + + vec_partials = [] + gen_speeds = list(gen_speeds) + for vel in vel_vecs: + partials = [Vector(0) for _ in gen_speeds] + for components, ref in vel.args: + mat, _ = linear_eq_to_matrix(components, gen_speeds) + for i in range(len(gen_speeds)): + for dim, direction in enumerate(ref): + if mat[dim, i] != 0: + partials[i] += direction * mat[dim, i] + + vec_partials.append(partials) + + return vec_partials + + +def dynamicsymbols(names, level=0, **assumptions): + """Uses symbols and Function for functions of time. + + Creates a SymPy UndefinedFunction, which is then initialized as a function + of a variable, the default being Symbol('t'). + + Parameters + ========== + + names : str + Names of the dynamic symbols you want to create; works the same way as + inputs to symbols + level : int + Level of differentiation of the returned function; d/dt once of t, + twice of t, etc. + assumptions : + - real(bool) : This is used to set the dynamicsymbol as real, + by default is False. + - positive(bool) : This is used to set the dynamicsymbol as positive, + by default is False. + - commutative(bool) : This is used to set the commutative property of + a dynamicsymbol, by default is True. + - integer(bool) : This is used to set the dynamicsymbol as integer, + by default is False. + + Examples + ======== + + >>> from sympy.physics.vector import dynamicsymbols + >>> from sympy import diff, Symbol + >>> q1 = dynamicsymbols('q1') + >>> q1 + q1(t) + >>> q2 = dynamicsymbols('q2', real=True) + >>> q2.is_real + True + >>> q3 = dynamicsymbols('q3', positive=True) + >>> q3.is_positive + True + >>> q4, q5 = dynamicsymbols('q4,q5', commutative=False) + >>> bool(q4*q5 != q5*q4) + True + >>> q6 = dynamicsymbols('q6', integer=True) + >>> q6.is_integer + True + >>> diff(q1, Symbol('t')) + Derivative(q1(t), t) + + """ + esses = symbols(names, cls=Function, **assumptions) + t = dynamicsymbols._t + if iterable(esses): + esses = [reduce(diff, [t] * level, e(t)) for e in esses] + return esses + else: + return reduce(diff, [t] * level, esses(t)) + + +dynamicsymbols._t = Symbol('t') # type: ignore +dynamicsymbols._str = '\'' # type: ignore diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/vector/printing.py b/wemm/lib/python3.10/site-packages/sympy/physics/vector/printing.py new file mode 100644 index 0000000000000000000000000000000000000000..2b589f673329e1e598b9b568fba6c07b8abe67bc --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/vector/printing.py @@ -0,0 +1,371 @@ +from sympy.core.function import Derivative +from sympy.core.function import UndefinedFunction, AppliedUndef +from sympy.core.symbol import Symbol +from sympy.interactive.printing import init_printing +from sympy.printing.latex import LatexPrinter +from sympy.printing.pretty.pretty import PrettyPrinter +from sympy.printing.pretty.pretty_symbology import center_accent +from sympy.printing.str import StrPrinter +from sympy.printing.precedence import PRECEDENCE + +__all__ = ['vprint', 'vsstrrepr', 'vsprint', 'vpprint', 'vlatex', + 'init_vprinting'] + + +class VectorStrPrinter(StrPrinter): + """String Printer for vector expressions. """ + + def _print_Derivative(self, e): + from sympy.physics.vector.functions import dynamicsymbols + t = dynamicsymbols._t + if (bool(sum(i == t for i in e.variables)) & + isinstance(type(e.args[0]), UndefinedFunction)): + ol = str(e.args[0].func) + for i, v in enumerate(e.variables): + ol += dynamicsymbols._str + return ol + else: + return StrPrinter().doprint(e) + + def _print_Function(self, e): + from sympy.physics.vector.functions import dynamicsymbols + t = dynamicsymbols._t + if isinstance(type(e), UndefinedFunction): + return StrPrinter().doprint(e).replace("(%s)" % t, '') + return e.func.__name__ + "(%s)" % self.stringify(e.args, ", ") + + +class VectorStrReprPrinter(VectorStrPrinter): + """String repr printer for vector expressions.""" + def _print_str(self, s): + return repr(s) + + +class VectorLatexPrinter(LatexPrinter): + """Latex Printer for vector expressions. """ + + def _print_Function(self, expr, exp=None): + from sympy.physics.vector.functions import dynamicsymbols + func = expr.func.__name__ + t = dynamicsymbols._t + + if (hasattr(self, '_print_' + func) and not + isinstance(type(expr), UndefinedFunction)): + return getattr(self, '_print_' + func)(expr, exp) + elif isinstance(type(expr), UndefinedFunction) and (expr.args == (t,)): + # treat this function like a symbol + expr = Symbol(func) + if exp is not None: + # copied from LatexPrinter._helper_print_standard_power, which + # we can't call because we only have exp as a string. + base = self.parenthesize(expr, PRECEDENCE['Pow']) + base = self.parenthesize_super(base) + return r"%s^{%s}" % (base, exp) + else: + return super()._print(expr) + else: + return super()._print_Function(expr, exp) + + def _print_Derivative(self, der_expr): + from sympy.physics.vector.functions import dynamicsymbols + # make sure it is in the right form + der_expr = der_expr.doit() + if not isinstance(der_expr, Derivative): + return r"\left(%s\right)" % self.doprint(der_expr) + + # check if expr is a dynamicsymbol + t = dynamicsymbols._t + expr = der_expr.expr + red = expr.atoms(AppliedUndef) + syms = der_expr.variables + test1 = not all(True for i in red if i.free_symbols == {t}) + test2 = not all(t == i for i in syms) + if test1 or test2: + return super()._print_Derivative(der_expr) + + # done checking + dots = len(syms) + base = self._print_Function(expr) + base_split = base.split('_', 1) + base = base_split[0] + if dots == 1: + base = r"\dot{%s}" % base + elif dots == 2: + base = r"\ddot{%s}" % base + elif dots == 3: + base = r"\dddot{%s}" % base + elif dots == 4: + base = r"\ddddot{%s}" % base + else: # Fallback to standard printing + return super()._print_Derivative(der_expr) + if len(base_split) != 1: + base += '_' + base_split[1] + return base + + +class VectorPrettyPrinter(PrettyPrinter): + """Pretty Printer for vectorialexpressions. """ + + def _print_Derivative(self, deriv): + from sympy.physics.vector.functions import dynamicsymbols + # XXX use U('PARTIAL DIFFERENTIAL') here ? + t = dynamicsymbols._t + dot_i = 0 + syms = list(reversed(deriv.variables)) + + while len(syms) > 0: + if syms[-1] == t: + syms.pop() + dot_i += 1 + else: + return super()._print_Derivative(deriv) + + if not (isinstance(type(deriv.expr), UndefinedFunction) and + (deriv.expr.args == (t,))): + return super()._print_Derivative(deriv) + else: + pform = self._print_Function(deriv.expr) + + # the following condition would happen with some sort of non-standard + # dynamic symbol I guess, so we'll just print the SymPy way + if len(pform.picture) > 1: + return super()._print_Derivative(deriv) + + # There are only special symbols up to fourth-order derivatives + if dot_i >= 5: + return super()._print_Derivative(deriv) + + # Deal with special symbols + dots = {0: "", + 1: "\N{COMBINING DOT ABOVE}", + 2: "\N{COMBINING DIAERESIS}", + 3: "\N{COMBINING THREE DOTS ABOVE}", + 4: "\N{COMBINING FOUR DOTS ABOVE}"} + + d = pform.__dict__ + # if unicode is false then calculate number of apostrophes needed and + # add to output + if not self._use_unicode: + apostrophes = "" + for i in range(0, dot_i): + apostrophes += "'" + d['picture'][0] += apostrophes + "(t)" + else: + d['picture'] = [center_accent(d['picture'][0], dots[dot_i])] + return pform + + def _print_Function(self, e): + from sympy.physics.vector.functions import dynamicsymbols + t = dynamicsymbols._t + # XXX works only for applied functions + func = e.func + args = e.args + func_name = func.__name__ + pform = self._print_Symbol(Symbol(func_name)) + # If this function is an Undefined function of t, it is probably a + # dynamic symbol, so we'll skip the (t). The rest of the code is + # identical to the normal PrettyPrinter code + if not (isinstance(func, UndefinedFunction) and (args == (t,))): + return super()._print_Function(e) + return pform + + +def vprint(expr, **settings): + r"""Function for printing of expressions generated in the + sympy.physics vector package. + + Extends SymPy's StrPrinter, takes the same setting accepted by SymPy's + :func:`~.sstr`, and is equivalent to ``print(sstr(foo))``. + + Parameters + ========== + + expr : valid SymPy object + SymPy expression to print. + settings : args + Same as the settings accepted by SymPy's sstr(). + + Examples + ======== + + >>> from sympy.physics.vector import vprint, dynamicsymbols + >>> u1 = dynamicsymbols('u1') + >>> print(u1) + u1(t) + >>> vprint(u1) + u1 + + """ + + outstr = vsprint(expr, **settings) + + import builtins + if (outstr != 'None'): + builtins._ = outstr + print(outstr) + + +def vsstrrepr(expr, **settings): + """Function for displaying expression representation's with vector + printing enabled. + + Parameters + ========== + + expr : valid SymPy object + SymPy expression to print. + settings : args + Same as the settings accepted by SymPy's sstrrepr(). + + """ + p = VectorStrReprPrinter(settings) + return p.doprint(expr) + + +def vsprint(expr, **settings): + r"""Function for displaying expressions generated in the + sympy.physics vector package. + + Returns the output of vprint() as a string. + + Parameters + ========== + + expr : valid SymPy object + SymPy expression to print + settings : args + Same as the settings accepted by SymPy's sstr(). + + Examples + ======== + + >>> from sympy.physics.vector import vsprint, dynamicsymbols + >>> u1, u2 = dynamicsymbols('u1 u2') + >>> u2d = dynamicsymbols('u2', level=1) + >>> print("%s = %s" % (u1, u2 + u2d)) + u1(t) = u2(t) + Derivative(u2(t), t) + >>> print("%s = %s" % (vsprint(u1), vsprint(u2 + u2d))) + u1 = u2 + u2' + + """ + + string_printer = VectorStrPrinter(settings) + return string_printer.doprint(expr) + + +def vpprint(expr, **settings): + r"""Function for pretty printing of expressions generated in the + sympy.physics vector package. + + Mainly used for expressions not inside a vector; the output of running + scripts and generating equations of motion. Takes the same options as + SymPy's :func:`~.pretty_print`; see that function for more information. + + Parameters + ========== + + expr : valid SymPy object + SymPy expression to pretty print + settings : args + Same as those accepted by SymPy's pretty_print. + + + """ + + pp = VectorPrettyPrinter(settings) + + # Note that this is copied from sympy.printing.pretty.pretty_print: + + # XXX: this is an ugly hack, but at least it works + use_unicode = pp._settings['use_unicode'] + from sympy.printing.pretty.pretty_symbology import pretty_use_unicode + uflag = pretty_use_unicode(use_unicode) + + try: + return pp.doprint(expr) + finally: + pretty_use_unicode(uflag) + + +def vlatex(expr, **settings): + r"""Function for printing latex representation of sympy.physics.vector + objects. + + For latex representation of Vectors, Dyadics, and dynamicsymbols. Takes the + same options as SymPy's :func:`~.latex`; see that function for more + information; + + Parameters + ========== + + expr : valid SymPy object + SymPy expression to represent in LaTeX form + settings : args + Same as latex() + + Examples + ======== + + >>> from sympy.physics.vector import vlatex, ReferenceFrame, dynamicsymbols + >>> N = ReferenceFrame('N') + >>> q1, q2 = dynamicsymbols('q1 q2') + >>> q1d, q2d = dynamicsymbols('q1 q2', 1) + >>> q1dd, q2dd = dynamicsymbols('q1 q2', 2) + >>> vlatex(N.x + N.y) + '\\mathbf{\\hat{n}_x} + \\mathbf{\\hat{n}_y}' + >>> vlatex(q1 + q2) + 'q_{1} + q_{2}' + >>> vlatex(q1d) + '\\dot{q}_{1}' + >>> vlatex(q1 * q2d) + 'q_{1} \\dot{q}_{2}' + >>> vlatex(q1dd * q1 / q1d) + '\\frac{q_{1} \\ddot{q}_{1}}{\\dot{q}_{1}}' + + """ + latex_printer = VectorLatexPrinter(settings) + + return latex_printer.doprint(expr) + + +def init_vprinting(**kwargs): + """Initializes time derivative printing for all SymPy objects, i.e. any + functions of time will be displayed in a more compact notation. The main + benefit of this is for printing of time derivatives; instead of + displaying as ``Derivative(f(t),t)``, it will display ``f'``. This is + only actually needed for when derivatives are present and are not in a + physics.vector.Vector or physics.vector.Dyadic object. This function is a + light wrapper to :func:`~.init_printing`. Any keyword + arguments for it are valid here. + + {0} + + Examples + ======== + + >>> from sympy import Function, symbols + >>> t, x = symbols('t, x') + >>> omega = Function('omega') + >>> omega(x).diff() + Derivative(omega(x), x) + >>> omega(t).diff() + Derivative(omega(t), t) + + Now use the string printer: + + >>> from sympy.physics.vector import init_vprinting + >>> init_vprinting(pretty_print=False) + >>> omega(x).diff() + Derivative(omega(x), x) + >>> omega(t).diff() + omega' + + """ + kwargs['str_printer'] = vsstrrepr + kwargs['pretty_printer'] = vpprint + kwargs['latex_printer'] = vlatex + init_printing(**kwargs) + + +params = init_printing.__doc__.split('Examples\n ========')[0] # type: ignore +init_vprinting.__doc__ = init_vprinting.__doc__.format(params) # type: ignore diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/__init__.py b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/__init__.py new file mode 100644 index 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b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/__pycache__/test_printing.cpython-310.pyc differ diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_dyadic.py b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_dyadic.py new file mode 100644 index 0000000000000000000000000000000000000000..ab365b4687162ccbd3b21dd9709b84dbcdec8aa0 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_dyadic.py @@ -0,0 +1,123 @@ +from sympy.core.numbers import (Float, pi) +from sympy.core.symbol import symbols +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix +from sympy.physics.vector import ReferenceFrame, dynamicsymbols, outer +from sympy.physics.vector.dyadic import _check_dyadic +from sympy.testing.pytest import raises + +A = ReferenceFrame('A') + + +def test_dyadic(): + d1 = A.x | A.x + d2 = A.y | A.y + d3 = A.x | A.y + assert d1 * 0 == 0 + assert d1 != 0 + assert d1 * 2 == 2 * A.x | A.x + assert d1 / 2. == 0.5 * d1 + assert d1 & (0 * d1) == 0 + assert d1 & d2 == 0 + assert d1 & A.x == A.x + assert d1 ^ A.x == 0 + assert d1 ^ A.y == A.x | A.z + assert d1 ^ A.z == - A.x | A.y + assert d2 ^ A.x == - A.y | A.z + assert A.x ^ d1 == 0 + assert A.y ^ d1 == - A.z | A.x + assert A.z ^ d1 == A.y | A.x + assert A.x & d1 == A.x + assert A.y & d1 == 0 + assert A.y & d2 == A.y + assert d1 & d3 == A.x | A.y + assert d3 & d1 == 0 + assert d1.dt(A) == 0 + q = dynamicsymbols('q') + qd = dynamicsymbols('q', 1) + B = A.orientnew('B', 'Axis', [q, A.z]) + assert d1.express(B) == d1.express(B, B) + assert d1.express(B) == ((cos(q)**2) * (B.x | B.x) + (-sin(q) * cos(q)) * + (B.x | B.y) + (-sin(q) * cos(q)) * (B.y | B.x) + (sin(q)**2) * + (B.y | B.y)) + assert d1.express(B, A) == (cos(q)) * (B.x | A.x) + (-sin(q)) * (B.y | A.x) + assert d1.express(A, B) == (cos(q)) * (A.x | B.x) + (-sin(q)) * (A.x | B.y) + assert d1.dt(B) == (-qd) * (A.y | A.x) + (-qd) * (A.x | A.y) + + assert d1.to_matrix(A) == Matrix([[1, 0, 0], [0, 0, 0], [0, 0, 0]]) + assert d1.to_matrix(A, B) == Matrix([[cos(q), -sin(q), 0], + [0, 0, 0], + [0, 0, 0]]) + assert d3.to_matrix(A) == Matrix([[0, 1, 0], [0, 0, 0], [0, 0, 0]]) + a, b, c, d, e, f = symbols('a, b, c, d, e, f') + v1 = a * A.x + b * A.y + c * A.z + v2 = d * A.x + e * A.y + f * A.z + d4 = v1.outer(v2) + assert d4.to_matrix(A) == Matrix([[a * d, a * e, a * f], + [b * d, b * e, b * f], + [c * d, c * e, c * f]]) + d5 = v1.outer(v1) + C = A.orientnew('C', 'Axis', [q, A.x]) + for expected, actual in zip(C.dcm(A) * d5.to_matrix(A) * C.dcm(A).T, + d5.to_matrix(C)): + assert (expected - actual).simplify() == 0 + + raises(TypeError, lambda: d1.applyfunc(0)) + + +def test_dyadic_simplify(): + x, y, z, k, n, m, w, f, s, A = symbols('x, y, z, k, n, m, w, f, s, A') + N = ReferenceFrame('N') + + dy = N.x | N.x + test1 = (1 / x + 1 / y) * dy + assert (N.x & test1 & N.x) != (x + y) / (x * y) + test1 = test1.simplify() + assert (N.x & test1 & N.x) == (x + y) / (x * y) + + test2 = (A**2 * s**4 / (4 * pi * k * m**3)) * dy + test2 = test2.simplify() + assert (N.x & test2 & N.x) == (A**2 * s**4 / (4 * pi * k * m**3)) + + test3 = ((4 + 4 * x - 2 * (2 + 2 * x)) / (2 + 2 * x)) * dy + test3 = test3.simplify() + assert (N.x & test3 & N.x) == 0 + + test4 = ((-4 * x * y**2 - 2 * y**3 - 2 * x**2 * y) / (x + y)**2) * dy + test4 = test4.simplify() + assert (N.x & test4 & N.x) == -2 * y + + +def test_dyadic_subs(): + N = ReferenceFrame('N') + s = symbols('s') + a = s*(N.x | N.x) + assert a.subs({s: 2}) == 2*(N.x | N.x) + + +def test_check_dyadic(): + raises(TypeError, lambda: _check_dyadic(0)) + + +def test_dyadic_evalf(): + N = ReferenceFrame('N') + a = pi * (N.x | N.x) + assert a.evalf(3) == Float('3.1416', 3) * (N.x | N.x) + s = symbols('s') + a = 5 * s * pi* (N.x | N.x) + assert a.evalf(2) == Float('5', 2) * Float('3.1416', 2) * s * (N.x | N.x) + assert a.evalf(9, subs={s: 5.124}) == Float('80.48760378', 9) * (N.x | N.x) + + +def test_dyadic_xreplace(): + x, y, z = symbols('x y z') + N = ReferenceFrame('N') + D = outer(N.x, N.x) + v = x*y * D + assert v.xreplace({x : cos(x)}) == cos(x)*y * D + assert v.xreplace({x*y : pi}) == pi * D + v = (x*y)**z * D + assert v.xreplace({(x*y)**z : 1}) == D + assert v.xreplace({x:1, z:0}) == D + raises(TypeError, lambda: v.xreplace()) + raises(TypeError, lambda: v.xreplace([x, y])) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_fieldfunctions.py b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_fieldfunctions.py new file mode 100644 index 0000000000000000000000000000000000000000..4e5c67aad44ca972dac6e455c57b60a74bae207a --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_fieldfunctions.py @@ -0,0 +1,133 @@ +from sympy.core.singleton import S +from sympy.core.symbol import Symbol +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.physics.vector import ReferenceFrame, Vector, Point, \ + dynamicsymbols +from sympy.physics.vector.fieldfunctions import divergence, \ + gradient, curl, is_conservative, is_solenoidal, \ + scalar_potential, scalar_potential_difference +from sympy.testing.pytest import raises + +R = ReferenceFrame('R') +q = dynamicsymbols('q') +P = R.orientnew('P', 'Axis', [q, R.z]) + + +def test_curl(): + assert curl(Vector(0), R) == Vector(0) + assert curl(R.x, R) == Vector(0) + assert curl(2*R[1]**2*R.y, R) == Vector(0) + assert curl(R[0]*R[1]*R.z, R) == R[0]*R.x - R[1]*R.y + assert curl(R[0]*R[1]*R[2] * (R.x+R.y+R.z), R) == \ + (-R[0]*R[1] + R[0]*R[2])*R.x + (R[0]*R[1] - R[1]*R[2])*R.y + \ + (-R[0]*R[2] + R[1]*R[2])*R.z + assert curl(2*R[0]**2*R.y, R) == 4*R[0]*R.z + assert curl(P[0]**2*R.x + P.y, R) == \ + - 2*(R[0]*cos(q) + R[1]*sin(q))*sin(q)*R.z + assert curl(P[0]*R.y, P) == cos(q)*P.z + + +def test_divergence(): + assert divergence(Vector(0), R) is S.Zero + assert divergence(R.x, R) is S.Zero + assert divergence(R[0]**2*R.x, R) == 2*R[0] + assert divergence(R[0]*R[1]*R[2] * (R.x+R.y+R.z), R) == \ + R[0]*R[1] + R[0]*R[2] + R[1]*R[2] + assert divergence((1/(R[0]*R[1]*R[2])) * (R.x+R.y+R.z), R) == \ + -1/(R[0]*R[1]*R[2]**2) - 1/(R[0]*R[1]**2*R[2]) - \ + 1/(R[0]**2*R[1]*R[2]) + v = P[0]*P.x + P[1]*P.y + P[2]*P.z + assert divergence(v, P) == 3 + assert divergence(v, R).simplify() == 3 + assert divergence(P[0]*R.x + R[0]*P.x, R) == 2*cos(q) + + +def test_gradient(): + a = Symbol('a') + assert gradient(0, R) == Vector(0) + assert gradient(R[0], R) == R.x + assert gradient(R[0]*R[1]*R[2], R) == \ + R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z + assert gradient(2*R[0]**2, R) == 4*R[0]*R.x + assert gradient(a*sin(R[1])/R[0], R) == \ + - a*sin(R[1])/R[0]**2*R.x + a*cos(R[1])/R[0]*R.y + assert gradient(P[0]*P[1], R) == \ + ((-R[0]*sin(q) + R[1]*cos(q))*cos(q) - (R[0]*cos(q) + R[1]*sin(q))*sin(q))*R.x + \ + ((-R[0]*sin(q) + R[1]*cos(q))*sin(q) + (R[0]*cos(q) + R[1]*sin(q))*cos(q))*R.y + assert gradient(P[0]*R[2], P) == P[2]*P.x + P[0]*P.z + + +scalar_field = 2*R[0]**2*R[1]*R[2] +grad_field = gradient(scalar_field, R) +vector_field = R[1]**2*R.x + 3*R[0]*R.y + 5*R[1]*R[2]*R.z +curl_field = curl(vector_field, R) + + +def test_conservative(): + assert is_conservative(0) is True + assert is_conservative(R.x) is True + assert is_conservative(2 * R.x + 3 * R.y + 4 * R.z) is True + assert is_conservative(R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z) is \ + True + assert is_conservative(R[0] * R.y) is False + assert is_conservative(grad_field) is True + assert is_conservative(curl_field) is False + assert is_conservative(4*R[0]*R[1]*R[2]*R.x + 2*R[0]**2*R[2]*R.y) is \ + False + assert is_conservative(R[2]*P.x + P[0]*R.z) is True + + +def test_solenoidal(): + assert is_solenoidal(0) is True + assert is_solenoidal(R.x) is True + assert is_solenoidal(2 * R.x + 3 * R.y + 4 * R.z) is True + assert is_solenoidal(R[1]*R[2]*R.x + R[0]*R[2]*R.y + R[0]*R[1]*R.z) is \ + True + assert is_solenoidal(R[1] * R.y) is False + assert is_solenoidal(grad_field) is False + assert is_solenoidal(curl_field) is True + assert is_solenoidal((-2*R[1] + 3)*R.z) is True + assert is_solenoidal(cos(q)*R.x + sin(q)*R.y + cos(q)*P.z) is True + assert is_solenoidal(R[2]*P.x + P[0]*R.z) is True + + +def test_scalar_potential(): + assert scalar_potential(0, R) == 0 + assert scalar_potential(R.x, R) == R[0] + assert scalar_potential(R.y, R) == R[1] + assert scalar_potential(R.z, R) == R[2] + assert scalar_potential(R[1]*R[2]*R.x + R[0]*R[2]*R.y + \ + R[0]*R[1]*R.z, R) == R[0]*R[1]*R[2] + assert scalar_potential(grad_field, R) == scalar_field + assert scalar_potential(R[2]*P.x + P[0]*R.z, R) == \ + R[0]*R[2]*cos(q) + R[1]*R[2]*sin(q) + assert scalar_potential(R[2]*P.x + P[0]*R.z, P) == P[0]*P[2] + raises(ValueError, lambda: scalar_potential(R[0] * R.y, R)) + + +def test_scalar_potential_difference(): + origin = Point('O') + point1 = origin.locatenew('P1', 1*R.x + 2*R.y + 3*R.z) + point2 = origin.locatenew('P2', 4*R.x + 5*R.y + 6*R.z) + genericpointR = origin.locatenew('RP', R[0]*R.x + R[1]*R.y + R[2]*R.z) + genericpointP = origin.locatenew('PP', P[0]*P.x + P[1]*P.y + P[2]*P.z) + assert scalar_potential_difference(S.Zero, R, point1, point2, \ + origin) == 0 + assert scalar_potential_difference(scalar_field, R, origin, \ + genericpointR, origin) == \ + scalar_field + assert scalar_potential_difference(grad_field, R, origin, \ + genericpointR, origin) == \ + scalar_field + assert scalar_potential_difference(grad_field, R, point1, point2, + origin) == 948 + assert scalar_potential_difference(R[1]*R[2]*R.x + R[0]*R[2]*R.y + \ + R[0]*R[1]*R.z, R, point1, + genericpointR, origin) == \ + R[0]*R[1]*R[2] - 6 + potential_diff_P = 2*P[2]*(P[0]*sin(q) + P[1]*cos(q))*\ + (P[0]*cos(q) - P[1]*sin(q))**2 + assert scalar_potential_difference(grad_field, P, origin, \ + genericpointP, \ + origin).simplify() == \ + potential_diff_P diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_frame.py b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_frame.py new file mode 100644 index 0000000000000000000000000000000000000000..8e2d0234c7d2d9f91fdb5421c5a92f05495006c6 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_frame.py @@ -0,0 +1,761 @@ +from sympy.core.numbers import pi +from sympy.core.symbol import symbols +from sympy.simplify import trigsimp +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.matrices.dense import (eye, zeros) +from sympy.matrices.immutable import ImmutableDenseMatrix as Matrix +from sympy.simplify.simplify import simplify +from sympy.physics.vector import (ReferenceFrame, Vector, CoordinateSym, + dynamicsymbols, time_derivative, express, + dot) +from sympy.physics.vector.frame import _check_frame +from sympy.physics.vector.vector import VectorTypeError +from sympy.testing.pytest import raises +import warnings +import pickle + + +def test_dict_list(): + + A = ReferenceFrame('A') + B = ReferenceFrame('B') + C = ReferenceFrame('C') + D = ReferenceFrame('D') + E = ReferenceFrame('E') + F = ReferenceFrame('F') + + B.orient_axis(A, A.x, 1.0) + C.orient_axis(B, B.x, 1.0) + D.orient_axis(C, C.x, 1.0) + + assert D._dict_list(A, 0) == [D, C, B, A] + + E.orient_axis(D, D.x, 1.0) + + assert C._dict_list(A, 0) == [C, B, A] + assert C._dict_list(E, 0) == [C, D, E] + + # only 0, 1, 2 permitted for second argument + raises(ValueError, lambda: C._dict_list(E, 5)) + # no connecting path + raises(ValueError, lambda: F._dict_list(A, 0)) + + +def test_coordinate_vars(): + """Tests the coordinate variables functionality""" + A = ReferenceFrame('A') + assert CoordinateSym('Ax', A, 0) == A[0] + assert CoordinateSym('Ax', A, 1) == A[1] + assert CoordinateSym('Ax', A, 2) == A[2] + raises(ValueError, lambda: CoordinateSym('Ax', A, 3)) + q = dynamicsymbols('q') + qd = dynamicsymbols('q', 1) + assert isinstance(A[0], CoordinateSym) and \ + isinstance(A[0], CoordinateSym) and \ + isinstance(A[0], CoordinateSym) + assert A.variable_map(A) == {A[0]:A[0], A[1]:A[1], A[2]:A[2]} + assert A[0].frame == A + B = A.orientnew('B', 'Axis', [q, A.z]) + assert B.variable_map(A) == {B[2]: A[2], B[1]: -A[0]*sin(q) + A[1]*cos(q), + B[0]: A[0]*cos(q) + A[1]*sin(q)} + assert A.variable_map(B) == {A[0]: B[0]*cos(q) - B[1]*sin(q), + A[1]: B[0]*sin(q) + B[1]*cos(q), A[2]: B[2]} + assert time_derivative(B[0], A) == -A[0]*sin(q)*qd + A[1]*cos(q)*qd + assert time_derivative(B[1], A) == -A[0]*cos(q)*qd - A[1]*sin(q)*qd + assert time_derivative(B[2], A) == 0 + assert express(B[0], A, variables=True) == A[0]*cos(q) + A[1]*sin(q) + assert express(B[1], A, variables=True) == -A[0]*sin(q) + A[1]*cos(q) + assert express(B[2], A, variables=True) == A[2] + assert time_derivative(A[0]*A.x + A[1]*A.y + A[2]*A.z, B) == A[1]*qd*A.x - A[0]*qd*A.y + assert time_derivative(B[0]*B.x + B[1]*B.y + B[2]*B.z, A) == - B[1]*qd*B.x + B[0]*qd*B.y + assert express(B[0]*B[1]*B[2], A, variables=True) == \ + A[2]*(-A[0]*sin(q) + A[1]*cos(q))*(A[0]*cos(q) + A[1]*sin(q)) + assert (time_derivative(B[0]*B[1]*B[2], A) - + (A[2]*(-A[0]**2*cos(2*q) - + 2*A[0]*A[1]*sin(2*q) + + A[1]**2*cos(2*q))*qd)).trigsimp() == 0 + assert express(B[0]*B.x + B[1]*B.y + B[2]*B.z, A) == \ + (B[0]*cos(q) - B[1]*sin(q))*A.x + (B[0]*sin(q) + \ + B[1]*cos(q))*A.y + B[2]*A.z + assert express(B[0]*B.x + B[1]*B.y + B[2]*B.z, A, + variables=True).simplify() == A[0]*A.x + A[1]*A.y + A[2]*A.z + assert express(A[0]*A.x + A[1]*A.y + A[2]*A.z, B) == \ + (A[0]*cos(q) + A[1]*sin(q))*B.x + \ + (-A[0]*sin(q) + A[1]*cos(q))*B.y + A[2]*B.z + assert express(A[0]*A.x + A[1]*A.y + A[2]*A.z, B, + variables=True).simplify() == B[0]*B.x + B[1]*B.y + B[2]*B.z + N = B.orientnew('N', 'Axis', [-q, B.z]) + assert ({k: v.simplify() for k, v in N.variable_map(A).items()} == + {N[0]: A[0], N[2]: A[2], N[1]: A[1]}) + C = A.orientnew('C', 'Axis', [q, A.x + A.y + A.z]) + mapping = A.variable_map(C) + assert trigsimp(mapping[A[0]]) == (2*C[0]*cos(q)/3 + C[0]/3 - + 2*C[1]*sin(q + pi/6)/3 + + C[1]/3 - 2*C[2]*cos(q + pi/3)/3 + + C[2]/3) + assert trigsimp(mapping[A[1]]) == -2*C[0]*cos(q + pi/3)/3 + \ + C[0]/3 + 2*C[1]*cos(q)/3 + C[1]/3 - 2*C[2]*sin(q + pi/6)/3 + C[2]/3 + assert trigsimp(mapping[A[2]]) == -2*C[0]*sin(q + pi/6)/3 + C[0]/3 - \ + 2*C[1]*cos(q + pi/3)/3 + C[1]/3 + 2*C[2]*cos(q)/3 + C[2]/3 + + +def test_ang_vel(): + q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4') + q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1) + N = ReferenceFrame('N') + A = N.orientnew('A', 'Axis', [q1, N.z]) + B = A.orientnew('B', 'Axis', [q2, A.x]) + C = B.orientnew('C', 'Axis', [q3, B.y]) + D = N.orientnew('D', 'Axis', [q4, N.y]) + u1, u2, u3 = dynamicsymbols('u1 u2 u3') + assert A.ang_vel_in(N) == (q1d)*A.z + assert B.ang_vel_in(N) == (q2d)*B.x + (q1d)*A.z + assert C.ang_vel_in(N) == (q3d)*C.y + (q2d)*B.x + (q1d)*A.z + + A2 = N.orientnew('A2', 'Axis', [q4, N.y]) + assert N.ang_vel_in(N) == 0 + assert N.ang_vel_in(A) == -q1d*N.z + assert N.ang_vel_in(B) == -q1d*A.z - q2d*B.x + assert N.ang_vel_in(C) == -q1d*A.z - q2d*B.x - q3d*B.y + assert N.ang_vel_in(A2) == -q4d*N.y + + assert A.ang_vel_in(N) == q1d*N.z + assert A.ang_vel_in(A) == 0 + assert A.ang_vel_in(B) == - q2d*B.x + assert A.ang_vel_in(C) == - q2d*B.x - q3d*B.y + assert A.ang_vel_in(A2) == q1d*N.z - q4d*N.y + + assert B.ang_vel_in(N) == q1d*A.z + q2d*A.x + assert B.ang_vel_in(A) == q2d*A.x + assert B.ang_vel_in(B) == 0 + assert B.ang_vel_in(C) == -q3d*B.y + assert B.ang_vel_in(A2) == q1d*A.z + q2d*A.x - q4d*N.y + + assert C.ang_vel_in(N) == q1d*A.z + q2d*A.x + q3d*B.y + assert C.ang_vel_in(A) == q2d*A.x + q3d*C.y + assert C.ang_vel_in(B) == q3d*B.y + assert C.ang_vel_in(C) == 0 + assert C.ang_vel_in(A2) == q1d*A.z + q2d*A.x + q3d*B.y - q4d*N.y + + assert A2.ang_vel_in(N) == q4d*A2.y + assert A2.ang_vel_in(A) == q4d*A2.y - q1d*N.z + assert A2.ang_vel_in(B) == q4d*N.y - q1d*A.z - q2d*A.x + assert A2.ang_vel_in(C) == q4d*N.y - q1d*A.z - q2d*A.x - q3d*B.y + assert A2.ang_vel_in(A2) == 0 + + C.set_ang_vel(N, u1*C.x + u2*C.y + u3*C.z) + assert C.ang_vel_in(N) == (u1)*C.x + (u2)*C.y + (u3)*C.z + assert N.ang_vel_in(C) == (-u1)*C.x + (-u2)*C.y + (-u3)*C.z + assert C.ang_vel_in(D) == (u1)*C.x + (u2)*C.y + (u3)*C.z + (-q4d)*D.y + assert D.ang_vel_in(C) == (-u1)*C.x + (-u2)*C.y + (-u3)*C.z + (q4d)*D.y + + q0 = dynamicsymbols('q0') + q0d = dynamicsymbols('q0', 1) + E = N.orientnew('E', 'Quaternion', (q0, q1, q2, q3)) + assert E.ang_vel_in(N) == ( + 2 * (q1d * q0 + q2d * q3 - q3d * q2 - q0d * q1) * E.x + + 2 * (q2d * q0 + q3d * q1 - q1d * q3 - q0d * q2) * E.y + + 2 * (q3d * q0 + q1d * q2 - q2d * q1 - q0d * q3) * E.z) + + F = N.orientnew('F', 'Body', (q1, q2, q3), 313) + assert F.ang_vel_in(N) == ((sin(q2)*sin(q3)*q1d + cos(q3)*q2d)*F.x + + (sin(q2)*cos(q3)*q1d - sin(q3)*q2d)*F.y + (cos(q2)*q1d + q3d)*F.z) + G = N.orientnew('G', 'Axis', (q1, N.x + N.y)) + assert G.ang_vel_in(N) == q1d * (N.x + N.y).normalize() + assert N.ang_vel_in(G) == -q1d * (N.x + N.y).normalize() + + +def test_dcm(): + q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4') + N = ReferenceFrame('N') + A = N.orientnew('A', 'Axis', [q1, N.z]) + B = A.orientnew('B', 'Axis', [q2, A.x]) + C = B.orientnew('C', 'Axis', [q3, B.y]) + D = N.orientnew('D', 'Axis', [q4, N.y]) + E = N.orientnew('E', 'Space', [q1, q2, q3], '123') + assert N.dcm(C) == Matrix([ + [- sin(q1) * sin(q2) * sin(q3) + cos(q1) * cos(q3), - sin(q1) * + cos(q2), sin(q1) * sin(q2) * cos(q3) + sin(q3) * cos(q1)], [sin(q1) * + cos(q3) + sin(q2) * sin(q3) * cos(q1), cos(q1) * cos(q2), sin(q1) * + sin(q3) - sin(q2) * cos(q1) * cos(q3)], [- sin(q3) * cos(q2), sin(q2), + cos(q2) * cos(q3)]]) + # This is a little touchy. Is it ok to use simplify in assert? + test_mat = D.dcm(C) - Matrix( + [[cos(q1) * cos(q3) * cos(q4) - sin(q3) * (- sin(q4) * cos(q2) + + sin(q1) * sin(q2) * cos(q4)), - sin(q2) * sin(q4) - sin(q1) * + cos(q2) * cos(q4), sin(q3) * cos(q1) * cos(q4) + cos(q3) * (- sin(q4) * + cos(q2) + sin(q1) * sin(q2) * cos(q4))], [sin(q1) * cos(q3) + + sin(q2) * sin(q3) * cos(q1), cos(q1) * cos(q2), sin(q1) * sin(q3) - + sin(q2) * cos(q1) * cos(q3)], [sin(q4) * cos(q1) * cos(q3) - + sin(q3) * (cos(q2) * cos(q4) + sin(q1) * sin(q2) * sin(q4)), sin(q2) * + cos(q4) - sin(q1) * sin(q4) * cos(q2), sin(q3) * sin(q4) * cos(q1) + + cos(q3) * (cos(q2) * cos(q4) + sin(q1) * sin(q2) * sin(q4))]]) + assert test_mat.expand() == zeros(3, 3) + assert E.dcm(N) == Matrix( + [[cos(q2)*cos(q3), sin(q3)*cos(q2), -sin(q2)], + [sin(q1)*sin(q2)*cos(q3) - sin(q3)*cos(q1), sin(q1)*sin(q2)*sin(q3) + + cos(q1)*cos(q3), sin(q1)*cos(q2)], [sin(q1)*sin(q3) + + sin(q2)*cos(q1)*cos(q3), - sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1), + cos(q1)*cos(q2)]]) + +def test_w_diff_dcm1(): + # Ref: + # Dynamics Theory and Applications, Kane 1985 + # Sec. 2.1 ANGULAR VELOCITY + A = ReferenceFrame('A') + B = ReferenceFrame('B') + + c11, c12, c13 = dynamicsymbols('C11 C12 C13') + c21, c22, c23 = dynamicsymbols('C21 C22 C23') + c31, c32, c33 = dynamicsymbols('C31 C32 C33') + + c11d, c12d, c13d = dynamicsymbols('C11 C12 C13', level=1) + c21d, c22d, c23d = dynamicsymbols('C21 C22 C23', level=1) + c31d, c32d, c33d = dynamicsymbols('C31 C32 C33', level=1) + + DCM = Matrix([ + [c11, c12, c13], + [c21, c22, c23], + [c31, c32, c33] + ]) + + B.orient(A, 'DCM', DCM) + b1a = (B.x).express(A) + b2a = (B.y).express(A) + b3a = (B.z).express(A) + + # Equation (2.1.1) + B.set_ang_vel(A, B.x*(dot((b3a).dt(A), B.y)) + + B.y*(dot((b1a).dt(A), B.z)) + + B.z*(dot((b2a).dt(A), B.x))) + + # Equation (2.1.21) + expr = ( (c12*c13d + c22*c23d + c32*c33d)*B.x + + (c13*c11d + c23*c21d + c33*c31d)*B.y + + (c11*c12d + c21*c22d + c31*c32d)*B.z) + assert B.ang_vel_in(A) - expr == 0 + +def test_w_diff_dcm2(): + q1, q2, q3 = dynamicsymbols('q1:4') + N = ReferenceFrame('N') + A = N.orientnew('A', 'axis', [q1, N.x]) + B = A.orientnew('B', 'axis', [q2, A.y]) + C = B.orientnew('C', 'axis', [q3, B.z]) + + DCM = C.dcm(N).T + D = N.orientnew('D', 'DCM', DCM) + + # Frames D and C are the same ReferenceFrame, + # since they have equal DCM respect to frame N. + # Therefore, D and C should have same angle velocity in N. + assert D.dcm(N) == C.dcm(N) == Matrix([ + [cos(q2)*cos(q3), sin(q1)*sin(q2)*cos(q3) + + sin(q3)*cos(q1), sin(q1)*sin(q3) - + sin(q2)*cos(q1)*cos(q3)], [-sin(q3)*cos(q2), + -sin(q1)*sin(q2)*sin(q3) + cos(q1)*cos(q3), + sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1)], + [sin(q2), -sin(q1)*cos(q2), cos(q1)*cos(q2)]]) + assert (D.ang_vel_in(N) - C.ang_vel_in(N)).express(N).simplify() == 0 + +def test_orientnew_respects_parent_class(): + class MyReferenceFrame(ReferenceFrame): + pass + B = MyReferenceFrame('B') + C = B.orientnew('C', 'Axis', [0, B.x]) + assert isinstance(C, MyReferenceFrame) + + +def test_orientnew_respects_input_indices(): + N = ReferenceFrame('N') + q1 = dynamicsymbols('q1') + A = N.orientnew('a', 'Axis', [q1, N.z]) + #modify default indices: + minds = [x+'1' for x in N.indices] + B = N.orientnew('b', 'Axis', [q1, N.z], indices=minds) + + assert N.indices == A.indices + assert B.indices == minds + +def test_orientnew_respects_input_latexs(): + N = ReferenceFrame('N') + q1 = dynamicsymbols('q1') + A = N.orientnew('a', 'Axis', [q1, N.z]) + + #build default and alternate latex_vecs: + def_latex_vecs = [(r"\mathbf{\hat{%s}_%s}" % (A.name.lower(), + A.indices[0])), (r"\mathbf{\hat{%s}_%s}" % + (A.name.lower(), A.indices[1])), + (r"\mathbf{\hat{%s}_%s}" % (A.name.lower(), + A.indices[2]))] + + name = 'b' + indices = [x+'1' for x in N.indices] + new_latex_vecs = [(r"\mathbf{\hat{%s}_{%s}}" % (name.lower(), + indices[0])), (r"\mathbf{\hat{%s}_{%s}}" % + (name.lower(), indices[1])), + (r"\mathbf{\hat{%s}_{%s}}" % (name.lower(), + indices[2]))] + + B = N.orientnew(name, 'Axis', [q1, N.z], latexs=new_latex_vecs) + + assert A.latex_vecs == def_latex_vecs + assert B.latex_vecs == new_latex_vecs + assert B.indices != indices + +def test_orientnew_respects_input_variables(): + N = ReferenceFrame('N') + q1 = dynamicsymbols('q1') + A = N.orientnew('a', 'Axis', [q1, N.z]) + + #build non-standard variable names + name = 'b' + new_variables = ['notb_'+x+'1' for x in N.indices] + B = N.orientnew(name, 'Axis', [q1, N.z], variables=new_variables) + + for j,var in enumerate(A.varlist): + assert var.name == A.name + '_' + A.indices[j] + + for j,var in enumerate(B.varlist): + assert var.name == new_variables[j] + +def test_issue_10348(): + u = dynamicsymbols('u:3') + I = ReferenceFrame('I') + I.orientnew('A', 'space', u, 'XYZ') + + +def test_issue_11503(): + A = ReferenceFrame("A") + A.orientnew("B", "Axis", [35, A.y]) + C = ReferenceFrame("C") + A.orient(C, "Axis", [70, C.z]) + + +def test_partial_velocity(): + + N = ReferenceFrame('N') + A = ReferenceFrame('A') + + u1, u2 = dynamicsymbols('u1, u2') + + A.set_ang_vel(N, u1 * A.x + u2 * N.y) + + assert N.partial_velocity(A, u1) == -A.x + assert N.partial_velocity(A, u1, u2) == (-A.x, -N.y) + + assert A.partial_velocity(N, u1) == A.x + assert A.partial_velocity(N, u1, u2) == (A.x, N.y) + + assert N.partial_velocity(N, u1) == 0 + assert A.partial_velocity(A, u1) == 0 + + +def test_issue_11498(): + A = ReferenceFrame('A') + B = ReferenceFrame('B') + + # Identity transformation + A.orient(B, 'DCM', eye(3)) + assert A.dcm(B) == Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) + assert B.dcm(A) == Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) + + # x -> y + # y -> -z + # z -> -x + A.orient(B, 'DCM', Matrix([[0, 1, 0], [0, 0, -1], [-1, 0, 0]])) + assert B.dcm(A) == Matrix([[0, 1, 0], [0, 0, -1], [-1, 0, 0]]) + assert A.dcm(B) == Matrix([[0, 0, -1], [1, 0, 0], [0, -1, 0]]) + assert B.dcm(A).T == A.dcm(B) + + +def test_reference_frame(): + raises(TypeError, lambda: ReferenceFrame(0)) + raises(TypeError, lambda: ReferenceFrame('N', 0)) + raises(ValueError, lambda: ReferenceFrame('N', [0, 1])) + raises(TypeError, lambda: ReferenceFrame('N', [0, 1, 2])) + raises(TypeError, lambda: ReferenceFrame('N', ['a', 'b', 'c'], 0)) + raises(ValueError, lambda: ReferenceFrame('N', ['a', 'b', 'c'], [0, 1])) + raises(TypeError, lambda: ReferenceFrame('N', ['a', 'b', 'c'], [0, 1, 2])) + raises(TypeError, lambda: ReferenceFrame('N', ['a', 'b', 'c'], + ['a', 'b', 'c'], 0)) + raises(ValueError, lambda: ReferenceFrame('N', ['a', 'b', 'c'], + ['a', 'b', 'c'], [0, 1])) + raises(TypeError, lambda: ReferenceFrame('N', ['a', 'b', 'c'], + ['a', 'b', 'c'], [0, 1, 2])) + N = ReferenceFrame('N') + assert N[0] == CoordinateSym('N_x', N, 0) + assert N[1] == CoordinateSym('N_y', N, 1) + assert N[2] == CoordinateSym('N_z', N, 2) + raises(ValueError, lambda: N[3]) + N = ReferenceFrame('N', ['a', 'b', 'c']) + assert N['a'] == N.x + assert N['b'] == N.y + assert N['c'] == N.z + raises(ValueError, lambda: N['d']) + assert str(N) == 'N' + + A = ReferenceFrame('A') + B = ReferenceFrame('B') + q0, q1, q2, q3 = symbols('q0 q1 q2 q3') + raises(TypeError, lambda: A.orient(B, 'DCM', 0)) + raises(TypeError, lambda: B.orient(N, 'Space', [q1, q2, q3], '222')) + raises(TypeError, lambda: B.orient(N, 'Axis', [q1, N.x + 2 * N.y], '222')) + raises(TypeError, lambda: B.orient(N, 'Axis', q1)) + raises(IndexError, lambda: B.orient(N, 'Axis', [q1])) + raises(TypeError, lambda: B.orient(N, 'Quaternion', [q0, q1, q2, q3], '222')) + raises(TypeError, lambda: B.orient(N, 'Quaternion', q0)) + raises(TypeError, lambda: B.orient(N, 'Quaternion', [q0, q1, q2])) + raises(NotImplementedError, lambda: B.orient(N, 'Foo', [q0, q1, q2])) + raises(TypeError, lambda: B.orient(N, 'Body', [q1, q2], '232')) + raises(TypeError, lambda: B.orient(N, 'Space', [q1, q2], '232')) + + N.set_ang_acc(B, 0) + assert N.ang_acc_in(B) == Vector(0) + N.set_ang_vel(B, 0) + assert N.ang_vel_in(B) == Vector(0) + + +def test_check_frame(): + raises(VectorTypeError, lambda: _check_frame(0)) + + +def test_dcm_diff_16824(): + # NOTE : This is a regression test for the bug introduced in PR 14758, + # identified in 16824, and solved by PR 16828. + + # This is the solution to Problem 2.2 on page 264 in Kane & Lenvinson's + # 1985 book. + + q1, q2, q3 = dynamicsymbols('q1:4') + + s1 = sin(q1) + c1 = cos(q1) + s2 = sin(q2) + c2 = cos(q2) + s3 = sin(q3) + c3 = cos(q3) + + dcm = Matrix([[c2*c3, s1*s2*c3 - s3*c1, c1*s2*c3 + s3*s1], + [c2*s3, s1*s2*s3 + c3*c1, c1*s2*s3 - c3*s1], + [-s2, s1*c2, c1*c2]]) + + A = ReferenceFrame('A') + B = ReferenceFrame('B') + B.orient(A, 'DCM', dcm) + + AwB = B.ang_vel_in(A) + + alpha2 = s3*c2*q1.diff() + c3*q2.diff() + beta2 = s1*c2*q3.diff() + c1*q2.diff() + + assert simplify(AwB.dot(A.y) - alpha2) == 0 + assert simplify(AwB.dot(B.y) - beta2) == 0 + +def test_orient_explicit(): + cxx, cyy, czz = dynamicsymbols('c_{xx}, c_{yy}, c_{zz}') + cxy, cxz, cyx = dynamicsymbols('c_{xy}, c_{xz}, c_{yx}') + cyz, czx, czy = dynamicsymbols('c_{yz}, c_{zx}, c_{zy}') + dcxx, dcyy, dczz = dynamicsymbols('c_{xx}, c_{yy}, c_{zz}', 1) + dcxy, dcxz, dcyx = dynamicsymbols('c_{xy}, c_{xz}, c_{yx}', 1) + dcyz, dczx, dczy = dynamicsymbols('c_{yz}, c_{zx}, c_{zy}', 1) + A = ReferenceFrame('A') + B = ReferenceFrame('B') + B_C_A = Matrix([[cxx, cxy, cxz], + [cyx, cyy, cyz], + [czx, czy, czz]]) + B_w_A = ((cyx*dczx + cyy*dczy + cyz*dczz)*B.x + + (czx*dcxx + czy*dcxy + czz*dcxz)*B.y + + (cxx*dcyx + cxy*dcyy + cxz*dcyz)*B.z) + A.orient_explicit(B, B_C_A) + assert B.dcm(A) == B_C_A + assert A.ang_vel_in(B) == B_w_A + assert B.ang_vel_in(A) == -B_w_A + +def test_orient_dcm(): + cxx, cyy, czz = dynamicsymbols('c_{xx}, c_{yy}, c_{zz}') + cxy, cxz, cyx = dynamicsymbols('c_{xy}, c_{xz}, c_{yx}') + cyz, czx, czy = dynamicsymbols('c_{yz}, c_{zx}, c_{zy}') + B_C_A = Matrix([[cxx, cxy, cxz], + [cyx, cyy, cyz], + [czx, czy, czz]]) + A = ReferenceFrame('A') + B = ReferenceFrame('B') + B.orient_dcm(A, B_C_A) + assert B.dcm(A) == Matrix([[cxx, cxy, cxz], + [cyx, cyy, cyz], + [czx, czy, czz]]) + +def test_orient_axis(): + A = ReferenceFrame('A') + B = ReferenceFrame('B') + A.orient_axis(B,-B.x, 1) + A1 = A.dcm(B) + A.orient_axis(B, B.x, -1) + A2 = A.dcm(B) + A.orient_axis(B, 1, -B.x) + A3 = A.dcm(B) + assert A1 == A2 + assert A2 == A3 + raises(TypeError, lambda: A.orient_axis(B, 1, 1)) + +def test_orient_body(): + A = ReferenceFrame('A') + B = ReferenceFrame('B') + B.orient_body_fixed(A, (1,1,0), 'XYX') + assert B.dcm(A) == Matrix([[cos(1), sin(1)**2, -sin(1)*cos(1)], [0, cos(1), sin(1)], [sin(1), -sin(1)*cos(1), cos(1)**2]]) + + +def test_orient_body_advanced(): + q1, q2, q3 = dynamicsymbols('q1:4') + c1, c2, c3 = symbols('c1:4') + u1, u2, u3 = dynamicsymbols('q1:4', 1) + + # Test with everything as dynamicsymbols + A, B = ReferenceFrame('A'), ReferenceFrame('B') + B.orient_body_fixed(A, (q1, q2, q3), 'zxy') + assert A.dcm(B) == Matrix([ + [-sin(q1) * sin(q2) * sin(q3) + cos(q1) * cos(q3), -sin(q1) * cos(q2), + sin(q1) * sin(q2) * cos(q3) + sin(q3) * cos(q1)], + [sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), cos(q1) * cos(q2), + sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)], + [-sin(q3) * cos(q2), sin(q2), cos(q2) * cos(q3)]]) + assert B.ang_vel_in(A).to_matrix(B) == Matrix([ + [-sin(q3) * cos(q2) * u1 + cos(q3) * u2], + [sin(q2) * u1 + u3], + [sin(q3) * u2 + cos(q2) * cos(q3) * u1]]) + + # Test with constant symbol + A, B = ReferenceFrame('A'), ReferenceFrame('B') + B.orient_body_fixed(A, (q1, c2, q3), 131) + assert A.dcm(B) == Matrix([ + [cos(c2), -sin(c2) * cos(q3), sin(c2) * sin(q3)], + [sin(c2) * cos(q1), -sin(q1) * sin(q3) + cos(c2) * cos(q1) * cos(q3), + -sin(q1) * cos(q3) - sin(q3) * cos(c2) * cos(q1)], + [sin(c2) * sin(q1), sin(q1) * cos(c2) * cos(q3) + sin(q3) * cos(q1), + -sin(q1) * sin(q3) * cos(c2) + cos(q1) * cos(q3)]]) + assert B.ang_vel_in(A).to_matrix(B) == Matrix([ + [cos(c2) * u1 + u3], + [-sin(c2) * cos(q3) * u1], + [sin(c2) * sin(q3) * u1]]) + + # Test all symbols not time dependent + A, B = ReferenceFrame('A'), ReferenceFrame('B') + B.orient_body_fixed(A, (c1, c2, c3), 123) + assert B.ang_vel_in(A) == Vector(0) + + +def test_orient_space_advanced(): + # space fixed is in the end like body fixed only in opposite order + q1, q2, q3 = dynamicsymbols('q1:4') + c1, c2, c3 = symbols('c1:4') + u1, u2, u3 = dynamicsymbols('q1:4', 1) + + # Test with everything as dynamicsymbols + A, B = ReferenceFrame('A'), ReferenceFrame('B') + B.orient_space_fixed(A, (q3, q2, q1), 'yxz') + assert A.dcm(B) == Matrix([ + [-sin(q1) * sin(q2) * sin(q3) + cos(q1) * cos(q3), -sin(q1) * cos(q2), + sin(q1) * sin(q2) * cos(q3) + sin(q3) * cos(q1)], + [sin(q1) * cos(q3) + sin(q2) * sin(q3) * cos(q1), cos(q1) * cos(q2), + sin(q1) * sin(q3) - sin(q2) * cos(q1) * cos(q3)], + [-sin(q3) * cos(q2), sin(q2), cos(q2) * cos(q3)]]) + assert B.ang_vel_in(A).to_matrix(B) == Matrix([ + [-sin(q3) * cos(q2) * u1 + cos(q3) * u2], + [sin(q2) * u1 + u3], + [sin(q3) * u2 + cos(q2) * cos(q3) * u1]]) + + # Test with constant symbol + A, B = ReferenceFrame('A'), ReferenceFrame('B') + B.orient_space_fixed(A, (q3, c2, q1), 131) + assert A.dcm(B) == Matrix([ + [cos(c2), -sin(c2) * cos(q3), sin(c2) * sin(q3)], + [sin(c2) * cos(q1), -sin(q1) * sin(q3) + cos(c2) * cos(q1) * cos(q3), + -sin(q1) * cos(q3) - sin(q3) * cos(c2) * cos(q1)], + [sin(c2) * sin(q1), sin(q1) * cos(c2) * cos(q3) + sin(q3) * cos(q1), + -sin(q1) * sin(q3) * cos(c2) + cos(q1) * cos(q3)]]) + assert B.ang_vel_in(A).to_matrix(B) == Matrix([ + [cos(c2) * u1 + u3], + [-sin(c2) * cos(q3) * u1], + [sin(c2) * sin(q3) * u1]]) + + # Test all symbols not time dependent + A, B = ReferenceFrame('A'), ReferenceFrame('B') + B.orient_space_fixed(A, (c1, c2, c3), 123) + assert B.ang_vel_in(A) == Vector(0) + + +def test_orient_body_simple_ang_vel(): + """This test ensures that the simplest form of that linear system solution + is returned, thus the == for the expression comparison.""" + + psi, theta, phi = dynamicsymbols('psi, theta, varphi') + t = dynamicsymbols._t + A = ReferenceFrame('A') + B = ReferenceFrame('B') + B.orient_body_fixed(A, (psi, theta, phi), 'ZXZ') + A_w_B = B.ang_vel_in(A) + assert A_w_B.args[0][1] == B + assert A_w_B.args[0][0][0] == (sin(theta)*sin(phi)*psi.diff(t) + + cos(phi)*theta.diff(t)) + assert A_w_B.args[0][0][1] == (sin(theta)*cos(phi)*psi.diff(t) - + sin(phi)*theta.diff(t)) + assert A_w_B.args[0][0][2] == cos(theta)*psi.diff(t) + phi.diff(t) + + +def test_orient_space(): + A = ReferenceFrame('A') + B = ReferenceFrame('B') + B.orient_space_fixed(A, (0,0,0), '123') + assert B.dcm(A) == Matrix([[1, 0, 0], [0, 1, 0], [0, 0, 1]]) + +def test_orient_quaternion(): + A = ReferenceFrame('A') + B = ReferenceFrame('B') + B.orient_quaternion(A, (0,0,0,0)) + assert B.dcm(A) == Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0]]) + +def test_looped_frame_warning(): + A = ReferenceFrame('A') + B = ReferenceFrame('B') + C = ReferenceFrame('C') + + a, b, c = symbols('a b c') + B.orient_axis(A, A.x, a) + C.orient_axis(B, B.x, b) + + with warnings.catch_warnings(record = True) as w: + warnings.simplefilter("always") + A.orient_axis(C, C.x, c) + assert issubclass(w[-1].category, UserWarning) + assert 'Loops are defined among the orientation of frames. ' + \ + 'This is likely not desired and may cause errors in your calculations.' in str(w[-1].message) + +def test_frame_dict(): + A = ReferenceFrame('A') + B = ReferenceFrame('B') + C = ReferenceFrame('C') + + a, b, c = symbols('a b c') + + B.orient_axis(A, A.x, a) + assert A._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(a), -sin(a)],[0, sin(a), cos(a)]])} + assert B._dcm_dict == {A: Matrix([[1, 0, 0],[0, cos(a), sin(a)],[0, -sin(a), cos(a)]])} + assert C._dcm_dict == {} + + B.orient_axis(C, C.x, b) + # Previous relation is not wiped + assert A._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(a), -sin(a)],[0, sin(a), cos(a)]])} + assert B._dcm_dict == {A: Matrix([[1, 0, 0],[0, cos(a), sin(a)],[0, -sin(a), cos(a)]]), \ + C: Matrix([[1, 0, 0],[0, cos(b), sin(b)],[0, -sin(b), cos(b)]])} + assert C._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(b), -sin(b)],[0, sin(b), cos(b)]])} + + A.orient_axis(B, B.x, c) + # Previous relation is updated + assert B._dcm_dict == {C: Matrix([[1, 0, 0],[0, cos(b), sin(b)],[0, -sin(b), cos(b)]]),\ + A: Matrix([[1, 0, 0],[0, cos(c), -sin(c)],[0, sin(c), cos(c)]])} + assert A._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(c), sin(c)],[0, -sin(c), cos(c)]])} + assert C._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(b), -sin(b)],[0, sin(b), cos(b)]])} + +def test_dcm_cache_dict(): + A = ReferenceFrame('A') + B = ReferenceFrame('B') + C = ReferenceFrame('C') + D = ReferenceFrame('D') + + a, b, c = symbols('a b c') + + B.orient_axis(A, A.x, a) + C.orient_axis(B, B.x, b) + D.orient_axis(C, C.x, c) + + assert D._dcm_dict == {C: Matrix([[1, 0, 0],[0, cos(c), sin(c)],[0, -sin(c), cos(c)]])} + assert C._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(b), sin(b)],[0, -sin(b), cos(b)]]), \ + D: Matrix([[1, 0, 0],[0, cos(c), -sin(c)],[0, sin(c), cos(c)]])} + assert B._dcm_dict == {A: Matrix([[1, 0, 0],[0, cos(a), sin(a)],[0, -sin(a), cos(a)]]), \ + C: Matrix([[1, 0, 0],[0, cos(b), -sin(b)],[0, sin(b), cos(b)]])} + assert A._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(a), -sin(a)],[0, sin(a), cos(a)]])} + + assert D._dcm_dict == D._dcm_cache + + D.dcm(A) # Check calculated dcm relation is stored in _dcm_cache and not in _dcm_dict + assert list(A._dcm_cache.keys()) == [A, B, D] + assert list(D._dcm_cache.keys()) == [C, A] + assert list(A._dcm_dict.keys()) == [B] + assert list(D._dcm_dict.keys()) == [C] + assert A._dcm_dict != A._dcm_cache + + A.orient_axis(B, B.x, b) # _dcm_cache of A is wiped out and new relation is stored. + assert A._dcm_dict == {B: Matrix([[1, 0, 0],[0, cos(b), sin(b)],[0, -sin(b), cos(b)]])} + assert A._dcm_dict == A._dcm_cache + assert B._dcm_dict == {C: Matrix([[1, 0, 0],[0, cos(b), -sin(b)],[0, sin(b), cos(b)]]), \ + A: Matrix([[1, 0, 0],[0, cos(b), -sin(b)],[0, sin(b), cos(b)]])} + +def test_xx_dyad(): + N = ReferenceFrame('N') + F = ReferenceFrame('F', indices=['1', '2', '3']) + assert N.xx == Vector.outer(N.x, N.x) + assert F.xx == Vector.outer(F.x, F.x) + +def test_xy_dyad(): + N = ReferenceFrame('N') + F = ReferenceFrame('F', indices=['1', '2', '3']) + assert N.xy == Vector.outer(N.x, N.y) + assert F.xy == Vector.outer(F.x, F.y) + +def test_xz_dyad(): + N = ReferenceFrame('N') + F = ReferenceFrame('F', indices=['1', '2', '3']) + assert N.xz == Vector.outer(N.x, N.z) + assert F.xz == Vector.outer(F.x, F.z) + +def test_yx_dyad(): + N = ReferenceFrame('N') + F = ReferenceFrame('F', indices=['1', '2', '3']) + assert N.yx == Vector.outer(N.y, N.x) + assert F.yx == Vector.outer(F.y, F.x) + +def test_yy_dyad(): + N = ReferenceFrame('N') + F = ReferenceFrame('F', indices=['1', '2', '3']) + assert N.yy == Vector.outer(N.y, N.y) + assert F.yy == Vector.outer(F.y, F.y) + +def test_yz_dyad(): + N = ReferenceFrame('N') + F = ReferenceFrame('F', indices=['1', '2', '3']) + assert N.yz == Vector.outer(N.y, N.z) + assert F.yz == Vector.outer(F.y, F.z) + +def test_zx_dyad(): + N = ReferenceFrame('N') + F = ReferenceFrame('F', indices=['1', '2', '3']) + assert N.zx == Vector.outer(N.z, N.x) + assert F.zx == Vector.outer(F.z, F.x) + +def test_zy_dyad(): + N = ReferenceFrame('N') + F = ReferenceFrame('F', indices=['1', '2', '3']) + assert N.zy == Vector.outer(N.z, N.y) + assert F.zy == Vector.outer(F.z, F.y) + +def test_zz_dyad(): + N = ReferenceFrame('N') + F = ReferenceFrame('F', indices=['1', '2', '3']) + assert N.zz == Vector.outer(N.z, N.z) + assert F.zz == Vector.outer(F.z, F.z) + +def test_unit_dyadic(): + N = ReferenceFrame('N') + F = ReferenceFrame('F', indices=['1', '2', '3']) + assert N.u == N.xx + N.yy + N.zz + assert F.u == F.xx + F.yy + F.zz + + +def test_pickle_frame(): + N = ReferenceFrame('N') + A = ReferenceFrame('A') + A.orient_axis(N, N.x, 1) + A_C_N = A.dcm(N) + N1 = pickle.loads(pickle.dumps(N)) + A1 = tuple(N1._dcm_dict.keys())[0] + assert A1.dcm(N1) == A_C_N diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_functions.py b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_functions.py new file mode 100644 index 0000000000000000000000000000000000000000..ff938da980c4bbd51d378b30fd5310a88e528e97 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_functions.py @@ -0,0 +1,509 @@ +from sympy.core.numbers import pi +from sympy.core.singleton import S +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (cos, sin) +from sympy.integrals.integrals import Integral +from sympy.physics.vector import Dyadic, Point, ReferenceFrame, Vector +from sympy.physics.vector.functions import (cross, dot, express, + time_derivative, + kinematic_equations, outer, + partial_velocity, + get_motion_params, dynamicsymbols) +from sympy.simplify import trigsimp +from sympy.testing.pytest import raises + +q1, q2, q3, q4, q5 = symbols('q1 q2 q3 q4 q5') +N = ReferenceFrame('N') +A = N.orientnew('A', 'Axis', [q1, N.z]) +B = A.orientnew('B', 'Axis', [q2, A.x]) +C = B.orientnew('C', 'Axis', [q3, B.y]) + + +def test_dot(): + assert dot(A.x, A.x) == 1 + assert dot(A.x, A.y) == 0 + assert dot(A.x, A.z) == 0 + + assert dot(A.y, A.x) == 0 + assert dot(A.y, A.y) == 1 + assert dot(A.y, A.z) == 0 + + assert dot(A.z, A.x) == 0 + assert dot(A.z, A.y) == 0 + assert dot(A.z, A.z) == 1 + + +def test_dot_different_frames(): + assert dot(N.x, A.x) == cos(q1) + assert dot(N.x, A.y) == -sin(q1) + assert dot(N.x, A.z) == 0 + assert dot(N.y, A.x) == sin(q1) + assert dot(N.y, A.y) == cos(q1) + assert dot(N.y, A.z) == 0 + assert dot(N.z, A.x) == 0 + assert dot(N.z, A.y) == 0 + assert dot(N.z, A.z) == 1 + + assert trigsimp(dot(N.x, A.x + A.y)) == sqrt(2)*cos(q1 + pi/4) + assert trigsimp(dot(N.x, A.x + A.y)) == trigsimp(dot(A.x + A.y, N.x)) + + assert dot(A.x, C.x) == cos(q3) + assert dot(A.x, C.y) == 0 + assert dot(A.x, C.z) == sin(q3) + assert dot(A.y, C.x) == sin(q2)*sin(q3) + assert dot(A.y, C.y) == cos(q2) + assert dot(A.y, C.z) == -sin(q2)*cos(q3) + assert dot(A.z, C.x) == -cos(q2)*sin(q3) + assert dot(A.z, C.y) == sin(q2) + assert dot(A.z, C.z) == cos(q2)*cos(q3) + + +def test_cross(): + assert cross(A.x, A.x) == 0 + assert cross(A.x, A.y) == A.z + assert cross(A.x, A.z) == -A.y + + assert cross(A.y, A.x) == -A.z + assert cross(A.y, A.y) == 0 + assert cross(A.y, A.z) == A.x + + assert cross(A.z, A.x) == A.y + assert cross(A.z, A.y) == -A.x + assert cross(A.z, A.z) == 0 + + +def test_cross_different_frames(): + assert cross(N.x, A.x) == sin(q1)*A.z + assert cross(N.x, A.y) == cos(q1)*A.z + assert cross(N.x, A.z) == -sin(q1)*A.x - cos(q1)*A.y + assert cross(N.y, A.x) == -cos(q1)*A.z + assert cross(N.y, A.y) == sin(q1)*A.z + assert cross(N.y, A.z) == cos(q1)*A.x - sin(q1)*A.y + assert cross(N.z, A.x) == A.y + assert cross(N.z, A.y) == -A.x + assert cross(N.z, A.z) == 0 + + assert cross(N.x, A.x) == sin(q1)*A.z + assert cross(N.x, A.y) == cos(q1)*A.z + assert cross(N.x, A.x + A.y) == sin(q1)*A.z + cos(q1)*A.z + assert cross(A.x + A.y, N.x) == -sin(q1)*A.z - cos(q1)*A.z + + assert cross(A.x, C.x) == sin(q3)*C.y + assert cross(A.x, C.y) == -sin(q3)*C.x + cos(q3)*C.z + assert cross(A.x, C.z) == -cos(q3)*C.y + assert cross(C.x, A.x) == -sin(q3)*C.y + assert cross(C.y, A.x).express(C).simplify() == sin(q3)*C.x - cos(q3)*C.z + assert cross(C.z, A.x) == cos(q3)*C.y + +def test_operator_match(): + """Test that the output of dot, cross, outer functions match + operator behavior. + """ + A = ReferenceFrame('A') + v = A.x + A.y + d = v | v + zerov = Vector(0) + zerod = Dyadic(0) + + # dot products + assert d & d == dot(d, d) + assert d & zerod == dot(d, zerod) + assert zerod & d == dot(zerod, d) + assert d & v == dot(d, v) + assert v & d == dot(v, d) + assert d & zerov == dot(d, zerov) + assert zerov & d == dot(zerov, d) + raises(TypeError, lambda: dot(d, S.Zero)) + raises(TypeError, lambda: dot(S.Zero, d)) + raises(TypeError, lambda: dot(d, 0)) + raises(TypeError, lambda: dot(0, d)) + assert v & v == dot(v, v) + assert v & zerov == dot(v, zerov) + assert zerov & v == dot(zerov, v) + raises(TypeError, lambda: dot(v, S.Zero)) + raises(TypeError, lambda: dot(S.Zero, v)) + raises(TypeError, lambda: dot(v, 0)) + raises(TypeError, lambda: dot(0, v)) + + # cross products + raises(TypeError, lambda: cross(d, d)) + raises(TypeError, lambda: cross(d, zerod)) + raises(TypeError, lambda: cross(zerod, d)) + assert d ^ v == cross(d, v) + assert v ^ d == cross(v, d) + assert d ^ zerov == cross(d, zerov) + assert zerov ^ d == cross(zerov, d) + assert zerov ^ d == cross(zerov, d) + raises(TypeError, lambda: cross(d, S.Zero)) + raises(TypeError, lambda: cross(S.Zero, d)) + raises(TypeError, lambda: cross(d, 0)) + raises(TypeError, lambda: cross(0, d)) + assert v ^ v == cross(v, v) + assert v ^ zerov == cross(v, zerov) + assert zerov ^ v == cross(zerov, v) + raises(TypeError, lambda: cross(v, S.Zero)) + raises(TypeError, lambda: cross(S.Zero, v)) + raises(TypeError, lambda: cross(v, 0)) + raises(TypeError, lambda: cross(0, v)) + + # outer products + raises(TypeError, lambda: outer(d, d)) + raises(TypeError, lambda: outer(d, zerod)) + raises(TypeError, lambda: outer(zerod, d)) + raises(TypeError, lambda: outer(d, v)) + raises(TypeError, lambda: outer(v, d)) + raises(TypeError, lambda: outer(d, zerov)) + raises(TypeError, lambda: outer(zerov, d)) + raises(TypeError, lambda: outer(zerov, d)) + raises(TypeError, lambda: outer(d, S.Zero)) + raises(TypeError, lambda: outer(S.Zero, d)) + raises(TypeError, lambda: outer(d, 0)) + raises(TypeError, lambda: outer(0, d)) + assert v | v == outer(v, v) + assert v | zerov == outer(v, zerov) + assert zerov | v == outer(zerov, v) + raises(TypeError, lambda: outer(v, S.Zero)) + raises(TypeError, lambda: outer(S.Zero, v)) + raises(TypeError, lambda: outer(v, 0)) + raises(TypeError, lambda: outer(0, v)) + + +def test_express(): + assert express(Vector(0), N) == Vector(0) + assert express(S.Zero, N) is S.Zero + assert express(A.x, C) == cos(q3)*C.x + sin(q3)*C.z + assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \ + sin(q2)*cos(q3)*C.z + assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \ + cos(q2)*cos(q3)*C.z + assert express(A.x, N) == cos(q1)*N.x + sin(q1)*N.y + assert express(A.y, N) == -sin(q1)*N.x + cos(q1)*N.y + assert express(A.z, N) == N.z + assert express(A.x, A) == A.x + assert express(A.y, A) == A.y + assert express(A.z, A) == A.z + assert express(A.x, B) == B.x + assert express(A.y, B) == cos(q2)*B.y - sin(q2)*B.z + assert express(A.z, B) == sin(q2)*B.y + cos(q2)*B.z + assert express(A.x, C) == cos(q3)*C.x + sin(q3)*C.z + assert express(A.y, C) == sin(q2)*sin(q3)*C.x + cos(q2)*C.y - \ + sin(q2)*cos(q3)*C.z + assert express(A.z, C) == -sin(q3)*cos(q2)*C.x + sin(q2)*C.y + \ + cos(q2)*cos(q3)*C.z + # Check to make sure UnitVectors get converted properly + assert express(N.x, N) == N.x + assert express(N.y, N) == N.y + assert express(N.z, N) == N.z + assert express(N.x, A) == (cos(q1)*A.x - sin(q1)*A.y) + assert express(N.y, A) == (sin(q1)*A.x + cos(q1)*A.y) + assert express(N.z, A) == A.z + assert express(N.x, B) == (cos(q1)*B.x - sin(q1)*cos(q2)*B.y + + sin(q1)*sin(q2)*B.z) + assert express(N.y, B) == (sin(q1)*B.x + cos(q1)*cos(q2)*B.y - + sin(q2)*cos(q1)*B.z) + assert express(N.z, B) == (sin(q2)*B.y + cos(q2)*B.z) + assert express(N.x, C) == ( + (cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*C.x - + sin(q1)*cos(q2)*C.y + + (sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*C.z) + assert express(N.y, C) == ( + (sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.x + + cos(q1)*cos(q2)*C.y + + (sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.z) + assert express(N.z, C) == (-sin(q3)*cos(q2)*C.x + sin(q2)*C.y + + cos(q2)*cos(q3)*C.z) + + assert express(A.x, N) == (cos(q1)*N.x + sin(q1)*N.y) + assert express(A.y, N) == (-sin(q1)*N.x + cos(q1)*N.y) + assert express(A.z, N) == N.z + assert express(A.x, A) == A.x + assert express(A.y, A) == A.y + assert express(A.z, A) == A.z + assert express(A.x, B) == B.x + assert express(A.y, B) == (cos(q2)*B.y - sin(q2)*B.z) + assert express(A.z, B) == (sin(q2)*B.y + cos(q2)*B.z) + assert express(A.x, C) == (cos(q3)*C.x + sin(q3)*C.z) + assert express(A.y, C) == (sin(q2)*sin(q3)*C.x + cos(q2)*C.y - + sin(q2)*cos(q3)*C.z) + assert express(A.z, C) == (-sin(q3)*cos(q2)*C.x + sin(q2)*C.y + + cos(q2)*cos(q3)*C.z) + + assert express(B.x, N) == (cos(q1)*N.x + sin(q1)*N.y) + assert express(B.y, N) == (-sin(q1)*cos(q2)*N.x + + cos(q1)*cos(q2)*N.y + sin(q2)*N.z) + assert express(B.z, N) == (sin(q1)*sin(q2)*N.x - + sin(q2)*cos(q1)*N.y + cos(q2)*N.z) + assert express(B.x, A) == A.x + assert express(B.y, A) == (cos(q2)*A.y + sin(q2)*A.z) + assert express(B.z, A) == (-sin(q2)*A.y + cos(q2)*A.z) + assert express(B.x, B) == B.x + assert express(B.y, B) == B.y + assert express(B.z, B) == B.z + assert express(B.x, C) == (cos(q3)*C.x + sin(q3)*C.z) + assert express(B.y, C) == C.y + assert express(B.z, C) == (-sin(q3)*C.x + cos(q3)*C.z) + + assert express(C.x, N) == ( + (cos(q1)*cos(q3) - sin(q1)*sin(q2)*sin(q3))*N.x + + (sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*N.y - + sin(q3)*cos(q2)*N.z) + assert express(C.y, N) == ( + -sin(q1)*cos(q2)*N.x + cos(q1)*cos(q2)*N.y + sin(q2)*N.z) + assert express(C.z, N) == ( + (sin(q3)*cos(q1) + sin(q1)*sin(q2)*cos(q3))*N.x + + (sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*N.y + + cos(q2)*cos(q3)*N.z) + assert express(C.x, A) == (cos(q3)*A.x + sin(q2)*sin(q3)*A.y - + sin(q3)*cos(q2)*A.z) + assert express(C.y, A) == (cos(q2)*A.y + sin(q2)*A.z) + assert express(C.z, A) == (sin(q3)*A.x - sin(q2)*cos(q3)*A.y + + cos(q2)*cos(q3)*A.z) + assert express(C.x, B) == (cos(q3)*B.x - sin(q3)*B.z) + assert express(C.y, B) == B.y + assert express(C.z, B) == (sin(q3)*B.x + cos(q3)*B.z) + assert express(C.x, C) == C.x + assert express(C.y, C) == C.y + assert express(C.z, C) == C.z == (C.z) + + # Check to make sure Vectors get converted back to UnitVectors + assert N.x == express((cos(q1)*A.x - sin(q1)*A.y), N).simplify() + assert N.y == express((sin(q1)*A.x + cos(q1)*A.y), N).simplify() + assert N.x == express((cos(q1)*B.x - sin(q1)*cos(q2)*B.y + + sin(q1)*sin(q2)*B.z), N).simplify() + assert N.y == express((sin(q1)*B.x + cos(q1)*cos(q2)*B.y - + sin(q2)*cos(q1)*B.z), N).simplify() + assert N.z == express((sin(q2)*B.y + cos(q2)*B.z), N).simplify() + + """ + These don't really test our code, they instead test the auto simplification + (or lack thereof) of SymPy. + assert N.x == express(( + (cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*C.x - + sin(q1)*cos(q2)*C.y + + (sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*C.z), N) + assert N.y == express(( + (sin(q1)*cos(q3) + sin(q2)*sin(q3)*cos(q1))*C.x + + cos(q1)*cos(q2)*C.y + + (sin(q1)*sin(q3) - sin(q2)*cos(q1)*cos(q3))*C.z), N) + assert N.z == express((-sin(q3)*cos(q2)*C.x + sin(q2)*C.y + + cos(q2)*cos(q3)*C.z), N) + """ + + assert A.x == express((cos(q1)*N.x + sin(q1)*N.y), A).simplify() + assert A.y == express((-sin(q1)*N.x + cos(q1)*N.y), A).simplify() + + assert A.y == express((cos(q2)*B.y - sin(q2)*B.z), A).simplify() + assert A.z == express((sin(q2)*B.y + cos(q2)*B.z), A).simplify() + + assert A.x == express((cos(q3)*C.x + sin(q3)*C.z), A).simplify() + + # Tripsimp messes up here too. + #print express((sin(q2)*sin(q3)*C.x + cos(q2)*C.y - + # sin(q2)*cos(q3)*C.z), A) + assert A.y == express((sin(q2)*sin(q3)*C.x + cos(q2)*C.y - + sin(q2)*cos(q3)*C.z), A).simplify() + + assert A.z == express((-sin(q3)*cos(q2)*C.x + sin(q2)*C.y + + cos(q2)*cos(q3)*C.z), A).simplify() + assert B.x == express((cos(q1)*N.x + sin(q1)*N.y), B).simplify() + assert B.y == express((-sin(q1)*cos(q2)*N.x + + cos(q1)*cos(q2)*N.y + sin(q2)*N.z), B).simplify() + + assert B.z == express((sin(q1)*sin(q2)*N.x - + sin(q2)*cos(q1)*N.y + cos(q2)*N.z), B).simplify() + + assert B.y == express((cos(q2)*A.y + sin(q2)*A.z), B).simplify() + assert B.z == express((-sin(q2)*A.y + cos(q2)*A.z), B).simplify() + assert B.x == express((cos(q3)*C.x + sin(q3)*C.z), B).simplify() + assert B.z == express((-sin(q3)*C.x + cos(q3)*C.z), B).simplify() + + """ + assert C.x == express(( + (cos(q1)*cos(q3)-sin(q1)*sin(q2)*sin(q3))*N.x + + (sin(q1)*cos(q3)+sin(q2)*sin(q3)*cos(q1))*N.y - + sin(q3)*cos(q2)*N.z), C) + assert C.y == express(( + -sin(q1)*cos(q2)*N.x + cos(q1)*cos(q2)*N.y + sin(q2)*N.z), C) + assert C.z == express(( + (sin(q3)*cos(q1)+sin(q1)*sin(q2)*cos(q3))*N.x + + (sin(q1)*sin(q3)-sin(q2)*cos(q1)*cos(q3))*N.y + + cos(q2)*cos(q3)*N.z), C) + """ + assert C.x == express((cos(q3)*A.x + sin(q2)*sin(q3)*A.y - + sin(q3)*cos(q2)*A.z), C).simplify() + assert C.y == express((cos(q2)*A.y + sin(q2)*A.z), C).simplify() + assert C.z == express((sin(q3)*A.x - sin(q2)*cos(q3)*A.y + + cos(q2)*cos(q3)*A.z), C).simplify() + assert C.x == express((cos(q3)*B.x - sin(q3)*B.z), C).simplify() + assert C.z == express((sin(q3)*B.x + cos(q3)*B.z), C).simplify() + + +def test_time_derivative(): + #The use of time_derivative for calculations pertaining to scalar + #fields has been tested in test_coordinate_vars in test_essential.py + A = ReferenceFrame('A') + q = dynamicsymbols('q') + qd = dynamicsymbols('q', 1) + B = A.orientnew('B', 'Axis', [q, A.z]) + d = A.x | A.x + assert time_derivative(d, B) == (-qd) * (A.y | A.x) + \ + (-qd) * (A.x | A.y) + d1 = A.x | B.y + assert time_derivative(d1, A) == - qd*(A.x|B.x) + assert time_derivative(d1, B) == - qd*(A.y|B.y) + d2 = A.x | B.x + assert time_derivative(d2, A) == qd*(A.x|B.y) + assert time_derivative(d2, B) == - qd*(A.y|B.x) + d3 = A.x | B.z + assert time_derivative(d3, A) == 0 + assert time_derivative(d3, B) == - qd*(A.y|B.z) + q1, q2, q3, q4 = dynamicsymbols('q1 q2 q3 q4') + q1d, q2d, q3d, q4d = dynamicsymbols('q1 q2 q3 q4', 1) + q1dd, q2dd, q3dd, q4dd = dynamicsymbols('q1 q2 q3 q4', 2) + C = B.orientnew('C', 'Axis', [q4, B.x]) + v1 = q1 * A.z + v2 = q2*A.x + q3*B.y + v3 = q1*A.x + q2*A.y + q3*A.z + assert time_derivative(B.x, C) == 0 + assert time_derivative(B.y, C) == - q4d*B.z + assert time_derivative(B.z, C) == q4d*B.y + assert time_derivative(v1, B) == q1d*A.z + assert time_derivative(v1, C) == - q1*sin(q)*q4d*A.x + \ + q1*cos(q)*q4d*A.y + q1d*A.z + assert time_derivative(v2, A) == q2d*A.x - q3*qd*B.x + q3d*B.y + assert time_derivative(v2, C) == q2d*A.x - q2*qd*A.y + \ + q2*sin(q)*q4d*A.z + q3d*B.y - q3*q4d*B.z + assert time_derivative(v3, B) == (q2*qd + q1d)*A.x + \ + (-q1*qd + q2d)*A.y + q3d*A.z + assert time_derivative(d, C) == - qd*(A.y|A.x) + \ + sin(q)*q4d*(A.z|A.x) - qd*(A.x|A.y) + sin(q)*q4d*(A.x|A.z) + raises(ValueError, lambda: time_derivative(B.x, C, order=0.5)) + raises(ValueError, lambda: time_derivative(B.x, C, order=-1)) + + +def test_get_motion_methods(): + #Initialization + t = dynamicsymbols._t + s1, s2, s3 = symbols('s1 s2 s3') + S1, S2, S3 = symbols('S1 S2 S3') + S4, S5, S6 = symbols('S4 S5 S6') + t1, t2 = symbols('t1 t2') + a, b, c = dynamicsymbols('a b c') + ad, bd, cd = dynamicsymbols('a b c', 1) + a2d, b2d, c2d = dynamicsymbols('a b c', 2) + v0 = S1*N.x + S2*N.y + S3*N.z + v01 = S4*N.x + S5*N.y + S6*N.z + v1 = s1*N.x + s2*N.y + s3*N.z + v2 = a*N.x + b*N.y + c*N.z + v2d = ad*N.x + bd*N.y + cd*N.z + v2dd = a2d*N.x + b2d*N.y + c2d*N.z + #Test position parameter + assert get_motion_params(frame = N) == (0, 0, 0) + assert get_motion_params(N, position=v1) == (0, 0, v1) + assert get_motion_params(N, position=v2) == (v2dd, v2d, v2) + #Test velocity parameter + assert get_motion_params(N, velocity=v1) == (0, v1, v1 * t) + assert get_motion_params(N, velocity=v1, position=v0, timevalue1=t1) == \ + (0, v1, v0 + v1*(t - t1)) + answer = get_motion_params(N, velocity=v1, position=v2, timevalue1=t1) + answer_expected = (0, v1, v1*t - v1*t1 + v2.subs(t, t1)) + assert answer == answer_expected + + answer = get_motion_params(N, velocity=v2, position=v0, timevalue1=t1) + integral_vector = Integral(a, (t, t1, t))*N.x + Integral(b, (t, t1, t))*N.y \ + + Integral(c, (t, t1, t))*N.z + answer_expected = (v2d, v2, v0 + integral_vector) + assert answer == answer_expected + + #Test acceleration parameter + assert get_motion_params(N, acceleration=v1) == \ + (v1, v1 * t, v1 * t**2/2) + assert get_motion_params(N, acceleration=v1, velocity=v0, + position=v2, timevalue1=t1, timevalue2=t2) == \ + (v1, (v0 + v1*t - v1*t2), + -v0*t1 + v1*t**2/2 + v1*t2*t1 - \ + v1*t1**2/2 + t*(v0 - v1*t2) + \ + v2.subs(t, t1)) + assert get_motion_params(N, acceleration=v1, velocity=v0, + position=v01, timevalue1=t1, timevalue2=t2) == \ + (v1, v0 + v1*t - v1*t2, + -v0*t1 + v01 + v1*t**2/2 + \ + v1*t2*t1 - v1*t1**2/2 + \ + t*(v0 - v1*t2)) + answer = get_motion_params(N, acceleration=a*N.x, velocity=S1*N.x, + position=S2*N.x, timevalue1=t1, timevalue2=t2) + i1 = Integral(a, (t, t2, t)) + answer_expected = (a*N.x, (S1 + i1)*N.x, \ + (S2 + Integral(S1 + i1, (t, t1, t)))*N.x) + assert answer == answer_expected + + +def test_kin_eqs(): + q0, q1, q2, q3 = dynamicsymbols('q0 q1 q2 q3') + q0d, q1d, q2d, q3d = dynamicsymbols('q0 q1 q2 q3', 1) + u1, u2, u3 = dynamicsymbols('u1 u2 u3') + ke = kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', 313) + assert ke == kinematic_equations([u1,u2,u3], [q1,q2,q3], 'body', '313') + kds = kinematic_equations([u1, u2, u3], [q0, q1, q2, q3], 'quaternion') + assert kds == [-0.5 * q0 * u1 - 0.5 * q2 * u3 + 0.5 * q3 * u2 + q1d, + -0.5 * q0 * u2 + 0.5 * q1 * u3 - 0.5 * q3 * u1 + q2d, + -0.5 * q0 * u3 - 0.5 * q1 * u2 + 0.5 * q2 * u1 + q3d, + 0.5 * q1 * u1 + 0.5 * q2 * u2 + 0.5 * q3 * u3 + q0d] + raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2], 'quaternion')) + raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2, q3], 'quaternion', '123')) + raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2, q3], 'foo')) + raises(TypeError, lambda: kinematic_equations(u1, [q0, q1, q2, q3], 'quaternion')) + raises(TypeError, lambda: kinematic_equations([u1], [q0, q1, q2, q3], 'quaternion')) + raises(TypeError, lambda: kinematic_equations([u1, u2, u3], q0, 'quaternion')) + raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2, q3], 'body')) + raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2, q3], 'space')) + raises(ValueError, lambda: kinematic_equations([u1, u2, u3], [q0, q1, q2], 'body', '222')) + assert kinematic_equations([0, 0, 0], [q0, q1, q2], 'space') == [S.Zero, S.Zero, S.Zero] + + +def test_partial_velocity(): + q1, q2, q3, u1, u2, u3 = dynamicsymbols('q1 q2 q3 u1 u2 u3') + u4, u5 = dynamicsymbols('u4, u5') + r = symbols('r') + + N = ReferenceFrame('N') + Y = N.orientnew('Y', 'Axis', [q1, N.z]) + L = Y.orientnew('L', 'Axis', [q2, Y.x]) + R = L.orientnew('R', 'Axis', [q3, L.y]) + R.set_ang_vel(N, u1 * L.x + u2 * L.y + u3 * L.z) + + C = Point('C') + C.set_vel(N, u4 * L.x + u5 * (Y.z ^ L.x)) + Dmc = C.locatenew('Dmc', r * L.z) + Dmc.v2pt_theory(C, N, R) + + vel_list = [Dmc.vel(N), C.vel(N), R.ang_vel_in(N)] + u_list = [u1, u2, u3, u4, u5] + assert (partial_velocity(vel_list, u_list, N) == + [[- r*L.y, r*L.x, 0, L.x, cos(q2)*L.y - sin(q2)*L.z], + [0, 0, 0, L.x, cos(q2)*L.y - sin(q2)*L.z], + [L.x, L.y, L.z, 0, 0]]) + + # Make sure that partial velocities can be computed regardless if the + # orientation between frames is defined or not. + A = ReferenceFrame('A') + B = ReferenceFrame('B') + v = u4 * A.x + u5 * B.y + assert partial_velocity((v, ), (u4, u5), A) == [[A.x, B.y]] + + raises(TypeError, lambda: partial_velocity(Dmc.vel(N), u_list, N)) + raises(TypeError, lambda: partial_velocity(vel_list, u1, N)) + +def test_dynamicsymbols(): + #Tests to check the assumptions applied to dynamicsymbols + f1 = dynamicsymbols('f1') + f2 = dynamicsymbols('f2', real=True) + f3 = dynamicsymbols('f3', positive=True) + f4, f5 = dynamicsymbols('f4,f5', commutative=False) + f6 = dynamicsymbols('f6', integer=True) + assert f1.is_real is None + assert f2.is_real + assert f3.is_positive + assert f4*f5 != f5*f4 + assert f6.is_integer diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_output.py b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_output.py new file mode 100644 index 0000000000000000000000000000000000000000..e02f3e5962bc23bbb62929e343a5afac574a2570 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_output.py @@ -0,0 +1,75 @@ +from sympy.core.singleton import S +from sympy.physics.vector import Vector, ReferenceFrame, Dyadic +from sympy.testing.pytest import raises + +A = ReferenceFrame('A') + + +def test_output_type(): + A = ReferenceFrame('A') + v = A.x + A.y + d = v | v + zerov = Vector(0) + zerod = Dyadic(0) + + # dot products + assert isinstance(d & d, Dyadic) + assert isinstance(d & zerod, Dyadic) + assert isinstance(zerod & d, Dyadic) + assert isinstance(d & v, Vector) + assert isinstance(v & d, Vector) + assert isinstance(d & zerov, Vector) + assert isinstance(zerov & d, Vector) + raises(TypeError, lambda: d & S.Zero) + raises(TypeError, lambda: S.Zero & d) + raises(TypeError, lambda: d & 0) + raises(TypeError, lambda: 0 & d) + assert not isinstance(v & v, (Vector, Dyadic)) + assert not isinstance(v & zerov, (Vector, Dyadic)) + assert not isinstance(zerov & v, (Vector, Dyadic)) + raises(TypeError, lambda: v & S.Zero) + raises(TypeError, lambda: S.Zero & v) + raises(TypeError, lambda: v & 0) + raises(TypeError, lambda: 0 & v) + + # cross products + raises(TypeError, lambda: d ^ d) + raises(TypeError, lambda: d ^ zerod) + raises(TypeError, lambda: zerod ^ d) + assert isinstance(d ^ v, Dyadic) + assert isinstance(v ^ d, Dyadic) + assert isinstance(d ^ zerov, Dyadic) + assert isinstance(zerov ^ d, Dyadic) + assert isinstance(zerov ^ d, Dyadic) + raises(TypeError, lambda: d ^ S.Zero) + raises(TypeError, lambda: S.Zero ^ d) + raises(TypeError, lambda: d ^ 0) + raises(TypeError, lambda: 0 ^ d) + assert isinstance(v ^ v, Vector) + assert isinstance(v ^ zerov, Vector) + assert isinstance(zerov ^ v, Vector) + raises(TypeError, lambda: v ^ S.Zero) + raises(TypeError, lambda: S.Zero ^ v) + raises(TypeError, lambda: v ^ 0) + raises(TypeError, lambda: 0 ^ v) + + # outer products + raises(TypeError, lambda: d | d) + raises(TypeError, lambda: d | zerod) + raises(TypeError, lambda: zerod | d) + raises(TypeError, lambda: d | v) + raises(TypeError, lambda: v | d) + raises(TypeError, lambda: d | zerov) + raises(TypeError, lambda: zerov | d) + raises(TypeError, lambda: zerov | d) + raises(TypeError, lambda: d | S.Zero) + raises(TypeError, lambda: S.Zero | d) + raises(TypeError, lambda: d | 0) + raises(TypeError, lambda: 0 | d) + assert isinstance(v | v, Dyadic) + assert isinstance(v | zerov, Dyadic) + assert isinstance(zerov | v, Dyadic) + raises(TypeError, lambda: v | S.Zero) + raises(TypeError, lambda: S.Zero | v) + raises(TypeError, lambda: v | 0) + raises(TypeError, lambda: 0 | v) diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_point.py b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_point.py new file mode 100644 index 0000000000000000000000000000000000000000..0e0c8b092ef61c590d3c713cef25feb3e64051c6 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_point.py @@ -0,0 +1,382 @@ +from sympy.physics.vector import dynamicsymbols, Point, ReferenceFrame +from sympy.testing.pytest import raises, ignore_warnings +import warnings + +def test_point_v1pt_theorys(): + q, q2 = dynamicsymbols('q q2') + qd, q2d = dynamicsymbols('q q2', 1) + qdd, q2dd = dynamicsymbols('q q2', 2) + N = ReferenceFrame('N') + B = ReferenceFrame('B') + B.set_ang_vel(N, qd * B.z) + O = Point('O') + P = O.locatenew('P', B.x) + P.set_vel(B, 0) + O.set_vel(N, 0) + assert P.v1pt_theory(O, N, B) == qd * B.y + O.set_vel(N, N.x) + assert P.v1pt_theory(O, N, B) == N.x + qd * B.y + P.set_vel(B, B.z) + assert P.v1pt_theory(O, N, B) == B.z + N.x + qd * B.y + + +def test_point_a1pt_theorys(): + q, q2 = dynamicsymbols('q q2') + qd, q2d = dynamicsymbols('q q2', 1) + qdd, q2dd = dynamicsymbols('q q2', 2) + N = ReferenceFrame('N') + B = ReferenceFrame('B') + B.set_ang_vel(N, qd * B.z) + O = Point('O') + P = O.locatenew('P', B.x) + P.set_vel(B, 0) + O.set_vel(N, 0) + assert P.a1pt_theory(O, N, B) == -(qd**2) * B.x + qdd * B.y + P.set_vel(B, q2d * B.z) + assert P.a1pt_theory(O, N, B) == -(qd**2) * B.x + qdd * B.y + q2dd * B.z + O.set_vel(N, q2d * B.x) + assert P.a1pt_theory(O, N, B) == ((q2dd - qd**2) * B.x + (q2d * qd + qdd) * B.y + + q2dd * B.z) + + +def test_point_v2pt_theorys(): + q = dynamicsymbols('q') + qd = dynamicsymbols('q', 1) + N = ReferenceFrame('N') + B = N.orientnew('B', 'Axis', [q, N.z]) + O = Point('O') + P = O.locatenew('P', 0) + O.set_vel(N, 0) + assert P.v2pt_theory(O, N, B) == 0 + P = O.locatenew('P', B.x) + assert P.v2pt_theory(O, N, B) == (qd * B.z ^ B.x) + O.set_vel(N, N.x) + assert P.v2pt_theory(O, N, B) == N.x + qd * B.y + + +def test_point_a2pt_theorys(): + q = dynamicsymbols('q') + qd = dynamicsymbols('q', 1) + qdd = dynamicsymbols('q', 2) + N = ReferenceFrame('N') + B = N.orientnew('B', 'Axis', [q, N.z]) + O = Point('O') + P = O.locatenew('P', 0) + O.set_vel(N, 0) + assert P.a2pt_theory(O, N, B) == 0 + P.set_pos(O, B.x) + assert P.a2pt_theory(O, N, B) == (-qd**2) * B.x + (qdd) * B.y + + +def test_point_funcs(): + q, q2 = dynamicsymbols('q q2') + qd, q2d = dynamicsymbols('q q2', 1) + qdd, q2dd = dynamicsymbols('q q2', 2) + N = ReferenceFrame('N') + B = ReferenceFrame('B') + B.set_ang_vel(N, 5 * B.y) + O = Point('O') + P = O.locatenew('P', q * B.x + q2 * B.y) + assert P.pos_from(O) == q * B.x + q2 * B.y + P.set_vel(B, qd * B.x + q2d * B.y) + assert P.vel(B) == qd * B.x + q2d * B.y + O.set_vel(N, 0) + assert O.vel(N) == 0 + assert P.a1pt_theory(O, N, B) == ((-25 * q + qdd) * B.x + (q2dd) * B.y + + (-10 * qd) * B.z) + + B = N.orientnew('B', 'Axis', [q, N.z]) + O = Point('O') + P = O.locatenew('P', 10 * B.x) + O.set_vel(N, 5 * N.x) + assert O.vel(N) == 5 * N.x + assert P.a2pt_theory(O, N, B) == (-10 * qd**2) * B.x + (10 * qdd) * B.y + + B.set_ang_vel(N, 5 * B.y) + O = Point('O') + P = O.locatenew('P', q * B.x + q2 * B.y) + P.set_vel(B, qd * B.x + q2d * B.y) + O.set_vel(N, 0) + assert P.v1pt_theory(O, N, B) == qd * B.x + q2d * B.y - 5 * q * B.z + + +def test_point_pos(): + q = dynamicsymbols('q') + N = ReferenceFrame('N') + B = N.orientnew('B', 'Axis', [q, N.z]) + O = Point('O') + P = O.locatenew('P', 10 * N.x + 5 * B.x) + assert P.pos_from(O) == 10 * N.x + 5 * B.x + Q = P.locatenew('Q', 10 * N.y + 5 * B.y) + assert Q.pos_from(P) == 10 * N.y + 5 * B.y + assert Q.pos_from(O) == 10 * N.x + 10 * N.y + 5 * B.x + 5 * B.y + assert O.pos_from(Q) == -10 * N.x - 10 * N.y - 5 * B.x - 5 * B.y + +def test_point_partial_velocity(): + + N = ReferenceFrame('N') + A = ReferenceFrame('A') + + p = Point('p') + + u1, u2 = dynamicsymbols('u1, u2') + + p.set_vel(N, u1 * A.x + u2 * N.y) + + assert p.partial_velocity(N, u1) == A.x + assert p.partial_velocity(N, u1, u2) == (A.x, N.y) + raises(ValueError, lambda: p.partial_velocity(A, u1)) + +def test_point_vel(): #Basic functionality + q1, q2 = dynamicsymbols('q1 q2') + N = ReferenceFrame('N') + B = ReferenceFrame('B') + Q = Point('Q') + O = Point('O') + Q.set_pos(O, q1 * N.x) + raises(ValueError , lambda: Q.vel(N)) # Velocity of O in N is not defined + O.set_vel(N, q2 * N.y) + assert O.vel(N) == q2 * N.y + raises(ValueError , lambda : O.vel(B)) #Velocity of O is not defined in B + +def test_auto_point_vel(): + t = dynamicsymbols._t + q1, q2 = dynamicsymbols('q1 q2') + N = ReferenceFrame('N') + B = ReferenceFrame('B') + O = Point('O') + Q = Point('Q') + Q.set_pos(O, q1 * N.x) + O.set_vel(N, q2 * N.y) + assert Q.vel(N) == q1.diff(t) * N.x + q2 * N.y # Velocity of Q using O + P1 = Point('P1') + P1.set_pos(O, q1 * B.x) + P2 = Point('P2') + P2.set_pos(P1, q2 * B.z) + raises(ValueError, lambda : P2.vel(B)) # O's velocity is defined in different frame, and no + #point in between has its velocity defined + raises(ValueError, lambda: P2.vel(N)) # Velocity of O not defined in N + +def test_auto_point_vel_multiple_point_path(): + t = dynamicsymbols._t + q1, q2 = dynamicsymbols('q1 q2') + B = ReferenceFrame('B') + P = Point('P') + P.set_vel(B, q1 * B.x) + P1 = Point('P1') + P1.set_pos(P, q2 * B.y) + P1.set_vel(B, q1 * B.z) + P2 = Point('P2') + P2.set_pos(P1, q1 * B.z) + P3 = Point('P3') + P3.set_pos(P2, 10 * q1 * B.y) + assert P3.vel(B) == 10 * q1.diff(t) * B.y + (q1 + q1.diff(t)) * B.z + +def test_auto_vel_dont_overwrite(): + t = dynamicsymbols._t + q1, q2, u1 = dynamicsymbols('q1, q2, u1') + N = ReferenceFrame('N') + P = Point('P1') + P.set_vel(N, u1 * N.x) + P1 = Point('P1') + P1.set_pos(P, q2 * N.y) + assert P1.vel(N) == q2.diff(t) * N.y + u1 * N.x + assert P.vel(N) == u1 * N.x + P1.set_vel(N, u1 * N.z) + assert P1.vel(N) == u1 * N.z + +def test_auto_point_vel_if_tree_has_vel_but_inappropriate_pos_vector(): + q1, q2 = dynamicsymbols('q1 q2') + B = ReferenceFrame('B') + S = ReferenceFrame('S') + P = Point('P') + P.set_vel(B, q1 * B.x) + P1 = Point('P1') + P1.set_pos(P, S.y) + raises(ValueError, lambda : P1.vel(B)) # P1.pos_from(P) can't be expressed in B + raises(ValueError, lambda : P1.vel(S)) # P.vel(S) not defined + +def test_auto_point_vel_shortest_path(): + t = dynamicsymbols._t + q1, q2, u1, u2 = dynamicsymbols('q1 q2 u1 u2') + B = ReferenceFrame('B') + P = Point('P') + P.set_vel(B, u1 * B.x) + P1 = Point('P1') + P1.set_pos(P, q2 * B.y) + P1.set_vel(B, q1 * B.z) + P2 = Point('P2') + P2.set_pos(P1, q1 * B.z) + P3 = Point('P3') + P3.set_pos(P2, 10 * q1 * B.y) + P4 = Point('P4') + P4.set_pos(P3, q1 * B.x) + O = Point('O') + O.set_vel(B, u2 * B.y) + O1 = Point('O1') + O1.set_pos(O, q2 * B.z) + P4.set_pos(O1, q1 * B.x + q2 * B.z) + with warnings.catch_warnings(): #There are two possible paths in this point tree, thus a warning is raised + warnings.simplefilter('error') + with ignore_warnings(UserWarning): + assert P4.vel(B) == q1.diff(t) * B.x + u2 * B.y + 2 * q2.diff(t) * B.z + +def test_auto_point_vel_connected_frames(): + t = dynamicsymbols._t + q, q1, q2, u = dynamicsymbols('q q1 q2 u') + N = ReferenceFrame('N') + B = ReferenceFrame('B') + O = Point('O') + O.set_vel(N, u * N.x) + P = Point('P') + P.set_pos(O, q1 * N.x + q2 * B.y) + raises(ValueError, lambda: P.vel(N)) + N.orient(B, 'Axis', (q, B.x)) + assert P.vel(N) == (u + q1.diff(t)) * N.x + q2.diff(t) * B.y - q2 * q.diff(t) * B.z + +def test_auto_point_vel_multiple_paths_warning_arises(): + q, u = dynamicsymbols('q u') + N = ReferenceFrame('N') + O = Point('O') + P = Point('P') + Q = Point('Q') + R = Point('R') + P.set_vel(N, u * N.x) + Q.set_vel(N, u *N.y) + R.set_vel(N, u * N.z) + O.set_pos(P, q * N.z) + O.set_pos(Q, q * N.y) + O.set_pos(R, q * N.x) + with warnings.catch_warnings(): #There are two possible paths in this point tree, thus a warning is raised + warnings.simplefilter("error") + raises(UserWarning ,lambda: O.vel(N)) + +def test_auto_vel_cyclic_warning_arises(): + P = Point('P') + P1 = Point('P1') + P2 = Point('P2') + P3 = Point('P3') + N = ReferenceFrame('N') + P.set_vel(N, N.x) + P1.set_pos(P, N.x) + P2.set_pos(P1, N.y) + P3.set_pos(P2, N.z) + P1.set_pos(P3, N.x + N.y) + with warnings.catch_warnings(): #The path is cyclic at P1, thus a warning is raised + warnings.simplefilter("error") + raises(UserWarning ,lambda: P2.vel(N)) + +def test_auto_vel_cyclic_warning_msg(): + P = Point('P') + P1 = Point('P1') + P2 = Point('P2') + P3 = Point('P3') + N = ReferenceFrame('N') + P.set_vel(N, N.x) + P1.set_pos(P, N.x) + P2.set_pos(P1, N.y) + P3.set_pos(P2, N.z) + P1.set_pos(P3, N.x + N.y) + with warnings.catch_warnings(record = True) as w: #The path is cyclic at P1, thus a warning is raised + warnings.simplefilter("always") + P2.vel(N) + msg = str(w[-1].message).replace("\n", " ") + assert issubclass(w[-1].category, UserWarning) + assert 'Kinematic loops are defined among the positions of points. This is likely not desired and may cause errors in your calculations.' in msg + +def test_auto_vel_multiple_path_warning_msg(): + N = ReferenceFrame('N') + O = Point('O') + P = Point('P') + Q = Point('Q') + P.set_vel(N, N.x) + Q.set_vel(N, N.y) + O.set_pos(P, N.z) + O.set_pos(Q, N.y) + with warnings.catch_warnings(record = True) as w: #There are two possible paths in this point tree, thus a warning is raised + warnings.simplefilter("always") + O.vel(N) + msg = str(w[-1].message).replace("\n", " ") + assert issubclass(w[-1].category, UserWarning) + assert 'Velocity' in msg + assert 'automatically calculated based on point' in msg + assert 'Velocities from these points are not necessarily the same. This may cause errors in your calculations.' in msg + +def test_auto_vel_derivative(): + q1, q2 = dynamicsymbols('q1:3') + u1, u2 = dynamicsymbols('u1:3', 1) + A = ReferenceFrame('A') + B = ReferenceFrame('B') + C = ReferenceFrame('C') + B.orient_axis(A, A.z, q1) + B.set_ang_vel(A, u1 * A.z) + C.orient_axis(B, B.z, q2) + C.set_ang_vel(B, u2 * B.z) + + Am = Point('Am') + Am.set_vel(A, 0) + Bm = Point('Bm') + Bm.set_pos(Am, B.x) + Bm.set_vel(B, 0) + Bm.set_vel(C, 0) + Cm = Point('Cm') + Cm.set_pos(Bm, C.x) + Cm.set_vel(C, 0) + temp = Cm._vel_dict.copy() + assert Cm.vel(A) == (u1 * B.y + (u1 + u2) * C.y) + Cm._vel_dict = temp + Cm.v2pt_theory(Bm, B, C) + assert Cm.vel(A) == (u1 * B.y + (u1 + u2) * C.y) + +def test_auto_point_acc_zero_vel(): + N = ReferenceFrame('N') + O = Point('O') + O.set_vel(N, 0) + assert O.acc(N) == 0 * N.x + +def test_auto_point_acc_compute_vel(): + t = dynamicsymbols._t + q1 = dynamicsymbols('q1') + N = ReferenceFrame('N') + A = ReferenceFrame('A') + A.orient_axis(N, N.z, q1) + + O = Point('O') + O.set_vel(N, 0) + P = Point('P') + P.set_pos(O, A.x) + assert P.acc(N) == -q1.diff(t) ** 2 * A.x + q1.diff(t, 2) * A.y + +def test_auto_acc_derivative(): + # Tests whether the Point.acc method gives the correct acceleration of the + # end point of two linkages in series, while getting minimal information. + q1, q2 = dynamicsymbols('q1:3') + u1, u2 = dynamicsymbols('q1:3', 1) + v1, v2 = dynamicsymbols('q1:3', 2) + A = ReferenceFrame('A') + B = ReferenceFrame('B') + C = ReferenceFrame('C') + B.orient_axis(A, A.z, q1) + C.orient_axis(B, B.z, q2) + + Am = Point('Am') + Am.set_vel(A, 0) + Bm = Point('Bm') + Bm.set_pos(Am, B.x) + Bm.set_vel(B, 0) + Bm.set_vel(C, 0) + Cm = Point('Cm') + Cm.set_pos(Bm, C.x) + Cm.set_vel(C, 0) + + # Copy dictionaries to later check the calculation using the 2pt_theories + Bm_vel_dict, Cm_vel_dict = Bm._vel_dict.copy(), Cm._vel_dict.copy() + Bm_acc_dict, Cm_acc_dict = Bm._acc_dict.copy(), Cm._acc_dict.copy() + check = -u1 ** 2 * B.x + v1 * B.y - (u1 + u2) ** 2 * C.x + (v1 + v2) * C.y + assert Cm.acc(A) == check + Bm._vel_dict, Cm._vel_dict = Bm_vel_dict, Cm_vel_dict + Bm._acc_dict, Cm._acc_dict = Bm_acc_dict, Cm_acc_dict + Bm.v2pt_theory(Am, A, B) + Cm.v2pt_theory(Bm, A, C) + Bm.a2pt_theory(Am, A, B) + assert Cm.a2pt_theory(Bm, A, C) == check diff --git a/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_printing.py b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_printing.py new file mode 100644 index 0000000000000000000000000000000000000000..0930fe9d0bc6e2fcc60b34f37215fdb19e32fdc4 --- /dev/null +++ b/wemm/lib/python3.10/site-packages/sympy/physics/vector/tests/test_printing.py @@ -0,0 +1,353 @@ +# -*- coding: utf-8 -*- + +from sympy.core.function import Function +from sympy.core.symbol import symbols +from sympy.functions.elementary.miscellaneous import sqrt +from sympy.functions.elementary.trigonometric import (asin, cos, sin) +from sympy.physics.vector import ReferenceFrame, dynamicsymbols, Dyadic +from sympy.physics.vector.printing import (VectorLatexPrinter, vpprint, + vsprint, vsstrrepr, vlatex) + + +a, b, c = symbols('a, b, c') +alpha, omega, beta = dynamicsymbols('alpha, omega, beta') + +A = ReferenceFrame('A') +N = ReferenceFrame('N') + +v = a ** 2 * N.x + b * N.y + c * sin(alpha) * N.z +w = alpha * N.x + sin(omega) * N.y + alpha * beta * N.z +ww = alpha * N.x + asin(omega) * N.y - alpha.diff() * beta * N.z +o = a/b * N.x + (c+b)/a * N.y + c**2/b * N.z + +y = a ** 2 * (N.x | N.y) + b * (N.y | N.y) + c * sin(alpha) * (N.z | N.y) +x = alpha * (N.x | N.x) + sin(omega) * (N.y | N.z) + alpha * beta * (N.z | N.x) +xx = N.x | (-N.y - N.z) +xx2 = N.x | (N.y + N.z) + +def ascii_vpretty(expr): + return vpprint(expr, use_unicode=False, wrap_line=False) + + +def unicode_vpretty(expr): + return vpprint(expr, use_unicode=True, wrap_line=False) + + +def test_latex_printer(): + r = Function('r')('t') + assert VectorLatexPrinter().doprint(r ** 2) == "r^{2}" + r2 = Function('r^2')('t') + assert VectorLatexPrinter().doprint(r2.diff()) == r'\dot{r^{2}}' + ra = Function('r__a')('t') + assert VectorLatexPrinter().doprint(ra.diff().diff()) == r'\ddot{r^{a}}' + + +def test_vector_pretty_print(): + + # TODO : The unit vectors should print with subscripts but they just + # print as `n_x` instead of making `x` a subscript with unicode. + + # TODO : The pretty print division does not print correctly here: + # w = alpha * N.x + sin(omega) * N.y + alpha / beta * N.z + + expected = """\ + 2 \n\ +a n_x + b n_y + c*sin(alpha) n_z\ +""" + uexpected = """\ + 2 \n\ +a n_x + b n_y + c⋅sin(α) n_z\ +""" + + assert ascii_vpretty(v) == expected + assert unicode_vpretty(v) == uexpected + + expected = 'alpha n_x + sin(omega) n_y + alpha*beta n_z' + uexpected = 'α n_x + sin(ω) n_y + α⋅β n_z' + + assert ascii_vpretty(w) == expected + assert unicode_vpretty(w) == uexpected + + expected = """\ + 2 \n\ +a b + c c \n\ +- n_x + ----- n_y + -- n_z\n\ +b a b \ +""" + uexpected = """\ + 2 \n\ +a b + c c \n\ +─ n_x + ───── n_y + ── n_z\n\ +b a b \ +""" + + assert ascii_vpretty(o) == expected + assert unicode_vpretty(o) == uexpected + + # https://github.com/sympy/sympy/issues/26731 + assert ascii_vpretty(-A.x) == '-a_x' + assert unicode_vpretty(-A.x) == '-a_x' + + # https://github.com/sympy/sympy/issues/26799 + assert ascii_vpretty(0*A.x) == '0' + assert unicode_vpretty(0*A.x) == '0' + + +def test_vector_latex(): + + a, b, c, d, omega = symbols('a, b, c, d, omega') + + v = (a ** 2 + b / c) * A.x + sqrt(d) * A.y + cos(omega) * A.z + + assert vlatex(v) == (r'(a^{2} + \frac{b}{c})\mathbf{\hat{a}_x} + ' + r'\sqrt{d}\mathbf{\hat{a}_y} + ' + r'\cos{\left(\omega \right)}' + r'\mathbf{\hat{a}_z}') + + theta, omega, alpha, q = dynamicsymbols('theta, omega, alpha, q') + + v = theta * A.x + omega * omega * A.y + (q * alpha) * A.z + + assert vlatex(v) == (r'\theta\mathbf{\hat{a}_x} + ' + r'\omega^{2}\mathbf{\hat{a}_y} + ' + r'\alpha q\mathbf{\hat{a}_z}') + + phi1, phi2, phi3 = dynamicsymbols('phi1, phi2, phi3') + theta1, theta2, theta3 = symbols('theta1, theta2, theta3') + + v = (sin(theta1) * A.x + + cos(phi1) * cos(phi2) * A.y + + cos(theta1 + phi3) * A.z) + + assert vlatex(v) == (r'\sin{\left(\theta_{1} \right)}' + r'\mathbf{\hat{a}_x} + \cos{' + r'\left(\phi_{1} \right)} \cos{' + r'\left(\phi_{2} \right)}\mathbf{\hat{a}_y} + ' + r'\cos{\left(\theta_{1} + ' + r'\phi_{3} \right)}\mathbf{\hat{a}_z}') + + N = ReferenceFrame('N') + + a, b, c, d, omega = symbols('a, b, c, d, omega') + + v = (a ** 2 + b / c) * N.x + sqrt(d) * N.y + cos(omega) * N.z + + expected = (r'(a^{2} + \frac{b}{c})\mathbf{\hat{n}_x} + ' + r'\sqrt{d}\mathbf{\hat{n}_y} + ' + r'\cos{\left(\omega \right)}' + r'\mathbf{\hat{n}_z}') + + assert vlatex(v) == expected + + # Try custom unit vectors. + + N = ReferenceFrame('N', latexs=(r'\hat{i}', r'\hat{j}', r'\hat{k}')) + + v = (a ** 2 + b / c) * N.x + sqrt(d) * N.y + cos(omega) * N.z + + expected = (r'(a^{2} + \frac{b}{c})\hat{i} + ' + r'\sqrt{d}\hat{j} + ' + r'\cos{\left(\omega \right)}\hat{k}') + assert vlatex(v) == expected + + expected = r'\alpha\mathbf{\hat{n}_x} + \operatorname{asin}{\left(\omega ' \ + r'\right)}\mathbf{\hat{n}_y} - \beta \dot{\alpha}\mathbf{\hat{n}_z}' + assert vlatex(ww) == expected + + expected = r'- \mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_y} - ' \ + r'\mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_z}' + assert vlatex(xx) == expected + + expected = r'\mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_y} + ' \ + r'\mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_z}' + assert vlatex(xx2) == expected + + +def test_vector_latex_arguments(): + assert vlatex(N.x * 3.0, full_prec=False) == r'3.0\mathbf{\hat{n}_x}' + assert vlatex(N.x * 3.0, full_prec=True) == r'3.00000000000000\mathbf{\hat{n}_x}' + + +def test_vector_latex_with_functions(): + + N = ReferenceFrame('N') + + omega, alpha = dynamicsymbols('omega, alpha') + + v = omega.diff() * N.x + + assert vlatex(v) == r'\dot{\omega}\mathbf{\hat{n}_x}' + + v = omega.diff() ** alpha * N.x + + assert vlatex(v) == (r'\dot{\omega}^{\alpha}' + r'\mathbf{\hat{n}_x}') + + +def test_dyadic_pretty_print(): + + expected = """\ + 2 +a n_x|n_y + b n_y|n_y + c*sin(alpha) n_z|n_y\ +""" + + uexpected = """\ + 2 +a n_x⊗n_y + b n_y⊗n_y + c⋅sin(α) n_z⊗n_y\ +""" + assert ascii_vpretty(y) == expected + assert unicode_vpretty(y) == uexpected + + expected = 'alpha n_x|n_x + sin(omega) n_y|n_z + alpha*beta n_z|n_x' + uexpected = 'α n_x⊗n_x + sin(ω) n_y⊗n_z + α⋅β n_z⊗n_x' + assert ascii_vpretty(x) == expected + assert unicode_vpretty(x) == uexpected + + assert ascii_vpretty(Dyadic([])) == '0' + assert unicode_vpretty(Dyadic([])) == '0' + + assert ascii_vpretty(xx) == '- n_x|n_y - n_x|n_z' + assert unicode_vpretty(xx) == '- n_x⊗n_y - n_x⊗n_z' + + assert ascii_vpretty(xx2) == 'n_x|n_y + n_x|n_z' + assert unicode_vpretty(xx2) == 'n_x⊗n_y + n_x⊗n_z' + + +def test_dyadic_latex(): + + expected = (r'a^{2}\mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_y} + ' + r'b\mathbf{\hat{n}_y}\otimes \mathbf{\hat{n}_y} + ' + r'c \sin{\left(\alpha \right)}' + r'\mathbf{\hat{n}_z}\otimes \mathbf{\hat{n}_y}') + + assert vlatex(y) == expected + + expected = (r'\alpha\mathbf{\hat{n}_x}\otimes \mathbf{\hat{n}_x} + ' + r'\sin{\left(\omega \right)}\mathbf{\hat{n}_y}' + r'\otimes \mathbf{\hat{n}_z} + ' + r'\alpha \beta\mathbf{\hat{n}_z}\otimes \mathbf{\hat{n}_x}') + + assert vlatex(x) == expected + + assert vlatex(Dyadic([])) == '0' + + +def test_dyadic_str(): + assert vsprint(Dyadic([])) == '0' + assert vsprint(y) == 'a**2*(N.x|N.y) + b*(N.y|N.y) + c*sin(alpha)*(N.z|N.y)' + assert vsprint(x) == 'alpha*(N.x|N.x) + sin(omega)*(N.y|N.z) + alpha*beta*(N.z|N.x)' + assert vsprint(ww) == "alpha*N.x + asin(omega)*N.y - beta*alpha'*N.z" + assert vsprint(xx) == '- (N.x|N.y) - (N.x|N.z)' + assert vsprint(xx2) == '(N.x|N.y) + (N.x|N.z)' + + +def test_vlatex(): # vlatex is broken #12078 + from sympy.physics.vector import vlatex + + x = symbols('x') + J = symbols('J') + + f = Function('f') + g = Function('g') + h = Function('h') + + expected = r'J \left(\frac{d}{d x} g{\left(x \right)} - \frac{d}{d x} h{\left(x \right)}\right)' + + expr = J*f(x).diff(x).subs(f(x), g(x)-h(x)) + + assert vlatex(expr) == expected + + +def test_issue_13354(): + """ + Test for proper pretty printing of physics vectors with ADD + instances in arguments. + + Test is exactly the one suggested in the original bug report by + @moorepants. + """ + + a, b, c = symbols('a, b, c') + A = ReferenceFrame('A') + v = a * A.x + b * A.y + c * A.z + w = b * A.x + c * A.y + a * A.z + z = w + v + + expected = """(a + b) a_x + (b + c) a_y + (a + c) a_z""" + + assert ascii_vpretty(z) == expected + + +def test_vector_derivative_printing(): + # First order + v = omega.diff() * N.x + assert unicode_vpretty(v) == 'ω̇ n_x' + assert ascii_vpretty(v) == "omega'(t) n_x" + + # Second order + v = omega.diff().diff() * N.x + + assert vlatex(v) == r'\ddot{\omega}\mathbf{\hat{n}_x}' + assert unicode_vpretty(v) == 'ω̈ n_x' + assert ascii_vpretty(v) == "omega''(t) n_x" + + # Third order + v = omega.diff().diff().diff() * N.x + + assert vlatex(v) == r'\dddot{\omega}\mathbf{\hat{n}_x}' + assert unicode_vpretty(v) == 'ω⃛ n_x' + assert ascii_vpretty(v) == "omega'''(t) n_x" + + # Fourth order + v = omega.diff().diff().diff().diff() * N.x + + assert vlatex(v) == r'\ddddot{\omega}\mathbf{\hat{n}_x}' + assert unicode_vpretty(v) == 'ω⃜ n_x' + assert ascii_vpretty(v) == "omega''''(t) n_x" + + # Fifth order + v = omega.diff().diff().diff().diff().diff() * N.x + + assert vlatex(v) == r'\frac{d^{5}}{d t^{5}} \omega\mathbf{\hat{n}_x}' + expected = '''\ + 5 \n\ +d \n\ +---(omega) n_x\n\ + 5 \n\ +dt \ +''' + uexpected = '''\ + 5 \n\ +d \n\ +───(ω) n_x\n\ + 5 \n\ +dt \ +''' + assert unicode_vpretty(v) == uexpected + assert ascii_vpretty(v) == expected + + +def test_vector_str_printing(): + assert vsprint(w) == 'alpha*N.x + sin(omega)*N.y + alpha*beta*N.z' + assert vsprint(omega.diff() * N.x) == "omega'*N.x" + assert vsstrrepr(w) == 'alpha*N.x + sin(omega)*N.y + alpha*beta*N.z' + + +def test_vector_str_arguments(): + assert vsprint(N.x * 3.0, full_prec=False) == '3.0*N.x' + assert vsprint(N.x * 3.0, full_prec=True) == '3.00000000000000*N.x' + + +def test_issue_14041(): + import sympy.physics.mechanics as me + + A_frame = me.ReferenceFrame('A') + thetad, phid = me.dynamicsymbols('theta, phi', 1) + L = symbols('L') + + assert vlatex(L*(phid + thetad)**2*A_frame.x) == \ + r"L \left(\dot{\phi} + \dot{\theta}\right)^{2}\mathbf{\hat{a}_x}" + assert vlatex((phid + thetad)**2*A_frame.x) == \ + r"\left(\dot{\phi} + \dot{\theta}\right)^{2}\mathbf{\hat{a}_x}" + assert vlatex((phid*thetad)**a*A_frame.x) == \ + r"\left(\dot{\phi} \dot{\theta}\right)^{a}\mathbf{\hat{a}_x}"