""" 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. - `chebval` -- evaluate a Chebyshev series at given points. 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. - `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 of 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 __all__ = ['chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline', 'chebadd', 'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebval', 'chebder', 'chebint', 'cheb2poly', 'poly2cheb', 'chebfromroots', 'chebvander', 'chebfit', 'chebtrim', 'chebroots', 'chebpts1', 'chebpts2', 'Chebyshev'] import numpy as np import numpy.linalg as la import polyutils as pu import warnings from polytemplate import polytemplate chebtrim = pu.trimcoef # # A collection of functions for manipulating z-series. These are private # functions and do minimal error checking. # def _cseries_to_zseries(cs) : """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 ---------- cs : 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 = cs.size zs = np.zeros(2*n-1, dtype=cs.dtype) zs[n-1:] = cs/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 ------- cs : 1-d ndarray Chebyshev coefficients, ordered from low to high. """ n = (zs.size + 1)//2 cs = zs[n-1:].copy() cs[1:n] *= 2 return cs 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 symmetic/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 symmetic/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 uneeded 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 ------- cs : 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(cs) : """ 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 ---------- cs : 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 [cs] = pu.as_series([cs]) n = len(cs) if n < 3: return cs else: c0 = cs[-2] c1 = cs[-1] for i in range(n - 3, -1, -1) : tmp = c0 c0 = polysub(cs[i], 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 the given roots. Return the array of coefficients for the C-series whose roots (a.k.a. "zeros") are given by *roots*. The returned array of coefficients is ordered from lowest order "term" to highest, and zeros of multiplicity greater than one must be included in *roots* a number of times equal to their multiplicity (e.g., if `2` is a root of multiplicity three, then [2,2,2] must be in *roots*). Parameters ---------- roots : array_like Sequence containing the roots. Returns ------- out : ndarray 1-d array of the C-series' coefficients, ordered from low to high. If all roots are real, ``out.dtype`` is a float type; otherwise, ``out.dtype`` is a complex type, even if all the coefficients in the result are real (see Examples below). See Also -------- polyfromroots Notes ----- What is returned are the :math:`c_i` such that: .. math:: \\sum_{i=0}^{n} c_i*T_i(x) = \\prod_{i=0}^{n} (x - roots[i]) where ``n == len(roots)`` and :math:`T_i(x)` is the `i`-th Chebyshev (basis) polynomial over the domain `[-1,1]`. Note that, unlike `polyfromroots`, due to the nature of the C-series basis set, the above identity *does not* imply :math:`c_n = 1` identically (see Examples). 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) prd = np.array([1], dtype=roots.dtype) for r in roots: prd = chebsub(chebmulx(prd), r*prd) return prd 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(cs): """Multiply a Chebyshev series by x. Multiply the polynomial `cs` by x, where x is the independent variable. Parameters ---------- cs : 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 """ # cs is a trimmed copy [cs] = pu.as_series([cs]) # The zero series needs special treatment if len(cs) == 1 and cs[0] == 0: return cs prd = np.empty(len(cs) + 1, dtype=cs.dtype) prd[0] = cs[0]*0 prd[1] = cs[0] if len(cs) > 1: tmp = cs[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 "re-project" the product onto said basis set, which typically produces "un-intuitive" (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 "re-project" the results onto said basis set, which typically produces "un-intuitive" (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(cs, pow, maxpower=16) : """Raise a Chebyshev series to a power. Returns the Chebyshev series `cs` raised to the power `pow`. The arguement `cs` 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 ---------- cs : array_like 1d 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 umanageable size. Default is 16 Returns ------- coef : ndarray Chebyshev series of power. See Also -------- chebadd, chebsub, chebmul, chebdiv Examples -------- """ # cs is a trimmed copy [cs] = pu.as_series([cs]) 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=cs.dtype) elif power == 1 : return cs else : # This can be made more efficient by using powers of two # in the usual way. zs = _cseries_to_zseries(cs) prd = zs for i in range(2, power + 1) : prd = np.convolve(prd, zs) return _zseries_to_cseries(prd) def chebder(cs, m=1, scl=1) : """ Differentiate a Chebyshev series. Returns the series `cs` differentiated `m` times. At each iteration the result is multiplied by `scl` (the scaling factor is for use in a linear change of variable). The argument `cs` is the sequence 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 ---------- cs: array_like 1-d array of Chebyshev series coefficients ordered from low to high. 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) Returns ------- der : ndarray Chebyshev series of the derivative. See Also -------- chebint Notes ----- In general, the result of differentiating a C-series needs to be "re-projected" onto the C-series basis set. Thus, typically, the result of this function is "un-intuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> cs = (1,2,3,4) >>> C.chebder(cs) array([ 14., 12., 24.]) >>> C.chebder(cs,3) array([ 96.]) >>> C.chebder(cs,scl=-1) array([-14., -12., -24.]) >>> C.chebder(cs,2,-1) array([ 12., 96.]) """ cnt = int(m) 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" # cs is a trimmed copy [cs] = pu.as_series([cs]) if cnt == 0: return cs elif cnt >= len(cs): return cs[:1]*0 else : zs = _cseries_to_zseries(cs) for i in range(cnt): zs = _zseries_der(zs)*scl return _zseries_to_cseries(zs) def chebint(cs, m=1, k=[], lbnd=0, scl=1): """ Integrate a Chebyshev series. Returns, as a C-series, the input C-series `cs`, integrated `m` times from `lbnd` to `x`. 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 `cs` is a sequence of coefficients, from lowest order C-series "term" to highest, e.g., [1,2,3] represents the series :math:`T_0(x) + 2T_1(x) + 3T_2(x)`. Parameters ---------- cs : array_like 1-d array of C-series coefficients, ordered from low to high. 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) 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 "re-projected" onto the C-series basis set. Thus, typically, the result of this function is "un-intuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import chebyshev as C >>> cs = (1,2,3) >>> C.chebint(cs) array([ 0.5, -0.5, 0.5, 0.5]) >>> C.chebint(cs,3) array([ 0.03125 , -0.1875 , 0.04166667, -0.05208333, 0.01041667, 0.00625 ]) >>> C.chebint(cs, k=3) array([ 3.5, -0.5, 0.5, 0.5]) >>> C.chebint(cs,lbnd=-2) array([ 8.5, -0.5, 0.5, 0.5]) >>> C.chebint(cs,scl=-2) array([-1., 1., -1., -1.]) """ cnt = int(m) if not np.iterable(k): k = [k] 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" # cs is a trimmed copy [cs] = pu.as_series([cs]) if cnt == 0: return cs k = list(k) + [0]*(cnt - len(k)) for i in range(cnt) : n = len(cs) cs *= scl if n == 1 and cs[0] == 0: cs[0] += k[i] else: zs = _cseries_to_zseries(cs) zs = _zseries_int(zs) cs = _zseries_to_cseries(zs) cs[0] += k[i] - chebval(lbnd, cs) return cs def chebval(x, cs): """Evaluate a Chebyshev series. If `cs` is of length `n`, this function returns : ``p(x) = cs[0]*T_0(x) + cs[1]*T_1(x) + ... + cs[n-1]*T_{n-1}(x)`` If x is a sequence or array then p(x) will have the same shape as x. If r is a ring_like object that supports multiplication and addition by the values in `cs`, then an object of the same type is returned. Parameters ---------- x : array_like, ring_like Array of numbers or objects that support multiplication and addition with themselves and with the elements of `cs`. cs : array_like 1-d array of Chebyshev coefficients ordered from low to high. Returns ------- values : ndarray, ring_like If the return is an ndarray then it has the same shape as `x`. See Also -------- chebfit Examples -------- Notes ----- The evaluation uses Clenshaw recursion, aka synthetic division. Examples -------- """ # cs is a trimmed copy [cs] = pu.as_series([cs]) if isinstance(x, tuple) or isinstance(x, list) : x = np.asarray(x) if len(cs) == 1 : c0 = cs[0] c1 = 0 elif len(cs) == 2 : c0 = cs[0] c1 = cs[1] else : x2 = 2*x c0 = cs[-2] c1 = cs[-1] for i in range(3, len(cs) + 1) : tmp = c0 c0 = cs[-i] - c1 c1 = tmp + c1*x2 return c0 + c1*x def chebvander(x, deg) : """Vandermonde matrix of given degree. Returns the Vandermonde matrix of degree `deg` and sample points `x`. This isn't a true Vandermonde matrix because `x` can be an arbitrary ndarray and the Chebyshev polynomials aren't powers. If ``V`` is the returned matrix and `x` is a 2d array, then the elements of ``V`` are ``V[i,j,k] = T_k(x[i,j])``, where ``T_k`` is the Chebyshev polynomial of degree ``k``. Parameters ---------- x : array_like Array of points. The values are converted to double or complex doubles. If x is scalar it is converted to a 1D array. deg : integer Degree of the resulting matrix. Returns ------- vander : Vandermonde matrix. The shape of the returned matrix is ``x.shape + (deg+1,)``. The last index is the degree. """ 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 v = np.empty((ideg + 1,) + x.shape, dtype=x.dtype) # 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 chebfit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Chebyshev series to data. Fit a Chebyshev series ``p(x) = p[0] * T_{0}(x) + ... + p[deg] * T_{deg}(x)`` of degree `deg` to points `(x, y)`. Returns a vector of coefficients `p` that minimises the squared error. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points ``(x[i], y[i])``. y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int Degree of the fitting polynomial 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] : present when `full` = True Residuals of the least-squares fit, the effective rank of the scaled Vandermonde matrix and its singular values, and the specified 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 -------- chebval : Evaluates a Chebyshev series. chebvander : Vandermonde matrix of Chebyshev series. polyfit : least squares fit using polynomials. linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution are the coefficients ``c[i]`` of the Chebyshev series ``T(x)`` that minimizes the squared error ``E = \\sum_j |y_j - T(x_j)|^2``. This problem is solved by setting up as the overdetermined matrix equation ``V(x)*c = y``, where ``V`` is the Vandermonde matrix of `x`, the elements of ``c`` are the coefficients to be solved for, and the elements of `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 coeficient values may be poorly determined. Using a lower order fit will usually get rid of the warning. The `rcond` parameter can also be set to a value smaller than its default, but the resulting fit may be spurious and have large contributions from roundoff error. Fits using Chebyshev series are usually better conditioned than fits using power series, but much can depend on the distribution of the sample points and the smoothness of the data. If the quality of the fit is inadequate splines may be a good alternative. References ---------- .. [1] Wikipedia, "Curve fitting", http://en.wikipedia.org/wiki/Curve_fitting Examples -------- """ order = int(deg) + 1 x = np.asarray(x) + 0.0 y = np.asarray(y) + 0.0 # check arguments. if deg < 0 : raise ValueError, "expected deg >= 0" if x.ndim != 1: raise TypeError, "expected 1D vector for x" if x.size == 0: raise TypeError, "expected non-empty vector for x" if y.ndim < 1 or y.ndim > 2 : raise TypeError, "expected 1D or 2D array for y" if len(x) != len(y): raise TypeError, "expected x and y to have same length" # set up the least squares matrices lhs = chebvander(x, deg) rhs = y 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 if rhs.ndim == 2: lhs *= w[:, np.newaxis] rhs *= w[:, np.newaxis] else: lhs *= w[:, np.newaxis] rhs *= w # set rcond if rcond is None : rcond = len(x)*np.finfo(x.dtype).eps # scale the design matrix and solve the least squares equation scl = np.sqrt((lhs*lhs).sum(0)) c, resids, rank, s = la.lstsq(lhs/scl, rhs, rcond) c = (c.T/scl).T # warn on rank reduction if rank != order and not full: msg = "The fit may be poorly conditioned" warnings.warn(msg, pu.RankWarning) if full : return c, [resids, rank, s, rcond] else : return c def chebroots(cs): """ Compute the roots of a Chebyshev series. Return the roots (a.k.a "zeros") of the C-series represented by `cs`, which is the sequence of the C-series' coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the C-series ``T_0 + 2*T_1 + 3*T_2``. Parameters ---------- cs : array_like 1-d array of C-series coefficients ordered from low to high. Returns ------- out : ndarray Array of the roots. If all the roots are real, then so is the dtype of ``out``; otherwise, ``out``'s dtype is complex. See Also -------- polyroots Notes ----- Algorithm(s) used: Remember: because the C-series basis set is different from the "standard" basis set, the results of this function *may* not be what one is expecting. Examples -------- >>> import numpy.polynomial as P >>> import numpy.polynomial.chebyshev as C >>> P.polyroots((-1,1,-1,1)) # x^3 - x^2 + x - 1 has two complex roots array([ -4.99600361e-16-1.j, -4.99600361e-16+1.j, 1.00000e+00+0.j]) >>> C.chebroots((-1,1,-1,1)) # T3 - T2 + T1 - T0 has only real roots array([ -5.00000000e-01, 2.60860684e-17, 1.00000000e+00]) """ # cs is a trimmed copy [cs] = pu.as_series([cs]) if len(cs) <= 1 : return np.array([], dtype=cs.dtype) if len(cs) == 2 : return np.array([-cs[0]/cs[1]]) n = len(cs) - 1 cs /= cs[-1] cmat = np.zeros((n,n), dtype=cs.dtype) cmat[1, 0] = 1 for i in range(1, n): cmat[i - 1, i] = .5 if i != n - 1: cmat[i + 1, i] = .5 else: cmat[:, i] -= cs[:-1]*.5 roots = la.eigvals(cmat) roots.sort() return roots def chebpts1(npts): """Chebyshev points of the first kind. Chebyshev points of the first kind are the set ``{cos(x_k)}``, where ``x_k = 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 second kind. 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. Chebyshev points of the second kind are the set ``{cos(x_k)}``, where ``x_k = pi*/(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 # exec polytemplate.substitute(name='Chebyshev', nick='cheb', domain='[-1,1]')