| | """ |
| | ================================================== |
| | Legendre Series (:mod:`numpy.polynomial.legendre`) |
| | ================================================== |
| | |
| | This module provides a number of objects (mostly functions) useful for |
| | dealing with Legendre series, including a `Legendre` class that |
| | encapsulates the usual arithmetic operations. (General information |
| | on how this module represents and works with such polynomials is in the |
| | docstring for its "parent" sub-package, `numpy.polynomial`). |
| | |
| | Classes |
| | ------- |
| | .. autosummary:: |
| | :toctree: generated/ |
| | |
| | Legendre |
| | |
| | Constants |
| | --------- |
| | |
| | .. autosummary:: |
| | :toctree: generated/ |
| | |
| | legdomain |
| | legzero |
| | legone |
| | legx |
| | |
| | Arithmetic |
| | ---------- |
| | |
| | .. autosummary:: |
| | :toctree: generated/ |
| | |
| | legadd |
| | legsub |
| | legmulx |
| | legmul |
| | legdiv |
| | legpow |
| | legval |
| | legval2d |
| | legval3d |
| | leggrid2d |
| | leggrid3d |
| | |
| | Calculus |
| | -------- |
| | |
| | .. autosummary:: |
| | :toctree: generated/ |
| | |
| | legder |
| | legint |
| | |
| | Misc Functions |
| | -------------- |
| | |
| | .. autosummary:: |
| | :toctree: generated/ |
| | |
| | legfromroots |
| | legroots |
| | legvander |
| | legvander2d |
| | legvander3d |
| | leggauss |
| | legweight |
| | legcompanion |
| | legfit |
| | legtrim |
| | legline |
| | leg2poly |
| | poly2leg |
| | |
| | See also |
| | -------- |
| | numpy.polynomial |
| | |
| | """ |
| | import numpy as np |
| | import numpy.linalg as la |
| | from numpy.core.multiarray import normalize_axis_index |
| |
|
| | from . import polyutils as pu |
| | from ._polybase import ABCPolyBase |
| |
|
| | __all__ = [ |
| | 'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd', |
| | 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder', |
| | 'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander', |
| | 'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d', |
| | 'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion', |
| | 'leggauss', 'legweight'] |
| |
|
| | legtrim = pu.trimcoef |
| |
|
| |
|
| | def poly2leg(pol): |
| | """ |
| | Convert a polynomial to a Legendre series. |
| | |
| | Convert an array representing the coefficients of a polynomial (relative |
| | to the "standard" basis) ordered from lowest degree to highest, to an |
| | array of the coefficients of the equivalent Legendre series, ordered |
| | from lowest to highest degree. |
| | |
| | Parameters |
| | ---------- |
| | pol : array_like |
| | 1-D array containing the polynomial coefficients |
| | |
| | Returns |
| | ------- |
| | c : ndarray |
| | 1-D array containing the coefficients of the equivalent Legendre |
| | series. |
| | |
| | See Also |
| | -------- |
| | leg2poly |
| | |
| | Notes |
| | ----- |
| | The easy way to do conversions between polynomial basis sets |
| | is to use the convert method of a class instance. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy import polynomial as P |
| | >>> p = P.Polynomial(np.arange(4)) |
| | >>> p |
| | Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) |
| | >>> c = P.Legendre(P.legendre.poly2leg(p.coef)) |
| | >>> c |
| | Legendre([ 1. , 3.25, 1. , 0.75], domain=[-1, 1], window=[-1, 1]) # may vary |
| | |
| | """ |
| | [pol] = pu.as_series([pol]) |
| | deg = len(pol) - 1 |
| | res = 0 |
| | for i in range(deg, -1, -1): |
| | res = legadd(legmulx(res), pol[i]) |
| | return res |
| |
|
| |
|
| | def leg2poly(c): |
| | """ |
| | Convert a Legendre series to a polynomial. |
| | |
| | Convert an array representing the coefficients of a Legendre series, |
| | ordered from lowest degree to highest, to an array of the coefficients |
| | of the equivalent polynomial (relative to the "standard" basis) ordered |
| | from lowest to highest degree. |
| | |
| | Parameters |
| | ---------- |
| | c : array_like |
| | 1-D array containing the Legendre series coefficients, ordered |
| | from lowest order term to highest. |
| | |
| | Returns |
| | ------- |
| | pol : ndarray |
| | 1-D array containing the coefficients of the equivalent polynomial |
| | (relative to the "standard" basis) ordered from lowest order term |
| | to highest. |
| | |
| | See Also |
| | -------- |
| | poly2leg |
| | |
| | Notes |
| | ----- |
| | The easy way to do conversions between polynomial basis sets |
| | is to use the convert method of a class instance. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy import polynomial as P |
| | >>> c = P.Legendre(range(4)) |
| | >>> c |
| | Legendre([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) |
| | >>> p = c.convert(kind=P.Polynomial) |
| | >>> p |
| | Polynomial([-1. , -3.5, 3. , 7.5], domain=[-1., 1.], window=[-1., 1.]) |
| | >>> P.legendre.leg2poly(range(4)) |
| | array([-1. , -3.5, 3. , 7.5]) |
| | |
| | |
| | """ |
| | from .polynomial import polyadd, polysub, polymulx |
| |
|
| | [c] = pu.as_series([c]) |
| | n = len(c) |
| | if n < 3: |
| | return c |
| | else: |
| | c0 = c[-2] |
| | c1 = c[-1] |
| | |
| | for i in range(n - 1, 1, -1): |
| | tmp = c0 |
| | c0 = polysub(c[i - 2], (c1*(i - 1))/i) |
| | c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i) |
| | return polyadd(c0, polymulx(c1)) |
| |
|
| | |
| | |
| | |
| | |
| |
|
| | |
| | legdomain = np.array([-1, 1]) |
| |
|
| | |
| | legzero = np.array([0]) |
| |
|
| | |
| | legone = np.array([1]) |
| |
|
| | |
| | legx = np.array([0, 1]) |
| |
|
| |
|
| | def legline(off, scl): |
| | """ |
| | Legendre series whose graph is a straight line. |
| | |
| | |
| | |
| | Parameters |
| | ---------- |
| | off, scl : scalars |
| | The specified line is given by ``off + scl*x``. |
| | |
| | Returns |
| | ------- |
| | y : ndarray |
| | This module's representation of the Legendre series for |
| | ``off + scl*x``. |
| | |
| | See Also |
| | -------- |
| | numpy.polynomial.polynomial.polyline |
| | numpy.polynomial.chebyshev.chebline |
| | numpy.polynomial.laguerre.lagline |
| | numpy.polynomial.hermite.hermline |
| | numpy.polynomial.hermite_e.hermeline |
| | |
| | Examples |
| | -------- |
| | >>> import numpy.polynomial.legendre as L |
| | >>> L.legline(3,2) |
| | array([3, 2]) |
| | >>> L.legval(-3, L.legline(3,2)) # should be -3 |
| | -3.0 |
| | |
| | """ |
| | if scl != 0: |
| | return np.array([off, scl]) |
| | else: |
| | return np.array([off]) |
| |
|
| |
|
| | def legfromroots(roots): |
| | """ |
| | Generate a Legendre series with given roots. |
| | |
| | The function returns the coefficients of the polynomial |
| | |
| | .. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), |
| | |
| | in Legendre form, where the `r_n` are the roots specified in `roots`. |
| | If a zero has multiplicity n, then it must appear in `roots` n times. |
| | For instance, if 2 is a root of multiplicity three and 3 is a root of |
| | multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The |
| | roots can appear in any order. |
| | |
| | If the returned coefficients are `c`, then |
| | |
| | .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x) |
| | |
| | The coefficient of the last term is not generally 1 for monic |
| | polynomials in Legendre form. |
| | |
| | Parameters |
| | ---------- |
| | roots : array_like |
| | Sequence containing the roots. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | 1-D array of coefficients. If all roots are real then `out` is a |
| | real array, if some of the roots are complex, then `out` is complex |
| | even if all the coefficients in the result are real (see Examples |
| | below). |
| | |
| | See Also |
| | -------- |
| | numpy.polynomial.polynomial.polyfromroots |
| | numpy.polynomial.chebyshev.chebfromroots |
| | numpy.polynomial.laguerre.lagfromroots |
| | numpy.polynomial.hermite.hermfromroots |
| | numpy.polynomial.hermite_e.hermefromroots |
| | |
| | Examples |
| | -------- |
| | >>> import numpy.polynomial.legendre as L |
| | >>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis |
| | array([ 0. , -0.4, 0. , 0.4]) |
| | >>> j = complex(0,1) |
| | >>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis |
| | array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) # may vary |
| | |
| | """ |
| | return pu._fromroots(legline, legmul, roots) |
| |
|
| |
|
| | def legadd(c1, c2): |
| | """ |
| | Add one Legendre series to another. |
| | |
| | Returns the sum of two Legendre series `c1` + `c2`. The arguments |
| | are sequences of coefficients ordered from lowest order term to |
| | highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. |
| | |
| | Parameters |
| | ---------- |
| | c1, c2 : array_like |
| | 1-D arrays of Legendre series coefficients ordered from low to |
| | high. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | Array representing the Legendre series of their sum. |
| | |
| | See Also |
| | -------- |
| | legsub, legmulx, legmul, legdiv, legpow |
| | |
| | Notes |
| | ----- |
| | Unlike multiplication, division, etc., the sum of two Legendre series |
| | is a Legendre series (without having to "reproject" the result onto |
| | the basis set) so addition, just like that of "standard" polynomials, |
| | is simply "component-wise." |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial import legendre as L |
| | >>> c1 = (1,2,3) |
| | >>> c2 = (3,2,1) |
| | >>> L.legadd(c1,c2) |
| | array([4., 4., 4.]) |
| | |
| | """ |
| | return pu._add(c1, c2) |
| |
|
| |
|
| | def legsub(c1, c2): |
| | """ |
| | Subtract one Legendre series from another. |
| | |
| | Returns the difference of two Legendre series `c1` - `c2`. The |
| | sequences of coefficients are from lowest order term to highest, i.e., |
| | [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. |
| | |
| | Parameters |
| | ---------- |
| | c1, c2 : array_like |
| | 1-D arrays of Legendre series coefficients ordered from low to |
| | high. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | Of Legendre series coefficients representing their difference. |
| | |
| | See Also |
| | -------- |
| | legadd, legmulx, legmul, legdiv, legpow |
| | |
| | Notes |
| | ----- |
| | Unlike multiplication, division, etc., the difference of two Legendre |
| | series is a Legendre series (without having to "reproject" the result |
| | onto the basis set) so subtraction, just like that of "standard" |
| | polynomials, is simply "component-wise." |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial import legendre as L |
| | >>> c1 = (1,2,3) |
| | >>> c2 = (3,2,1) |
| | >>> L.legsub(c1,c2) |
| | array([-2., 0., 2.]) |
| | >>> L.legsub(c2,c1) # -C.legsub(c1,c2) |
| | array([ 2., 0., -2.]) |
| | |
| | """ |
| | return pu._sub(c1, c2) |
| |
|
| |
|
| | def legmulx(c): |
| | """Multiply a Legendre series by x. |
| | |
| | Multiply the Legendre series `c` by x, where x is the independent |
| | variable. |
| | |
| | |
| | Parameters |
| | ---------- |
| | c : array_like |
| | 1-D array of Legendre series coefficients ordered from low to |
| | high. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | Array representing the result of the multiplication. |
| | |
| | See Also |
| | -------- |
| | legadd, legmul, legdiv, legpow |
| | |
| | Notes |
| | ----- |
| | The multiplication uses the recursion relationship for Legendre |
| | polynomials in the form |
| | |
| | .. math:: |
| | |
| | xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1) |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial import legendre as L |
| | >>> L.legmulx([1,2,3]) |
| | array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary |
| | |
| | """ |
| | |
| | [c] = pu.as_series([c]) |
| | |
| | if len(c) == 1 and c[0] == 0: |
| | return c |
| |
|
| | prd = np.empty(len(c) + 1, dtype=c.dtype) |
| | prd[0] = c[0]*0 |
| | prd[1] = c[0] |
| | for i in range(1, len(c)): |
| | j = i + 1 |
| | k = i - 1 |
| | s = i + j |
| | prd[j] = (c[i]*j)/s |
| | prd[k] += (c[i]*i)/s |
| | return prd |
| |
|
| |
|
| | def legmul(c1, c2): |
| | """ |
| | Multiply one Legendre series by another. |
| | |
| | Returns the product of two Legendre series `c1` * `c2`. The arguments |
| | are sequences of coefficients, from lowest order "term" to highest, |
| | e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. |
| | |
| | Parameters |
| | ---------- |
| | c1, c2 : array_like |
| | 1-D arrays of Legendre series coefficients ordered from low to |
| | high. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | Of Legendre series coefficients representing their product. |
| | |
| | See Also |
| | -------- |
| | legadd, legsub, legmulx, legdiv, legpow |
| | |
| | Notes |
| | ----- |
| | In general, the (polynomial) product of two C-series results in terms |
| | that are not in the Legendre polynomial basis set. Thus, to express |
| | the product as a Legendre series, it is necessary to "reproject" the |
| | product onto said basis set, which may produce "unintuitive" (but |
| | correct) results; see Examples section below. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial import legendre as L |
| | >>> c1 = (1,2,3) |
| | >>> c2 = (3,2) |
| | >>> L.legmul(c1,c2) # multiplication requires "reprojection" |
| | array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) # may vary |
| | |
| | """ |
| | |
| | [c1, c2] = pu.as_series([c1, c2]) |
| |
|
| | if len(c1) > len(c2): |
| | c = c2 |
| | xs = c1 |
| | else: |
| | c = c1 |
| | xs = c2 |
| |
|
| | if len(c) == 1: |
| | c0 = c[0]*xs |
| | c1 = 0 |
| | elif len(c) == 2: |
| | c0 = c[0]*xs |
| | c1 = c[1]*xs |
| | else: |
| | nd = len(c) |
| | c0 = c[-2]*xs |
| | c1 = c[-1]*xs |
| | for i in range(3, len(c) + 1): |
| | tmp = c0 |
| | nd = nd - 1 |
| | c0 = legsub(c[-i]*xs, (c1*(nd - 1))/nd) |
| | c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd) |
| | return legadd(c0, legmulx(c1)) |
| |
|
| |
|
| | def legdiv(c1, c2): |
| | """ |
| | Divide one Legendre series by another. |
| | |
| | Returns the quotient-with-remainder of two Legendre series |
| | `c1` / `c2`. The arguments are sequences of coefficients from lowest |
| | order "term" to highest, e.g., [1,2,3] represents the series |
| | ``P_0 + 2*P_1 + 3*P_2``. |
| | |
| | Parameters |
| | ---------- |
| | c1, c2 : array_like |
| | 1-D arrays of Legendre series coefficients ordered from low to |
| | high. |
| | |
| | Returns |
| | ------- |
| | quo, rem : ndarrays |
| | Of Legendre series coefficients representing the quotient and |
| | remainder. |
| | |
| | See Also |
| | -------- |
| | legadd, legsub, legmulx, legmul, legpow |
| | |
| | Notes |
| | ----- |
| | In general, the (polynomial) division of one Legendre series by another |
| | results in quotient and remainder terms that are not in the Legendre |
| | polynomial basis set. Thus, to express these results as a Legendre |
| | series, it is necessary to "reproject" the results onto the Legendre |
| | basis set, which may produce "unintuitive" (but correct) results; see |
| | Examples section below. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial import legendre as L |
| | >>> c1 = (1,2,3) |
| | >>> c2 = (3,2,1) |
| | >>> L.legdiv(c1,c2) # quotient "intuitive," remainder not |
| | (array([3.]), array([-8., -4.])) |
| | >>> c2 = (0,1,2,3) |
| | >>> L.legdiv(c2,c1) # neither "intuitive" |
| | (array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) # may vary |
| | |
| | """ |
| | return pu._div(legmul, c1, c2) |
| |
|
| |
|
| | def legpow(c, pow, maxpower=16): |
| | """Raise a Legendre series to a power. |
| | |
| | Returns the Legendre series `c` raised to the power `pow`. The |
| | argument `c` is a sequence of coefficients ordered from low to high. |
| | i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.`` |
| | |
| | Parameters |
| | ---------- |
| | c : array_like |
| | 1-D array of Legendre series coefficients ordered from low to |
| | high. |
| | pow : integer |
| | Power to which the series will be raised |
| | maxpower : integer, optional |
| | Maximum power allowed. This is mainly to limit growth of the series |
| | to unmanageable size. Default is 16 |
| | |
| | Returns |
| | ------- |
| | coef : ndarray |
| | Legendre series of power. |
| | |
| | See Also |
| | -------- |
| | legadd, legsub, legmulx, legmul, legdiv |
| | |
| | """ |
| | return pu._pow(legmul, c, pow, maxpower) |
| |
|
| |
|
| | def legder(c, m=1, scl=1, axis=0): |
| | """ |
| | Differentiate a Legendre series. |
| | |
| | Returns the Legendre series coefficients `c` differentiated `m` times |
| | along `axis`. At each iteration the result is multiplied by `scl` (the |
| | scaling factor is for use in a linear change of variable). The argument |
| | `c` is an array of coefficients from low to high degree along each |
| | axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2`` |
| | while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + |
| | 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is |
| | ``y``. |
| | |
| | Parameters |
| | ---------- |
| | c : array_like |
| | Array of Legendre series coefficients. If c is multidimensional the |
| | different axis correspond to different variables with the degree in |
| | each axis given by the corresponding index. |
| | m : int, optional |
| | Number of derivatives taken, must be non-negative. (Default: 1) |
| | scl : scalar, optional |
| | Each differentiation is multiplied by `scl`. The end result is |
| | multiplication by ``scl**m``. This is for use in a linear change of |
| | variable. (Default: 1) |
| | axis : int, optional |
| | Axis over which the derivative is taken. (Default: 0). |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | Returns |
| | ------- |
| | der : ndarray |
| | Legendre series of the derivative. |
| | |
| | See Also |
| | -------- |
| | legint |
| | |
| | Notes |
| | ----- |
| | In general, the result of differentiating a Legendre series does not |
| | resemble the same operation on a power series. Thus the result of this |
| | function may be "unintuitive," albeit correct; see Examples section |
| | below. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial import legendre as L |
| | >>> c = (1,2,3,4) |
| | >>> L.legder(c) |
| | array([ 6., 9., 20.]) |
| | >>> L.legder(c, 3) |
| | array([60.]) |
| | >>> L.legder(c, scl=-1) |
| | array([ -6., -9., -20.]) |
| | >>> L.legder(c, 2,-1) |
| | array([ 9., 60.]) |
| | |
| | """ |
| | c = np.array(c, ndmin=1, copy=True) |
| | if c.dtype.char in '?bBhHiIlLqQpP': |
| | c = c.astype(np.double) |
| | cnt = pu._deprecate_as_int(m, "the order of derivation") |
| | iaxis = pu._deprecate_as_int(axis, "the axis") |
| | if cnt < 0: |
| | raise ValueError("The order of derivation must be non-negative") |
| | iaxis = normalize_axis_index(iaxis, c.ndim) |
| |
|
| | if cnt == 0: |
| | return c |
| |
|
| | c = np.moveaxis(c, iaxis, 0) |
| | n = len(c) |
| | if cnt >= n: |
| | c = c[:1]*0 |
| | else: |
| | for i in range(cnt): |
| | n = n - 1 |
| | c *= scl |
| | der = np.empty((n,) + c.shape[1:], dtype=c.dtype) |
| | for j in range(n, 2, -1): |
| | der[j - 1] = (2*j - 1)*c[j] |
| | c[j - 2] += c[j] |
| | if n > 1: |
| | der[1] = 3*c[2] |
| | der[0] = c[1] |
| | c = der |
| | c = np.moveaxis(c, 0, iaxis) |
| | return c |
| |
|
| |
|
| | def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0): |
| | """ |
| | Integrate a Legendre series. |
| | |
| | Returns the Legendre series coefficients `c` integrated `m` times from |
| | `lbnd` along `axis`. At each iteration the resulting series is |
| | **multiplied** by `scl` and an integration constant, `k`, is added. |
| | The scaling factor is for use in a linear change of variable. ("Buyer |
| | beware": note that, depending on what one is doing, one may want `scl` |
| | to be the reciprocal of what one might expect; for more information, |
| | see the Notes section below.) The argument `c` is an array of |
| | coefficients from low to high degree along each axis, e.g., [1,2,3] |
| | represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]] |
| | represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + |
| | 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. |
| | |
| | Parameters |
| | ---------- |
| | c : array_like |
| | Array of Legendre series coefficients. If c is multidimensional the |
| | different axis correspond to different variables with the degree in |
| | each axis given by the corresponding index. |
| | m : int, optional |
| | Order of integration, must be positive. (Default: 1) |
| | k : {[], list, scalar}, optional |
| | Integration constant(s). The value of the first integral at |
| | ``lbnd`` is the first value in the list, the value of the second |
| | integral at ``lbnd`` is the second value, etc. If ``k == []`` (the |
| | default), all constants are set to zero. If ``m == 1``, a single |
| | scalar can be given instead of a list. |
| | lbnd : scalar, optional |
| | The lower bound of the integral. (Default: 0) |
| | scl : scalar, optional |
| | Following each integration the result is *multiplied* by `scl` |
| | before the integration constant is added. (Default: 1) |
| | axis : int, optional |
| | Axis over which the integral is taken. (Default: 0). |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | Returns |
| | ------- |
| | S : ndarray |
| | Legendre series coefficient array of the integral. |
| | |
| | Raises |
| | ------ |
| | ValueError |
| | If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or |
| | ``np.ndim(scl) != 0``. |
| | |
| | See Also |
| | -------- |
| | legder |
| | |
| | Notes |
| | ----- |
| | Note that the result of each integration is *multiplied* by `scl`. |
| | Why is this important to note? Say one is making a linear change of |
| | variable :math:`u = ax + b` in an integral relative to `x`. Then |
| | :math:`dx = du/a`, so one will need to set `scl` equal to |
| | :math:`1/a` - perhaps not what one would have first thought. |
| | |
| | Also note that, in general, the result of integrating a C-series needs |
| | to be "reprojected" onto the C-series basis set. Thus, typically, |
| | the result of this function is "unintuitive," albeit correct; see |
| | Examples section below. |
| | |
| | Examples |
| | -------- |
| | >>> from numpy.polynomial import legendre as L |
| | >>> c = (1,2,3) |
| | >>> L.legint(c) |
| | array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary |
| | >>> L.legint(c, 3) |
| | array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, # may vary |
| | -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) |
| | >>> L.legint(c, k=3) |
| | array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary |
| | >>> L.legint(c, lbnd=-2) |
| | array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary |
| | >>> L.legint(c, scl=2) |
| | array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) # may vary |
| | |
| | """ |
| | c = np.array(c, ndmin=1, copy=True) |
| | if c.dtype.char in '?bBhHiIlLqQpP': |
| | c = c.astype(np.double) |
| | if not np.iterable(k): |
| | k = [k] |
| | cnt = pu._deprecate_as_int(m, "the order of integration") |
| | iaxis = pu._deprecate_as_int(axis, "the axis") |
| | if cnt < 0: |
| | raise ValueError("The order of integration must be non-negative") |
| | if len(k) > cnt: |
| | raise ValueError("Too many integration constants") |
| | if np.ndim(lbnd) != 0: |
| | raise ValueError("lbnd must be a scalar.") |
| | if np.ndim(scl) != 0: |
| | raise ValueError("scl must be a scalar.") |
| | iaxis = normalize_axis_index(iaxis, c.ndim) |
| |
|
| | if cnt == 0: |
| | return c |
| |
|
| | c = np.moveaxis(c, iaxis, 0) |
| | k = list(k) + [0]*(cnt - len(k)) |
| | for i in range(cnt): |
| | n = len(c) |
| | c *= scl |
| | if n == 1 and np.all(c[0] == 0): |
| | c[0] += k[i] |
| | else: |
| | tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype) |
| | tmp[0] = c[0]*0 |
| | tmp[1] = c[0] |
| | if n > 1: |
| | tmp[2] = c[1]/3 |
| | for j in range(2, n): |
| | t = c[j]/(2*j + 1) |
| | tmp[j + 1] = t |
| | tmp[j - 1] -= t |
| | tmp[0] += k[i] - legval(lbnd, tmp) |
| | c = tmp |
| | c = np.moveaxis(c, 0, iaxis) |
| | return c |
| |
|
| |
|
| | def legval(x, c, tensor=True): |
| | """ |
| | Evaluate a Legendre series at points x. |
| | |
| | If `c` is of length `n + 1`, this function returns the value: |
| | |
| | .. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x) |
| | |
| | The parameter `x` is converted to an array only if it is a tuple or a |
| | list, otherwise it is treated as a scalar. In either case, either `x` |
| | or its elements must support multiplication and addition both with |
| | themselves and with the elements of `c`. |
| | |
| | If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If |
| | `c` is multidimensional, then the shape of the result depends on the |
| | value of `tensor`. If `tensor` is true the shape will be c.shape[1:] + |
| | x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that |
| | scalars have shape (,). |
| | |
| | Trailing zeros in the coefficients will be used in the evaluation, so |
| | they should be avoided if efficiency is a concern. |
| | |
| | Parameters |
| | ---------- |
| | x : array_like, compatible object |
| | If `x` is a list or tuple, it is converted to an ndarray, otherwise |
| | it is left unchanged and treated as a scalar. In either case, `x` |
| | or its elements must support addition and multiplication with |
| | themselves and with the elements of `c`. |
| | c : array_like |
| | Array of coefficients ordered so that the coefficients for terms of |
| | degree n are contained in c[n]. If `c` is multidimensional the |
| | remaining indices enumerate multiple polynomials. In the two |
| | dimensional case the coefficients may be thought of as stored in |
| | the columns of `c`. |
| | tensor : boolean, optional |
| | If True, the shape of the coefficient array is extended with ones |
| | on the right, one for each dimension of `x`. Scalars have dimension 0 |
| | for this action. The result is that every column of coefficients in |
| | `c` is evaluated for every element of `x`. If False, `x` is broadcast |
| | over the columns of `c` for the evaluation. This keyword is useful |
| | when `c` is multidimensional. The default value is True. |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | Returns |
| | ------- |
| | values : ndarray, algebra_like |
| | The shape of the return value is described above. |
| | |
| | See Also |
| | -------- |
| | legval2d, leggrid2d, legval3d, leggrid3d |
| | |
| | Notes |
| | ----- |
| | The evaluation uses Clenshaw recursion, aka synthetic division. |
| | |
| | """ |
| | c = np.array(c, ndmin=1, copy=False) |
| | if c.dtype.char in '?bBhHiIlLqQpP': |
| | c = c.astype(np.double) |
| | if isinstance(x, (tuple, list)): |
| | x = np.asarray(x) |
| | if isinstance(x, np.ndarray) and tensor: |
| | c = c.reshape(c.shape + (1,)*x.ndim) |
| |
|
| | if len(c) == 1: |
| | c0 = c[0] |
| | c1 = 0 |
| | elif len(c) == 2: |
| | c0 = c[0] |
| | c1 = c[1] |
| | else: |
| | nd = len(c) |
| | c0 = c[-2] |
| | c1 = c[-1] |
| | for i in range(3, len(c) + 1): |
| | tmp = c0 |
| | nd = nd - 1 |
| | c0 = c[-i] - (c1*(nd - 1))/nd |
| | c1 = tmp + (c1*x*(2*nd - 1))/nd |
| | return c0 + c1*x |
| |
|
| |
|
| | def legval2d(x, y, c): |
| | """ |
| | Evaluate a 2-D Legendre series at points (x, y). |
| | |
| | This function returns the values: |
| | |
| | .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y) |
| | |
| | The parameters `x` and `y` are converted to arrays only if they are |
| | tuples or a lists, otherwise they are treated as a scalars and they |
| | must have the same shape after conversion. In either case, either `x` |
| | and `y` or their elements must support multiplication and addition both |
| | with themselves and with the elements of `c`. |
| | |
| | If `c` is a 1-D array a one is implicitly appended to its shape to make |
| | it 2-D. The shape of the result will be c.shape[2:] + x.shape. |
| | |
| | Parameters |
| | ---------- |
| | x, y : array_like, compatible objects |
| | The two dimensional series is evaluated at the points `(x, y)`, |
| | where `x` and `y` must have the same shape. If `x` or `y` is a list |
| | or tuple, it is first converted to an ndarray, otherwise it is left |
| | unchanged and if it isn't an ndarray it is treated as a scalar. |
| | c : array_like |
| | Array of coefficients ordered so that the coefficient of the term |
| | of multi-degree i,j is contained in ``c[i,j]``. If `c` has |
| | dimension greater than two the remaining indices enumerate multiple |
| | sets of coefficients. |
| | |
| | Returns |
| | ------- |
| | values : ndarray, compatible object |
| | The values of the two dimensional Legendre series at points formed |
| | from pairs of corresponding values from `x` and `y`. |
| | |
| | See Also |
| | -------- |
| | legval, leggrid2d, legval3d, leggrid3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._valnd(legval, c, x, y) |
| |
|
| |
|
| | def leggrid2d(x, y, c): |
| | """ |
| | Evaluate a 2-D Legendre series on the Cartesian product of x and y. |
| | |
| | This function returns the values: |
| | |
| | .. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b) |
| | |
| | where the points `(a, b)` consist of all pairs formed by taking |
| | `a` from `x` and `b` from `y`. The resulting points form a grid with |
| | `x` in the first dimension and `y` in the second. |
| | |
| | The parameters `x` and `y` are converted to arrays only if they are |
| | tuples or a lists, otherwise they are treated as a scalars. In either |
| | case, either `x` and `y` or their elements must support multiplication |
| | and addition both with themselves and with the elements of `c`. |
| | |
| | If `c` has fewer than two dimensions, ones are implicitly appended to |
| | its shape to make it 2-D. The shape of the result will be c.shape[2:] + |
| | x.shape + y.shape. |
| | |
| | Parameters |
| | ---------- |
| | x, y : array_like, compatible objects |
| | The two dimensional series is evaluated at the points in the |
| | Cartesian product of `x` and `y`. If `x` or `y` is a list or |
| | tuple, it is first converted to an ndarray, otherwise it is left |
| | unchanged and, if it isn't an ndarray, it is treated as a scalar. |
| | c : array_like |
| | Array of coefficients ordered so that the coefficient of the term of |
| | multi-degree i,j is contained in `c[i,j]`. If `c` has dimension |
| | greater than two the remaining indices enumerate multiple sets of |
| | coefficients. |
| | |
| | Returns |
| | ------- |
| | values : ndarray, compatible object |
| | The values of the two dimensional Chebyshev series at points in the |
| | Cartesian product of `x` and `y`. |
| | |
| | See Also |
| | -------- |
| | legval, legval2d, legval3d, leggrid3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._gridnd(legval, c, x, y) |
| |
|
| |
|
| | def legval3d(x, y, z, c): |
| | """ |
| | Evaluate a 3-D Legendre series at points (x, y, z). |
| | |
| | This function returns the values: |
| | |
| | .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z) |
| | |
| | The parameters `x`, `y`, and `z` are converted to arrays only if |
| | they are tuples or a lists, otherwise they are treated as a scalars and |
| | they must have the same shape after conversion. In either case, either |
| | `x`, `y`, and `z` or their elements must support multiplication and |
| | addition both with themselves and with the elements of `c`. |
| | |
| | If `c` has fewer than 3 dimensions, ones are implicitly appended to its |
| | shape to make it 3-D. The shape of the result will be c.shape[3:] + |
| | x.shape. |
| | |
| | Parameters |
| | ---------- |
| | x, y, z : array_like, compatible object |
| | The three dimensional series is evaluated at the points |
| | `(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If |
| | any of `x`, `y`, or `z` is a list or tuple, it is first converted |
| | to an ndarray, otherwise it is left unchanged and if it isn't an |
| | ndarray it is treated as a scalar. |
| | c : array_like |
| | Array of coefficients ordered so that the coefficient of the term of |
| | multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension |
| | greater than 3 the remaining indices enumerate multiple sets of |
| | coefficients. |
| | |
| | Returns |
| | ------- |
| | values : ndarray, compatible object |
| | The values of the multidimensional polynomial on points formed with |
| | triples of corresponding values from `x`, `y`, and `z`. |
| | |
| | See Also |
| | -------- |
| | legval, legval2d, leggrid2d, leggrid3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._valnd(legval, c, x, y, z) |
| |
|
| |
|
| | def leggrid3d(x, y, z, c): |
| | """ |
| | Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z. |
| | |
| | This function returns the values: |
| | |
| | .. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c) |
| | |
| | where the points `(a, b, c)` consist of all triples formed by taking |
| | `a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form |
| | a grid with `x` in the first dimension, `y` in the second, and `z` in |
| | the third. |
| | |
| | The parameters `x`, `y`, and `z` are converted to arrays only if they |
| | are tuples or a lists, otherwise they are treated as a scalars. In |
| | either case, either `x`, `y`, and `z` or their elements must support |
| | multiplication and addition both with themselves and with the elements |
| | of `c`. |
| | |
| | If `c` has fewer than three dimensions, ones are implicitly appended to |
| | its shape to make it 3-D. The shape of the result will be c.shape[3:] + |
| | x.shape + y.shape + z.shape. |
| | |
| | Parameters |
| | ---------- |
| | x, y, z : array_like, compatible objects |
| | The three dimensional series is evaluated at the points in the |
| | Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a |
| | list or tuple, it is first converted to an ndarray, otherwise it is |
| | left unchanged and, if it isn't an ndarray, it is treated as a |
| | scalar. |
| | c : array_like |
| | Array of coefficients ordered so that the coefficients for terms of |
| | degree i,j are contained in ``c[i,j]``. If `c` has dimension |
| | greater than two the remaining indices enumerate multiple sets of |
| | coefficients. |
| | |
| | Returns |
| | ------- |
| | values : ndarray, compatible object |
| | The values of the two dimensional polynomial at points in the Cartesian |
| | product of `x` and `y`. |
| | |
| | See Also |
| | -------- |
| | legval, legval2d, leggrid2d, legval3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._gridnd(legval, c, x, y, z) |
| |
|
| |
|
| | def legvander(x, deg): |
| | """Pseudo-Vandermonde matrix of given degree. |
| | |
| | Returns the pseudo-Vandermonde matrix of degree `deg` and sample points |
| | `x`. The pseudo-Vandermonde matrix is defined by |
| | |
| | .. math:: V[..., i] = L_i(x) |
| | |
| | where `0 <= i <= deg`. The leading indices of `V` index the elements of |
| | `x` and the last index is the degree of the Legendre polynomial. |
| | |
| | If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the |
| | array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and |
| | ``legval(x, c)`` are the same up to roundoff. This equivalence is |
| | useful both for least squares fitting and for the evaluation of a large |
| | number of Legendre series of the same degree and sample points. |
| | |
| | Parameters |
| | ---------- |
| | x : array_like |
| | Array of points. The dtype is converted to float64 or complex128 |
| | depending on whether any of the elements are complex. If `x` is |
| | scalar it is converted to a 1-D array. |
| | deg : int |
| | Degree of the resulting matrix. |
| | |
| | Returns |
| | ------- |
| | vander : ndarray |
| | The pseudo-Vandermonde matrix. The shape of the returned matrix is |
| | ``x.shape + (deg + 1,)``, where The last index is the degree of the |
| | corresponding Legendre polynomial. The dtype will be the same as |
| | the converted `x`. |
| | |
| | """ |
| | ideg = pu._deprecate_as_int(deg, "deg") |
| | if ideg < 0: |
| | raise ValueError("deg must be non-negative") |
| |
|
| | x = np.array(x, copy=False, ndmin=1) + 0.0 |
| | dims = (ideg + 1,) + x.shape |
| | dtyp = x.dtype |
| | v = np.empty(dims, dtype=dtyp) |
| | |
| | |
| | v[0] = x*0 + 1 |
| | if ideg > 0: |
| | v[1] = x |
| | for i in range(2, ideg + 1): |
| | v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i |
| | return np.moveaxis(v, 0, -1) |
| |
|
| |
|
| | def legvander2d(x, y, deg): |
| | """Pseudo-Vandermonde matrix of given degrees. |
| | |
| | Returns the pseudo-Vandermonde matrix of degrees `deg` and sample |
| | points `(x, y)`. The pseudo-Vandermonde matrix is defined by |
| | |
| | .. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y), |
| | |
| | where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of |
| | `V` index the points `(x, y)` and the last index encodes the degrees of |
| | the Legendre polynomials. |
| | |
| | If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V` |
| | correspond to the elements of a 2-D coefficient array `c` of shape |
| | (xdeg + 1, ydeg + 1) in the order |
| | |
| | .. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ... |
| | |
| | and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same |
| | up to roundoff. This equivalence is useful both for least squares |
| | fitting and for the evaluation of a large number of 2-D Legendre |
| | series of the same degrees and sample points. |
| | |
| | Parameters |
| | ---------- |
| | x, y : array_like |
| | Arrays of point coordinates, all of the same shape. The dtypes |
| | will be converted to either float64 or complex128 depending on |
| | whether any of the elements are complex. Scalars are converted to |
| | 1-D arrays. |
| | deg : list of ints |
| | List of maximum degrees of the form [x_deg, y_deg]. |
| | |
| | Returns |
| | ------- |
| | vander2d : ndarray |
| | The shape of the returned matrix is ``x.shape + (order,)``, where |
| | :math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same |
| | as the converted `x` and `y`. |
| | |
| | See Also |
| | -------- |
| | legvander, legvander3d, legval2d, legval3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._vander_nd_flat((legvander, legvander), (x, y), deg) |
| |
|
| |
|
| | def legvander3d(x, y, z, deg): |
| | """Pseudo-Vandermonde matrix of given degrees. |
| | |
| | Returns the pseudo-Vandermonde matrix of degrees `deg` and sample |
| | points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`, |
| | then The pseudo-Vandermonde matrix is defined by |
| | |
| | .. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z), |
| | |
| | where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading |
| | indices of `V` index the points `(x, y, z)` and the last index encodes |
| | the degrees of the Legendre polynomials. |
| | |
| | If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns |
| | of `V` correspond to the elements of a 3-D coefficient array `c` of |
| | shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order |
| | |
| | .. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},... |
| | |
| | and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the |
| | same up to roundoff. This equivalence is useful both for least squares |
| | fitting and for the evaluation of a large number of 3-D Legendre |
| | series of the same degrees and sample points. |
| | |
| | Parameters |
| | ---------- |
| | x, y, z : array_like |
| | Arrays of point coordinates, all of the same shape. The dtypes will |
| | be converted to either float64 or complex128 depending on whether |
| | any of the elements are complex. Scalars are converted to 1-D |
| | arrays. |
| | deg : list of ints |
| | List of maximum degrees of the form [x_deg, y_deg, z_deg]. |
| | |
| | Returns |
| | ------- |
| | vander3d : ndarray |
| | The shape of the returned matrix is ``x.shape + (order,)``, where |
| | :math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will |
| | be the same as the converted `x`, `y`, and `z`. |
| | |
| | See Also |
| | -------- |
| | legvander, legvander3d, legval2d, legval3d |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg) |
| |
|
| |
|
| | def legfit(x, y, deg, rcond=None, full=False, w=None): |
| | """ |
| | Least squares fit of Legendre series to data. |
| | |
| | Return the coefficients of a Legendre series of degree `deg` that is the |
| | least squares fit to the data values `y` given at points `x`. If `y` is |
| | 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple |
| | fits are done, one for each column of `y`, and the resulting |
| | coefficients are stored in the corresponding columns of a 2-D return. |
| | The fitted polynomial(s) are in the form |
| | |
| | .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x), |
| | |
| | where `n` is `deg`. |
| | |
| | Parameters |
| | ---------- |
| | x : array_like, shape (M,) |
| | x-coordinates of the M sample points ``(x[i], y[i])``. |
| | y : array_like, shape (M,) or (M, K) |
| | y-coordinates of the sample points. Several data sets of sample |
| | points sharing the same x-coordinates can be fitted at once by |
| | passing in a 2D-array that contains one dataset per column. |
| | deg : int or 1-D array_like |
| | Degree(s) of the fitting polynomials. If `deg` is a single integer |
| | all terms up to and including the `deg`'th term are included in the |
| | fit. For NumPy versions >= 1.11.0 a list of integers specifying the |
| | degrees of the terms to include may be used instead. |
| | rcond : float, optional |
| | Relative condition number of the fit. Singular values smaller than |
| | this relative to the largest singular value will be ignored. The |
| | default value is len(x)*eps, where eps is the relative precision of |
| | the float type, about 2e-16 in most cases. |
| | full : bool, optional |
| | Switch determining nature of return value. When it is False (the |
| | default) just the coefficients are returned, when True diagnostic |
| | information from the singular value decomposition is also returned. |
| | w : array_like, shape (`M`,), optional |
| | Weights. If not None, the weight ``w[i]`` applies to the unsquared |
| | residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are |
| | chosen so that the errors of the products ``w[i]*y[i]`` all have the |
| | same variance. When using inverse-variance weighting, use |
| | ``w[i] = 1/sigma(y[i])``. The default value is None. |
| | |
| | .. versionadded:: 1.5.0 |
| | |
| | Returns |
| | ------- |
| | coef : ndarray, shape (M,) or (M, K) |
| | Legendre coefficients ordered from low to high. If `y` was |
| | 2-D, the coefficients for the data in column k of `y` are in |
| | column `k`. If `deg` is specified as a list, coefficients for |
| | terms not included in the fit are set equal to zero in the |
| | returned `coef`. |
| | |
| | [residuals, rank, singular_values, rcond] : list |
| | These values are only returned if ``full == True`` |
| | |
| | - residuals -- sum of squared residuals of the least squares fit |
| | - rank -- the numerical rank of the scaled Vandermonde matrix |
| | - singular_values -- singular values of the scaled Vandermonde matrix |
| | - rcond -- value of `rcond`. |
| | |
| | For more details, see `numpy.linalg.lstsq`. |
| | |
| | Warns |
| | ----- |
| | RankWarning |
| | The rank of the coefficient matrix in the least-squares fit is |
| | deficient. The warning is only raised if ``full == False``. The |
| | warnings can be turned off by |
| | |
| | >>> import warnings |
| | >>> warnings.simplefilter('ignore', np.RankWarning) |
| | |
| | See Also |
| | -------- |
| | numpy.polynomial.polynomial.polyfit |
| | numpy.polynomial.chebyshev.chebfit |
| | numpy.polynomial.laguerre.lagfit |
| | numpy.polynomial.hermite.hermfit |
| | numpy.polynomial.hermite_e.hermefit |
| | legval : Evaluates a Legendre series. |
| | legvander : Vandermonde matrix of Legendre series. |
| | legweight : Legendre weight function (= 1). |
| | numpy.linalg.lstsq : Computes a least-squares fit from the matrix. |
| | scipy.interpolate.UnivariateSpline : Computes spline fits. |
| | |
| | Notes |
| | ----- |
| | The solution is the coefficients of the Legendre series `p` that |
| | minimizes the sum of the weighted squared errors |
| | |
| | .. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2, |
| | |
| | where :math:`w_j` are the weights. This problem is solved by setting up |
| | as the (typically) overdetermined matrix equation |
| | |
| | .. math:: V(x) * c = w * y, |
| | |
| | where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the |
| | coefficients to be solved for, `w` are the weights, and `y` are the |
| | observed values. This equation is then solved using the singular value |
| | decomposition of `V`. |
| | |
| | If some of the singular values of `V` are so small that they are |
| | neglected, then a `RankWarning` will be issued. This means that the |
| | coefficient values may be poorly determined. Using a lower order fit |
| | will usually get rid of the warning. The `rcond` parameter can also be |
| | set to a value smaller than its default, but the resulting fit may be |
| | spurious and have large contributions from roundoff error. |
| | |
| | Fits using Legendre series are usually better conditioned than fits |
| | using power series, but much can depend on the distribution of the |
| | sample points and the smoothness of the data. If the quality of the fit |
| | is inadequate splines may be a good alternative. |
| | |
| | References |
| | ---------- |
| | .. [1] Wikipedia, "Curve fitting", |
| | https://en.wikipedia.org/wiki/Curve_fitting |
| | |
| | Examples |
| | -------- |
| | |
| | """ |
| | return pu._fit(legvander, x, y, deg, rcond, full, w) |
| |
|
| |
|
| | def legcompanion(c): |
| | """Return the scaled companion matrix of c. |
| | |
| | The basis polynomials are scaled so that the companion matrix is |
| | symmetric when `c` is an Legendre basis polynomial. This provides |
| | better eigenvalue estimates than the unscaled case and for basis |
| | polynomials the eigenvalues are guaranteed to be real if |
| | `numpy.linalg.eigvalsh` is used to obtain them. |
| | |
| | Parameters |
| | ---------- |
| | c : array_like |
| | 1-D array of Legendre series coefficients ordered from low to high |
| | degree. |
| | |
| | Returns |
| | ------- |
| | mat : ndarray |
| | Scaled companion matrix of dimensions (deg, deg). |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | |
| | [c] = pu.as_series([c]) |
| | if len(c) < 2: |
| | raise ValueError('Series must have maximum degree of at least 1.') |
| | if len(c) == 2: |
| | return np.array([[-c[0]/c[1]]]) |
| |
|
| | n = len(c) - 1 |
| | mat = np.zeros((n, n), dtype=c.dtype) |
| | scl = 1./np.sqrt(2*np.arange(n) + 1) |
| | top = mat.reshape(-1)[1::n+1] |
| | bot = mat.reshape(-1)[n::n+1] |
| | top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n] |
| | bot[...] = top |
| | mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1)) |
| | return mat |
| |
|
| |
|
| | def legroots(c): |
| | """ |
| | Compute the roots of a Legendre series. |
| | |
| | Return the roots (a.k.a. "zeros") of the polynomial |
| | |
| | .. math:: p(x) = \\sum_i c[i] * L_i(x). |
| | |
| | Parameters |
| | ---------- |
| | c : 1-D array_like |
| | 1-D array of coefficients. |
| | |
| | Returns |
| | ------- |
| | out : ndarray |
| | Array of the roots of the series. If all the roots are real, |
| | then `out` is also real, otherwise it is complex. |
| | |
| | See Also |
| | -------- |
| | numpy.polynomial.polynomial.polyroots |
| | numpy.polynomial.chebyshev.chebroots |
| | numpy.polynomial.laguerre.lagroots |
| | numpy.polynomial.hermite.hermroots |
| | numpy.polynomial.hermite_e.hermeroots |
| | |
| | Notes |
| | ----- |
| | The root estimates are obtained as the eigenvalues of the companion |
| | matrix, Roots far from the origin of the complex plane may have large |
| | errors due to the numerical instability of the series for such values. |
| | Roots with multiplicity greater than 1 will also show larger errors as |
| | the value of the series near such points is relatively insensitive to |
| | errors in the roots. Isolated roots near the origin can be improved by |
| | a few iterations of Newton's method. |
| | |
| | The Legendre series basis polynomials aren't powers of ``x`` so the |
| | results of this function may seem unintuitive. |
| | |
| | Examples |
| | -------- |
| | >>> import numpy.polynomial.legendre as leg |
| | >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots |
| | array([-0.85099543, -0.11407192, 0.51506735]) # may vary |
| | |
| | """ |
| | |
| | [c] = pu.as_series([c]) |
| | if len(c) < 2: |
| | return np.array([], dtype=c.dtype) |
| | if len(c) == 2: |
| | return np.array([-c[0]/c[1]]) |
| |
|
| | |
| | m = legcompanion(c)[::-1,::-1] |
| | r = la.eigvals(m) |
| | r.sort() |
| | return r |
| |
|
| |
|
| | def leggauss(deg): |
| | """ |
| | Gauss-Legendre quadrature. |
| | |
| | Computes the sample points and weights for Gauss-Legendre quadrature. |
| | These sample points and weights will correctly integrate polynomials of |
| | degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with |
| | the weight function :math:`f(x) = 1`. |
| | |
| | Parameters |
| | ---------- |
| | deg : int |
| | Number of sample points and weights. It must be >= 1. |
| | |
| | Returns |
| | ------- |
| | x : ndarray |
| | 1-D ndarray containing the sample points. |
| | y : ndarray |
| | 1-D ndarray containing the weights. |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | The results have only been tested up to degree 100, higher degrees may |
| | be problematic. The weights are determined by using the fact that |
| | |
| | .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k)) |
| | |
| | where :math:`c` is a constant independent of :math:`k` and :math:`x_k` |
| | is the k'th root of :math:`L_n`, and then scaling the results to get |
| | the right value when integrating 1. |
| | |
| | """ |
| | ideg = pu._deprecate_as_int(deg, "deg") |
| | if ideg <= 0: |
| | raise ValueError("deg must be a positive integer") |
| |
|
| | |
| | |
| | c = np.array([0]*deg + [1]) |
| | m = legcompanion(c) |
| | x = la.eigvalsh(m) |
| |
|
| | |
| | dy = legval(x, c) |
| | df = legval(x, legder(c)) |
| | x -= dy/df |
| |
|
| | |
| | |
| | fm = legval(x, c[1:]) |
| | fm /= np.abs(fm).max() |
| | df /= np.abs(df).max() |
| | w = 1/(fm * df) |
| |
|
| | |
| | w = (w + w[::-1])/2 |
| | x = (x - x[::-1])/2 |
| |
|
| | |
| | w *= 2. / w.sum() |
| |
|
| | return x, w |
| |
|
| |
|
| | def legweight(x): |
| | """ |
| | Weight function of the Legendre polynomials. |
| | |
| | The weight function is :math:`1` and the interval of integration is |
| | :math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not |
| | normalized, with respect to this weight function. |
| | |
| | Parameters |
| | ---------- |
| | x : array_like |
| | Values at which the weight function will be computed. |
| | |
| | Returns |
| | ------- |
| | w : ndarray |
| | The weight function at `x`. |
| | |
| | Notes |
| | ----- |
| | |
| | .. versionadded:: 1.7.0 |
| | |
| | """ |
| | w = x*0.0 + 1.0 |
| | return w |
| |
|
| | |
| | |
| | |
| |
|
| | class Legendre(ABCPolyBase): |
| | """A Legendre series class. |
| | |
| | The Legendre class provides the standard Python numerical methods |
| | '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the |
| | attributes and methods listed in the `ABCPolyBase` documentation. |
| | |
| | Parameters |
| | ---------- |
| | coef : array_like |
| | Legendre coefficients in order of increasing degree, i.e., |
| | ``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``. |
| | domain : (2,) array_like, optional |
| | Domain to use. The interval ``[domain[0], domain[1]]`` is mapped |
| | to the interval ``[window[0], window[1]]`` by shifting and scaling. |
| | The default value is [-1, 1]. |
| | window : (2,) array_like, optional |
| | Window, see `domain` for its use. The default value is [-1, 1]. |
| | |
| | .. versionadded:: 1.6.0 |
| | symbol : str, optional |
| | Symbol used to represent the independent variable in string |
| | representations of the polynomial expression, e.g. for printing. |
| | The symbol must be a valid Python identifier. Default value is 'x'. |
| | |
| | .. versionadded:: 1.24 |
| | |
| | """ |
| | |
| | _add = staticmethod(legadd) |
| | _sub = staticmethod(legsub) |
| | _mul = staticmethod(legmul) |
| | _div = staticmethod(legdiv) |
| | _pow = staticmethod(legpow) |
| | _val = staticmethod(legval) |
| | _int = staticmethod(legint) |
| | _der = staticmethod(legder) |
| | _fit = staticmethod(legfit) |
| | _line = staticmethod(legline) |
| | _roots = staticmethod(legroots) |
| | _fromroots = staticmethod(legfromroots) |
| |
|
| | |
| | domain = np.array(legdomain) |
| | window = np.array(legdomain) |
| | basis_name = 'P' |
| |
|
