|
""" |
|
Objects for dealing with Chebyshev series. |
|
|
|
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`). |
|
|
|
Constants |
|
--------- |
|
- `chebdomain` -- Chebyshev series default domain, [-1,1]. |
|
- `chebzero` -- (Coefficients of the) Chebyshev series that evaluates |
|
identically to 0. |
|
- `chebone` -- (Coefficients of the) Chebyshev series that evaluates |
|
identically to 1. |
|
- `chebx` -- (Coefficients of the) Chebyshev series for the identity map, |
|
``f(x) = x``. |
|
|
|
Arithmetic |
|
---------- |
|
- `chebadd` -- add two Chebyshev series. |
|
- `chebsub` -- subtract one Chebyshev series from another. |
|
- `chebmul` -- multiply two Chebyshev series. |
|
- `chebdiv` -- divide one Chebyshev series by another. |
|
- `chebpow` -- raise a Chebyshev series to an positive integer power |
|
- `chebval` -- evaluate a Chebyshev series at given points. |
|
- `chebval2d` -- evaluate a 2D Chebyshev series at given points. |
|
- `chebval3d` -- evaluate a 3D Chebyshev series at given points. |
|
- `chebgrid2d` -- evaluate a 2D Chebyshev series on a Cartesian product. |
|
- `chebgrid3d` -- evaluate a 3D Chebyshev series on a Cartesian product. |
|
|
|
Calculus |
|
-------- |
|
- `chebder` -- differentiate a Chebyshev series. |
|
- `chebint` -- integrate a Chebyshev series. |
|
|
|
Misc Functions |
|
-------------- |
|
- `chebfromroots` -- create a Chebyshev series with specified roots. |
|
- `chebroots` -- find the roots of a Chebyshev series. |
|
- `chebvander` -- Vandermonde-like matrix for Chebyshev polynomials. |
|
- `chebvander2d` -- Vandermonde-like matrix for 2D power series. |
|
- `chebvander3d` -- Vandermonde-like matrix for 3D power series. |
|
- `chebgauss` -- Gauss-Chebyshev quadrature, points and weights. |
|
- `chebweight` -- Chebyshev weight function. |
|
- `chebcompanion` -- symmetrized companion matrix in Chebyshev form. |
|
- `chebfit` -- least-squares fit returning a Chebyshev series. |
|
- `chebpts1` -- Chebyshev points of the first kind. |
|
- `chebpts2` -- Chebyshev points of the second kind. |
|
- `chebtrim` -- trim leading coefficients from a Chebyshev series. |
|
- `chebline` -- Chebyshev series representing given straight line. |
|
- `cheb2poly` -- convert a Chebyshev series to a polynomial. |
|
- `poly2cheb` -- convert a polynomial to a Chebyshev series. |
|
|
|
Classes |
|
------- |
|
- `Chebyshev` -- A Chebyshev series class. |
|
|
|
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 |
|
(preprint: http://www.math.hmc.edu/~benjamin/papers/CombTrig.pdf, pg. 4) |
|
|
|
""" |
|
from __future__ import division, absolute_import, print_function |
|
|
|
import warnings |
|
import numpy as np |
|
import numpy.linalg as la |
|
|
|
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'] |
|
|
|
chebtrim = pu.trimcoef |
|
|
|
|
|
|
|
|
|
|
|
|
|
def _cseries_to_zseries(c): |
|
"""Covert Chebyshev series to z-series. |
|
|
|
Covert 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): |
|
"""Covert z-series to a Chebyshev series. |
|
|
|
Covert 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() |
|
len1 = len(z1) |
|
len2 = len(z2) |
|
if len2 == 1: |
|
z1 /= z2 |
|
return z1, z1[:1]*0 |
|
elif len1 < len2: |
|
return z1[:1]*0, z1 |
|
else: |
|
dlen = len1 - len2 |
|
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+len2] -= tmp |
|
z1[j:j+len2] -= tmp |
|
i += 1 |
|
j -= 1 |
|
r = z1[i] |
|
quo[i] = r |
|
tmp = r*z2 |
|
z1[i:i+len2] -= tmp |
|
quo /= scl |
|
rem = z1[i+1:i-1+len2].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 |
|
|
|
|
|
|
|
|
|
|
|
|
|
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.], [-1., 1.]) |
|
>>> c = p.convert(kind=P.Chebyshev) |
|
>>> c |
|
Chebyshev([ 1. , 3.25, 1. , 0.75], [-1., 1.]) |
|
>>> P.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.], [-1., 1.]) |
|
>>> p = c.convert(kind=P.Polynomial) |
|
>>> p |
|
Polynomial([ -2., -8., 4., 12.], [-1., 1.]) |
|
>>> P.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] |
|
|
|
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)) |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
chebdomain = np.array([-1, 1]) |
|
|
|
|
|
chebzero = np.array([0]) |
|
|
|
|
|
chebone = np.array([1]) |
|
|
|
|
|
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 |
|
-------- |
|
polyline |
|
|
|
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 |
|
-------- |
|
polyfromroots, legfromroots, lagfromroots, hermfromroots, |
|
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+0.j, 0.5+0.j]) |
|
|
|
""" |
|
if len(roots) == 0: |
|
return np.ones(1) |
|
else: |
|
[roots] = pu.as_series([roots], trim=False) |
|
roots.sort() |
|
p = [chebline(-r, 1) for r in roots] |
|
n = len(p) |
|
while n > 1: |
|
m, r = divmod(n, 2) |
|
tmp = [chebmul(p[i], p[i+m]) for i in range(m)] |
|
if r: |
|
tmp[0] = chebmul(tmp[0], p[-1]) |
|
p = tmp |
|
n = m |
|
return p[0] |
|
|
|
|
|
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, 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.]) |
|
|
|
""" |
|
|
|
[c1, c2] = pu.as_series([c1, c2]) |
|
if len(c1) > len(c2): |
|
c1[:c2.size] += c2 |
|
ret = c1 |
|
else: |
|
c2[:c1.size] += c1 |
|
ret = c2 |
|
return pu.trimseq(ret) |
|
|
|
|
|
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, 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.]) |
|
|
|
""" |
|
|
|
[c1, c2] = pu.as_series([c1, c2]) |
|
if len(c1) > len(c2): |
|
c1[:c2.size] -= c2 |
|
ret = c1 |
|
else: |
|
c2 = -c2 |
|
c2[:c1.size] += c1 |
|
ret = c2 |
|
return pu.trimseq(ret) |
|
|
|
|
|
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 |
|
|
|
""" |
|
|
|
[c] = pu.as_series([c]) |
|
|
|
if len(c) == 1 and c[0] == 0: |
|
return c |
|
|
|
prd = np.empty(len(c) + 1, dtype=c.dtype) |
|
prd[0] = c[0]*0 |
|
prd[1] = c[0] |
|
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, 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] = 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, 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] = pu.as_series([c1, c2]) |
|
if c2[-1] == 0: |
|
raise ZeroDivisionError() |
|
|
|
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, chebmul, chebdiv |
|
|
|
Examples |
|
-------- |
|
|
|
""" |
|
|
|
[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: |
|
|
|
|
|
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=1) |
|
if c.dtype.char in '?bBhHiIlLqQpP': |
|
c = c.astype(np.double) |
|
cnt, iaxis = [int(t) for t in [m, axis]] |
|
|
|
if cnt != m: |
|
raise ValueError("The order of derivation must be integer") |
|
if cnt < 0: |
|
raise ValueError("The order of derivation must be non-negative") |
|
if iaxis != axis: |
|
raise ValueError("The axis must be integer") |
|
if not -c.ndim <= iaxis < c.ndim: |
|
raise ValueError("The axis is out of range") |
|
if iaxis < 0: |
|
iaxis += c.ndim |
|
|
|
if cnt == 0: |
|
return c |
|
|
|
c = np.rollaxis(c, iaxis) |
|
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.rollaxis(c, 0, iaxis + 1) |
|
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.isscalar(lbnd) == False``, or |
|
``np.isscalar(scl) == False``. |
|
|
|
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, |
|
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=1) |
|
if c.dtype.char in '?bBhHiIlLqQpP': |
|
c = c.astype(np.double) |
|
if not np.iterable(k): |
|
k = [k] |
|
cnt, iaxis = [int(t) for t in [m, axis]] |
|
|
|
if cnt != m: |
|
raise ValueError("The order of integration must be integer") |
|
if cnt < 0: |
|
raise ValueError("The order of integration must be non-negative") |
|
if len(k) > cnt: |
|
raise ValueError("Too many integration constants") |
|
if iaxis != axis: |
|
raise ValueError("The axis must be integer") |
|
if not -c.ndim <= iaxis < c.ndim: |
|
raise ValueError("The axis is out of range") |
|
if iaxis < 0: |
|
iaxis += c.ndim |
|
|
|
if cnt == 0: |
|
return c |
|
|
|
c = np.rollaxis(c, iaxis) |
|
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): |
|
t = c[j]/(2*j + 1) |
|
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.rollaxis(c, 0, iaxis + 1) |
|
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 |
|
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. |
|
|
|
Examples |
|
-------- |
|
|
|
""" |
|
c = np.array(c, ndmin=1, copy=1) |
|
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 |
|
|
|
""" |
|
try: |
|
x, y = np.array((x, y), copy=0) |
|
except: |
|
raise ValueError('x, y are incompatible') |
|
|
|
c = chebval(x, c) |
|
c = chebval(y, c, tensor=False) |
|
return c |
|
|
|
|
|
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 |
|
|
|
""" |
|
c = chebval(x, c) |
|
c = chebval(y, c) |
|
return c |
|
|
|
|
|
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 |
|
|
|
""" |
|
try: |
|
x, y, z = np.array((x, y, z), copy=0) |
|
except: |
|
raise ValueError('x, y, z are incompatible') |
|
|
|
c = chebval(x, c) |
|
c = chebval(y, c, tensor=False) |
|
c = chebval(z, c, tensor=False) |
|
return c |
|
|
|
|
|
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 |
|
|
|
""" |
|
c = chebval(x, c) |
|
c = chebval(y, c) |
|
c = chebval(z, c) |
|
return c |
|
|
|
|
|
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 = int(deg) |
|
if ideg != deg: |
|
raise ValueError("deg must be integer") |
|
if ideg < 0: |
|
raise ValueError("deg must be non-negative") |
|
|
|
x = np.array(x, copy=0, ndmin=1) + 0.0 |
|
dims = (ideg + 1,) + x.shape |
|
dtyp = x.dtype |
|
v = np.empty(dims, dtype=dtyp) |
|
|
|
v[0] = x*0 + 1 |
|
if ideg > 0: |
|
x2 = 2*x |
|
v[1] = x |
|
for i in range(2, ideg + 1): |
|
v[i] = v[i-1]*x2 - v[i-2] |
|
return np.rollaxis(v, 0, v.ndim) |
|
|
|
|
|
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]*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 |
|
|
|
""" |
|
ideg = [int(d) for d in deg] |
|
is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] |
|
if is_valid != [1, 1]: |
|
raise ValueError("degrees must be non-negative integers") |
|
degx, degy = ideg |
|
x, y = np.array((x, y), copy=0) + 0.0 |
|
|
|
vx = chebvander(x, degx) |
|
vy = chebvander(y, degy) |
|
v = vx[..., None]*vy[..., None,:] |
|
return v.reshape(v.shape[:-2] + (-1,)) |
|
|
|
|
|
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 |
|
|
|
""" |
|
ideg = [int(d) for d in deg] |
|
is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] |
|
if is_valid != [1, 1, 1]: |
|
raise ValueError("degrees must be non-negative integers") |
|
degx, degy, degz = ideg |
|
x, y, z = np.array((x, y, z), copy=0) + 0.0 |
|
|
|
vx = chebvander(x, degx) |
|
vy = chebvander(y, degy) |
|
vz = chebvander(z, degz) |
|
v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:] |
|
return v.reshape(v.shape[:-3] + (-1,)) |
|
|
|
|
|
def chebfit(x, y, deg, rcond=None, full=False, w=None): |
|
""" |
|
Least squares fit of Chebyshev 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 * 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 |
|
Degree of the fitting series |
|
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 contribution of each point |
|
``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the |
|
weights are chosen so that the errors of the products ``w[i]*y[i]`` |
|
all have the same variance. 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 |
|
|
|
resid -- sum of squared residuals of the least squares fit |
|
rank -- the numerical rank of the scaled Vandermonde matrix |
|
sv -- singular values of the scaled Vandermonde matrix |
|
rcond -- value of `rcond`. |
|
|
|
For more details, see `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', RankWarning) |
|
|
|
See Also |
|
-------- |
|
polyfit, legfit, lagfit, hermfit, hermefit |
|
chebval : Evaluates a Chebyshev series. |
|
chebvander : Vandermonde matrix of Chebyshev series. |
|
chebweight : Chebyshev weight function. |
|
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", |
|
http://en.wikipedia.org/wiki/Curve_fitting |
|
|
|
Examples |
|
-------- |
|
|
|
""" |
|
order = int(deg) + 1 |
|
x = np.asarray(x) + 0.0 |
|
y = np.asarray(y) + 0.0 |
|
|
|
|
|
if deg < 0: |
|
raise ValueError("expected deg >= 0") |
|
if x.ndim != 1: |
|
raise TypeError("expected 1D vector for x") |
|
if x.size == 0: |
|
raise TypeError("expected non-empty vector for x") |
|
if y.ndim < 1 or y.ndim > 2: |
|
raise TypeError("expected 1D or 2D array for y") |
|
if len(x) != len(y): |
|
raise TypeError("expected x and y to have same length") |
|
|
|
|
|
lhs = chebvander(x, deg).T |
|
rhs = y.T |
|
if w is not None: |
|
w = np.asarray(w) + 0.0 |
|
if w.ndim != 1: |
|
raise TypeError("expected 1D vector for w") |
|
if len(x) != len(w): |
|
raise TypeError("expected x and w to have same length") |
|
|
|
|
|
lhs = lhs * w |
|
rhs = rhs * w |
|
|
|
|
|
if rcond is None: |
|
rcond = len(x)*np.finfo(x.dtype).eps |
|
|
|
|
|
if issubclass(lhs.dtype.type, np.complexfloating): |
|
scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1)) |
|
else: |
|
scl = np.sqrt(np.square(lhs).sum(1)) |
|
scl[scl == 0] = 1 |
|
|
|
|
|
c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond) |
|
c = (c.T/scl).T |
|
|
|
|
|
if rank != order and not full: |
|
msg = "The fit may be poorly conditioned" |
|
warnings.warn(msg, pu.RankWarning) |
|
|
|
if full: |
|
return c, [resids, rank, s, rcond] |
|
else: |
|
return c |
|
|
|
|
|
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 aa 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] = 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 |
|
-------- |
|
polyroots, legroots, lagroots, hermroots, 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]) |
|
|
|
""" |
|
|
|
[c] = pu.as_series([c]) |
|
if len(c) < 2: |
|
return np.array([], dtype=c.dtype) |
|
if len(c) == 2: |
|
return np.array([-c[0]/c[1]]) |
|
|
|
m = chebcompanion(c) |
|
r = la.eigvals(m) |
|
r.sort() |
|
return r |
|
|
|
|
|
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 = int(deg) |
|
if ideg != deg or ideg < 1: |
|
raise ValueError("deg must be a non-negative 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 = np.linspace(-np.pi, 0, _npts, endpoint=False) + np.pi/(2*_npts) |
|
return np.cos(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)]``. |
|
|
|
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) |
|
|
|
|
|
|
|
|
|
|
|
|
|
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 |
|
|
|
""" |
|
|
|
_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) |
|
|
|
|
|
nickname = 'cheb' |
|
domain = np.array(chebdomain) |
|
window = np.array(chebdomain) |
|
|
