| """ |
| Singularities |
| ============= |
| |
| This module implements algorithms for finding singularities for a function |
| and identifying types of functions. |
| |
| The differential calculus methods in this module include methods to identify |
| the following function types in the given ``Interval``: |
| - Increasing |
| - Strictly Increasing |
| - Decreasing |
| - Strictly Decreasing |
| - Monotonic |
| |
| """ |
|
|
| from sympy.core.power import Pow |
| from sympy.core.singleton import S |
| from sympy.core.symbol import Symbol |
| from sympy.core.sympify import sympify |
| from sympy.functions.elementary.exponential import log |
| from sympy.functions.elementary.trigonometric import sec, csc, cot, tan, cos |
| from sympy.functions.elementary.hyperbolic import ( |
| sech, csch, coth, tanh, cosh, asech, acsch, atanh, acoth) |
| from sympy.utilities.misc import filldedent |
|
|
|
|
| def singularities(expression, symbol, domain=None): |
| """ |
| Find singularities of a given function. |
| |
| Parameters |
| ========== |
| |
| expression : Expr |
| The target function in which singularities need to be found. |
| symbol : Symbol |
| The symbol over the values of which the singularity in |
| expression in being searched for. |
| |
| Returns |
| ======= |
| |
| Set |
| A set of values for ``symbol`` for which ``expression`` has a |
| singularity. An ``EmptySet`` is returned if ``expression`` has no |
| singularities for any given value of ``Symbol``. |
| |
| Raises |
| ====== |
| |
| NotImplementedError |
| Methods for determining the singularities of this function have |
| not been developed. |
| |
| Notes |
| ===== |
| |
| This function does not find non-isolated singularities |
| nor does it find branch points of the expression. |
| |
| Currently supported functions are: |
| - univariate continuous (real or complex) functions |
| |
| References |
| ========== |
| |
| .. [1] https://en.wikipedia.org/wiki/Mathematical_singularity |
| |
| Examples |
| ======== |
| |
| >>> from sympy import singularities, Symbol, log |
| >>> x = Symbol('x', real=True) |
| >>> y = Symbol('y', real=False) |
| >>> singularities(x**2 + x + 1, x) |
| EmptySet |
| >>> singularities(1/(x + 1), x) |
| {-1} |
| >>> singularities(1/(y**2 + 1), y) |
| {-I, I} |
| >>> singularities(1/(y**3 + 1), y) |
| {-1, 1/2 - sqrt(3)*I/2, 1/2 + sqrt(3)*I/2} |
| >>> singularities(log(x), x) |
| {0} |
| |
| """ |
| from sympy.solvers.solveset import solveset |
|
|
| if domain is None: |
| domain = S.Reals if symbol.is_real else S.Complexes |
| try: |
| sings = S.EmptySet |
| e = expression.rewrite([sec, csc, cot, tan], cos) |
| e = e.rewrite([sech, csch, coth, tanh], cosh) |
| for i in e.atoms(Pow): |
| if i.exp.is_infinite: |
| raise NotImplementedError |
| if i.exp.is_negative: |
| |
| sings += solveset(i.base, symbol, domain) |
| for i in expression.atoms(log, asech, acsch): |
| sings += solveset(i.args[0], symbol, domain) |
| for i in expression.atoms(atanh, acoth): |
| sings += solveset(i.args[0] - 1, symbol, domain) |
| sings += solveset(i.args[0] + 1, symbol, domain) |
| return sings |
| except NotImplementedError: |
| raise NotImplementedError(filldedent(''' |
| Methods for determining the singularities |
| of this function have not been developed.''')) |
|
|
|
|
| |
| |
| |
|
|
|
|
| def monotonicity_helper(expression, predicate, interval=S.Reals, symbol=None): |
| """ |
| Helper function for functions checking function monotonicity. |
| |
| Parameters |
| ========== |
| |
| expression : Expr |
| The target function which is being checked |
| predicate : function |
| The property being tested for. The function takes in an integer |
| and returns a boolean. The integer input is the derivative and |
| the boolean result should be true if the property is being held, |
| and false otherwise. |
| interval : Set, optional |
| The range of values in which we are testing, defaults to all reals. |
| symbol : Symbol, optional |
| The symbol present in expression which gets varied over the given range. |
| |
| It returns a boolean indicating whether the interval in which |
| the function's derivative satisfies given predicate is a superset |
| of the given interval. |
| |
| Returns |
| ======= |
| |
| Boolean |
| True if ``predicate`` is true for all the derivatives when ``symbol`` |
| is varied in ``range``, False otherwise. |
| |
| """ |
| from sympy.solvers.solveset import solveset |
|
|
| expression = sympify(expression) |
| free = expression.free_symbols |
|
|
| if symbol is None: |
| if len(free) > 1: |
| raise NotImplementedError( |
| 'The function has not yet been implemented' |
| ' for all multivariate expressions.' |
| ) |
|
|
| variable = symbol or (free.pop() if free else Symbol('x')) |
| derivative = expression.diff(variable) |
| predicate_interval = solveset(predicate(derivative), variable, S.Reals) |
| return interval.is_subset(predicate_interval) |
|
|
|
|
| def is_increasing(expression, interval=S.Reals, symbol=None): |
| """ |
| Return whether the function is increasing in the given interval. |
| |
| Parameters |
| ========== |
| |
| expression : Expr |
| The target function which is being checked. |
| interval : Set, optional |
| The range of values in which we are testing (defaults to set of |
| all real numbers). |
| symbol : Symbol, optional |
| The symbol present in expression which gets varied over the given range. |
| |
| Returns |
| ======= |
| |
| Boolean |
| True if ``expression`` is increasing (either strictly increasing or |
| constant) in the given ``interval``, False otherwise. |
| |
| Examples |
| ======== |
| |
| >>> from sympy import is_increasing |
| >>> from sympy.abc import x, y |
| >>> from sympy import S, Interval, oo |
| >>> is_increasing(x**3 - 3*x**2 + 4*x, S.Reals) |
| True |
| >>> is_increasing(-x**2, Interval(-oo, 0)) |
| True |
| >>> is_increasing(-x**2, Interval(0, oo)) |
| False |
| >>> is_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval(-2, 3)) |
| False |
| >>> is_increasing(x**2 + y, Interval(1, 2), x) |
| True |
| |
| """ |
| return monotonicity_helper(expression, lambda x: x >= 0, interval, symbol) |
|
|
|
|
| def is_strictly_increasing(expression, interval=S.Reals, symbol=None): |
| """ |
| Return whether the function is strictly increasing in the given interval. |
| |
| Parameters |
| ========== |
| |
| expression : Expr |
| The target function which is being checked. |
| interval : Set, optional |
| The range of values in which we are testing (defaults to set of |
| all real numbers). |
| symbol : Symbol, optional |
| The symbol present in expression which gets varied over the given range. |
| |
| Returns |
| ======= |
| |
| Boolean |
| True if ``expression`` is strictly increasing in the given ``interval``, |
| False otherwise. |
| |
| Examples |
| ======== |
| |
| >>> from sympy import is_strictly_increasing |
| >>> from sympy.abc import x, y |
| >>> from sympy import Interval, oo |
| >>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.Ropen(-oo, -2)) |
| True |
| >>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.Lopen(3, oo)) |
| True |
| >>> is_strictly_increasing(4*x**3 - 6*x**2 - 72*x + 30, Interval.open(-2, 3)) |
| False |
| >>> is_strictly_increasing(-x**2, Interval(0, oo)) |
| False |
| >>> is_strictly_increasing(-x**2 + y, Interval(-oo, 0), x) |
| False |
| |
| """ |
| return monotonicity_helper(expression, lambda x: x > 0, interval, symbol) |
|
|
|
|
| def is_decreasing(expression, interval=S.Reals, symbol=None): |
| """ |
| Return whether the function is decreasing in the given interval. |
| |
| Parameters |
| ========== |
| |
| expression : Expr |
| The target function which is being checked. |
| interval : Set, optional |
| The range of values in which we are testing (defaults to set of |
| all real numbers). |
| symbol : Symbol, optional |
| The symbol present in expression which gets varied over the given range. |
| |
| Returns |
| ======= |
| |
| Boolean |
| True if ``expression`` is decreasing (either strictly decreasing or |
| constant) in the given ``interval``, False otherwise. |
| |
| Examples |
| ======== |
| |
| >>> from sympy import is_decreasing |
| >>> from sympy.abc import x, y |
| >>> from sympy import S, Interval, oo |
| >>> is_decreasing(1/(x**2 - 3*x), Interval.open(S(3)/2, 3)) |
| True |
| >>> is_decreasing(1/(x**2 - 3*x), Interval.open(1.5, 3)) |
| True |
| >>> is_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo)) |
| True |
| >>> is_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, S(3)/2)) |
| False |
| >>> is_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, 1.5)) |
| False |
| >>> is_decreasing(-x**2, Interval(-oo, 0)) |
| False |
| >>> is_decreasing(-x**2 + y, Interval(-oo, 0), x) |
| False |
| |
| """ |
| return monotonicity_helper(expression, lambda x: x <= 0, interval, symbol) |
|
|
|
|
| def is_strictly_decreasing(expression, interval=S.Reals, symbol=None): |
| """ |
| Return whether the function is strictly decreasing in the given interval. |
| |
| Parameters |
| ========== |
| |
| expression : Expr |
| The target function which is being checked. |
| interval : Set, optional |
| The range of values in which we are testing (defaults to set of |
| all real numbers). |
| symbol : Symbol, optional |
| The symbol present in expression which gets varied over the given range. |
| |
| Returns |
| ======= |
| |
| Boolean |
| True if ``expression`` is strictly decreasing in the given ``interval``, |
| False otherwise. |
| |
| Examples |
| ======== |
| |
| >>> from sympy import is_strictly_decreasing |
| >>> from sympy.abc import x, y |
| >>> from sympy import S, Interval, oo |
| >>> is_strictly_decreasing(1/(x**2 - 3*x), Interval.Lopen(3, oo)) |
| True |
| >>> is_strictly_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, S(3)/2)) |
| False |
| >>> is_strictly_decreasing(1/(x**2 - 3*x), Interval.Ropen(-oo, 1.5)) |
| False |
| >>> is_strictly_decreasing(-x**2, Interval(-oo, 0)) |
| False |
| >>> is_strictly_decreasing(-x**2 + y, Interval(-oo, 0), x) |
| False |
| |
| """ |
| return monotonicity_helper(expression, lambda x: x < 0, interval, symbol) |
|
|
|
|
| def is_monotonic(expression, interval=S.Reals, symbol=None): |
| """ |
| Return whether the function is monotonic in the given interval. |
| |
| Parameters |
| ========== |
| |
| expression : Expr |
| The target function which is being checked. |
| interval : Set, optional |
| The range of values in which we are testing (defaults to set of |
| all real numbers). |
| symbol : Symbol, optional |
| The symbol present in expression which gets varied over the given range. |
| |
| Returns |
| ======= |
| |
| Boolean |
| True if ``expression`` is monotonic in the given ``interval``, |
| False otherwise. |
| |
| Raises |
| ====== |
| |
| NotImplementedError |
| Monotonicity check has not been implemented for the queried function. |
| |
| Examples |
| ======== |
| |
| >>> from sympy import is_monotonic |
| >>> from sympy.abc import x, y |
| >>> from sympy import S, Interval, oo |
| >>> is_monotonic(1/(x**2 - 3*x), Interval.open(S(3)/2, 3)) |
| True |
| >>> is_monotonic(1/(x**2 - 3*x), Interval.open(1.5, 3)) |
| True |
| >>> is_monotonic(1/(x**2 - 3*x), Interval.Lopen(3, oo)) |
| True |
| >>> is_monotonic(x**3 - 3*x**2 + 4*x, S.Reals) |
| True |
| >>> is_monotonic(-x**2, S.Reals) |
| False |
| >>> is_monotonic(x**2 + y + 1, Interval(1, 2), x) |
| True |
| |
| """ |
| from sympy.solvers.solveset import solveset |
|
|
| expression = sympify(expression) |
|
|
| free = expression.free_symbols |
| if symbol is None and len(free) > 1: |
| raise NotImplementedError( |
| 'is_monotonic has not yet been implemented' |
| ' for all multivariate expressions.' |
| ) |
|
|
| variable = symbol or (free.pop() if free else Symbol('x')) |
| turning_points = solveset(expression.diff(variable), variable, interval) |
| return interval.intersection(turning_points) is S.EmptySet |
|
|
