numpy.polynomial.chebyshev

module documentation

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

Constants

Arithmetic

Calculus

Misc Functions

Notes

The implementations of multiplication, division, integration, and differentiation use the algebraic identities [1]:

T_n(x) = (zⁿ + z^− n)/(2) z(dx)/(dz) = (z − z^− 1)/(2).

where

x = (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)

Class	`Chebyshev`	A Chebyshev series class.
Function	`cheb2poly`	Convert a Chebyshev series to a polynomial.
Function	`chebadd`	Add one Chebyshev series to another.
Function	`chebcompanion`	Return the scaled companion matrix of c.
Function	`chebder`	Differentiate a Chebyshev series.
Function	`chebdiv`	Divide one Chebyshev series by another.
Function	`chebfit`	Least squares fit of Chebyshev series to data.
Function	`chebfromroots`	Generate a Chebyshev series with given roots.
Function	`chebgauss`	Gauss-Chebyshev quadrature.
Function	`chebgrid2d`	Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.
Function	`chebgrid3d`	Evaluate a 3-D Chebyshev series on the Cartesian product of x, y, and z.
Function	`chebint`	Integrate a Chebyshev series.
Function	`chebinterpolate`	Interpolate a function at the Chebyshev points of the first kind.
Function	`chebline`	Chebyshev series whose graph is a straight line.
Function	`chebmul`	Multiply one Chebyshev series by another.
Function	`chebmulx`	Multiply a Chebyshev series by x.
Function	`chebpow`	Raise a Chebyshev series to a power.
Function	`chebpts1`	Chebyshev points of the first kind.
Function	`chebpts2`	Chebyshev points of the second kind.
Function	`chebroots`	Compute the roots of a Chebyshev series.
Function	`chebsub`	Subtract one Chebyshev series from another.
Function	`chebval`	Evaluate a Chebyshev series at points x.
Function	`chebval2d`	Evaluate a 2-D Chebyshev series at points (x, y).
Function	`chebval3d`	Evaluate a 3-D Chebyshev series at points (x, y, z).
Function	`chebvander`	Pseudo-Vandermonde matrix of given degree.
Function	`chebvander2d`	Pseudo-Vandermonde matrix of given degrees.
Function	`chebvander3d`	Pseudo-Vandermonde matrix of given degrees.
Function	`chebweight`	The weight function of the Chebyshev polynomials.
Function	`poly2cheb`	Convert a polynomial to a Chebyshev series.
Variable	`chebdomain`	Undocumented
Variable	`chebone`	Undocumented
Variable	`chebx`	Undocumented
Variable	`chebzero`	Undocumented
Function	`_cseries_to_zseries`	Convert Chebyshev series to z-series.
Function	`_zseries_der`	Differentiate a z-series.
Function	`_zseries_div`	Divide the first z-series by the second.
Function	`_zseries_int`	Integrate a z-series.
Function	`_zseries_mul`	Multiply two z-series.
Function	`_zseries_to_cseries`	Convert z-series to a Chebyshev series.

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.

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.])

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.

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.])

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

New in version 1.7.0.

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).

New in version 1.7.0.

Returns

der : ndarray: Chebyshev series of the derivative.

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.])

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.

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.]))

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

p(x) = c₀ + c₁*T₁(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.

New in version 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)

Notes

The solution is the coefficients of the Chebyshev series p that minimizes the sum of the weighted squared errors

E = ∑_jw²_j*|y_j − p(x_j)|²,

where w_j are the weights. This problem is solved by setting up as the (typically) overdetermined matrix equation

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

def chebfromroots(roots):

Generate a Chebyshev series with given roots.

The function returns the coefficients of the polynomial

p(x) = (x − r₀)*(x − r₁)*...*(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

p(x) = c₀ + c₁*T₁(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).

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])

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 2*deg − 1 or less over the interval [ − 1, 1] with the weight function f(x) = 1 ⁄ √(1 − x²).

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

New in version 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

x_i = cos(π(2i − 1) ⁄ (2n))

w_i = π ⁄ n

def chebgrid2d(x, y, c):

Evaluate a 2-D Chebyshev series on the Cartesian product of x and y.

This function returns the values:

p(a, b) = ∑_i, jc_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.

Notes

New in version 1.7.0.

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:

p(a, b, c) = ∑_i, j, kc_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.

Notes

New in version 1.7.0.

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).

New in version 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.

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 u = ax + b in an integral relative to x. Then dx = du ⁄ a, so one will need to set scl equal to 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.])

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.

New in version 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.

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.

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

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.

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])

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

New in version 1.5.0.

Examples

>>> from numpy.polynomial import chebyshev as C
>>> C.chebmulx([1,2,3])
array([1. , 2.5, 1. , 1.5])

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.

Examples

>>> from numpy.polynomial import chebyshev as C
>>> C.chebpow([1, 2, 3, 4], 2)
array([15.5, 22. , 16. , ..., 12.5, 12. ,  8. ])

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.

Notes

New in version 1.5.0.

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

New in version 1.5.0.

def chebroots(c):

Compute the roots of a Chebyshev series.

Return the roots (a.k.a. "zeros") of the polynomial

p(x) = ∑_ic[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.

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

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.

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.])

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:

p(x) = c₀*T₀(x) + c₁*T₁(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.

New in version 1.7.0.

Returns

values : ndarray, algebra_like: The shape of the return value is described above.

Notes

The evaluation uses Clenshaw recursion, aka synthetic division.

def chebval2d(x, y, c):

Evaluate a 2-D Chebyshev series at points (x, y).

This function returns the values:

p(x, y) = ∑_i, jc_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.

Notes

New in version 1.7.0.

def chebval3d(x, y, z, c):

Evaluate a 3-D Chebyshev series at points (x, y, z).

This function returns the values:

p(x, y, z) = ∑_i, j, kc_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.

Notes

New in version 1.7.0.

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

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.

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

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

c₀₀, c₀₁, c₀₂..., c₁₀, c₁₁, c₁₂...

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 order = (deg[0] + 1)*(deg[1] + 1). The dtype will be the same as the converted x and y.

Notes

New in version 1.7.0.

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

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

c₀₀₀, c₀₀₁, c₀₀₂, ..., c₀₁₀, c₀₁₁, c₀₁₂, ...

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 order = (deg[0] + 1)*(deg[1] + 1)*(deg[2] + 1). The dtype will be the same as the converted x, y, and z.

Notes

New in version 1.7.0.

def chebweight(x):

The weight function of the Chebyshev polynomials.

The weight function is 1 ⁄ √(1 − x²) and the interval of integration is [ − 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

New in version 1.7.0.

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.

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])

chebdomain =

Undocumented

chebone =

Undocumented

chebx =

Undocumented

chebzero =

Undocumented

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.

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.

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.

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

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

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.

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`numpy.polynomial.chebyshev`