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1741 lines
53 KiB
Python
1741 lines
53 KiB
Python
"""
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==============================================================
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Hermite Series, "Physicists" (:mod:`numpy.polynomial.hermite`)
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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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from . import polyutils as pu
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from ._polybase import ABCPolyBase
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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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def poly2herm(pol):
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"""
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poly2herm(pol)
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Convert a polynomial to a Hermite series.
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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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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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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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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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[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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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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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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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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See Also
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--------
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poly2herm
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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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from .polynomial import polyadd, polymulx, polysub
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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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# i is the current degree of c1
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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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#
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# These are constant arrays are of integer type so as to be compatible
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# with the widest range of other types, such as Decimal.
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#
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# Hermite
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hermdomain = np.array([-1., 1.])
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# Hermite coefficients representing zero.
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hermzero = np.array([0])
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# Hermite coefficients representing one.
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hermone = np.array([1])
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# Hermite coefficients representing the identity x.
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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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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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The function returns the coefficients of the polynomial
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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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.. 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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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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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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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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-----
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Unlike multiplication, division, etc., the difference of two Hermite
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series is a Hermite series (without having to "reproject" the result
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onto the basis set) so subtraction, just like that of "standard"
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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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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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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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--------
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hermadd, hermsub, hermmul, hermdiv, hermpow
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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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xP_i(x) = (P_{i + 1}(x)/2 + i*P_{i - 1}(x))
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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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# c is a trimmed copy
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[c] = pu.as_series([c])
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# The zero series needs special treatment
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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
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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 product.
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|
|
See Also
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--------
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hermadd, hermsub, hermmulx, hermdiv, hermpow
|
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|
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Notes
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-----
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In general, the (polynomial) product of two C-series results in terms
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that are not in the Hermite polynomial basis set. Thus, to express
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the product as a Hermite series, it is necessary to "reproject" the
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product onto said basis set, which may produce "unintuitive" (but
|
|
correct) results; see Examples section below.
|
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|
|
Examples
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--------
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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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# s1, s2 are trimmed copies
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[c1, c2] = pu.as_series([c1, c2])
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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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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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|
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def hermdiv(c1, c2):
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"""
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Divide one Hermite series by another.
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Returns the quotient-with-remainder of two Hermite series
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`c1` / `c2`. The arguments are sequences of coefficients from lowest
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order "term" to highest, e.g., [1,2,3] represents the series
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``P_0 + 2*P_1 + 3*P_2``.
|
|
|
|
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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[quo, rem] : ndarrays
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Of Hermite series coefficients representing the quotient and
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remainder.
|
|
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|
See Also
|
|
--------
|
|
hermadd, hermsub, hermmulx, hermmul, hermpow
|
|
|
|
Notes
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-----
|
|
In general, the (polynomial) division of one Hermite series by another
|
|
results in quotient and remainder terms that are not in the Hermite
|
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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
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>>> hermdiv([ 52., 29., 52., 7., 6.], [0, 1, 2])
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(array([1., 2., 3.]), array([0.]))
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>>> hermdiv([ 54., 31., 52., 7., 6.], [0, 1, 2])
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(array([1., 2., 3.]), array([2., 2.]))
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>>> hermdiv([ 53., 30., 52., 7., 6.], [0, 1, 2])
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(array([1., 2., 3.]), array([1., 1.]))
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"""
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return pu._div(hermmul, c1, c2)
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|
|
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def hermpow(c, pow, maxpower=16):
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"""Raise a Hermite series to a power.
|
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|
|
Returns the Hermite series `c` raised to the power `pow`. The
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argument `c` is a sequence of coefficients ordered from low to high.
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i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.``
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|
|
Parameters
|
|
----------
|
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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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|
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)
|
|
|
|
|
|
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).
|
|
|
|
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).
|
|
|
|
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.
|
|
|
|
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
|
|
|
|
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
|
|
|
|
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
|
|
|
|
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
|
|
|
|
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
|
|
|
|
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
|
|
|
|
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).
|
|
|
|
Examples
|
|
--------
|
|
>>> from numpy.polynomial.hermite import hermcompanion
|
|
>>> hermcompanion([1, 0, 1])
|
|
array([[0. , 0.35355339],
|
|
[0.70710678, 0. ]])
|
|
|
|
"""
|
|
# c is a trimmed copy
|
|
[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
|
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numpy.polynomial.chebyshev.chebroots
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numpy.polynomial.hermite_e.hermeroots
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Notes
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-----
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The root estimates are obtained as the eigenvalues of the companion
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matrix, Roots far from the origin of the complex plane may have large
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errors due to the numerical instability of the series for such
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values. Roots with multiplicity greater than 1 will also show larger
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errors as the value of the series near such points is relatively
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insensitive to errors in the roots. Isolated roots near the origin can
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be improved by a few iterations of Newton's method.
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The Hermite series basis polynomials aren't powers of `x` so the
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results of this function may seem unintuitive.
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Examples
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--------
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>>> from numpy.polynomial.hermite import hermroots, hermfromroots
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>>> coef = hermfromroots([-1, 0, 1])
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>>> coef
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array([0. , 0.25 , 0. , 0.125])
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>>> hermroots(coef)
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array([-1.00000000e+00, -1.38777878e-17, 1.00000000e+00])
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"""
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# c is a trimmed copy
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[c] = pu.as_series([c])
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if len(c) <= 1:
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return np.array([], dtype=c.dtype)
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if len(c) == 2:
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return np.array([-.5 * c[0] / c[1]])
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# rotated companion matrix reduces error
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m = hermcompanion(c)[::-1, ::-1]
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r = la.eigvals(m)
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r.sort()
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return r
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def _normed_hermite_n(x, n):
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"""
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Evaluate a normalized Hermite polynomial.
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Compute the value of the normalized Hermite polynomial of degree ``n``
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at the points ``x``.
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Parameters
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----------
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x : ndarray of double.
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Points at which to evaluate the function
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n : int
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Degree of the normalized Hermite function to be evaluated.
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Returns
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-------
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values : ndarray
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The shape of the return value is described above.
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Notes
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-----
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This function is needed for finding the Gauss points and integration
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weights for high degrees. The values of the standard Hermite functions
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overflow when n >= 207.
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"""
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if n == 0:
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return np.full(x.shape, 1 / np.sqrt(np.sqrt(np.pi)))
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c0 = 0.
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c1 = 1. / np.sqrt(np.sqrt(np.pi))
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nd = float(n)
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for i in range(n - 1):
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tmp = c0
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c0 = -c1 * np.sqrt((nd - 1.) / nd)
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c1 = tmp + c1 * x * np.sqrt(2. / nd)
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nd = nd - 1.0
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return c0 + c1 * x * np.sqrt(2)
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def hermgauss(deg):
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"""
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Gauss-Hermite quadrature.
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Computes the sample points and weights for Gauss-Hermite quadrature.
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These sample points and weights will correctly integrate polynomials of
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degree :math:`2*deg - 1` or less over the interval :math:`[-\\inf, \\inf]`
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with the weight function :math:`f(x) = \\exp(-x^2)`.
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Parameters
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----------
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deg : int
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Number of sample points and weights. It must be >= 1.
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Returns
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-------
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x : ndarray
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1-D ndarray containing the sample points.
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y : ndarray
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1-D ndarray containing the weights.
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Notes
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-----
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The results have only been tested up to degree 100, higher degrees may
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be problematic. The weights are determined by using the fact that
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.. math:: w_k = c / (H'_n(x_k) * H_{n-1}(x_k))
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where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
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is the k'th root of :math:`H_n`, and then scaling the results to get
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the right value when integrating 1.
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Examples
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--------
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>>> from numpy.polynomial.hermite import hermgauss
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>>> hermgauss(2)
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(array([-0.70710678, 0.70710678]), array([0.88622693, 0.88622693]))
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"""
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ideg = pu._as_int(deg, "deg")
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if ideg <= 0:
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raise ValueError("deg must be a positive integer")
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# first approximation of roots. We use the fact that the companion
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# matrix is symmetric in this case in order to obtain better zeros.
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c = np.array([0] * deg + [1], dtype=np.float64)
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m = hermcompanion(c)
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x = la.eigvalsh(m)
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# improve roots by one application of Newton
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dy = _normed_hermite_n(x, ideg)
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df = _normed_hermite_n(x, ideg - 1) * np.sqrt(2 * ideg)
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x -= dy / df
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# compute the weights. We scale the factor to avoid possible numerical
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# overflow.
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fm = _normed_hermite_n(x, ideg - 1)
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fm /= np.abs(fm).max()
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w = 1 / (fm * fm)
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# for Hermite we can also symmetrize
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w = (w + w[::-1]) / 2
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x = (x - x[::-1]) / 2
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# scale w to get the right value
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w *= np.sqrt(np.pi) / w.sum()
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return x, w
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def hermweight(x):
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"""
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Weight function of the Hermite polynomials.
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The weight function is :math:`\\exp(-x^2)` and the interval of
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integration is :math:`[-\\inf, \\inf]`. the Hermite polynomials are
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orthogonal, but not normalized, with respect to this weight function.
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Parameters
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----------
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x : array_like
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Values at which the weight function will be computed.
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Returns
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-------
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w : ndarray
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The weight function at `x`.
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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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#
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# Hermite series class
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#
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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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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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# Virtual Functions
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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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# Virtual properties
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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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