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편집 파일: chebyshev.py

""" 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 numpy.core.multiarray import normalize_axis_index from . import polyutils as pu from ._polybase import ABCPolyBase __all__ = [ 'chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline', 'chebadd', 'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow', 'chebval', 'chebder', 'chebint', 'cheb2poly', 'poly2cheb', 'chebfromroots', 'chebvander', 'chebfit', 'chebtrim', 'chebroots', 'chebpts1', 'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d', 'chebgrid2d', 'chebgrid3d', 'chebvander2d', 'chebvander3d', 'chebcompanion', 'chebgauss', 'chebweight'] chebtrim = pu.trimcoef # # A collection of functions for manipulating z-series. These are private # functions and do minimal error checking. # def _cseries_to_zseries(c): """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 # # Chebyshev series functions # def poly2cheb(pol): """ Convert a polynomial to a Chebyshev series. Convert an array representing the coefficients of a polynomial (relative to the "standard" basis) ordered from lowest degree to highest, to an array of the coefficients of the equivalent Chebyshev series, ordered from lowest to highest degree. Parameters ---------- pol : array_like 1-D array containing the polynomial coefficients Returns ------- c : ndarray 1-D array containing the coefficients of the equivalent Chebyshev series. See Also -------- cheb2poly Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> from numpy import polynomial as P >>> p = P.Polynomial(range(4)) >>> p Polynomial([ 0., 1., 2., 3.], [-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] # i is the current degree of c1 for i in range(n - 1, 1, -1): tmp = c0 c0 = polysub(c[i - 2], c1) c1 = polyadd(tmp, polymulx(c1)*2) return polyadd(c0, polymulx(c1)) # # These are constant arrays are of integer type so as to be compatible # with the widest range of other types, such as Decimal. # # Chebyshev default domain. chebdomain = np.array([-1, 1]) # Chebyshev coefficients representing zero. chebzero = np.array([0]) # Chebyshev coefficients representing one. chebone = np.array([1]) # Chebyshev coefficients representing the identity x. chebx = np.array([0, 1]) def chebline(off, scl): """ Chebyshev series whose graph is a straight line. Parameters ---------- off, scl : scalars The specified line is given by ``off + scl*x``. Returns ------- y : ndarray This module's representation of the Chebyshev series for ``off + scl*x``. See Also -------- 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 are trimmed copies [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 are trimmed copies [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 is a trimmed copy [c] = pu.as_series([c]) # The zero series needs special treatment if len(c) == 1 and c[0] == 0: return c prd = np.empty(len(c) + 1, dtype=c.dtype) prd[0] = c[0]*0 prd[1] = c[0] if len(c) > 1: tmp = c[1:]/2 prd[2:] = tmp prd[0:-2] += tmp return prd def chebmul(c1, c2): """ Multiply one Chebyshev series by another. Returns the product of two Chebyshev series `c1` * `c2`. The arguments are sequences of coefficients, from lowest order "term" to highest, e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Chebyshev series coefficients ordered from low to high. Returns ------- out : ndarray Of Chebyshev series coefficients representing their product. See Also -------- chebadd, chebsub, chebdiv, chebpow Notes ----- In general, the (polynomial) product of two C-series results in terms that are not in the Chebyshev polynomial basis set. Thus, to express the product as a C-series, it is typically necessary to "reproject" the product onto said basis set, which typically produces "unintuitive live" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> C.chebmul(c1,c2) # multiplication requires "reprojection" array([ 6.5, 12. , 12. , 4. , 1.5]) """ # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) z1 = _cseries_to_zseries(c1) z2 = _cseries_to_zseries(c2) prd = _zseries_mul(z1, z2) ret = _zseries_to_cseries(prd) return pu.trimseq(ret) def chebdiv(c1, c2): """ Divide one Chebyshev series by another. Returns the quotient-with-remainder of two Chebyshev series `c1` / `c2`. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series ``T_0 + 2*T_1 + 3*T_2``. Parameters ---------- c1, c2 : array_like 1-D arrays of Chebyshev series coefficients ordered from low to high. Returns ------- [quo, rem] : ndarrays Of Chebyshev series coefficients representing the quotient and remainder. See Also -------- chebadd, chebsub, chebmul, chebpow Notes ----- In general, the (polynomial) division of one C-series by another results in quotient and remainder terms that are not in the Chebyshev polynomial basis set. Thus, to express these results as C-series, it is typically necessary to "reproject" the results onto said basis set, which typically produces "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> C.chebdiv(c1,c2) # quotient "intuitive," remainder not (array([ 3.]), array([-8., -4.])) >>> c2 = (0,1,2,3) >>> C.chebdiv(c2,c1) # neither "intuitive" (array([ 0., 2.]), array([-2., -4.])) """ # c1, c2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if c2[-1] == 0: raise ZeroDivisionError() 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 is a trimmed copy [c] = pu.as_series([c]) power = int(pow) if power != pow or power < 0: raise ValueError("Power must be a non-negative integer.") elif maxpower is not None and power > maxpower: raise ValueError("Power is too large") elif power == 0: return np.array([1], dtype=c.dtype) elif power == 1: return c else: # This can be made more efficient by using powers of two # in the usual way. zs = _cseries_to_zseries(c) prd = zs for i in range(2, power + 1): prd = np.convolve(prd, zs) return _zseries_to_cseries(prd) def chebder(c, m=1, scl=1, axis=0): """ Differentiate a Chebyshev series. Returns the Chebyshev series coefficients `c` differentiated `m` times along `axis`. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `c` is an array of coefficients from low to high degree along each axis, e.g., [1,2,3] represents the series ``1*T_0 + 2*T_1 + 3*T_2`` while [[1,2],[1,2]] represents ``1*T_0(x)*T_0(y) + 1*T_1(x)*T_0(y) + 2*T_0(x)*T_1(y) + 2*T_1(x)*T_1(y)`` if axis=0 is ``x`` and axis=1 is ``y``. Parameters ---------- c : array_like Array of Chebyshev series coefficients. If c is multidimensional the different axis correspond to different variables with the degree in each axis given by the corresponding index. m : int, optional Number of derivatives taken, must be non-negative. (Default: 1) scl : scalar, optional Each differentiation is multiplied by `scl`. The end result is multiplication by ``scl**m``. This is for use in a linear change of variable. (Default: 1) axis : int, optional Axis over which the derivative is taken. (Default: 0). .. versionadded:: 1.7.0 Returns ------- der : ndarray Chebyshev series of the derivative. See Also -------- chebint Notes ----- In general, the result of differentiating a C-series needs to be "reprojected" onto the C-series basis set. Thus, typically, the result of this function is "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> c = (1,2,3,4) >>> C.chebder(c) array([ 14., 12., 24.]) >>> C.chebder(c,3) array([ 96.]) >>> C.chebder(c,scl=-1) array([-14., -12., -24.]) >>> C.chebder(c,2,-1) array([ 12., 96.]) """ c = np.array(c, ndmin=1, copy=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") iaxis = normalize_axis_index(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") iaxis = normalize_axis_index(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) # Use forward recursion to generate the entries. v[0] = x*0 + 1 if ideg > 0: x2 = 2*x v[1] = x for i in range(2, ideg + 1): v[i] = v[i-1]*x2 - v[i-2] return np.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] + 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 Chebyshev series of degree `deg` that is the least squares fit to the data values `y` given at points `x`. If `y` is 1-D the returned coefficients will also be 1-D. If `y` is 2-D multiple fits are done, one for each column of `y`, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * T_1(x) + ... + c_n * T_n(x), where `n` is `deg`. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int or 1-D array_like Degree(s) of the fitting polynomials. If `deg` is a single integer all terms up to and including the `deg`'th term are included in the fit. For NumPy versions >= 1.11.0 a list of integers specifying the degrees of the terms to include may be used instead. rcond : float, optional Relative condition number of the fit. Singular values smaller than this relative to the largest singular value will be ignored. The default value is len(x)*eps, where eps is the relative precision of the float type, about 2e-16 in most cases. full : bool, optional Switch determining nature of return value. When it is False (the default) just the coefficients are returned, when True diagnostic information from the singular value decomposition is also returned. w : array_like, shape (`M`,), optional Weights. If not None, the 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 -------- """ x = np.asarray(x) + 0.0 y = np.asarray(y) + 0.0 deg = np.asarray(deg) # check arguments. if deg.ndim > 1 or deg.dtype.kind not in 'iu' or deg.size == 0: raise TypeError("deg must be an int or non-empty 1-D array of int") if deg.min() < 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") if deg.ndim == 0: lmax = deg order = lmax + 1 van = chebvander(x, lmax) else: deg = np.sort(deg) lmax = deg[-1] order = len(deg) van = chebvander(x, lmax)[:, deg] # set up the least squares matrices in transposed form lhs = van.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") # apply weights. Don't use inplace operations as they # can cause problems with NA. lhs = lhs * w rhs = rhs * w # set rcond if rcond is None: rcond = len(x)*np.finfo(x.dtype).eps # Determine the norms of the design matrix columns. 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 # Solve the least squares problem. c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond) c = (c.T/scl).T # Expand c to include non-fitted coefficients which are set to zero if deg.ndim > 0: if c.ndim == 2: cc = np.zeros((lmax + 1, c.shape[1]), dtype=c.dtype) else: cc = np.zeros(lmax + 1, dtype=c.dtype) cc[deg] = c c = cc # warn on rank reduction if rank != order and not full: msg = "The fit may be poorly conditioned" warnings.warn(msg, pu.RankWarning, stacklevel=2) 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 a Chebyshev basis polynomial. This provides better eigenvalue estimates than the unscaled case and for basis polynomials the eigenvalues are guaranteed to be real if `numpy.linalg.eigvalsh` is used to obtain them. Parameters ---------- c : array_like 1-D array of Chebyshev series coefficients ordered from low to high degree. Returns ------- mat : ndarray Scaled companion matrix of dimensions (deg, deg). Notes ----- .. versionadded:: 1.7.0 """ # c is a trimmed copy [c] = pu.as_series([c]) if len(c) < 2: raise ValueError('Series must have maximum degree of at least 1.') if len(c) == 2: return np.array([[-c[0]/c[1]]]) n = len(c) - 1 mat = np.zeros((n, n), dtype=c.dtype) scl = np.array([1.] + [np.sqrt(.5)]*(n-1)) top = mat.reshape(-1)[1::n+1] bot = mat.reshape(-1)[n::n+1] top[0] = np.sqrt(.5) top[1:] = 1/2 bot[...] = top mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*.5 return mat def chebroots(c): """ Compute the roots of a Chebyshev series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * T_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the roots of the series. If all the roots are real, then `out` is also real, otherwise it is complex. See Also -------- 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 is a trimmed copy [c] = pu.as_series([c]) if len(c) < 2: return np.array([], dtype=c.dtype) if len(c) == 2: return np.array([-c[0]/c[1]]) 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) # # Chebyshev series class # class Chebyshev(ABCPolyBase): """A Chebyshev series class. The Chebyshev class provides the standard Python numerical methods '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the methods listed below. Parameters ---------- coef : array_like Chebyshev coefficients in order of increasing degree, i.e., ``(1, 2, 3)`` gives ``1*T_0(x) + 2*T_1(x) + 3*T_2(x)``. domain : (2,) array_like, optional Domain to use. The interval ``[domain[0], domain[1]]`` is mapped to the interval ``[window[0], window[1]]`` by shifting and scaling. The default value is [-1, 1]. window : (2,) array_like, optional Window, see `domain` for its use. The default value is [-1, 1]. .. versionadded:: 1.6.0 """ # Virtual Functions _add = staticmethod(chebadd) _sub = staticmethod(chebsub) _mul = staticmethod(chebmul) _div = staticmethod(chebdiv) _pow = staticmethod(chebpow) _val = staticmethod(chebval) _int = staticmethod(chebint) _der = staticmethod(chebder) _fit = staticmethod(chebfit) _line = staticmethod(chebline) _roots = staticmethod(chebroots) _fromroots = staticmethod(chebfromroots) # Virtual properties nickname = 'cheb' domain = np.array(chebdomain) window = np.array(chebdomain)