1665 lines
50 KiB
Python
1665 lines
50 KiB
Python
"""
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==================================================
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Legendre Series (:mod:`numpy.polynomial.legendre`)
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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 Legendre series, including a `Legendre` 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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Legendre
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Constants
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---------
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.. autosummary::
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:toctree: generated/
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legdomain
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legzero
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legone
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legx
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Arithmetic
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----------
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.. autosummary::
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:toctree: generated/
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legadd
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legsub
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legmulx
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legmul
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legdiv
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legpow
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legval
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legval2d
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legval3d
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leggrid2d
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leggrid3d
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Calculus
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--------
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.. autosummary::
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:toctree: generated/
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legder
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legint
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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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legfromroots
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legroots
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legvander
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legvander2d
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legvander3d
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leggauss
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legweight
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legcompanion
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legfit
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legtrim
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legline
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leg2poly
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poly2leg
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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.core.multiarray 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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'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd',
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'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder',
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'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander',
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'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d',
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'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion',
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'leggauss', 'legweight']
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legtrim = pu.trimcoef
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def poly2leg(pol):
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"""
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Convert a polynomial to a Legendre 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 Legendre 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 Legendre
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series.
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See Also
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--------
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leg2poly
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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 import polynomial as P
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>>> p = P.Polynomial(np.arange(4))
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>>> p
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Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
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>>> c = P.Legendre(P.legendre.poly2leg(p.coef))
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>>> c
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Legendre([ 1. , 3.25, 1. , 0.75], domain=[-1, 1], window=[-1, 1]) # may vary
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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 = legadd(legmulx(res), pol[i])
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return res
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def leg2poly(c):
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"""
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Convert a Legendre series to a polynomial.
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Convert an array representing the coefficients of a Legendre 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 Legendre 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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poly2leg
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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 import polynomial as P
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>>> c = P.Legendre(range(4))
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>>> c
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Legendre([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
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>>> p = c.convert(kind=P.Polynomial)
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>>> p
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Polynomial([-1. , -3.5, 3. , 7.5], domain=[-1., 1.], window=[-1., 1.])
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>>> P.legendre.leg2poly(range(4))
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array([-1. , -3.5, 3. , 7.5])
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"""
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from .polynomial import polyadd, polysub, polymulx
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[c] = pu.as_series([c])
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n = len(c)
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if n < 3:
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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*(i - 1))/i)
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c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i)
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return polyadd(c0, polymulx(c1))
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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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# Legendre
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legdomain = np.array([-1, 1])
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# Legendre coefficients representing zero.
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legzero = np.array([0])
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# Legendre coefficients representing one.
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legone = np.array([1])
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# Legendre coefficients representing the identity x.
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legx = np.array([0, 1])
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def legline(off, scl):
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"""
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Legendre 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 Legendre 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.laguerre.lagline
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numpy.polynomial.hermite.hermline
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numpy.polynomial.hermite_e.hermeline
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Examples
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--------
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>>> import numpy.polynomial.legendre as L
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>>> L.legline(3,2)
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array([3, 2])
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>>> L.legval(-3, L.legline(3,2)) # should be -3
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-3.0
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"""
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if scl != 0:
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return np.array([off, scl])
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else:
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return np.array([off])
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def legfromroots(roots):
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"""
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Generate a Legendre 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 Legendre form, where the `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 * L_1(x) + ... + c_n * L_n(x)
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The coefficient of the last term is not generally 1 for monic
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polynomials in Legendre 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.chebyshev.chebfromroots
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numpy.polynomial.laguerre.lagfromroots
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numpy.polynomial.hermite.hermfromroots
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numpy.polynomial.hermite_e.hermefromroots
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Examples
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--------
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>>> import numpy.polynomial.legendre as L
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>>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis
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array([ 0. , -0.4, 0. , 0.4])
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>>> j = complex(0,1)
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>>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis
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array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) # may vary
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"""
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return pu._fromroots(legline, legmul, roots)
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def legadd(c1, c2):
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"""
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Add one Legendre series to another.
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Returns the sum of two Legendre 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 Legendre 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 Legendre series of their sum.
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See Also
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--------
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legsub, legmulx, legmul, legdiv, legpow
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Notes
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-----
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Unlike multiplication, division, etc., the sum of two Legendre series
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is a Legendre 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 import legendre as L
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>>> c1 = (1,2,3)
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>>> c2 = (3,2,1)
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>>> L.legadd(c1,c2)
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array([4., 4., 4.])
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"""
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return pu._add(c1, c2)
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def legsub(c1, c2):
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"""
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Subtract one Legendre series from another.
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Returns the difference of two Legendre 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 Legendre 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 Legendre series coefficients representing their difference.
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See Also
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--------
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legadd, legmulx, legmul, legdiv, legpow
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Notes
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-----
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Unlike multiplication, division, etc., the difference of two Legendre
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series is a Legendre 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 import legendre as L
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>>> c1 = (1,2,3)
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>>> c2 = (3,2,1)
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>>> L.legsub(c1,c2)
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array([-2., 0., 2.])
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>>> L.legsub(c2,c1) # -C.legsub(c1,c2)
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array([ 2., 0., -2.])
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"""
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return pu._sub(c1, c2)
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def legmulx(c):
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"""Multiply a Legendre series by x.
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Multiply the Legendre 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 Legendre 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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legadd, legmul, legdiv, legpow
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Notes
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-----
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The multiplication uses the recursion relationship for Legendre
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polynomials in the form
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.. math::
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xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1)
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Examples
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--------
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>>> from numpy.polynomial import legendre as L
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>>> L.legmulx([1,2,3])
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array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary
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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]
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for i in range(1, len(c)):
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j = i + 1
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k = i - 1
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s = i + j
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prd[j] = (c[i]*j)/s
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prd[k] += (c[i]*i)/s
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return prd
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def legmul(c1, c2):
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"""
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Multiply one Legendre series by another.
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Returns the product of two Legendre 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 Legendre 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 Legendre series coefficients representing their product.
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See Also
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--------
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legadd, legsub, legmulx, legdiv, legpow
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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 Legendre polynomial basis set. Thus, to express
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the product as a Legendre series, it is necessary to "reproject" the
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product onto said basis set, which may produce "unintuitive" (but
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correct) results; see Examples section below.
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Examples
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--------
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>>> from numpy.polynomial import legendre as L
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>>> c1 = (1,2,3)
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>>> c2 = (3,2)
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>>> L.legmul(c1,c2) # multiplication requires "reprojection"
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array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) # may vary
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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 = legsub(c[-i]*xs, (c1*(nd - 1))/nd)
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c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd)
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return legadd(c0, legmulx(c1))
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def legdiv(c1, c2):
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"""
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Divide one Legendre series by another.
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Returns the quotient-with-remainder of two Legendre 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
|
|
``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 Legendre 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 Legendre series coefficients representing the quotient and
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remainder.
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See Also
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--------
|
|
legadd, legsub, legmulx, legmul, legpow
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|
|
Notes
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-----
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In general, the (polynomial) division of one Legendre series by another
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results in quotient and remainder terms that are not in the Legendre
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polynomial basis set. Thus, to express these results as a Legendre
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series, it is necessary to "reproject" the results onto the Legendre
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basis set, which may produce "unintuitive" (but correct) results; see
|
|
Examples section below.
|
|
|
|
Examples
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|
--------
|
|
>>> from numpy.polynomial import legendre as L
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>>> c1 = (1,2,3)
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>>> c2 = (3,2,1)
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>>> L.legdiv(c1,c2) # quotient "intuitive," remainder not
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(array([3.]), array([-8., -4.]))
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>>> c2 = (0,1,2,3)
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>>> L.legdiv(c2,c1) # neither "intuitive"
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(array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) # may vary
|
|
|
|
"""
|
|
return pu._div(legmul, c1, c2)
|
|
|
|
|
|
def legpow(c, pow, maxpower=16):
|
|
"""Raise a Legendre series to a power.
|
|
|
|
Returns the Legendre series `c` raised to the power `pow`. The
|
|
argument `c` is a sequence of coefficients ordered from low to high.
|
|
i.e., [1,2,3] is the series ``P_0 + 2*P_1 + 3*P_2.``
|
|
|
|
Parameters
|
|
----------
|
|
c : array_like
|
|
1-D array of Legendre series coefficients ordered from low to
|
|
high.
|
|
pow : integer
|
|
Power to which the series will be raised
|
|
maxpower : integer, optional
|
|
Maximum power allowed. This is mainly to limit growth of the series
|
|
to unmanageable size. Default is 16
|
|
|
|
Returns
|
|
-------
|
|
coef : ndarray
|
|
Legendre series of power.
|
|
|
|
See Also
|
|
--------
|
|
legadd, legsub, legmulx, legmul, legdiv
|
|
|
|
"""
|
|
return pu._pow(legmul, c, pow, maxpower)
|
|
|
|
|
|
def legder(c, m=1, scl=1, axis=0):
|
|
"""
|
|
Differentiate a Legendre series.
|
|
|
|
Returns the Legendre series coefficients `c` differentiated `m` times
|
|
along `axis`. At each iteration the result is multiplied by `scl` (the
|
|
scaling factor is for use in a linear change of variable). The argument
|
|
`c` is an array of coefficients from low to high degree along each
|
|
axis, e.g., [1,2,3] represents the series ``1*L_0 + 2*L_1 + 3*L_2``
|
|
while [[1,2],[1,2]] represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) +
|
|
2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is
|
|
``y``.
|
|
|
|
Parameters
|
|
----------
|
|
c : array_like
|
|
Array of Legendre series coefficients. If c is multidimensional the
|
|
different axis correspond to different variables with the degree in
|
|
each axis given by the corresponding index.
|
|
m : int, optional
|
|
Number of derivatives taken, must be non-negative. (Default: 1)
|
|
scl : scalar, optional
|
|
Each differentiation is multiplied by `scl`. The end result is
|
|
multiplication by ``scl**m``. This is for use in a linear change of
|
|
variable. (Default: 1)
|
|
axis : int, optional
|
|
Axis over which the derivative is taken. (Default: 0).
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
Returns
|
|
-------
|
|
der : ndarray
|
|
Legendre series of the derivative.
|
|
|
|
See Also
|
|
--------
|
|
legint
|
|
|
|
Notes
|
|
-----
|
|
In general, the result of differentiating a Legendre series does not
|
|
resemble the same operation on a power series. Thus the result of this
|
|
function may be "unintuitive," albeit correct; see Examples section
|
|
below.
|
|
|
|
Examples
|
|
--------
|
|
>>> from numpy.polynomial import legendre as L
|
|
>>> c = (1,2,3,4)
|
|
>>> L.legder(c)
|
|
array([ 6., 9., 20.])
|
|
>>> L.legder(c, 3)
|
|
array([60.])
|
|
>>> L.legder(c, scl=-1)
|
|
array([ -6., -9., -20.])
|
|
>>> L.legder(c, 2,-1)
|
|
array([ 9., 60.])
|
|
|
|
"""
|
|
c = np.array(c, ndmin=1, copy=True)
|
|
if c.dtype.char in '?bBhHiIlLqQpP':
|
|
c = c.astype(np.double)
|
|
cnt = pu._deprecate_as_int(m, "the order of derivation")
|
|
iaxis = pu._deprecate_as_int(axis, "the axis")
|
|
if cnt < 0:
|
|
raise ValueError("The order of derivation must be non-negative")
|
|
iaxis = normalize_axis_index(iaxis, c.ndim)
|
|
|
|
if cnt == 0:
|
|
return c
|
|
|
|
c = np.moveaxis(c, iaxis, 0)
|
|
n = len(c)
|
|
if cnt >= n:
|
|
c = c[:1]*0
|
|
else:
|
|
for i in range(cnt):
|
|
n = n - 1
|
|
c *= scl
|
|
der = np.empty((n,) + c.shape[1:], dtype=c.dtype)
|
|
for j in range(n, 2, -1):
|
|
der[j - 1] = (2*j - 1)*c[j]
|
|
c[j - 2] += c[j]
|
|
if n > 1:
|
|
der[1] = 3*c[2]
|
|
der[0] = c[1]
|
|
c = der
|
|
c = np.moveaxis(c, 0, iaxis)
|
|
return c
|
|
|
|
|
|
def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
|
|
"""
|
|
Integrate a Legendre series.
|
|
|
|
Returns the Legendre series coefficients `c` integrated `m` times from
|
|
`lbnd` along `axis`. At each iteration the resulting series is
|
|
**multiplied** by `scl` and an integration constant, `k`, is added.
|
|
The scaling factor is for use in a linear change of variable. ("Buyer
|
|
beware": note that, depending on what one is doing, one may want `scl`
|
|
to be the reciprocal of what one might expect; for more information,
|
|
see the Notes section below.) The argument `c` is an array of
|
|
coefficients from low to high degree along each axis, e.g., [1,2,3]
|
|
represents the series ``L_0 + 2*L_1 + 3*L_2`` while [[1,2],[1,2]]
|
|
represents ``1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) +
|
|
2*L_1(x)*L_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
|
|
|
|
Parameters
|
|
----------
|
|
c : array_like
|
|
Array of Legendre series coefficients. If c is multidimensional the
|
|
different axis correspond to different variables with the degree in
|
|
each axis given by the corresponding index.
|
|
m : int, optional
|
|
Order of integration, must be positive. (Default: 1)
|
|
k : {[], list, scalar}, optional
|
|
Integration constant(s). The value of the first integral at
|
|
``lbnd`` is the first value in the list, the value of the second
|
|
integral at ``lbnd`` is the second value, etc. If ``k == []`` (the
|
|
default), all constants are set to zero. If ``m == 1``, a single
|
|
scalar can be given instead of a list.
|
|
lbnd : scalar, optional
|
|
The lower bound of the integral. (Default: 0)
|
|
scl : scalar, optional
|
|
Following each integration the result is *multiplied* by `scl`
|
|
before the integration constant is added. (Default: 1)
|
|
axis : int, optional
|
|
Axis over which the integral is taken. (Default: 0).
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
Returns
|
|
-------
|
|
S : ndarray
|
|
Legendre series coefficient array of the integral.
|
|
|
|
Raises
|
|
------
|
|
ValueError
|
|
If ``m < 0``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
|
|
``np.ndim(scl) != 0``.
|
|
|
|
See Also
|
|
--------
|
|
legder
|
|
|
|
Notes
|
|
-----
|
|
Note that the result of each integration is *multiplied* by `scl`.
|
|
Why is this important to note? Say one is making a linear change of
|
|
variable :math:`u = ax + b` in an integral relative to `x`. Then
|
|
:math:`dx = du/a`, so one will need to set `scl` equal to
|
|
:math:`1/a` - perhaps not what one would have first thought.
|
|
|
|
Also note that, in general, the result of integrating a C-series needs
|
|
to be "reprojected" onto the C-series basis set. Thus, typically,
|
|
the result of this function is "unintuitive," albeit correct; see
|
|
Examples section below.
|
|
|
|
Examples
|
|
--------
|
|
>>> from numpy.polynomial import legendre as L
|
|
>>> c = (1,2,3)
|
|
>>> L.legint(c)
|
|
array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary
|
|
>>> L.legint(c, 3)
|
|
array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, # may vary
|
|
-1.73472348e-18, 1.90476190e-02, 9.52380952e-03])
|
|
>>> L.legint(c, k=3)
|
|
array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary
|
|
>>> L.legint(c, lbnd=-2)
|
|
array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary
|
|
>>> L.legint(c, scl=2)
|
|
array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) # may vary
|
|
|
|
"""
|
|
c = np.array(c, ndmin=1, copy=True)
|
|
if c.dtype.char in '?bBhHiIlLqQpP':
|
|
c = c.astype(np.double)
|
|
if not np.iterable(k):
|
|
k = [k]
|
|
cnt = pu._deprecate_as_int(m, "the order of integration")
|
|
iaxis = pu._deprecate_as_int(axis, "the axis")
|
|
if cnt < 0:
|
|
raise ValueError("The order of integration must be non-negative")
|
|
if len(k) > cnt:
|
|
raise ValueError("Too many integration constants")
|
|
if np.ndim(lbnd) != 0:
|
|
raise ValueError("lbnd must be a scalar.")
|
|
if np.ndim(scl) != 0:
|
|
raise ValueError("scl must be a scalar.")
|
|
iaxis = normalize_axis_index(iaxis, c.ndim)
|
|
|
|
if cnt == 0:
|
|
return c
|
|
|
|
c = np.moveaxis(c, iaxis, 0)
|
|
k = list(k) + [0]*(cnt - len(k))
|
|
for i in range(cnt):
|
|
n = len(c)
|
|
c *= scl
|
|
if n == 1 and np.all(c[0] == 0):
|
|
c[0] += k[i]
|
|
else:
|
|
tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype)
|
|
tmp[0] = c[0]*0
|
|
tmp[1] = c[0]
|
|
if n > 1:
|
|
tmp[2] = c[1]/3
|
|
for j in range(2, n):
|
|
t = c[j]/(2*j + 1)
|
|
tmp[j + 1] = t
|
|
tmp[j - 1] -= t
|
|
tmp[0] += k[i] - legval(lbnd, tmp)
|
|
c = tmp
|
|
c = np.moveaxis(c, 0, iaxis)
|
|
return c
|
|
|
|
|
|
def legval(x, c, tensor=True):
|
|
"""
|
|
Evaluate a Legendre series at points x.
|
|
|
|
If `c` is of length `n + 1`, this function returns the value:
|
|
|
|
.. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_n(x)
|
|
|
|
The parameter `x` is converted to an array only if it is a tuple or a
|
|
list, otherwise it is treated as a scalar. In either case, either `x`
|
|
or its elements must support multiplication and addition both with
|
|
themselves and with the elements of `c`.
|
|
|
|
If `c` is a 1-D array, then `p(x)` will have the same shape as `x`. If
|
|
`c` is multidimensional, then the shape of the result depends on the
|
|
value of `tensor`. If `tensor` is true the shape will be c.shape[1:] +
|
|
x.shape. If `tensor` is false the shape will be c.shape[1:]. Note that
|
|
scalars have shape (,).
|
|
|
|
Trailing zeros in the coefficients will be used in the evaluation, so
|
|
they should be avoided if efficiency is a concern.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like, compatible object
|
|
If `x` is a list or tuple, it is converted to an ndarray, otherwise
|
|
it is left unchanged and treated as a scalar. In either case, `x`
|
|
or its elements must support addition and multiplication with
|
|
themselves and with the elements of `c`.
|
|
c : array_like
|
|
Array of coefficients ordered so that the coefficients for terms of
|
|
degree n are contained in c[n]. If `c` is multidimensional the
|
|
remaining indices enumerate multiple polynomials. In the two
|
|
dimensional case the coefficients may be thought of as stored in
|
|
the columns of `c`.
|
|
tensor : boolean, optional
|
|
If True, the shape of the coefficient array is extended with ones
|
|
on the right, one for each dimension of `x`. Scalars have dimension 0
|
|
for this action. The result is that every column of coefficients in
|
|
`c` is evaluated for every element of `x`. If False, `x` is broadcast
|
|
over the columns of `c` for the evaluation. This keyword is useful
|
|
when `c` is multidimensional. The default value is True.
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
Returns
|
|
-------
|
|
values : ndarray, algebra_like
|
|
The shape of the return value is described above.
|
|
|
|
See Also
|
|
--------
|
|
legval2d, leggrid2d, legval3d, leggrid3d
|
|
|
|
Notes
|
|
-----
|
|
The evaluation uses Clenshaw recursion, aka synthetic division.
|
|
|
|
"""
|
|
c = np.array(c, ndmin=1, copy=False)
|
|
if c.dtype.char in '?bBhHiIlLqQpP':
|
|
c = c.astype(np.double)
|
|
if isinstance(x, (tuple, list)):
|
|
x = np.asarray(x)
|
|
if isinstance(x, np.ndarray) and tensor:
|
|
c = c.reshape(c.shape + (1,)*x.ndim)
|
|
|
|
if len(c) == 1:
|
|
c0 = c[0]
|
|
c1 = 0
|
|
elif len(c) == 2:
|
|
c0 = c[0]
|
|
c1 = c[1]
|
|
else:
|
|
nd = len(c)
|
|
c0 = c[-2]
|
|
c1 = c[-1]
|
|
for i in range(3, len(c) + 1):
|
|
tmp = c0
|
|
nd = nd - 1
|
|
c0 = c[-i] - (c1*(nd - 1))/nd
|
|
c1 = tmp + (c1*x*(2*nd - 1))/nd
|
|
return c0 + c1*x
|
|
|
|
|
|
def legval2d(x, y, c):
|
|
"""
|
|
Evaluate a 2-D Legendre series at points (x, y).
|
|
|
|
This function returns the values:
|
|
|
|
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_j(y)
|
|
|
|
The parameters `x` and `y` are converted to arrays only if they are
|
|
tuples or a lists, otherwise they are treated as a scalars and they
|
|
must have the same shape after conversion. In either case, either `x`
|
|
and `y` or their elements must support multiplication and addition both
|
|
with themselves and with the elements of `c`.
|
|
|
|
If `c` is a 1-D array a one is implicitly appended to its shape to make
|
|
it 2-D. The shape of the result will be c.shape[2:] + x.shape.
|
|
|
|
Parameters
|
|
----------
|
|
x, y : array_like, compatible objects
|
|
The two dimensional series is evaluated at the points `(x, y)`,
|
|
where `x` and `y` must have the same shape. If `x` or `y` is a list
|
|
or tuple, it is first converted to an ndarray, otherwise it is left
|
|
unchanged and if it isn't an ndarray it is treated as a scalar.
|
|
c : array_like
|
|
Array of coefficients ordered so that the coefficient of the term
|
|
of multi-degree i,j is contained in ``c[i,j]``. If `c` has
|
|
dimension greater than two the remaining indices enumerate multiple
|
|
sets of coefficients.
|
|
|
|
Returns
|
|
-------
|
|
values : ndarray, compatible object
|
|
The values of the two dimensional Legendre series at points formed
|
|
from pairs of corresponding values from `x` and `y`.
|
|
|
|
See Also
|
|
--------
|
|
legval, leggrid2d, legval3d, leggrid3d
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
"""
|
|
return pu._valnd(legval, c, x, y)
|
|
|
|
|
|
def leggrid2d(x, y, c):
|
|
"""
|
|
Evaluate a 2-D Legendre series on the Cartesian product of x and y.
|
|
|
|
This function returns the values:
|
|
|
|
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * L_i(a) * L_j(b)
|
|
|
|
where the points `(a, b)` consist of all pairs formed by taking
|
|
`a` from `x` and `b` from `y`. The resulting points form a grid with
|
|
`x` in the first dimension and `y` in the second.
|
|
|
|
The parameters `x` and `y` are converted to arrays only if they are
|
|
tuples or a lists, otherwise they are treated as a scalars. In either
|
|
case, either `x` and `y` or their elements must support multiplication
|
|
and addition both with themselves and with the elements of `c`.
|
|
|
|
If `c` has fewer than two dimensions, ones are implicitly appended to
|
|
its shape to make it 2-D. The shape of the result will be c.shape[2:] +
|
|
x.shape + y.shape.
|
|
|
|
Parameters
|
|
----------
|
|
x, y : array_like, compatible objects
|
|
The two dimensional series is evaluated at the points in the
|
|
Cartesian product of `x` and `y`. If `x` or `y` is a list or
|
|
tuple, it is first converted to an ndarray, otherwise it is left
|
|
unchanged and, if it isn't an ndarray, it is treated as a scalar.
|
|
c : array_like
|
|
Array of coefficients ordered so that the coefficient of the term of
|
|
multi-degree i,j is contained in `c[i,j]`. If `c` has dimension
|
|
greater than two the remaining indices enumerate multiple sets of
|
|
coefficients.
|
|
|
|
Returns
|
|
-------
|
|
values : ndarray, compatible object
|
|
The values of the two dimensional Chebyshev series at points in the
|
|
Cartesian product of `x` and `y`.
|
|
|
|
See Also
|
|
--------
|
|
legval, legval2d, legval3d, leggrid3d
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
"""
|
|
return pu._gridnd(legval, c, x, y)
|
|
|
|
|
|
def legval3d(x, y, z, c):
|
|
"""
|
|
Evaluate a 3-D Legendre series at points (x, y, z).
|
|
|
|
This function returns the values:
|
|
|
|
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_k(z)
|
|
|
|
The parameters `x`, `y`, and `z` are converted to arrays only if
|
|
they are tuples or a lists, otherwise they are treated as a scalars and
|
|
they must have the same shape after conversion. In either case, either
|
|
`x`, `y`, and `z` or their elements must support multiplication and
|
|
addition both with themselves and with the elements of `c`.
|
|
|
|
If `c` has fewer than 3 dimensions, ones are implicitly appended to its
|
|
shape to make it 3-D. The shape of the result will be c.shape[3:] +
|
|
x.shape.
|
|
|
|
Parameters
|
|
----------
|
|
x, y, z : array_like, compatible object
|
|
The three dimensional series is evaluated at the points
|
|
`(x, y, z)`, where `x`, `y`, and `z` must have the same shape. If
|
|
any of `x`, `y`, or `z` is a list or tuple, it is first converted
|
|
to an ndarray, otherwise it is left unchanged and if it isn't an
|
|
ndarray it is treated as a scalar.
|
|
c : array_like
|
|
Array of coefficients ordered so that the coefficient of the term of
|
|
multi-degree i,j,k is contained in ``c[i,j,k]``. If `c` has dimension
|
|
greater than 3 the remaining indices enumerate multiple sets of
|
|
coefficients.
|
|
|
|
Returns
|
|
-------
|
|
values : ndarray, compatible object
|
|
The values of the multidimensional polynomial on points formed with
|
|
triples of corresponding values from `x`, `y`, and `z`.
|
|
|
|
See Also
|
|
--------
|
|
legval, legval2d, leggrid2d, leggrid3d
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
"""
|
|
return pu._valnd(legval, c, x, y, z)
|
|
|
|
|
|
def leggrid3d(x, y, z, c):
|
|
"""
|
|
Evaluate a 3-D Legendre series on the Cartesian product of x, y, and z.
|
|
|
|
This function returns the values:
|
|
|
|
.. math:: p(a,b,c) = \\sum_{i,j,k} c_{i,j,k} * L_i(a) * L_j(b) * L_k(c)
|
|
|
|
where the points `(a, b, c)` consist of all triples formed by taking
|
|
`a` from `x`, `b` from `y`, and `c` from `z`. The resulting points form
|
|
a grid with `x` in the first dimension, `y` in the second, and `z` in
|
|
the third.
|
|
|
|
The parameters `x`, `y`, and `z` are converted to arrays only if they
|
|
are tuples or a lists, otherwise they are treated as a scalars. In
|
|
either case, either `x`, `y`, and `z` or their elements must support
|
|
multiplication and addition both with themselves and with the elements
|
|
of `c`.
|
|
|
|
If `c` has fewer than three dimensions, ones are implicitly appended to
|
|
its shape to make it 3-D. The shape of the result will be c.shape[3:] +
|
|
x.shape + y.shape + z.shape.
|
|
|
|
Parameters
|
|
----------
|
|
x, y, z : array_like, compatible objects
|
|
The three dimensional series is evaluated at the points in the
|
|
Cartesian product of `x`, `y`, and `z`. If `x`,`y`, or `z` is a
|
|
list or tuple, it is first converted to an ndarray, otherwise it is
|
|
left unchanged and, if it isn't an ndarray, it is treated as a
|
|
scalar.
|
|
c : array_like
|
|
Array of coefficients ordered so that the coefficients for terms of
|
|
degree i,j are contained in ``c[i,j]``. If `c` has dimension
|
|
greater than two the remaining indices enumerate multiple sets of
|
|
coefficients.
|
|
|
|
Returns
|
|
-------
|
|
values : ndarray, compatible object
|
|
The values of the two dimensional polynomial at points in the Cartesian
|
|
product of `x` and `y`.
|
|
|
|
See Also
|
|
--------
|
|
legval, legval2d, leggrid2d, legval3d
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
"""
|
|
return pu._gridnd(legval, c, x, y, z)
|
|
|
|
|
|
def legvander(x, deg):
|
|
"""Pseudo-Vandermonde matrix of given degree.
|
|
|
|
Returns the pseudo-Vandermonde matrix of degree `deg` and sample points
|
|
`x`. The pseudo-Vandermonde matrix is defined by
|
|
|
|
.. math:: V[..., i] = L_i(x)
|
|
|
|
where `0 <= i <= deg`. The leading indices of `V` index the elements of
|
|
`x` and the last index is the degree of the Legendre polynomial.
|
|
|
|
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
|
|
array ``V = legvander(x, n)``, then ``np.dot(V, c)`` and
|
|
``legval(x, c)`` are the same up to roundoff. This equivalence is
|
|
useful both for least squares fitting and for the evaluation of a large
|
|
number of Legendre series of the same degree and sample points.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Array of points. The dtype is converted to float64 or complex128
|
|
depending on whether any of the elements are complex. If `x` is
|
|
scalar it is converted to a 1-D array.
|
|
deg : int
|
|
Degree of the resulting matrix.
|
|
|
|
Returns
|
|
-------
|
|
vander : ndarray
|
|
The pseudo-Vandermonde matrix. The shape of the returned matrix is
|
|
``x.shape + (deg + 1,)``, where The last index is the degree of the
|
|
corresponding Legendre polynomial. The dtype will be the same as
|
|
the converted `x`.
|
|
|
|
"""
|
|
ideg = pu._deprecate_as_int(deg, "deg")
|
|
if ideg < 0:
|
|
raise ValueError("deg must be non-negative")
|
|
|
|
x = np.array(x, copy=False, ndmin=1) + 0.0
|
|
dims = (ideg + 1,) + x.shape
|
|
dtyp = x.dtype
|
|
v = np.empty(dims, dtype=dtyp)
|
|
# Use forward recursion to generate the entries. This is not as accurate
|
|
# as reverse recursion in this application but it is more efficient.
|
|
v[0] = x*0 + 1
|
|
if ideg > 0:
|
|
v[1] = x
|
|
for i in range(2, ideg + 1):
|
|
v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i
|
|
return np.moveaxis(v, 0, -1)
|
|
|
|
|
|
def legvander2d(x, y, deg):
|
|
"""Pseudo-Vandermonde matrix of given degrees.
|
|
|
|
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
|
|
points `(x, y)`. The pseudo-Vandermonde matrix is defined by
|
|
|
|
.. math:: V[..., (deg[1] + 1)*i + j] = L_i(x) * L_j(y),
|
|
|
|
where `0 <= i <= deg[0]` and `0 <= j <= deg[1]`. The leading indices of
|
|
`V` index the points `(x, y)` and the last index encodes the degrees of
|
|
the Legendre polynomials.
|
|
|
|
If ``V = legvander2d(x, y, [xdeg, ydeg])``, then the columns of `V`
|
|
correspond to the elements of a 2-D coefficient array `c` of shape
|
|
(xdeg + 1, ydeg + 1) in the order
|
|
|
|
.. math:: c_{00}, c_{01}, c_{02} ... , c_{10}, c_{11}, c_{12} ...
|
|
|
|
and ``np.dot(V, c.flat)`` and ``legval2d(x, y, c)`` will be the same
|
|
up to roundoff. This equivalence is useful both for least squares
|
|
fitting and for the evaluation of a large number of 2-D Legendre
|
|
series of the same degrees and sample points.
|
|
|
|
Parameters
|
|
----------
|
|
x, y : array_like
|
|
Arrays of point coordinates, all of the same shape. The dtypes
|
|
will be converted to either float64 or complex128 depending on
|
|
whether any of the elements are complex. Scalars are converted to
|
|
1-D arrays.
|
|
deg : list of ints
|
|
List of maximum degrees of the form [x_deg, y_deg].
|
|
|
|
Returns
|
|
-------
|
|
vander2d : ndarray
|
|
The shape of the returned matrix is ``x.shape + (order,)``, where
|
|
:math:`order = (deg[0]+1)*(deg[1]+1)`. The dtype will be the same
|
|
as the converted `x` and `y`.
|
|
|
|
See Also
|
|
--------
|
|
legvander, legvander3d, legval2d, legval3d
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
"""
|
|
return pu._vander_nd_flat((legvander, legvander), (x, y), deg)
|
|
|
|
|
|
def legvander3d(x, y, z, deg):
|
|
"""Pseudo-Vandermonde matrix of given degrees.
|
|
|
|
Returns the pseudo-Vandermonde matrix of degrees `deg` and sample
|
|
points `(x, y, z)`. If `l, m, n` are the given degrees in `x, y, z`,
|
|
then The pseudo-Vandermonde matrix is defined by
|
|
|
|
.. math:: V[..., (m+1)(n+1)i + (n+1)j + k] = L_i(x)*L_j(y)*L_k(z),
|
|
|
|
where `0 <= i <= l`, `0 <= j <= m`, and `0 <= j <= n`. The leading
|
|
indices of `V` index the points `(x, y, z)` and the last index encodes
|
|
the degrees of the Legendre polynomials.
|
|
|
|
If ``V = legvander3d(x, y, z, [xdeg, ydeg, zdeg])``, then the columns
|
|
of `V` correspond to the elements of a 3-D coefficient array `c` of
|
|
shape (xdeg + 1, ydeg + 1, zdeg + 1) in the order
|
|
|
|
.. math:: c_{000}, c_{001}, c_{002},... , c_{010}, c_{011}, c_{012},...
|
|
|
|
and ``np.dot(V, c.flat)`` and ``legval3d(x, y, z, c)`` will be the
|
|
same up to roundoff. This equivalence is useful both for least squares
|
|
fitting and for the evaluation of a large number of 3-D Legendre
|
|
series of the same degrees and sample points.
|
|
|
|
Parameters
|
|
----------
|
|
x, y, z : array_like
|
|
Arrays of point coordinates, all of the same shape. The dtypes will
|
|
be converted to either float64 or complex128 depending on whether
|
|
any of the elements are complex. Scalars are converted to 1-D
|
|
arrays.
|
|
deg : list of ints
|
|
List of maximum degrees of the form [x_deg, y_deg, z_deg].
|
|
|
|
Returns
|
|
-------
|
|
vander3d : ndarray
|
|
The shape of the returned matrix is ``x.shape + (order,)``, where
|
|
:math:`order = (deg[0]+1)*(deg[1]+1)*(deg[2]+1)`. The dtype will
|
|
be the same as the converted `x`, `y`, and `z`.
|
|
|
|
See Also
|
|
--------
|
|
legvander, legvander3d, legval2d, legval3d
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
"""
|
|
return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg)
|
|
|
|
|
|
def legfit(x, y, deg, rcond=None, full=False, w=None):
|
|
"""
|
|
Least squares fit of Legendre series to data.
|
|
|
|
Return the coefficients of a Legendre series of degree `deg` that is the
|
|
least squares fit to the data values `y` given at points `x`. If `y` is
|
|
1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple
|
|
fits are done, one for each column of `y`, and the resulting
|
|
coefficients are stored in the corresponding columns of a 2-D return.
|
|
The fitted polynomial(s) are in the form
|
|
|
|
.. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x),
|
|
|
|
where `n` is `deg`.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like, shape (M,)
|
|
x-coordinates of the M sample points ``(x[i], y[i])``.
|
|
y : array_like, shape (M,) or (M, K)
|
|
y-coordinates of the sample points. Several data sets of sample
|
|
points sharing the same x-coordinates can be fitted at once by
|
|
passing in a 2D-array that contains one dataset per column.
|
|
deg : int or 1-D array_like
|
|
Degree(s) of the fitting polynomials. If `deg` is a single integer
|
|
all terms up to and including the `deg`'th term are included in the
|
|
fit. For NumPy versions >= 1.11.0 a list of integers specifying the
|
|
degrees of the terms to include may be used instead.
|
|
rcond : float, optional
|
|
Relative condition number of the fit. Singular values smaller than
|
|
this relative to the largest singular value will be ignored. The
|
|
default value is len(x)*eps, where eps is the relative precision of
|
|
the float type, about 2e-16 in most cases.
|
|
full : bool, optional
|
|
Switch determining nature of return value. When it is False (the
|
|
default) just the coefficients are returned, when True diagnostic
|
|
information from the singular value decomposition is also returned.
|
|
w : array_like, shape (`M`,), optional
|
|
Weights. If not None, the weight ``w[i]`` applies to the unsquared
|
|
residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are
|
|
chosen so that the errors of the products ``w[i]*y[i]`` all have the
|
|
same variance. When using inverse-variance weighting, use
|
|
``w[i] = 1/sigma(y[i])``. The default value is None.
|
|
|
|
.. versionadded:: 1.5.0
|
|
|
|
Returns
|
|
-------
|
|
coef : ndarray, shape (M,) or (M, K)
|
|
Legendre coefficients ordered from low to high. If `y` was
|
|
2-D, the coefficients for the data in column k of `y` are in
|
|
column `k`. If `deg` is specified as a list, coefficients for
|
|
terms not included in the fit are set equal to zero in the
|
|
returned `coef`.
|
|
|
|
[residuals, rank, singular_values, rcond] : list
|
|
These values are only returned if ``full == True``
|
|
|
|
- residuals -- sum of squared residuals of the least squares fit
|
|
- rank -- the numerical rank of the scaled Vandermonde matrix
|
|
- singular_values -- singular values of the scaled Vandermonde matrix
|
|
- rcond -- value of `rcond`.
|
|
|
|
For more details, see `numpy.linalg.lstsq`.
|
|
|
|
Warns
|
|
-----
|
|
RankWarning
|
|
The rank of the coefficient matrix in the least-squares fit is
|
|
deficient. The warning is only raised if ``full == False``. The
|
|
warnings can be turned off by
|
|
|
|
>>> import warnings
|
|
>>> warnings.simplefilter('ignore', np.RankWarning)
|
|
|
|
See Also
|
|
--------
|
|
numpy.polynomial.polynomial.polyfit
|
|
numpy.polynomial.chebyshev.chebfit
|
|
numpy.polynomial.laguerre.lagfit
|
|
numpy.polynomial.hermite.hermfit
|
|
numpy.polynomial.hermite_e.hermefit
|
|
legval : Evaluates a Legendre series.
|
|
legvander : Vandermonde matrix of Legendre series.
|
|
legweight : Legendre weight function (= 1).
|
|
numpy.linalg.lstsq : Computes a least-squares fit from the matrix.
|
|
scipy.interpolate.UnivariateSpline : Computes spline fits.
|
|
|
|
Notes
|
|
-----
|
|
The solution is the coefficients of the Legendre series `p` that
|
|
minimizes the sum of the weighted squared errors
|
|
|
|
.. math:: E = \\sum_j w_j^2 * |y_j - p(x_j)|^2,
|
|
|
|
where :math:`w_j` are the weights. This problem is solved by setting up
|
|
as the (typically) overdetermined matrix equation
|
|
|
|
.. math:: V(x) * c = w * y,
|
|
|
|
where `V` is the weighted pseudo Vandermonde matrix of `x`, `c` are the
|
|
coefficients to be solved for, `w` are the weights, and `y` are the
|
|
observed values. This equation is then solved using the singular value
|
|
decomposition of `V`.
|
|
|
|
If some of the singular values of `V` are so small that they are
|
|
neglected, then a `RankWarning` will be issued. This means that the
|
|
coefficient values may be poorly determined. Using a lower order fit
|
|
will usually get rid of the warning. The `rcond` parameter can also be
|
|
set to a value smaller than its default, but the resulting fit may be
|
|
spurious and have large contributions from roundoff error.
|
|
|
|
Fits using Legendre series are usually better conditioned than fits
|
|
using power series, but much can depend on the distribution of the
|
|
sample points and the smoothness of the data. If the quality of the fit
|
|
is inadequate splines may be a good alternative.
|
|
|
|
References
|
|
----------
|
|
.. [1] Wikipedia, "Curve fitting",
|
|
https://en.wikipedia.org/wiki/Curve_fitting
|
|
|
|
Examples
|
|
--------
|
|
|
|
"""
|
|
return pu._fit(legvander, x, y, deg, rcond, full, w)
|
|
|
|
|
|
def legcompanion(c):
|
|
"""Return the scaled companion matrix of c.
|
|
|
|
The basis polynomials are scaled so that the companion matrix is
|
|
symmetric when `c` is an Legendre basis polynomial. This provides
|
|
better eigenvalue estimates than the unscaled case and for basis
|
|
polynomials the eigenvalues are guaranteed to be real if
|
|
`numpy.linalg.eigvalsh` is used to obtain them.
|
|
|
|
Parameters
|
|
----------
|
|
c : array_like
|
|
1-D array of Legendre series coefficients ordered from low to high
|
|
degree.
|
|
|
|
Returns
|
|
-------
|
|
mat : ndarray
|
|
Scaled companion matrix of dimensions (deg, deg).
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
"""
|
|
# c 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([[-c[0]/c[1]]])
|
|
|
|
n = len(c) - 1
|
|
mat = np.zeros((n, n), dtype=c.dtype)
|
|
scl = 1./np.sqrt(2*np.arange(n) + 1)
|
|
top = mat.reshape(-1)[1::n+1]
|
|
bot = mat.reshape(-1)[n::n+1]
|
|
top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n]
|
|
bot[...] = top
|
|
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1))
|
|
return mat
|
|
|
|
|
|
def legroots(c):
|
|
"""
|
|
Compute the roots of a Legendre series.
|
|
|
|
Return the roots (a.k.a. "zeros") of the polynomial
|
|
|
|
.. math:: p(x) = \\sum_i c[i] * L_i(x).
|
|
|
|
Parameters
|
|
----------
|
|
c : 1-D array_like
|
|
1-D array of coefficients.
|
|
|
|
Returns
|
|
-------
|
|
out : ndarray
|
|
Array of the roots of the series. If all the roots are real,
|
|
then `out` is also real, otherwise it is complex.
|
|
|
|
See Also
|
|
--------
|
|
numpy.polynomial.polynomial.polyroots
|
|
numpy.polynomial.chebyshev.chebroots
|
|
numpy.polynomial.laguerre.lagroots
|
|
numpy.polynomial.hermite.hermroots
|
|
numpy.polynomial.hermite_e.hermeroots
|
|
|
|
Notes
|
|
-----
|
|
The root estimates are obtained as the eigenvalues of the companion
|
|
matrix, Roots far from the origin of the complex plane may have large
|
|
errors due to the numerical instability of the series for such values.
|
|
Roots with multiplicity greater than 1 will also show larger errors as
|
|
the value of the series near such points is relatively insensitive to
|
|
errors in the roots. Isolated roots near the origin can be improved by
|
|
a few iterations of Newton's method.
|
|
|
|
The Legendre series basis polynomials aren't powers of ``x`` so the
|
|
results of this function may seem unintuitive.
|
|
|
|
Examples
|
|
--------
|
|
>>> import numpy.polynomial.legendre as leg
|
|
>>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots
|
|
array([-0.85099543, -0.11407192, 0.51506735]) # may vary
|
|
|
|
"""
|
|
# c is a trimmed copy
|
|
[c] = pu.as_series([c])
|
|
if len(c) < 2:
|
|
return np.array([], dtype=c.dtype)
|
|
if len(c) == 2:
|
|
return np.array([-c[0]/c[1]])
|
|
|
|
# rotated companion matrix reduces error
|
|
m = legcompanion(c)[::-1,::-1]
|
|
r = la.eigvals(m)
|
|
r.sort()
|
|
return r
|
|
|
|
|
|
def leggauss(deg):
|
|
"""
|
|
Gauss-Legendre quadrature.
|
|
|
|
Computes the sample points and weights for Gauss-Legendre quadrature.
|
|
These sample points and weights will correctly integrate polynomials of
|
|
degree :math:`2*deg - 1` or less over the interval :math:`[-1, 1]` with
|
|
the weight function :math:`f(x) = 1`.
|
|
|
|
Parameters
|
|
----------
|
|
deg : int
|
|
Number of sample points and weights. It must be >= 1.
|
|
|
|
Returns
|
|
-------
|
|
x : ndarray
|
|
1-D ndarray containing the sample points.
|
|
y : ndarray
|
|
1-D ndarray containing the weights.
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
The results have only been tested up to degree 100, higher degrees may
|
|
be problematic. The weights are determined by using the fact that
|
|
|
|
.. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k))
|
|
|
|
where :math:`c` is a constant independent of :math:`k` and :math:`x_k`
|
|
is the k'th root of :math:`L_n`, and then scaling the results to get
|
|
the right value when integrating 1.
|
|
|
|
"""
|
|
ideg = pu._deprecate_as_int(deg, "deg")
|
|
if ideg <= 0:
|
|
raise ValueError("deg must be a positive integer")
|
|
|
|
# first approximation of roots. We use the fact that the companion
|
|
# matrix is symmetric in this case in order to obtain better zeros.
|
|
c = np.array([0]*deg + [1])
|
|
m = legcompanion(c)
|
|
x = la.eigvalsh(m)
|
|
|
|
# improve roots by one application of Newton
|
|
dy = legval(x, c)
|
|
df = legval(x, legder(c))
|
|
x -= dy/df
|
|
|
|
# compute the weights. We scale the factor to avoid possible numerical
|
|
# overflow.
|
|
fm = legval(x, c[1:])
|
|
fm /= np.abs(fm).max()
|
|
df /= np.abs(df).max()
|
|
w = 1/(fm * df)
|
|
|
|
# for Legendre we can also symmetrize
|
|
w = (w + w[::-1])/2
|
|
x = (x - x[::-1])/2
|
|
|
|
# scale w to get the right value
|
|
w *= 2. / w.sum()
|
|
|
|
return x, w
|
|
|
|
|
|
def legweight(x):
|
|
"""
|
|
Weight function of the Legendre polynomials.
|
|
|
|
The weight function is :math:`1` and the interval of integration is
|
|
:math:`[-1, 1]`. The Legendre polynomials are orthogonal, but not
|
|
normalized, with respect to this weight function.
|
|
|
|
Parameters
|
|
----------
|
|
x : array_like
|
|
Values at which the weight function will be computed.
|
|
|
|
Returns
|
|
-------
|
|
w : ndarray
|
|
The weight function at `x`.
|
|
|
|
Notes
|
|
-----
|
|
|
|
.. versionadded:: 1.7.0
|
|
|
|
"""
|
|
w = x*0.0 + 1.0
|
|
return w
|
|
|
|
#
|
|
# Legendre series class
|
|
#
|
|
|
|
class Legendre(ABCPolyBase):
|
|
"""A Legendre series class.
|
|
|
|
The Legendre class provides the standard Python numerical methods
|
|
'+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
|
|
attributes and methods listed in the `ABCPolyBase` documentation.
|
|
|
|
Parameters
|
|
----------
|
|
coef : array_like
|
|
Legendre coefficients in order of increasing degree, i.e.,
|
|
``(1, 2, 3)`` gives ``1*P_0(x) + 2*P_1(x) + 3*P_2(x)``.
|
|
domain : (2,) array_like, optional
|
|
Domain to use. The interval ``[domain[0], domain[1]]`` is mapped
|
|
to the interval ``[window[0], window[1]]`` by shifting and scaling.
|
|
The default value is [-1, 1].
|
|
window : (2,) array_like, optional
|
|
Window, see `domain` for its use. The default value is [-1, 1].
|
|
|
|
.. versionadded:: 1.6.0
|
|
symbol : str, optional
|
|
Symbol used to represent the independent variable in string
|
|
representations of the polynomial expression, e.g. for printing.
|
|
The symbol must be a valid Python identifier. Default value is 'x'.
|
|
|
|
.. versionadded:: 1.24
|
|
|
|
"""
|
|
# Virtual Functions
|
|
_add = staticmethod(legadd)
|
|
_sub = staticmethod(legsub)
|
|
_mul = staticmethod(legmul)
|
|
_div = staticmethod(legdiv)
|
|
_pow = staticmethod(legpow)
|
|
_val = staticmethod(legval)
|
|
_int = staticmethod(legint)
|
|
_der = staticmethod(legder)
|
|
_fit = staticmethod(legfit)
|
|
_line = staticmethod(legline)
|
|
_roots = staticmethod(legroots)
|
|
_fromroots = staticmethod(legfromroots)
|
|
|
|
# Virtual properties
|
|
domain = np.array(legdomain)
|
|
window = np.array(legdomain)
|
|
basis_name = 'P'
|