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""" |
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============================================================== |
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Hermite Series, "Physicists" (:mod:`numpy.polynomial.hermite`) |
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============================================================== |
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|
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This module provides a number of objects (mostly functions) useful for |
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dealing with Hermite series, including a `Hermite` class that |
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encapsulates the usual arithmetic operations. (General information |
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on how this module represents and works with such polynomials is in the |
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docstring for its "parent" sub-package, `numpy.polynomial`). |
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Classes |
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------- |
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.. autosummary:: |
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:toctree: generated/ |
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Hermite |
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Constants |
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--------- |
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.. autosummary:: |
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:toctree: generated/ |
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hermdomain |
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hermzero |
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hermone |
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hermx |
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Arithmetic |
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---------- |
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.. autosummary:: |
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:toctree: generated/ |
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hermadd |
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hermsub |
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hermmulx |
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hermmul |
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hermdiv |
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hermpow |
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hermval |
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hermval2d |
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hermval3d |
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hermgrid2d |
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hermgrid3d |
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Calculus |
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-------- |
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.. autosummary:: |
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:toctree: generated/ |
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hermder |
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hermint |
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Misc Functions |
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-------------- |
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.. autosummary:: |
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:toctree: generated/ |
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hermfromroots |
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hermroots |
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hermvander |
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hermvander2d |
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hermvander3d |
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hermgauss |
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hermweight |
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hermcompanion |
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hermfit |
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hermtrim |
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hermline |
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herm2poly |
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poly2herm |
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See also |
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-------- |
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`numpy.polynomial` |
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""" |
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import numpy as np |
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import numpy.linalg as la |
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from numpy.lib.array_utils import normalize_axis_index |
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|
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from . import polyutils as pu |
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from ._polybase import ABCPolyBase |
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|
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__all__ = [ |
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'hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd', |
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'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', 'hermval', |
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'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots', |
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'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite', |
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'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', 'hermvander2d', |
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'hermvander3d', 'hermcompanion', 'hermgauss', 'hermweight'] |
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hermtrim = pu.trimcoef |
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|
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def poly2herm(pol): |
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""" |
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poly2herm(pol) |
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|
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Convert a polynomial to a Hermite series. |
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|
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Convert an array representing the coefficients of a polynomial (relative |
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to the "standard" basis) ordered from lowest degree to highest, to an |
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array of the coefficients of the equivalent Hermite series, ordered |
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from lowest to highest degree. |
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|
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Parameters |
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---------- |
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pol : array_like |
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1-D array containing the polynomial coefficients |
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Returns |
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------- |
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c : ndarray |
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1-D array containing the coefficients of the equivalent Hermite |
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series. |
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|
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See Also |
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-------- |
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herm2poly |
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Notes |
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----- |
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The easy way to do conversions between polynomial basis sets |
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is to use the convert method of a class instance. |
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|
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Examples |
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-------- |
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>>> from numpy.polynomial.hermite import poly2herm |
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>>> poly2herm(np.arange(4)) |
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array([1. , 2.75 , 0.5 , 0.375]) |
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|
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""" |
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[pol] = pu.as_series([pol]) |
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deg = len(pol) - 1 |
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res = 0 |
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for i in range(deg, -1, -1): |
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res = hermadd(hermmulx(res), pol[i]) |
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return res |
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def herm2poly(c): |
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""" |
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Convert a Hermite series to a polynomial. |
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|
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Convert an array representing the coefficients of a Hermite series, |
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ordered from lowest degree to highest, to an array of the coefficients |
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of the equivalent polynomial (relative to the "standard" basis) ordered |
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from lowest to highest degree. |
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|
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Parameters |
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---------- |
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c : array_like |
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1-D array containing the Hermite series coefficients, ordered |
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from lowest order term to highest. |
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|
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Returns |
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------- |
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pol : ndarray |
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1-D array containing the coefficients of the equivalent polynomial |
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(relative to the "standard" basis) ordered from lowest order term |
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to highest. |
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|
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See Also |
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-------- |
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poly2herm |
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|
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Notes |
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----- |
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The easy way to do conversions between polynomial basis sets |
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is to use the convert method of a class instance. |
|
|
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Examples |
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-------- |
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>>> from numpy.polynomial.hermite import herm2poly |
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>>> herm2poly([ 1. , 2.75 , 0.5 , 0.375]) |
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array([0., 1., 2., 3.]) |
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|
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""" |
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from .polynomial import polyadd, polysub, polymulx |
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|
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[c] = pu.as_series([c]) |
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n = len(c) |
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if n == 1: |
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return c |
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if n == 2: |
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c[1] *= 2 |
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return c |
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else: |
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c0 = c[-2] |
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c1 = c[-1] |
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|
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for i in range(n - 1, 1, -1): |
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tmp = c0 |
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c0 = polysub(c[i - 2], c1*(2*(i - 1))) |
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c1 = polyadd(tmp, polymulx(c1)*2) |
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return polyadd(c0, polymulx(c1)*2) |
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hermdomain = np.array([-1., 1.]) |
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hermzero = np.array([0]) |
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hermone = np.array([1]) |
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hermx = np.array([0, 1/2]) |
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def hermline(off, scl): |
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""" |
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Hermite series whose graph is a straight line. |
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Parameters |
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---------- |
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off, scl : scalars |
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The specified line is given by ``off + scl*x``. |
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Returns |
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------- |
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y : ndarray |
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This module's representation of the Hermite series for |
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``off + scl*x``. |
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See Also |
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-------- |
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numpy.polynomial.polynomial.polyline |
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numpy.polynomial.chebyshev.chebline |
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numpy.polynomial.legendre.legline |
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numpy.polynomial.laguerre.lagline |
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numpy.polynomial.hermite_e.hermeline |
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|
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Examples |
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-------- |
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>>> from numpy.polynomial.hermite import hermline, hermval |
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>>> hermval(0,hermline(3, 2)) |
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3.0 |
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>>> hermval(1,hermline(3, 2)) |
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5.0 |
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""" |
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if scl != 0: |
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return np.array([off, scl/2]) |
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else: |
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return np.array([off]) |
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def hermfromroots(roots): |
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""" |
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Generate a Hermite series with given roots. |
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|
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The function returns the coefficients of the polynomial |
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|
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.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n), |
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in Hermite form, where the :math:`r_n` are the roots specified in `roots`. |
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If a zero has multiplicity n, then it must appear in `roots` n times. |
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For instance, if 2 is a root of multiplicity three and 3 is a root of |
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multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The |
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roots can appear in any order. |
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If the returned coefficients are `c`, then |
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|
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.. math:: p(x) = c_0 + c_1 * H_1(x) + ... + c_n * H_n(x) |
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The coefficient of the last term is not generally 1 for monic |
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polynomials in Hermite form. |
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Parameters |
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---------- |
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roots : array_like |
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Sequence containing the roots. |
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Returns |
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------- |
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out : ndarray |
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1-D array of coefficients. If all roots are real then `out` is a |
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real array, if some of the roots are complex, then `out` is complex |
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even if all the coefficients in the result are real (see Examples |
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below). |
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See Also |
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-------- |
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numpy.polynomial.polynomial.polyfromroots |
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numpy.polynomial.legendre.legfromroots |
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numpy.polynomial.laguerre.lagfromroots |
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numpy.polynomial.chebyshev.chebfromroots |
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numpy.polynomial.hermite_e.hermefromroots |
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|
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Examples |
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-------- |
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>>> from numpy.polynomial.hermite import hermfromroots, hermval |
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>>> coef = hermfromroots((-1, 0, 1)) |
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>>> hermval((-1, 0, 1), coef) |
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array([0., 0., 0.]) |
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>>> coef = hermfromroots((-1j, 1j)) |
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>>> hermval((-1j, 1j), coef) |
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array([0.+0.j, 0.+0.j]) |
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""" |
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return pu._fromroots(hermline, hermmul, roots) |
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def hermadd(c1, c2): |
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""" |
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Add one Hermite series to another. |
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Returns the sum of two Hermite series `c1` + `c2`. The arguments |
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are sequences of coefficients ordered from lowest order term to |
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highest, i.e., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. |
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Parameters |
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---------- |
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c1, c2 : array_like |
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1-D arrays of Hermite series coefficients ordered from low to |
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high. |
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Returns |
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------- |
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out : ndarray |
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Array representing the Hermite series of their sum. |
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See Also |
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-------- |
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hermsub, hermmulx, hermmul, hermdiv, hermpow |
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|
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Notes |
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----- |
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Unlike multiplication, division, etc., the sum of two Hermite series |
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is a Hermite series (without having to "reproject" the result onto |
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the basis set) so addition, just like that of "standard" polynomials, |
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is simply "component-wise." |
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Examples |
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-------- |
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>>> from numpy.polynomial.hermite import hermadd |
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>>> hermadd([1, 2, 3], [1, 2, 3, 4]) |
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array([2., 4., 6., 4.]) |
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|
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""" |
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return pu._add(c1, c2) |
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def hermsub(c1, c2): |
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""" |
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Subtract one Hermite series from another. |
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Returns the difference of two Hermite series `c1` - `c2`. The |
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sequences of coefficients are from lowest order term to highest, i.e., |
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[1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. |
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Parameters |
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---------- |
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c1, c2 : array_like |
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1-D arrays of Hermite series coefficients ordered from low to |
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high. |
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Returns |
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------- |
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out : ndarray |
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Of Hermite series coefficients representing their difference. |
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See Also |
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-------- |
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hermadd, hermmulx, hermmul, hermdiv, hermpow |
|
|
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Notes |
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----- |
|
Unlike multiplication, division, etc., the difference of two Hermite |
|
series is a Hermite series (without having to "reproject" the result |
|
onto the basis set) so subtraction, just like that of "standard" |
|
polynomials, is simply "component-wise." |
|
|
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Examples |
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-------- |
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>>> from numpy.polynomial.hermite import hermsub |
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>>> hermsub([1, 2, 3, 4], [1, 2, 3]) |
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array([0., 0., 0., 4.]) |
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|
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""" |
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return pu._sub(c1, c2) |
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def hermmulx(c): |
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"""Multiply a Hermite series by x. |
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|
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Multiply the Hermite series `c` by x, where x is the independent |
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variable. |
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Parameters |
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---------- |
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c : array_like |
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1-D array of Hermite series coefficients ordered from low to |
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high. |
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Returns |
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------- |
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out : ndarray |
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Array representing the result of the multiplication. |
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See Also |
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-------- |
|
hermadd, hermsub, hermmul, hermdiv, hermpow |
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|
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Notes |
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----- |
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The multiplication uses the recursion relationship for Hermite |
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polynomials in the form |
|
|
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.. math:: |
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|
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xP_i(x) = (P_{i + 1}(x)/2 + i*P_{i - 1}(x)) |
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|
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Examples |
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-------- |
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>>> from numpy.polynomial.hermite import hermmulx |
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>>> hermmulx([1, 2, 3]) |
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array([2. , 6.5, 1. , 1.5]) |
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|
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""" |
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[c] = pu.as_series([c]) |
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if len(c) == 1 and c[0] == 0: |
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return c |
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prd = np.empty(len(c) + 1, dtype=c.dtype) |
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prd[0] = c[0]*0 |
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prd[1] = c[0]/2 |
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for i in range(1, len(c)): |
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prd[i + 1] = c[i]/2 |
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prd[i - 1] += c[i]*i |
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return prd |
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def hermmul(c1, c2): |
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""" |
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Multiply one Hermite series by another. |
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Returns the product of two Hermite series `c1` * `c2`. The arguments |
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are sequences of coefficients, from lowest order "term" to highest, |
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e.g., [1,2,3] represents the series ``P_0 + 2*P_1 + 3*P_2``. |
|
|
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Parameters |
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---------- |
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c1, c2 : array_like |
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1-D arrays of Hermite series coefficients ordered from low to |
|
high. |
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|
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Returns |
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------- |
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out : ndarray |
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Of Hermite series coefficients representing their product. |
|
|
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See Also |
|
-------- |
|
hermadd, hermsub, hermmulx, hermdiv, hermpow |
|
|
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Notes |
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----- |
|
In general, the (polynomial) product of two C-series results in terms |
|
that are not in the Hermite polynomial basis set. Thus, to express |
|
the product as a Hermite series, it is necessary to "reproject" the |
|
product onto said basis set, which may produce "unintuitive" (but |
|
correct) results; see Examples section below. |
|
|
|
Examples |
|
-------- |
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>>> from numpy.polynomial.hermite import hermmul |
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>>> hermmul([1, 2, 3], [0, 1, 2]) |
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array([52., 29., 52., 7., 6.]) |
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|
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""" |
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|
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[c1, c2] = pu.as_series([c1, c2]) |
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|
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if len(c1) > len(c2): |
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c = c2 |
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xs = c1 |
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else: |
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c = c1 |
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xs = c2 |
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|
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if len(c) == 1: |
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c0 = c[0]*xs |
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c1 = 0 |
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elif len(c) == 2: |
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c0 = c[0]*xs |
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c1 = c[1]*xs |
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else: |
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nd = len(c) |
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c0 = c[-2]*xs |
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c1 = c[-1]*xs |
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for i in range(3, len(c) + 1): |
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tmp = c0 |
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nd = nd - 1 |
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c0 = hermsub(c[-i]*xs, c1*(2*(nd - 1))) |
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c1 = hermadd(tmp, hermmulx(c1)*2) |
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return hermadd(c0, hermmulx(c1)*2) |
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def hermdiv(c1, c2): |
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""" |
|
Divide one Hermite series by another. |
|
|
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Returns the quotient-with-remainder of two Hermite 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``. |
|
|
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Parameters |
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---------- |
|
c1, c2 : array_like |
|
1-D arrays of Hermite series coefficients ordered from low to |
|
high. |
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|
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Returns |
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------- |
|
[quo, rem] : ndarrays |
|
Of Hermite series coefficients representing the quotient and |
|
remainder. |
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|
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See Also |
|
-------- |
|
hermadd, hermsub, hermmulx, hermmul, hermpow |
|
|
|
Notes |
|
----- |
|
In general, the (polynomial) division of one Hermite series by another |
|
results in quotient and remainder terms that are not in the Hermite |
|
polynomial basis set. Thus, to express these results as a Hermite |
|
series, it is necessary to "reproject" the results onto the Hermite |
|
basis set, which may produce "unintuitive" (but correct) results; see |
|
Examples section below. |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermdiv |
|
>>> hermdiv([ 52., 29., 52., 7., 6.], [0, 1, 2]) |
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(array([1., 2., 3.]), array([0.])) |
|
>>> hermdiv([ 54., 31., 52., 7., 6.], [0, 1, 2]) |
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(array([1., 2., 3.]), array([2., 2.])) |
|
>>> hermdiv([ 53., 30., 52., 7., 6.], [0, 1, 2]) |
|
(array([1., 2., 3.]), array([1., 1.])) |
|
|
|
""" |
|
return pu._div(hermmul, c1, c2) |
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|
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def hermpow(c, pow, maxpower=16): |
|
"""Raise a Hermite series to a power. |
|
|
|
Returns the Hermite 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 Hermite 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 |
|
Hermite series of power. |
|
|
|
See Also |
|
-------- |
|
hermadd, hermsub, hermmulx, hermmul, hermdiv |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermpow |
|
>>> hermpow([1, 2, 3], 2) |
|
array([81., 52., 82., 12., 9.]) |
|
|
|
""" |
|
return pu._pow(hermmul, c, pow, maxpower) |
|
|
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|
|
def hermder(c, m=1, scl=1, axis=0): |
|
""" |
|
Differentiate a Hermite series. |
|
|
|
Returns the Hermite 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*H_0 + 2*H_1 + 3*H_2`` |
|
while [[1,2],[1,2]] represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + |
|
2*H_0(x)*H_1(y) + 2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is |
|
``y``. |
|
|
|
Parameters |
|
---------- |
|
c : array_like |
|
Array of Hermite 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 |
|
Hermite series of the derivative. |
|
|
|
See Also |
|
-------- |
|
hermint |
|
|
|
Notes |
|
----- |
|
In general, the result of differentiating a Hermite 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.hermite import hermder |
|
>>> hermder([ 1. , 0.5, 0.5, 0.5]) |
|
array([1., 2., 3.]) |
|
>>> hermder([-0.5, 1./2., 1./8., 1./12., 1./16.], m=2) |
|
array([1., 2., 3.]) |
|
|
|
""" |
|
c = np.array(c, ndmin=1, copy=True) |
|
if c.dtype.char in '?bBhHiIlLqQpP': |
|
c = c.astype(np.double) |
|
cnt = pu._as_int(m, "the order of derivation") |
|
iaxis = pu._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, 0, -1): |
|
der[j - 1] = (2*j)*c[j] |
|
c = der |
|
c = np.moveaxis(c, 0, iaxis) |
|
return c |
|
|
|
|
|
def hermint(c, m=1, k=[], lbnd=0, scl=1, axis=0): |
|
""" |
|
Integrate a Hermite series. |
|
|
|
Returns the Hermite 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 ``H_0 + 2*H_1 + 3*H_2`` while [[1,2],[1,2]] |
|
represents ``1*H_0(x)*H_0(y) + 1*H_1(x)*H_0(y) + 2*H_0(x)*H_1(y) + |
|
2*H_1(x)*H_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. |
|
|
|
Parameters |
|
---------- |
|
c : array_like |
|
Array of Hermite 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 |
|
Hermite series coefficients of the integral. |
|
|
|
Raises |
|
------ |
|
ValueError |
|
If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or |
|
``np.ndim(scl) != 0``. |
|
|
|
See Also |
|
-------- |
|
hermder |
|
|
|
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.hermite import hermint |
|
>>> hermint([1,2,3]) # integrate once, value 0 at 0. |
|
array([1. , 0.5, 0.5, 0.5]) |
|
>>> hermint([1,2,3], m=2) # integrate twice, value & deriv 0 at 0 |
|
array([-0.5 , 0.5 , 0.125 , 0.08333333, 0.0625 ]) # may vary |
|
>>> hermint([1,2,3], k=1) # integrate once, value 1 at 0. |
|
array([2. , 0.5, 0.5, 0.5]) |
|
>>> hermint([1,2,3], lbnd=-1) # integrate once, value 0 at -1 |
|
array([-2. , 0.5, 0.5, 0.5]) |
|
>>> hermint([1,2,3], m=2, k=[1,2], lbnd=-1) |
|
array([ 1.66666667, -0.5 , 0.125 , 0.08333333, 0.0625 ]) # 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._as_int(m, "the order of integration") |
|
iaxis = pu._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]/2 |
|
for j in range(1, n): |
|
tmp[j + 1] = c[j]/(2*(j + 1)) |
|
tmp[0] += k[i] - hermval(lbnd, tmp) |
|
c = tmp |
|
c = np.moveaxis(c, 0, iaxis) |
|
return c |
|
|
|
|
|
def hermval(x, c, tensor=True): |
|
""" |
|
Evaluate an Hermite series at points x. |
|
|
|
If `c` is of length ``n + 1``, this function returns the value: |
|
|
|
.. math:: p(x) = c_0 * H_0(x) + c_1 * H_1(x) + ... + c_n * H_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 |
|
-------- |
|
hermval2d, hermgrid2d, hermval3d, hermgrid3d |
|
|
|
Notes |
|
----- |
|
The evaluation uses Clenshaw recursion, aka synthetic division. |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermval |
|
>>> coef = [1,2,3] |
|
>>> hermval(1, coef) |
|
11.0 |
|
>>> hermval([[1,2],[3,4]], coef) |
|
array([[ 11., 51.], |
|
[115., 203.]]) |
|
|
|
""" |
|
c = np.array(c, ndmin=1, copy=None) |
|
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) |
|
|
|
x2 = x*2 |
|
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*(2*(nd - 1)) |
|
c1 = tmp + c1*x2 |
|
return c0 + c1*x2 |
|
|
|
|
|
def hermval2d(x, y, c): |
|
""" |
|
Evaluate a 2-D Hermite series at points (x, y). |
|
|
|
This function returns the values: |
|
|
|
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * H_i(x) * H_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 polynomial at points formed with |
|
pairs of corresponding values from `x` and `y`. |
|
|
|
See Also |
|
-------- |
|
hermval, hermgrid2d, hermval3d, hermgrid3d |
|
|
|
Notes |
|
----- |
|
|
|
.. versionadded:: 1.7.0 |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermval2d |
|
>>> x = [1, 2] |
|
>>> y = [4, 5] |
|
>>> c = [[1, 2, 3], [4, 5, 6]] |
|
>>> hermval2d(x, y, c) |
|
array([1035., 2883.]) |
|
|
|
""" |
|
return pu._valnd(hermval, c, x, y) |
|
|
|
|
|
def hermgrid2d(x, y, c): |
|
""" |
|
Evaluate a 2-D Hermite series on the Cartesian product of x and y. |
|
|
|
This function returns the values: |
|
|
|
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * H_i(a) * H_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. |
|
|
|
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 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 |
|
-------- |
|
hermval, hermval2d, hermval3d, hermgrid3d |
|
|
|
Notes |
|
----- |
|
|
|
.. versionadded:: 1.7.0 |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermgrid2d |
|
>>> x = [1, 2, 3] |
|
>>> y = [4, 5] |
|
>>> c = [[1, 2, 3], [4, 5, 6]] |
|
>>> hermgrid2d(x, y, c) |
|
array([[1035., 1599.], |
|
[1867., 2883.], |
|
[2699., 4167.]]) |
|
|
|
""" |
|
return pu._gridnd(hermval, c, x, y) |
|
|
|
|
|
def hermval3d(x, y, z, c): |
|
""" |
|
Evaluate a 3-D Hermite series at points (x, y, z). |
|
|
|
This function returns the values: |
|
|
|
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * H_i(x) * H_j(y) * H_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 |
|
-------- |
|
hermval, hermval2d, hermgrid2d, hermgrid3d |
|
|
|
Notes |
|
----- |
|
|
|
.. versionadded:: 1.7.0 |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermval3d |
|
>>> x = [1, 2] |
|
>>> y = [4, 5] |
|
>>> z = [6, 7] |
|
>>> c = [[[1, 2, 3], [4, 5, 6]], [[7, 8, 9], [10, 11, 12]]] |
|
>>> hermval3d(x, y, z, c) |
|
array([ 40077., 120131.]) |
|
|
|
""" |
|
return pu._valnd(hermval, c, x, y, z) |
|
|
|
|
|
def hermgrid3d(x, y, z, c): |
|
""" |
|
Evaluate a 3-D Hermite 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} * H_i(a) * H_j(b) * H_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 |
|
-------- |
|
hermval, hermval2d, hermgrid2d, hermval3d |
|
|
|
Notes |
|
----- |
|
|
|
.. versionadded:: 1.7.0 |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermgrid3d |
|
>>> x = [1, 2] |
|
>>> y = [4, 5] |
|
>>> z = [6, 7] |
|
>>> c = [[[1, 2, 3], [4, 5, 6]], [[7, 8, 9], [10, 11, 12]]] |
|
>>> hermgrid3d(x, y, z, c) |
|
array([[[ 40077., 54117.], |
|
[ 49293., 66561.]], |
|
[[ 72375., 97719.], |
|
[ 88975., 120131.]]]) |
|
|
|
""" |
|
return pu._gridnd(hermval, c, x, y, z) |
|
|
|
|
|
def hermvander(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] = H_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 Hermite polynomial. |
|
|
|
If `c` is a 1-D array of coefficients of length ``n + 1`` and `V` is the |
|
array ``V = hermvander(x, n)``, then ``np.dot(V, c)`` and |
|
``hermval(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 Hermite 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 Hermite polynomial. The dtype will be the same as |
|
the converted `x`. |
|
|
|
Examples |
|
-------- |
|
>>> import numpy as np |
|
>>> from numpy.polynomial.hermite import hermvander |
|
>>> x = np.array([-1, 0, 1]) |
|
>>> hermvander(x, 3) |
|
array([[ 1., -2., 2., 4.], |
|
[ 1., 0., -2., -0.], |
|
[ 1., 2., 2., -4.]]) |
|
|
|
""" |
|
ideg = pu._as_int(deg, "deg") |
|
if ideg < 0: |
|
raise ValueError("deg must be non-negative") |
|
|
|
x = np.array(x, copy=None, 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: |
|
x2 = x*2 |
|
v[1] = x2 |
|
for i in range(2, ideg + 1): |
|
v[i] = (v[i-1]*x2 - v[i-2]*(2*(i - 1))) |
|
return np.moveaxis(v, 0, -1) |
|
|
|
|
|
def hermvander2d(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] = H_i(x) * H_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 Hermite polynomials. |
|
|
|
If ``V = hermvander2d(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 ``hermval2d(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 Hermite |
|
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 |
|
-------- |
|
hermvander, hermvander3d, hermval2d, hermval3d |
|
|
|
Notes |
|
----- |
|
|
|
.. versionadded:: 1.7.0 |
|
|
|
Examples |
|
-------- |
|
>>> import numpy as np |
|
>>> from numpy.polynomial.hermite import hermvander2d |
|
>>> x = np.array([-1, 0, 1]) |
|
>>> y = np.array([-1, 0, 1]) |
|
>>> hermvander2d(x, y, [2, 2]) |
|
array([[ 1., -2., 2., -2., 4., -4., 2., -4., 4.], |
|
[ 1., 0., -2., 0., 0., -0., -2., -0., 4.], |
|
[ 1., 2., 2., 2., 4., 4., 2., 4., 4.]]) |
|
|
|
""" |
|
return pu._vander_nd_flat((hermvander, hermvander), (x, y), deg) |
|
|
|
|
|
def hermvander3d(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] = H_i(x)*H_j(y)*H_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 Hermite polynomials. |
|
|
|
If ``V = hermvander3d(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 ``hermval3d(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 Hermite |
|
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 |
|
-------- |
|
hermvander, hermvander3d, hermval2d, hermval3d |
|
|
|
Notes |
|
----- |
|
|
|
.. versionadded:: 1.7.0 |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermvander3d |
|
>>> x = np.array([-1, 0, 1]) |
|
>>> y = np.array([-1, 0, 1]) |
|
>>> z = np.array([-1, 0, 1]) |
|
>>> hermvander3d(x, y, z, [0, 1, 2]) |
|
array([[ 1., -2., 2., -2., 4., -4.], |
|
[ 1., 0., -2., 0., 0., -0.], |
|
[ 1., 2., 2., 2., 4., 4.]]) |
|
|
|
""" |
|
return pu._vander_nd_flat((hermvander, hermvander, hermvander), (x, y, z), deg) |
|
|
|
|
|
def hermfit(x, y, deg, rcond=None, full=False, w=None): |
|
""" |
|
Least squares fit of Hermite series to data. |
|
|
|
Return the coefficients of a Hermite 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 * H_1(x) + ... + c_n * H_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. |
|
|
|
Returns |
|
------- |
|
coef : ndarray, shape (M,) or (M, K) |
|
Hermite 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`. |
|
|
|
[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.exceptions.RankWarning) |
|
|
|
See Also |
|
-------- |
|
numpy.polynomial.chebyshev.chebfit |
|
numpy.polynomial.legendre.legfit |
|
numpy.polynomial.laguerre.lagfit |
|
numpy.polynomial.polynomial.polyfit |
|
numpy.polynomial.hermite_e.hermefit |
|
hermval : Evaluates a Hermite series. |
|
hermvander : Vandermonde matrix of Hermite series. |
|
hermweight : Hermite weight function |
|
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 Hermite 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 the :math:`w_j` are the weights. This problem is solved by |
|
setting up 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, `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 `~exceptions.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 Hermite series are probably most useful when the data can be |
|
approximated by ``sqrt(w(x)) * p(x)``, where ``w(x)`` is the Hermite |
|
weight. In that case the weight ``sqrt(w(x[i]))`` should be used |
|
together with data values ``y[i]/sqrt(w(x[i]))``. The weight function is |
|
available as `hermweight`. |
|
|
|
References |
|
---------- |
|
.. [1] Wikipedia, "Curve fitting", |
|
https://en.wikipedia.org/wiki/Curve_fitting |
|
|
|
Examples |
|
-------- |
|
>>> import numpy as np |
|
>>> from numpy.polynomial.hermite import hermfit, hermval |
|
>>> x = np.linspace(-10, 10) |
|
>>> rng = np.random.default_rng() |
|
>>> err = rng.normal(scale=1./10, size=len(x)) |
|
>>> y = hermval(x, [1, 2, 3]) + err |
|
>>> hermfit(x, y, 2) |
|
array([1.02294967, 2.00016403, 2.99994614]) # may vary |
|
|
|
""" |
|
return pu._fit(hermvander, x, y, deg, rcond, full, w) |
|
|
|
|
|
def hermcompanion(c): |
|
"""Return the scaled companion matrix of c. |
|
|
|
The basis polynomials are scaled so that the companion matrix is |
|
symmetric when `c` is an Hermite 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 Hermite series coefficients ordered from low to high |
|
degree. |
|
|
|
Returns |
|
------- |
|
mat : ndarray |
|
Scaled companion matrix of dimensions (deg, deg). |
|
|
|
Notes |
|
----- |
|
|
|
.. versionadded:: 1.7.0 |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermcompanion |
|
>>> hermcompanion([1, 0, 1]) |
|
array([[0. , 0.35355339], |
|
[0.70710678, 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([[-.5*c[0]/c[1]]]) |
|
|
|
n = len(c) - 1 |
|
mat = np.zeros((n, n), dtype=c.dtype) |
|
scl = np.hstack((1., 1./np.sqrt(2.*np.arange(n - 1, 0, -1)))) |
|
scl = np.multiply.accumulate(scl)[::-1] |
|
top = mat.reshape(-1)[1::n+1] |
|
bot = mat.reshape(-1)[n::n+1] |
|
top[...] = np.sqrt(.5*np.arange(1, n)) |
|
bot[...] = top |
|
mat[:, -1] -= scl*c[:-1]/(2.0*c[-1]) |
|
return mat |
|
|
|
|
|
def hermroots(c): |
|
""" |
|
Compute the roots of a Hermite series. |
|
|
|
Return the roots (a.k.a. "zeros") of the polynomial |
|
|
|
.. math:: p(x) = \\sum_i c[i] * H_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.legendre.legroots |
|
numpy.polynomial.laguerre.lagroots |
|
numpy.polynomial.chebyshev.chebroots |
|
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 Hermite series basis polynomials aren't powers of `x` so the |
|
results of this function may seem unintuitive. |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermroots, hermfromroots |
|
>>> coef = hermfromroots([-1, 0, 1]) |
|
>>> coef |
|
array([0. , 0.25 , 0. , 0.125]) |
|
>>> hermroots(coef) |
|
array([-1.00000000e+00, -1.38777878e-17, 1.00000000e+00]) |
|
|
|
""" |
|
|
|
[c] = pu.as_series([c]) |
|
if len(c) <= 1: |
|
return np.array([], dtype=c.dtype) |
|
if len(c) == 2: |
|
return np.array([-.5*c[0]/c[1]]) |
|
|
|
|
|
m = hermcompanion(c)[::-1,::-1] |
|
r = la.eigvals(m) |
|
r.sort() |
|
return r |
|
|
|
|
|
def _normed_hermite_n(x, n): |
|
""" |
|
Evaluate a normalized Hermite polynomial. |
|
|
|
Compute the value of the normalized Hermite polynomial of degree ``n`` |
|
at the points ``x``. |
|
|
|
|
|
Parameters |
|
---------- |
|
x : ndarray of double. |
|
Points at which to evaluate the function |
|
n : int |
|
Degree of the normalized Hermite function to be evaluated. |
|
|
|
Returns |
|
------- |
|
values : ndarray |
|
The shape of the return value is described above. |
|
|
|
Notes |
|
----- |
|
.. versionadded:: 1.10.0 |
|
|
|
This function is needed for finding the Gauss points and integration |
|
weights for high degrees. The values of the standard Hermite functions |
|
overflow when n >= 207. |
|
|
|
""" |
|
if n == 0: |
|
return np.full(x.shape, 1/np.sqrt(np.sqrt(np.pi))) |
|
|
|
c0 = 0. |
|
c1 = 1./np.sqrt(np.sqrt(np.pi)) |
|
nd = float(n) |
|
for i in range(n - 1): |
|
tmp = c0 |
|
c0 = -c1*np.sqrt((nd - 1.)/nd) |
|
c1 = tmp + c1*x*np.sqrt(2./nd) |
|
nd = nd - 1.0 |
|
return c0 + c1*x*np.sqrt(2) |
|
|
|
|
|
def hermgauss(deg): |
|
""" |
|
Gauss-Hermite quadrature. |
|
|
|
Computes the sample points and weights for Gauss-Hermite quadrature. |
|
These sample points and weights will correctly integrate polynomials of |
|
degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]` |
|
with the weight function :math:`f(x) = \\exp(-x^2)`. |
|
|
|
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 / (H'_n(x_k) * H_{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:`H_n`, and then scaling the results to get |
|
the right value when integrating 1. |
|
|
|
Examples |
|
-------- |
|
>>> from numpy.polynomial.hermite import hermgauss |
|
>>> hermgauss(2) |
|
(array([-0.70710678, 0.70710678]), array([0.88622693, 0.88622693])) |
|
|
|
""" |
|
ideg = pu._as_int(deg, "deg") |
|
if ideg <= 0: |
|
raise ValueError("deg must be a positive integer") |
|
|
|
|
|
|
|
c = np.array([0]*deg + [1], dtype=np.float64) |
|
m = hermcompanion(c) |
|
x = la.eigvalsh(m) |
|
|
|
|
|
dy = _normed_hermite_n(x, ideg) |
|
df = _normed_hermite_n(x, ideg - 1) * np.sqrt(2*ideg) |
|
x -= dy/df |
|
|
|
|
|
|
|
fm = _normed_hermite_n(x, ideg - 1) |
|
fm /= np.abs(fm).max() |
|
w = 1/(fm * fm) |
|
|
|
|
|
w = (w + w[::-1])/2 |
|
x = (x - x[::-1])/2 |
|
|
|
|
|
w *= np.sqrt(np.pi) / w.sum() |
|
|
|
return x, w |
|
|
|
|
|
def hermweight(x): |
|
""" |
|
Weight function of the Hermite polynomials. |
|
|
|
The weight function is :math:`\\exp(-x^2)` and the interval of |
|
integration is :math:`[-\\inf, \\inf]`. the Hermite 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 |
|
----- |
|
|
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.. versionadded:: 1.7.0 |
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Examples |
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-------- |
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>>> import numpy as np |
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>>> from numpy.polynomial.hermite import hermweight |
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>>> x = np.arange(-2, 2) |
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>>> hermweight(x) |
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array([0.01831564, 0.36787944, 1. , 0.36787944]) |
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""" |
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w = np.exp(-x**2) |
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return w |
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class Hermite(ABCPolyBase): |
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"""An Hermite series class. |
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The Hermite class provides the standard Python numerical methods |
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'+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the |
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attributes and methods listed below. |
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Parameters |
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---------- |
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coef : array_like |
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Hermite coefficients in order of increasing degree, i.e, |
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``(1, 2, 3)`` gives ``1*H_0(x) + 2*H_1(x) + 3*H_2(x)``. |
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domain : (2,) array_like, optional |
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Domain to use. The interval ``[domain[0], domain[1]]`` is mapped |
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to the interval ``[window[0], window[1]]`` by shifting and scaling. |
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The default value is [-1., 1.]. |
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window : (2,) array_like, optional |
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Window, see `domain` for its use. The default value is [-1., 1.]. |
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.. versionadded:: 1.6.0 |
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symbol : str, optional |
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Symbol used to represent the independent variable in string |
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representations of the polynomial expression, e.g. for printing. |
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The symbol must be a valid Python identifier. Default value is 'x'. |
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.. versionadded:: 1.24 |
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""" |
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_add = staticmethod(hermadd) |
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_sub = staticmethod(hermsub) |
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_mul = staticmethod(hermmul) |
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_div = staticmethod(hermdiv) |
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_pow = staticmethod(hermpow) |
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_val = staticmethod(hermval) |
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_int = staticmethod(hermint) |
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_der = staticmethod(hermder) |
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_fit = staticmethod(hermfit) |
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_line = staticmethod(hermline) |
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_roots = staticmethod(hermroots) |
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_fromroots = staticmethod(hermfromroots) |
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domain = np.array(hermdomain) |
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window = np.array(hermdomain) |
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basis_name = 'H' |
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