""" Legendre Series (:mod: numpy.polynomial.legendre) =================================================== .. currentmodule:: numpy.polynomial.polynomial This module provides a number of objects (mostly functions) useful for dealing with Legendre series, including a Legendre 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 --------- .. autosummary:: :toctree: generated/ legdomain Legendre series default domain, [-1,1]. legzero Legendre series that evaluates identically to 0. legone Legendre series that evaluates identically to 1. legx Legendre series for the identity map, f(x) = x. Arithmetic ---------- .. autosummary:: :toctree: generated/ legmulx multiply a Legendre series in P_i(x) by x. legadd add two Legendre series. legsub subtract one Legendre series from another. legmul multiply two Legendre series. legdiv divide one Legendre series by another. legpow raise a Legendre series to an positive integer power legval evaluate a Legendre series at given points. legval2d evaluate a 2D Legendre series at given points. legval3d evaluate a 3D Legendre series at given points. leggrid2d evaluate a 2D Legendre series on a Cartesian product. leggrid3d evaluate a 3D Legendre series on a Cartesian product. Calculus -------- .. autosummary:: :toctree: generated/ legder differentiate a Legendre series. legint integrate a Legendre series. Misc Functions -------------- .. autosummary:: :toctree: generated/ legfromroots create a Legendre series with specified roots. legroots find the roots of a Legendre series. legvander Vandermonde-like matrix for Legendre polynomials. legvander2d Vandermonde-like matrix for 2D power series. legvander3d Vandermonde-like matrix for 3D power series. leggauss Gauss-Legendre quadrature, points and weights. legweight Legendre weight function. legcompanion symmetrized companion matrix in Legendre form. legfit least-squares fit returning a Legendre series. legtrim trim leading coefficients from a Legendre series. legline Legendre series representing given straight line. leg2poly convert a Legendre series to a polynomial. poly2leg convert a polynomial to a Legendre series. Classes ------- Legendre A Legendre series class. See also -------- numpy.polynomial.polynomial numpy.polynomial.chebyshev numpy.polynomial.laguerre numpy.polynomial.hermite numpy.polynomial.hermite_e """ from __future__ import division, absolute_import, print_function import warnings import numpy as np import numpy.linalg as la from . import polyutils as pu from ._polybase import ABCPolyBase __all__ = [ 'legzero', 'legone', 'legx', 'legdomain', 'legline', 'legadd', 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder', 'legint', 'leg2poly', 'poly2leg', 'legfromroots', 'legvander', 'legfit', 'legtrim', 'legroots', 'Legendre', 'legval2d', 'legval3d', 'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d', 'legcompanion', 'leggauss', 'legweight'] legtrim = pu.trimcoef def poly2leg(pol): """ Convert a polynomial to a Legendre 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 Legendre 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 Legendre series. See Also -------- leg2poly 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(np.arange(4)) >>> p Polynomial([ 0., 1., 2., 3.], [-1., 1.]) >>> c = P.Legendre(P.poly2leg(p.coef)) >>> c Legendre([ 1. , 3.25, 1. , 0.75], [-1., 1.]) """ [pol] = pu.as_series([pol]) deg = len(pol) - 1 res = 0 for i in range(deg, -1, -1): res = legadd(legmulx(res), pol[i]) return res def leg2poly(c): """ Convert a Legendre series to a polynomial. Convert an array representing the coefficients of a Legendre 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 Legendre 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 -------- poly2leg Notes ----- The easy way to do conversions between polynomial basis sets is to use the convert method of a class instance. Examples -------- >>> c = P.Legendre(range(4)) >>> c Legendre([ 0., 1., 2., 3.], [-1., 1.]) >>> p = c.convert(kind=P.Polynomial) >>> p Polynomial([-1. , -3.5, 3. , 7.5], [-1., 1.]) >>> P.leg2poly(range(4)) array([-1. , -3.5, 3. , 7.5]) """ 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*(i - 1))/i) c1 = polyadd(tmp, (polymulx(c1)*(2*i - 1))/i) 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. # # Legendre legdomain = np.array([-1, 1]) # Legendre coefficients representing zero. legzero = np.array([0]) # Legendre coefficients representing one. legone = np.array([1]) # Legendre coefficients representing the identity x. legx = np.array([0, 1]) def legline(off, scl): """ Legendre 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 Legendre series for off + scl*x. See Also -------- polyline, chebline Examples -------- >>> import numpy.polynomial.legendre as L >>> L.legline(3,2) array([3, 2]) >>> L.legval(-3, L.legline(3,2)) # should be -3 -3.0 """ if scl != 0: return np.array([off, scl]) else: return np.array([off]) def legfromroots(roots): """ Generate a Legendre 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 Legendre 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 * L_1(x) + ... + c_n * L_n(x) The coefficient of the last term is not generally 1 for monic polynomials in Legendre 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, chebfromroots, lagfromroots, hermfromroots, hermefromroots. Examples -------- >>> import numpy.polynomial.legendre as L >>> L.legfromroots((-1,0,1)) # x^3 - x relative to the standard basis array([ 0. , -0.4, 0. , 0.4]) >>> j = complex(0,1) >>> L.legfromroots((-j,j)) # x^2 + 1 relative to the standard basis array([ 1.33333333+0.j, 0.00000000+0.j, 0.66666667+0.j]) """ if len(roots) == 0: return np.ones(1) else: [roots] = pu.as_series([roots], trim=False) roots.sort() p = [legline(-r, 1) for r in roots] n = len(p) while n > 1: m, r = divmod(n, 2) tmp = [legmul(p[i], p[i+m]) for i in range(m)] if r: tmp[0] = legmul(tmp[0], p[-1]) p = tmp n = m return p[0] def legadd(c1, c2): """ Add one Legendre series to another. Returns the sum of two Legendre series c1 + c2. The arguments are sequences of coefficients ordered from lowest order term to highest, i.e., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2. Parameters ---------- c1, c2 : array_like 1-D arrays of Legendre series coefficients ordered from low to high. Returns ------- out : ndarray Array representing the Legendre series of their sum. See Also -------- legsub, legmul, legdiv, legpow Notes ----- Unlike multiplication, division, etc., the sum of two Legendre series is a Legendre 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 legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legadd(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 legsub(c1, c2): """ Subtract one Legendre series from another. Returns the difference of two Legendre series c1 - c2. The sequences of coefficients are from lowest order term to highest, i.e., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2. Parameters ---------- c1, c2 : array_like 1-D arrays of Legendre series coefficients ordered from low to high. Returns ------- out : ndarray Of Legendre series coefficients representing their difference. See Also -------- legadd, legmul, legdiv, legpow Notes ----- Unlike multiplication, division, etc., the difference of two Legendre series is a Legendre 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 legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legsub(c1,c2) array([-2., 0., 2.]) >>> L.legsub(c2,c1) # -C.legsub(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 legmulx(c): """Multiply a Legendre series by x. Multiply the Legendre series c by x, where x is the independent variable. Parameters ---------- c : array_like 1-D array of Legendre series coefficients ordered from low to high. Returns ------- out : ndarray Array representing the result of the multiplication. Notes ----- The multiplication uses the recursion relationship for Legendre polynomials in the form .. math:: xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1) """ # 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] for i in range(1, len(c)): j = i + 1 k = i - 1 s = i + j prd[j] = (c[i]*j)/s prd[k] += (c[i]*i)/s return prd def legmul(c1, c2): """ Multiply one Legendre series by another. Returns the product of two Legendre series c1 * c2. The arguments are sequences of coefficients, from lowest order "term" to highest, e.g., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2. Parameters ---------- c1, c2 : array_like 1-D arrays of Legendre series coefficients ordered from low to high. Returns ------- out : ndarray Of Legendre series coefficients representing their product. See Also -------- legadd, legsub, legdiv, legpow Notes ----- In general, the (polynomial) product of two C-series results in terms that are not in the Legendre polynomial basis set. Thus, to express the product as a Legendre series, it is necessary to "reproject" the product onto said basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2) >>> P.legmul(c1,c2) # multiplication requires "reprojection" array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) """ # s1, s2 are trimmed copies [c1, c2] = pu.as_series([c1, c2]) if len(c1) > len(c2): c = c2 xs = c1 else: c = c1 xs = c2 if len(c) == 1: c0 = c[0]*xs c1 = 0 elif len(c) == 2: c0 = c[0]*xs c1 = c[1]*xs else: nd = len(c) c0 = c[-2]*xs c1 = c[-1]*xs for i in range(3, len(c) + 1): tmp = c0 nd = nd - 1 c0 = legsub(c[-i]*xs, (c1*(nd - 1))/nd) c1 = legadd(tmp, (legmulx(c1)*(2*nd - 1))/nd) return legadd(c0, legmulx(c1)) def legdiv(c1, c2): """ Divide one Legendre series by another. Returns the quotient-with-remainder of two Legendre series c1 / c2. The arguments are sequences of coefficients from lowest order "term" to highest, e.g., [1,2,3] represents the series P_0 + 2*P_1 + 3*P_2. Parameters ---------- c1, c2 : array_like 1-D arrays of Legendre series coefficients ordered from low to high. Returns ------- quo, rem : ndarrays Of Legendre series coefficients representing the quotient and remainder. See Also -------- legadd, legsub, legmul, legpow Notes ----- In general, the (polynomial) division of one Legendre series by another results in quotient and remainder terms that are not in the Legendre polynomial basis set. Thus, to express these results as a Legendre series, it is necessary to "reproject" the results onto the Legendre basis set, which may produce "unintuitive" (but correct) results; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> c1 = (1,2,3) >>> c2 = (3,2,1) >>> L.legdiv(c1,c2) # quotient "intuitive," remainder not (array([ 3.]), array([-8., -4.])) >>> c2 = (0,1,2,3) >>> L.legdiv(c2,c1) # neither "intuitive" (array([-0.07407407, 1.66666667]), array([-1.03703704, -2.51851852])) """ # 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: quo = np.empty(lc1 - lc2 + 1, dtype=c1.dtype) rem = c1 for i in range(lc1 - lc2, - 1, -1): p = legmul([0]*i + [1], c2) q = rem[-1]/p[-1] rem = rem[:-1] - q*p[:-1] quo[i] = q return quo, pu.trimseq(rem) def legpow(c, pow, maxpower=16): """Raise a Legendre series to a power. Returns the Legendre series c raised to the power pow. The arguement c is a sequence of coefficients ordered from low to high. i.e., [1,2,3] is the series P_0 + 2*P_1 + 3*P_2. Parameters ---------- c : array_like 1-D array of Legendre 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 Legendre series of power. See Also -------- legadd, legsub, legmul, legdiv 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. prd = c for i in range(2, power + 1): prd = legmul(prd, c) return prd def legder(c, m=1, scl=1, axis=0): """ Differentiate a Legendre series. Returns the Legendre 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*L_0 + 2*L_1 + 3*L_2 while [[1,2],[1,2]] represents 1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y) if axis=0 is x and axis=1 is y. Parameters ---------- c : array_like Array of Legendre 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 Legendre series of the derivative. See Also -------- legint Notes ----- In general, the result of differentiating a Legendre series does not resemble the same operation on a power series. Thus the result of this function may be "unintuitive," albeit correct; see Examples section below. Examples -------- >>> from numpy.polynomial import legendre as L >>> c = (1,2,3,4) >>> L.legder(c) array([ 6., 9., 20.]) >>> L.legder(c, 3) array([ 60.]) >>> L.legder(c, scl=-1) array([ -6., -9., -20.]) >>> L.legder(c, 2,-1) array([ 9., 60.]) """ c = np.array(c, ndmin=1, copy=1) if c.dtype.char in '?bBhHiIlLqQpP': c = c.astype(np.double) cnt, iaxis = [int(t) for t in [m, axis]] if cnt != m: raise ValueError("The order of derivation must be integer") if cnt < 0: raise ValueError("The order of derivation must be non-negative") if iaxis != axis: raise ValueError("The axis must be integer") if not -c.ndim <= iaxis < c.ndim: raise ValueError("The axis is out of range") if iaxis < 0: iaxis += c.ndim if cnt == 0: return c c = np.rollaxis(c, iaxis) n = len(c) if cnt >= n: c = c[:1]*0 else: for i in range(cnt): n = n - 1 c *= scl der = np.empty((n,) + c.shape[1:], dtype=c.dtype) for j in range(n, 2, -1): der[j - 1] = (2*j - 1)*c[j] c[j - 2] += c[j] if n > 1: der[1] = 3*c[2] der[0] = c[1] c = der c = np.rollaxis(c, 0, iaxis + 1) return c def legint(c, m=1, k=[], lbnd=0, scl=1, axis=0): """ Integrate a Legendre series. Returns the Legendre 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 L_0 + 2*L_1 + 3*L_2 while [[1,2],[1,2]] represents 1*L_0(x)*L_0(y) + 1*L_1(x)*L_0(y) + 2*L_0(x)*L_1(y) + 2*L_1(x)*L_1(y) if axis=0 is x and axis=1 is y. Parameters ---------- c : array_like Array of Legendre 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 lbnd is the first value in the list, the value of the second integral at lbnd 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 Legendre series coefficient array of the integral. Raises ------ ValueError If m < 0, len(k) > m, np.isscalar(lbnd) == False, or np.isscalar(scl) == False. See Also -------- legder 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 legendre as L >>> c = (1,2,3) >>> L.legint(c) array([ 0.33333333, 0.4 , 0.66666667, 0.6 ]) >>> L.legint(c, 3) array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) >>> L.legint(c, k=3) array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) >>> L.legint(c, lbnd=-2) array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) >>> L.legint(c, scl=2) array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) """ c = np.array(c, ndmin=1, copy=1) if c.dtype.char in '?bBhHiIlLqQpP': c = c.astype(np.double) if not np.iterable(k): k = [k] cnt, iaxis = [int(t) for t in [m, axis]] if cnt != m: raise ValueError("The order of integration must be integer") if cnt < 0: raise ValueError("The order of integration must be non-negative") if len(k) > cnt: raise ValueError("Too many integration constants") if iaxis != axis: raise ValueError("The axis must be integer") if not -c.ndim <= iaxis < c.ndim: raise ValueError("The axis is out of range") if iaxis < 0: iaxis += c.ndim if cnt == 0: return c c = np.rollaxis(c, iaxis) k = list(k) + [0]*(cnt - len(k)) for i in range(cnt): n = len(c) c *= scl if n == 1 and np.all(c[0] == 0): c[0] += k[i] else: tmp = np.empty((n + 1,) + c.shape[1:], dtype=c.dtype) tmp[0] = c[0]*0 tmp[1] = c[0] if n > 1: tmp[2] = c[1]/3 for j in range(2, n): t = c[j]/(2*j + 1) tmp[j + 1] = t tmp[j - 1] -= t tmp[0] += k[i] - legval(lbnd, tmp) c = tmp c = np.rollaxis(c, 0, iaxis + 1) return c def legval(x, c, tensor=True): """ Evaluate a Legendre series at points x. If c is of length n + 1, this function returns the value: .. math:: p(x) = c_0 * L_0(x) + c_1 * L_1(x) + ... + c_n * L_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 -------- legval2d, leggrid2d, legval3d, leggrid3d Notes ----- The evaluation uses Clenshaw recursion, aka synthetic division. Examples -------- """ c = np.array(c, ndmin=1, copy=0) 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: nd = len(c) c0 = c[-2] c1 = c[-1] for i in range(3, len(c) + 1): tmp = c0 nd = nd - 1 c0 = c[-i] - (c1*(nd - 1))/nd c1 = tmp + (c1*x*(2*nd - 1))/nd return c0 + c1*x def legval2d(x, y, c): """ Evaluate a 2-D Legendre series at points (x, y). This function returns the values: .. math:: p(x,y) = \\sum_{i,j} c_{i,j} * L_i(x) * L_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 two the remaining indices enumerate multiple sets of coefficients. Returns ------- values : ndarray, compatible object The values of the two dimensional Legendre series at points formed from pairs of corresponding values from x and y. See Also -------- legval, leggrid2d, legval3d, leggrid3d Notes ----- .. versionadded::1.7.0 """ try: x, y = np.array((x, y), copy=0) except: raise ValueError('x, y are incompatible') c = legval(x, c) c = legval(y, c, tensor=False) return c def leggrid2d(x, y, c): """ Evaluate a 2-D Legendre series on the Cartesian product of x and y. This function returns the values: .. math:: p(a,b) = \sum_{i,j} c_{i,j} * L_i(a) * L_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 -------- legval, legval2d, legval3d, leggrid3d Notes ----- .. versionadded::1.7.0 """ c = legval(x, c) c = legval(y, c) return c def legval3d(x, y, z, c): """ Evaluate a 3-D Legendre series at points (x, y, z). This function returns the values: .. math:: p(x,y,z) = \\sum_{i,j,k} c_{i,j,k} * L_i(x) * L_j(y) * L_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 -------- legval, legval2d, leggrid2d, leggrid3d 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 = legval(x, c) c = legval(y, c, tensor=False) c = legval(z, c, tensor=False) return c def leggrid3d(x, y, z, c): """ Evaluate a 3-D Legendre 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} * L_i(a) * L_j(b) * L_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 -------- legval, legval2d, leggrid2d, legval3d Notes ----- .. versionadded::1.7.0 """ c = legval(x, c) c = legval(y, c) c = legval(z, c) return c def legvander(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] = L_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 Legendre polynomial. If c is a 1-D array of coefficients of length n + 1 and V is the array V = legvander(x, n), then np.dot(V, c) and legval(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 Legendre 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 Legendre 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. This is not as accurate # as reverse recursion in this application but it is more efficient. v[0] = x*0 + 1 if ideg > 0: v[1] = x for i in range(2, ideg + 1): v[i] = (v[i-1]*x*(2*i - 1) - v[i-2]*(i - 1))/i return np.rollaxis(v, 0, v.ndim) def legvander2d(x, y, deg): """Pseudo-Vandermonde matrix of given degrees. Returns the pseudo-Vandermonde matrix of degrees deg and sample points (x, y). The pseudo-Vandermonde matrix is defined by .. math:: V[..., deg[1]*i + j] = L_i(x) * L_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 Legendre polynomials. If V = legvander2d(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 legval2d(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 Legendre 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 -------- legvander, legvander3d. legval2d, legval3d 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 = legvander(x, degx) vy = legvander(y, degy) v = vx[..., None]*vy[..., None,:] return v.reshape(v.shape[:-2] + (-1,)) def legvander3d(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] = L_i(x)*L_j(y)*L_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 Legendre polynomials. If V = legvander3d(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 legval3d(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 Legendre 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 -------- legvander, legvander3d. legval2d, legval3d 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 = legvander(x, degx) vy = legvander(y, degy) vz = legvander(z, degz) v = vx[..., None, None]*vy[..., None,:, None]*vz[..., None, None,:] return v.reshape(v.shape[:-3] + (-1,)) def legfit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Legendre series to data. Return the coefficients of a Legendre series of degree deg that is the least squares fit to the data values y given at points x. If y is 1-D the returned coefficients will also be 1-D. If y is 2-D multiple fits are done, one for each column of y, and the resulting coefficients are stored in the corresponding columns of a 2-D return. The fitted polynomial(s) are in the form .. math:: p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x), where n is deg. Parameters ---------- x : array_like, shape (M,) x-coordinates of the M sample points (x[i], y[i]). y : array_like, shape (M,) or (M, K) y-coordinates of the sample points. Several data sets of sample points sharing the same x-coordinates can be fitted at once by passing in a 2D-array that contains one dataset per column. deg : int Degree of the fitting 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) Legendre 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 -------- chebfit, polyfit, lagfit, hermfit, hermefit legval : Evaluates a Legendre series. legvander : Vandermonde matrix of Legendre series. legweight : Legendre weight function (= 1). linalg.lstsq : Computes a least-squares fit from the matrix. scipy.interpolate.UnivariateSpline : Computes spline fits. Notes ----- The solution is the coefficients of the Legendre 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 Legendre 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 in transposed form lhs = legvander(x, deg).T rhs = y.T if w is not None: w = np.asarray(w) + 0.0 if w.ndim != 1: raise TypeError("expected 1D vector for w") if len(x) != len(w): raise TypeError("expected x and w to have same length") # 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 # 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 legcompanion(c): """Return the scaled companion matrix of c. The basis polynomials are scaled so that the companion matrix is symmetric when c is an Legendre 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 Legendre 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 = 1./np.sqrt(2*np.arange(n) + 1) top = mat.reshape(-1)[1::n+1] bot = mat.reshape(-1)[n::n+1] top[...] = np.arange(1, n)*scl[:n-1]*scl[1:n] bot[...] = top mat[:, -1] -= (c[:-1]/c[-1])*(scl/scl[-1])*(n/(2*n - 1)) return mat def legroots(c): """ Compute the roots of a Legendre series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * L_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, chebroots, 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 Legendre series basis polynomials aren't powers of x so the results of this function may seem unintuitive. Examples -------- >>> import numpy.polynomial.legendre as leg >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots array([-0.85099543, -0.11407192, 0.51506735]) """ # 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 = legcompanion(c) r = la.eigvals(m) r.sort() return r def leggauss(deg): """ Gauss-Legendre quadrature. Computes the sample points and weights for Gauss-Legendre 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. 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. The weights are determined by using the fact that .. math:: w_k = c / (L'_n(x_k) * L_{n-1}(x_k)) where :math:c is a constant independent of :math:k and :math:x_k is the k'th root of :math:L_n, and then scaling the results to get the right value when integrating 1. """ ideg = int(deg) if ideg != deg or ideg < 1: raise ValueError("deg must be a non-negative integer") # first approximation of roots. We use the fact that the companion # matrix is symmetric in this case in order to obtain better zeros. c = np.array([0]*deg + [1]) m = legcompanion(c) x = la.eigvals(m) x.sort() # improve roots by one application of Newton dy = legval(x, c) df = legval(x, legder(c)) x -= dy/df # compute the weights. We scale the factor to avoid possible numerical # overflow. fm = legval(x, c[1:]) fm /= np.abs(fm).max() df /= np.abs(df).max() w = 1/(fm * df) # for Legendre we can also symmetrize w = (w + w[::-1])/2 x = (x - x[::-1])/2 # scale w to get the right value w *= 2. / w.sum() return x, w def legweight(x): """ Weight function of the Legendre polynomials. The weight function is :math:1 and the interval of integration is :math:[-1, 1]. The Legendre 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 = x*0.0 + 1.0 return w # # Legendre series class # class Legendre(ABCPolyBase): """A Legendre series class. The Legendre class provides the standard Python numerical methods '+', '-', '*', '//', '%', 'divmod', '**', and '()' as well as the attributes and methods listed in the ABCPolyBase documentation. Parameters ---------- coef : array_like Legendre coefficients in order of increasing degree, i.e., (1, 2, 3) gives 1*P_0(x) + 2*P_1(x) + 3*P_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(legadd) _sub = staticmethod(legsub) _mul = staticmethod(legmul) _div = staticmethod(legdiv) _pow = staticmethod(legpow) _val = staticmethod(legval) _int = staticmethod(legint) _der = staticmethod(legder) _fit = staticmethod(legfit) _line = staticmethod(legline) _roots = staticmethod(legroots) _fromroots = staticmethod(legfromroots) # Virtual properties nickname = 'leg' domain = np.array(legdomain) window = np.array(legdomain)