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"""
====================================================
Chebyshev Series (:mod:`numpy.polynomial.chebyshev`)
====================================================
This module provides a number of objects (mostly functions) useful for
dealing with Chebyshev series, including a `Chebyshev` class that
encapsulates the usual arithmetic operations. (General information
on how this module represents and works with such polynomials is in the
docstring for its "parent" sub-package, `numpy.polynomial`).
Classes
-------
.. autosummary::
:toctree: generated/
Chebyshev
Constants
---------
.. autosummary::
:toctree: generated/
chebdomain
chebzero
chebone
chebx
Arithmetic
----------
.. autosummary::
:toctree: generated/
chebadd
chebsub
chebmulx
chebmul
chebdiv
chebpow
chebval
chebval2d
chebval3d
chebgrid2d
chebgrid3d
Calculus
--------
.. autosummary::
:toctree: generated/
chebder
chebint
Misc Functions
--------------
.. autosummary::
:toctree: generated/
chebfromroots
chebroots
chebvander
chebvander2d
chebvander3d
chebgauss
chebweight
chebcompanion
chebfit
chebpts1
chebpts2
chebtrim
chebline
cheb2poly
poly2cheb
chebinterpolate
See also
--------
`numpy.polynomial`
Notes
-----
The implementations of multiplication, division, integration, and
differentiation use the algebraic identities [1]_:
.. math::
T_n(x) = \\frac{z^n + z^{-n}}{2} \\\\
z\\frac{dx}{dz} = \\frac{z - z^{-1}}{2}.
where
.. math:: x = \\frac{z + z^{-1}}{2}.
These identities allow a Chebyshev series to be expressed as a finite,
symmetric Laurent series. In this module, this sort of Laurent series
is referred to as a "z-series."
References
----------
.. [1] A. T. Benjamin, et al., "Combinatorial Trigonometry with Chebyshev
Polynomials," *Journal of Statistical Planning and Inference 14*, 2008
(https://web.archive.org/web/20080221202153/https://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4)
"""
import numpy as np
import numpy.linalg as la
from numpy.core.multiarray import normalize_axis_index
from . import polyutils as pu
from ._polybase import ABCPolyBase
__all__ = [
'chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline', 'chebadd',
'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow', 'chebval',
'chebder', 'chebint', 'cheb2poly', 'poly2cheb', 'chebfromroots',
'chebvander', 'chebfit', 'chebtrim', 'chebroots', 'chebpts1',
'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d', 'chebgrid2d',
'chebgrid3d', 'chebvander2d', 'chebvander3d', 'chebcompanion',
'chebgauss', 'chebweight', 'chebinterpolate']
chebtrim = pu.trimcoef
#
# A collection of functions for manipulating z-series. These are private
# functions and do minimal error checking.
#
def _cseries_to_zseries(c):
"""Convert Chebyshev series to z-series.
Convert a Chebyshev series to the equivalent z-series. The result is
never an empty array. The dtype of the return is the same as that of
the input. No checks are run on the arguments as this routine is for
internal use.
Parameters
----------
c : 1-D ndarray
Chebyshev coefficients, ordered from low to high
Returns
-------
zs : 1-D ndarray
Odd length symmetric z-series, ordered from low to high.
"""
n = c.size
zs = np.zeros(2*n-1, dtype=c.dtype)
zs[n-1:] = c/2
return zs + zs[::-1]
def _zseries_to_cseries(zs):
"""Convert z-series to a Chebyshev series.
Convert a z series to the equivalent Chebyshev series. The result is
never an empty array. The dtype of the return is the same as that of
the input. No checks are run on the arguments as this routine is for
internal use.
Parameters
----------
zs : 1-D ndarray
Odd length symmetric z-series, ordered from low to high.
Returns
-------
c : 1-D ndarray
Chebyshev coefficients, ordered from low to high.
"""
n = (zs.size + 1)//2
c = zs[n-1:].copy()
c[1:n] *= 2
return c
def _zseries_mul(z1, z2):
"""Multiply two z-series.
Multiply two z-series to produce a z-series.
Parameters
----------
z1, z2 : 1-D ndarray
The arrays must be 1-D but this is not checked.
Returns
-------
product : 1-D ndarray
The product z-series.
Notes
-----
This is simply convolution. If symmetric/anti-symmetric z-series are
denoted by S/A then the following rules apply:
S*S, A*A -> S
S*A, A*S -> A
"""
return np.convolve(z1, z2)
def _zseries_div(z1, z2):
"""Divide the first z-series by the second.
Divide `z1` by `z2` and return the quotient and remainder as z-series.
Warning: this implementation only applies when both z1 and z2 have the
same symmetry, which is sufficient for present purposes.
Parameters
----------
z1, z2 : 1-D ndarray
The arrays must be 1-D and have the same symmetry, but this is not
checked.
Returns
-------
(quotient, remainder) : 1-D ndarrays
Quotient and remainder as z-series.
Notes
-----
This is not the same as polynomial division on account of the desired form
of the remainder. If symmetric/anti-symmetric z-series are denoted by S/A
then the following rules apply:
S/S -> S,S
A/A -> S,A
The restriction to types of the same symmetry could be fixed but seems like
unneeded generality. There is no natural form for the remainder in the case
where there is no symmetry.
"""
z1 = z1.copy()
z2 = z2.copy()
lc1 = len(z1)
lc2 = len(z2)
if lc2 == 1:
z1 /= z2
return z1, z1[:1]*0
elif lc1 < lc2:
return z1[:1]*0, z1
else:
dlen = lc1 - lc2
scl = z2[0]
z2 /= scl
quo = np.empty(dlen + 1, dtype=z1.dtype)
i = 0
j = dlen
while i < j:
r = z1[i]
quo[i] = z1[i]
quo[dlen - i] = r
tmp = r*z2
z1[i:i+lc2] -= tmp
z1[j:j+lc2] -= tmp
i += 1
j -= 1
r = z1[i]
quo[i] = r
tmp = r*z2
z1[i:i+lc2] -= tmp
quo /= scl
rem = z1[i+1:i-1+lc2].copy()
return quo, rem
def _zseries_der(zs):
"""Differentiate a z-series.
The derivative is with respect to x, not z. This is achieved using the
chain rule and the value of dx/dz given in the module notes.
Parameters
----------
zs : z-series
The z-series to differentiate.
Returns
-------
derivative : z-series
The derivative
Notes
-----
The zseries for x (ns) has been multiplied by two in order to avoid
using floats that are incompatible with Decimal and likely other
specialized scalar types. This scaling has been compensated by
multiplying the value of zs by two also so that the two cancels in the
division.
"""
n = len(zs)//2
ns = np.array([-1, 0, 1], dtype=zs.dtype)
zs *= np.arange(-n, n+1)*2
d, r = _zseries_div(zs, ns)
return d
def _zseries_int(zs):
"""Integrate a z-series.
The integral is with respect to x, not z. This is achieved by a change
of variable using dx/dz given in the module notes.
Parameters
----------
zs : z-series
The z-series to integrate
Returns
-------
integral : z-series
The indefinite integral
Notes
-----
The zseries for x (ns) has been multiplied by two in order to avoid
using floats that are incompatible with Decimal and likely other
specialized scalar types. This scaling has been compensated by
dividing the resulting zs by two.
"""
n = 1 + len(zs)//2
ns = np.array([-1, 0, 1], dtype=zs.dtype)
zs = _zseries_mul(zs, ns)
div = np.arange(-n, n+1)*2
zs[:n] /= div[:n]
zs[n+1:] /= div[n+1:]
zs[n] = 0
return zs
#
# Chebyshev series functions
#
def poly2cheb(pol):
"""
Convert a polynomial to a Chebyshev series.
Convert an array representing the coefficients of a polynomial (relative
to the "standard" basis) ordered from lowest degree to highest, to an
array of the coefficients of the equivalent Chebyshev series, ordered
from lowest to highest degree.
Parameters
----------
pol : array_like
1-D array containing the polynomial coefficients
Returns
-------
c : ndarray
1-D array containing the coefficients of the equivalent Chebyshev
series.
See Also
--------
cheb2poly
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy import polynomial as P
>>> p = P.Polynomial(range(4))
>>> p
Polynomial([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
>>> c = p.convert(kind=P.Chebyshev)
>>> c
Chebyshev([1. , 3.25, 1. , 0.75], domain=[-1., 1.], window=[-1., 1.])
>>> P.chebyshev.poly2cheb(range(4))
array([1. , 3.25, 1. , 0.75])
"""
[pol] = pu.as_series([pol])
deg = len(pol) - 1
res = 0
for i in range(deg, -1, -1):
res = chebadd(chebmulx(res), pol[i])
return res
def cheb2poly(c):
"""
Convert a Chebyshev series to a polynomial.
Convert an array representing the coefficients of a Chebyshev series,
ordered from lowest degree to highest, to an array of the coefficients
of the equivalent polynomial (relative to the "standard" basis) ordered
from lowest to highest degree.
Parameters
----------
c : array_like
1-D array containing the Chebyshev series coefficients, ordered
from lowest order term to highest.
Returns
-------
pol : ndarray
1-D array containing the coefficients of the equivalent polynomial
(relative to the "standard" basis) ordered from lowest order term
to highest.
See Also
--------
poly2cheb
Notes
-----
The easy way to do conversions between polynomial basis sets
is to use the convert method of a class instance.
Examples
--------
>>> from numpy import polynomial as P
>>> c = P.Chebyshev(range(4))
>>> c
Chebyshev([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1])
>>> p = c.convert(kind=P.Polynomial)
>>> p
Polynomial([-2., -8., 4., 12.], domain=[-1., 1.], window=[-1., 1.])
>>> P.chebyshev.cheb2poly(range(4))
array([-2., -8., 4., 12.])
"""
from .polynomial import polyadd, polysub, polymulx
[c] = pu.as_series([c])
n = len(c)
if n < 3:
return c
else:
c0 = c[-2]
c1 = c[-1]
# i is the current degree of c1
for i in range(n - 1, 1, -1):
tmp = c0
c0 = polysub(c[i - 2], c1)
c1 = polyadd(tmp, polymulx(c1)*2)
return polyadd(c0, polymulx(c1))
#
# These are constant arrays are of integer type so as to be compatible
# with the widest range of other types, such as Decimal.
#
# Chebyshev default domain.
chebdomain = np.array([-1, 1])
# Chebyshev coefficients representing zero.
chebzero = np.array([0])
# Chebyshev coefficients representing one.
chebone = np.array([1])
# Chebyshev coefficients representing the identity x.
chebx = np.array([0, 1])
def chebline(off, scl):
"""
Chebyshev series whose graph is a straight line.
Parameters
----------
off, scl : scalars
The specified line is given by ``off + scl*x``.
Returns
-------
y : ndarray
This module's representation of the Chebyshev series for
``off + scl*x``.
See Also
--------
numpy.polynomial.polynomial.polyline
numpy.polynomial.legendre.legline
numpy.polynomial.laguerre.lagline
numpy.polynomial.hermite.hermline
numpy.polynomial.hermite_e.hermeline
Examples
--------
>>> import numpy.polynomial.chebyshev as C
>>> C.chebline(3,2)
array([3, 2])
>>> C.chebval(-3, C.chebline(3,2)) # should be -3
-3.0
"""
if scl != 0:
return np.array([off, scl])
else:
return np.array([off])
def chebfromroots(roots):
"""
Generate a Chebyshev series with given roots.
The function returns the coefficients of the polynomial
.. math:: p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),
in Chebyshev form, where the `r_n` are the roots specified in `roots`.
If a zero has multiplicity n, then it must appear in `roots` n times.
For instance, if 2 is a root of multiplicity three and 3 is a root of
multiplicity 2, then `roots` looks something like [2, 2, 2, 3, 3]. The
roots can appear in any order.
If the returned coefficients are `c`, then
.. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x)
The coefficient of the last term is not generally 1 for monic
polynomials in Chebyshev form.
Parameters
----------
roots : array_like
Sequence containing the roots.
Returns
-------
out : ndarray
1-D array of coefficients. If all roots are real then `out` is a
real array, if some of the roots are complex, then `out` is complex
even if all the coefficients in the result are real (see Examples
below).
See Also
--------
numpy.polynomial.polynomial.polyfromroots
numpy.polynomial.legendre.legfromroots
numpy.polynomial.laguerre.lagfromroots
numpy.polynomial.hermite.hermfromroots
numpy.polynomial.hermite_e.hermefromroots
Examples
--------
>>> import numpy.polynomial.chebyshev as C
>>> C.chebfromroots((-1,0,1)) # x^3 - x relative to the standard basis
array([ 0. , -0.25, 0. , 0.25])
>>> j = complex(0,1)
>>> C.chebfromroots((-j,j)) # x^2 + 1 relative to the standard basis
array([1.5+0.j, 0. +0.j, 0.5+0.j])
"""
return pu._fromroots(chebline, chebmul, roots)
def chebadd(c1, c2):
"""
Add one Chebyshev series to another.
Returns the sum of two Chebyshev series `c1` + `c2`. The arguments
are sequences of coefficients ordered from lowest order term to
highest, i.e., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the Chebyshev series of their sum.
See Also
--------
chebsub, chebmulx, chebmul, chebdiv, chebpow
Notes
-----
Unlike multiplication, division, etc., the sum of two Chebyshev series
is a Chebyshev series (without having to "reproject" the result onto
the basis set) so addition, just like that of "standard" polynomials,
is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebadd(c1,c2)
array([4., 4., 4.])
"""
return pu._add(c1, c2)
def chebsub(c1, c2):
"""
Subtract one Chebyshev series from another.
Returns the difference of two Chebyshev series `c1` - `c2`. The
sequences of coefficients are from lowest order term to highest, i.e.,
[1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Chebyshev series coefficients representing their difference.
See Also
--------
chebadd, chebmulx, chebmul, chebdiv, chebpow
Notes
-----
Unlike multiplication, division, etc., the difference of two Chebyshev
series is a Chebyshev series (without having to "reproject" the result
onto the basis set) so subtraction, just like that of "standard"
polynomials, is simply "component-wise."
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebsub(c1,c2)
array([-2., 0., 2.])
>>> C.chebsub(c2,c1) # -C.chebsub(c1,c2)
array([ 2., 0., -2.])
"""
return pu._sub(c1, c2)
def chebmulx(c):
"""Multiply a Chebyshev series by x.
Multiply the polynomial `c` by x, where x is the independent
variable.
Parameters
----------
c : array_like
1-D array of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Array representing the result of the multiplication.
Notes
-----
.. versionadded:: 1.5.0
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> C.chebmulx([1,2,3])
array([1. , 2.5, 1. , 1.5])
"""
# c is a trimmed copy
[c] = pu.as_series([c])
# The zero series needs special treatment
if len(c) == 1 and c[0] == 0:
return c
prd = np.empty(len(c) + 1, dtype=c.dtype)
prd[0] = c[0]*0
prd[1] = c[0]
if len(c) > 1:
tmp = c[1:]/2
prd[2:] = tmp
prd[0:-2] += tmp
return prd
def chebmul(c1, c2):
"""
Multiply one Chebyshev series by another.
Returns the product of two Chebyshev series `c1` * `c2`. The arguments
are sequences of coefficients, from lowest order "term" to highest,
e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
out : ndarray
Of Chebyshev series coefficients representing their product.
See Also
--------
chebadd, chebsub, chebmulx, chebdiv, chebpow
Notes
-----
In general, the (polynomial) product of two C-series results in terms
that are not in the Chebyshev polynomial basis set. Thus, to express
the product as a C-series, it is typically necessary to "reproject"
the product onto said basis set, which typically produces
"unintuitive live" (but correct) results; see Examples section below.
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebmul(c1,c2) # multiplication requires "reprojection"
array([ 6.5, 12. , 12. , 4. , 1.5])
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
prd = _zseries_mul(z1, z2)
ret = _zseries_to_cseries(prd)
return pu.trimseq(ret)
def chebdiv(c1, c2):
"""
Divide one Chebyshev series by another.
Returns the quotient-with-remainder of two Chebyshev series
`c1` / `c2`. The arguments are sequences of coefficients from lowest
order "term" to highest, e.g., [1,2,3] represents the series
``T_0 + 2*T_1 + 3*T_2``.
Parameters
----------
c1, c2 : array_like
1-D arrays of Chebyshev series coefficients ordered from low to
high.
Returns
-------
[quo, rem] : ndarrays
Of Chebyshev series coefficients representing the quotient and
remainder.
See Also
--------
chebadd, chebsub, chebmulx, chebmul, chebpow
Notes
-----
In general, the (polynomial) division of one C-series by another
results in quotient and remainder terms that are not in the Chebyshev
polynomial basis set. Thus, to express these results as C-series, it
is typically necessary to "reproject" the results onto said basis
set, which typically produces "unintuitive" (but correct) results;
see Examples section below.
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> c1 = (1,2,3)
>>> c2 = (3,2,1)
>>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not
(array([3.]), array([-8., -4.]))
>>> c2 = (0,1,2,3)
>>> C.chebdiv(c2,c1) # neither "intuitive"
(array([0., 2.]), array([-2., -4.]))
"""
# c1, c2 are trimmed copies
[c1, c2] = pu.as_series([c1, c2])
if c2[-1] == 0:
raise ZeroDivisionError()
# note: this is more efficient than `pu._div(chebmul, c1, c2)`
lc1 = len(c1)
lc2 = len(c2)
if lc1 < lc2:
return c1[:1]*0, c1
elif lc2 == 1:
return c1/c2[-1], c1[:1]*0
else:
z1 = _cseries_to_zseries(c1)
z2 = _cseries_to_zseries(c2)
quo, rem = _zseries_div(z1, z2)
quo = pu.trimseq(_zseries_to_cseries(quo))
rem = pu.trimseq(_zseries_to_cseries(rem))
return quo, rem
def chebpow(c, pow, maxpower=16):
"""Raise a Chebyshev series to a power.
Returns the Chebyshev 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 ``T_0 + 2*T_1 + 3*T_2.``
Parameters
----------
c : array_like
1-D array of Chebyshev 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
Chebyshev series of power.
See Also
--------
chebadd, chebsub, chebmulx, chebmul, chebdiv
Examples
--------
>>> from numpy.polynomial import chebyshev as C
>>> C.chebpow([1, 2, 3, 4], 2)
array([15.5, 22. , 16. , ..., 12.5, 12. , 8. ])
"""
# note: this is more efficient than `pu._pow(chebmul, c1, c2)`, as it
# avoids converting between z and c series repeatedly
# c is a trimmed copy
[c] = pu.as_series([c])
power = int(pow)
if power != pow or power < 0:
raise ValueError("Power must be a non-negative integer.")
elif maxpower is not None and power > maxpower:
raise ValueError("Power is too large")
elif power == 0:
return np.array([1], dtype=c.dtype)
elif power == 1:
return c
else:
# This can be made more efficient by using powers of two
# in the usual way.
zs = _cseries_to_zseries(c)
prd = zs
for i in range(2, power + 1):
prd = np.convolve(prd, zs)
return _zseries_to_cseries(prd)
def chebder(c, m=1, scl=1, axis=0):
"""
Differentiate a Chebyshev series.
Returns the Chebyshev 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*T_0 + 2*T_1 + 3*T_2``
while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) +
2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is
``y``.
Parameters
----------
c : array_like
Array of Chebyshev 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
Chebyshev series of the derivative.
See Also
--------
chebint
Notes
-----
In general, the result of differentiating 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 chebyshev as C
>>> c = (1,2,3,4)
>>> C.chebder(c)
array([14., 12., 24.])
>>> C.chebder(c,3)
array([96.])
>>> C.chebder(c,scl=-1)
array([-14., -12., -24.])
>>> C.chebder(c,2,-1)
array([12., 96.])
"""
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)*c[j]
c[j - 2] += (j*c[j])/(j - 2)
if n > 1:
der[1] = 4*c[2]
der[0] = c[1]
c = der
c = np.moveaxis(c, 0, iaxis)
return c
def chebint(c, m=1, k=[], lbnd=0, scl=1, axis=0):
"""
Integrate a Chebyshev series.
Returns the Chebyshev 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 ``T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]]
represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) +
2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``.
Parameters
----------
c : array_like
Array of Chebyshev 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 zero
is the first value in the list, the value of the second integral
at zero 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
C-series coefficients of the integral.
Raises
------
ValueError
If ``m < 1``, ``len(k) > m``, ``np.ndim(lbnd) != 0``, or
``np.ndim(scl) != 0``.
See Also
--------
chebder
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 chebyshev as C
>>> c = (1,2,3)
>>> C.chebint(c)
array([ 0.5, -0.5, 0.5, 0.5])
>>> C.chebint(c,3)
array([ 0.03125 , -0.1875 , 0.04166667, -0.05208333, 0.01041667, # may vary
0.00625 ])
>>> C.chebint(c, k=3)
array([ 3.5, -0.5, 0.5, 0.5])
>>> C.chebint(c,lbnd=-2)
array([ 8.5, -0.5, 0.5, 0.5])
>>> C.chebint(c,scl=-2)
array([-1., 1., -1., -1.])
"""
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]/4
for j in range(2, n):
tmp[j + 1] = c[j]/(2*(j + 1))
tmp[j - 1] -= c[j]/(2*(j - 1))
tmp[0] += k[i] - chebval(lbnd, tmp)
c = tmp
c = np.moveaxis(c, 0, iaxis)
return c
def chebval(x, c, tensor=True):
"""
Evaluate a Chebyshev series at points x.
If `c` is of length `n + 1`, this function returns the value:
.. math:: p(x) = c_0 * T_0(x) + c_1 * T_1(x) + ... + c_n * T_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
--------
chebval2d, chebgrid2d, chebval3d, chebgrid3d
Notes
-----
The evaluation uses Clenshaw recursion, aka synthetic division.
"""
c = np.array(c, ndmin=1, copy=True)
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:
x2 = 2*x
c0 = c[-2]
c1 = c[-1]
for i in range(3, len(c) + 1):
tmp = c0
c0 = c[-i] - c1
c1 = tmp + c1*x2
return c0 + c1*x
def chebval2d(x, y, c):
"""
Evaluate a 2-D Chebyshev series at points (x, y).
This function returns the values:
.. math:: p(x,y) = \\sum_{i,j} c_{i,j} * T_i(x) * T_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 2 the remaining indices enumerate multiple
sets of coefficients.
Returns
-------
values : ndarray, compatible object
The values of the two dimensional Chebyshev series at points formed
from pairs of corresponding values from `x` and `y`.
See Also
--------
chebval, chebgrid2d, chebval3d, chebgrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(chebval, c, x, y)
def chebgrid2d(x, y, c):
"""
Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.
This function returns the values:
.. math:: p(a,b) = \\sum_{i,j} c_{i,j} * T_i(a) * T_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
--------
chebval, chebval2d, chebval3d, chebgrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(chebval, c, x, y)
def chebval3d(x, y, z, c):
"""
Evaluate a 3-D Chebyshev series at points (x, y, z).
This function returns the values:
.. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * T_i(x) * T_j(y) * T_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
--------
chebval, chebval2d, chebgrid2d, chebgrid3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._valnd(chebval, c, x, y, z)
def chebgrid3d(x, y, z, c):
"""
Evaluate a 3-D Chebyshev 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} * T_i(a) * T_j(b) * T_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
--------
chebval, chebval2d, chebgrid2d, chebval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._gridnd(chebval, c, x, y, z)
def chebvander(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] = T_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 Chebyshev polynomial.
If `c` is a 1-D array of coefficients of length `n + 1` and `V` is the
matrix ``V = chebvander(x, n)``, then ``np.dot(V, c)`` and
``chebval(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 Chebyshev 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 Chebyshev 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.
v[0] = x*0 + 1
if ideg > 0:
x2 = 2*x
v[1] = x
for i in range(2, ideg + 1):
v[i] = v[i-1]*x2 - v[i-2]
return np.moveaxis(v, 0, -1)
def chebvander2d(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] = T_i(x) * T_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 Chebyshev polynomials.
If ``V = chebvander2d(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 ``chebval2d(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 Chebyshev
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
--------
chebvander, chebvander3d, chebval2d, chebval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((chebvander, chebvander), (x, y), deg)
def chebvander3d(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] = T_i(x)*T_j(y)*T_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 Chebyshev polynomials.
If ``V = chebvander3d(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 ``chebval3d(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 Chebyshev
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
--------
chebvander, chebvander3d, chebval2d, chebval3d
Notes
-----
.. versionadded:: 1.7.0
"""
return pu._vander_nd_flat((chebvander, chebvander, chebvander), (x, y, z), deg)
def chebfit(x, y, deg, rcond=None, full=False, w=None):
"""
Least squares fit of Chebyshev series to data.
Return the coefficients of a Chebyshev 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 * T_1(x) + ... + c_n * T_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)
Chebyshev 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.RankWarning)
See Also
--------
numpy.polynomial.polynomial.polyfit
numpy.polynomial.legendre.legfit
numpy.polynomial.laguerre.lagfit
numpy.polynomial.hermite.hermfit
numpy.polynomial.hermite_e.hermefit
chebval : Evaluates a Chebyshev series.
chebvander : Vandermonde matrix of Chebyshev series.
chebweight : Chebyshev 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 Chebyshev 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 Chebyshev 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(chebvander, x, y, deg, rcond, full, w)
def chebcompanion(c):
"""Return the scaled companion matrix of c.
The basis polynomials are scaled so that the companion matrix is
symmetric when `c` is a Chebyshev 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 Chebyshev 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 = np.array([1.] + [np.sqrt(.5)]*(n-1))
top = mat.reshape(-1)[1::n+1]
bot = mat.reshape(-1)[n::n+1]
top[0] = np.sqrt(.5)
top[1:] = 1/2
bot[...] = top
mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5
return mat
def chebroots(c):
"""
Compute the roots of a Chebyshev series.
Return the roots (a.k.a. "zeros") of the polynomial
.. math:: p(x) = \\sum_i c[i] * T_i(x).
Parameters
----------
c : 1-D array_like
1-D array of coefficients.
Returns
-------
out : ndarray
Array of the roots of the series. If all the roots are real,
then `out` is also real, otherwise it is complex.
See Also
--------
numpy.polynomial.polynomial.polyroots
numpy.polynomial.legendre.legroots
numpy.polynomial.laguerre.lagroots
numpy.polynomial.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 Chebyshev series basis polynomials aren't powers of `x` so the
results of this function may seem unintuitive.
Examples
--------
>>> import numpy.polynomial.chebyshev as cheb
>>> cheb.chebroots((-1, 1,-1, 1)) # T3 - T2 + T1 - T0 has real roots
array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00]) # 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 = chebcompanion(c)[::-1,::-1]
r = la.eigvals(m)
r.sort()
return r
def chebinterpolate(func, deg, args=()):
"""Interpolate a function at the Chebyshev points of the first kind.
Returns the Chebyshev series that interpolates `func` at the Chebyshev
points of the first kind in the interval [-1, 1]. The interpolating
series tends to a minmax approximation to `func` with increasing `deg`
if the function is continuous in the interval.
.. versionadded:: 1.14.0
Parameters
----------
func : function
The function to be approximated. It must be a function of a single
variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are
extra arguments passed in the `args` parameter.
deg : int
Degree of the interpolating polynomial
args : tuple, optional
Extra arguments to be used in the function call. Default is no extra
arguments.
Returns
-------
coef : ndarray, shape (deg + 1,)
Chebyshev coefficients of the interpolating series ordered from low to
high.
Examples
--------
>>> import numpy.polynomial.chebyshev as C
>>> C.chebfromfunction(lambda x: np.tanh(x) + 0.5, 8)
array([ 5.00000000e-01, 8.11675684e-01, -9.86864911e-17,
-5.42457905e-02, -2.71387850e-16, 4.51658839e-03,
2.46716228e-17, -3.79694221e-04, -3.26899002e-16])
Notes
-----
The Chebyshev polynomials used in the interpolation are orthogonal when
sampled at the Chebyshev points of the first kind. If it is desired to
constrain some of the coefficients they can simply be set to the desired
value after the interpolation, no new interpolation or fit is needed. This
is especially useful if it is known apriori that some of coefficients are
zero. For instance, if the function is even then the coefficients of the
terms of odd degree in the result can be set to zero.
"""
deg = np.asarray(deg)
# check arguments.
if deg.ndim > 0 or deg.dtype.kind not in 'iu' or deg.size == 0:
raise TypeError("deg must be an int")
if deg < 0:
raise ValueError("expected deg >= 0")
order = deg + 1
xcheb = chebpts1(order)
yfunc = func(xcheb, *args)
m = chebvander(xcheb, deg)
c = np.dot(m.T, yfunc)
c[0] /= order
c[1:] /= 0.5*order
return c
def chebgauss(deg):
"""
Gauss-Chebyshev quadrature.
Computes the sample points and weights for Gauss-Chebyshev 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/\\sqrt{1 - x^2}`.
Parameters
----------
deg : int
Number of sample points and weights. It must be >= 1.
Returns
-------
x : ndarray
1-D ndarray containing the sample points.
y : ndarray
1-D ndarray containing the weights.
Notes
-----
.. versionadded:: 1.7.0
The results have only been tested up to degree 100, higher degrees may
be problematic. For Gauss-Chebyshev there are closed form solutions for
the sample points and weights. If n = `deg`, then
.. math:: x_i = \\cos(\\pi (2 i - 1) / (2 n))
.. math:: w_i = \\pi / n
"""
ideg = pu._deprecate_as_int(deg, "deg")
if ideg <= 0:
raise ValueError("deg must be a positive integer")
x = np.cos(np.pi * np.arange(1, 2*ideg, 2) / (2.0*ideg))
w = np.ones(ideg)*(np.pi/ideg)
return x, w
def chebweight(x):
"""
The weight function of the Chebyshev polynomials.
The weight function is :math:`1/\\sqrt{1 - x^2}` and the interval of
integration is :math:`[-1, 1]`. The Chebyshev 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 = 1./(np.sqrt(1. + x) * np.sqrt(1. - x))
return w
def chebpts1(npts):
"""
Chebyshev points of the first kind.
The Chebyshev points of the first kind are the points ``cos(x)``,
where ``x = [pi*(k + .5)/npts for k in range(npts)]``.
Parameters
----------
npts : int
Number of sample points desired.
Returns
-------
pts : ndarray
The Chebyshev points of the first kind.
See Also
--------
chebpts2
Notes
-----
.. versionadded:: 1.5.0
"""
_npts = int(npts)
if _npts != npts:
raise ValueError("npts must be integer")
if _npts < 1:
raise ValueError("npts must be >= 1")
x = 0.5 * np.pi / _npts * np.arange(-_npts+1, _npts+1, 2)
return np.sin(x)
def chebpts2(npts):
"""
Chebyshev points of the second kind.
The Chebyshev points of the second kind are the points ``cos(x)``,
where ``x = [pi*k/(npts - 1) for k in range(npts)]`` sorted in ascending
order.
Parameters
----------
npts : int
Number of sample points desired.
Returns
-------
pts : ndarray
The Chebyshev points of the second kind.
Notes
-----
.. versionadded:: 1.5.0
"""
_npts = int(npts)
if _npts != npts:
raise ValueError("npts must be integer")
if _npts < 2:
raise ValueError("npts must be >= 2")
x = np.linspace(-np.pi, 0, _npts)
return np.cos(x)
#
# Chebyshev series class
#
class Chebyshev(ABCPolyBase):
"""A Chebyshev series class.
The Chebyshev class provides the standard Python numerical methods
'+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the
methods listed below.
Parameters
----------
coef : array_like
Chebyshev coefficients in order of increasing degree, i.e.,
``(1, 2, 3)`` gives ``1*T_0(x) + 2*T_1(x) + 3*T_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(chebadd)
_sub = staticmethod(chebsub)
_mul = staticmethod(chebmul)
_div = staticmethod(chebdiv)
_pow = staticmethod(chebpow)
_val = staticmethod(chebval)
_int = staticmethod(chebint)
_der = staticmethod(chebder)
_fit = staticmethod(chebfit)
_line = staticmethod(chebline)
_roots = staticmethod(chebroots)
_fromroots = staticmethod(chebfromroots)
@classmethod
def interpolate(cls, func, deg, domain=None, args=()):
"""Interpolate a function at the Chebyshev points of the first kind.
Returns the series that interpolates `func` at the Chebyshev points of
the first kind scaled and shifted to the `domain`. The resulting series
tends to a minmax approximation of `func` when the function is
continuous in the domain.
.. versionadded:: 1.14.0
Parameters
----------
func : function
The function to be interpolated. It must be a function of a single
variable of the form ``f(x, a, b, c...)``, where ``a, b, c...`` are
extra arguments passed in the `args` parameter.
deg : int
Degree of the interpolating polynomial.
domain : {None, [beg, end]}, optional
Domain over which `func` is interpolated. The default is None, in
which case the domain is [-1, 1].
args : tuple, optional
Extra arguments to be used in the function call. Default is no
extra arguments.
Returns
-------
polynomial : Chebyshev instance
Interpolating Chebyshev instance.
Notes
-----
See `numpy.polynomial.chebfromfunction` for more details.
"""
if domain is None:
domain = cls.domain
xfunc = lambda x: func(pu.mapdomain(x, cls.window, domain), *args)
coef = chebinterpolate(xfunc, deg)
return cls(coef, domain=domain)
# Virtual properties
domain = np.array(chebdomain)
window = np.array(chebdomain)
basis_name = 'T'
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