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diff --git a/.venv/lib/python3.12/site-packages/numpy/polynomial/legendre.py b/.venv/lib/python3.12/site-packages/numpy/polynomial/legendre.py new file mode 100644 index 00000000..8e9c19d9 --- /dev/null +++ b/.venv/lib/python3.12/site-packages/numpy/polynomial/legendre.py @@ -0,0 +1,1664 @@ +""" +================================================== +Legendre Series (:mod:`numpy.polynomial.legendre`) +================================================== + +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`). + +Classes +------- +.. autosummary:: + :toctree: generated/ + + Legendre + +Constants +--------- + +.. autosummary:: + :toctree: generated/ + + legdomain + legzero + legone + legx + +Arithmetic +---------- + +.. autosummary:: + :toctree: generated/ + + legadd + legsub + legmulx + legmul + legdiv + legpow + legval + legval2d + legval3d + leggrid2d + leggrid3d + +Calculus +-------- + +.. autosummary:: + :toctree: generated/ + + legder + legint + +Misc Functions +-------------- + +.. autosummary:: + :toctree: generated/ + + legfromroots + legroots + legvander + legvander2d + legvander3d + leggauss + legweight + legcompanion + legfit + legtrim + legline + leg2poly + poly2leg + +See also +-------- +numpy.polynomial + +""" +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__ = [ + '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.], domain=[-1, 1], window=[-1, 1]) + >>> c = P.Legendre(P.legendre.poly2leg(p.coef)) + >>> c + Legendre([ 1. , 3.25, 1. , 0.75], domain=[-1, 1], window=[-1, 1]) # may vary + + """ + [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 + -------- + >>> from numpy import polynomial as P + >>> c = P.Legendre(range(4)) + >>> c + Legendre([0., 1., 2., 3.], domain=[-1, 1], window=[-1, 1]) + >>> p = c.convert(kind=P.Polynomial) + >>> p + Polynomial([-1. , -3.5, 3. , 7.5], domain=[-1., 1.], window=[-1., 1.]) + >>> P.legendre.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 + -------- + numpy.polynomial.polynomial.polyline + numpy.polynomial.chebyshev.chebline + numpy.polynomial.laguerre.lagline + numpy.polynomial.hermite.hermline + numpy.polynomial.hermite_e.hermeline + + 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 + -------- + numpy.polynomial.polynomial.polyfromroots + numpy.polynomial.chebyshev.chebfromroots + numpy.polynomial.laguerre.lagfromroots + numpy.polynomial.hermite.hermfromroots + numpy.polynomial.hermite_e.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]) # may vary + + """ + return pu._fromroots(legline, legmul, roots) + + +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, legmulx, 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.]) + + """ + return pu._add(c1, c2) + + +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, legmulx, 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.]) + + """ + return pu._sub(c1, c2) + + +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. + + See Also + -------- + legadd, legmul, legdiv, legpow + + 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) + + Examples + -------- + >>> from numpy.polynomial import legendre as L + >>> L.legmulx([1,2,3]) + array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary + + """ + # 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, legmulx, 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) + >>> L.legmul(c1,c2) # multiplication requires "reprojection" + array([ 4.33333333, 10.4 , 11.66666667, 3.6 ]) # may vary + + """ + # 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, legmulx, 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])) # may vary + + """ + return pu._div(legmul, c1, c2) + + +def legpow(c, pow, maxpower=16): + """Raise a Legendre series to a power. + + Returns the Legendre 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 ``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, legmulx, legmul, legdiv + + """ + return pu._pow(legmul, c, pow, maxpower) + + +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=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + cnt = pu._deprecate_as_int(m, "the order of derivation") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of derivation must be non-negative") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + 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.moveaxis(c, 0, iaxis) + 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.ndim(lbnd) != 0``, or + ``np.ndim(scl) != 0``. + + 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 ]) # may vary + >>> L.legint(c, 3) + array([ 1.66666667e-02, -1.78571429e-02, 4.76190476e-02, # may vary + -1.73472348e-18, 1.90476190e-02, 9.52380952e-03]) + >>> L.legint(c, k=3) + array([ 3.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary + >>> L.legint(c, lbnd=-2) + array([ 7.33333333, 0.4 , 0.66666667, 0.6 ]) # may vary + >>> L.legint(c, scl=2) + array([ 0.66666667, 0.8 , 1.33333333, 1.2 ]) # may vary + + """ + c = np.array(c, ndmin=1, copy=True) + if c.dtype.char in '?bBhHiIlLqQpP': + c = c.astype(np.double) + if not np.iterable(k): + k = [k] + cnt = pu._deprecate_as_int(m, "the order of integration") + iaxis = pu._deprecate_as_int(axis, "the axis") + if cnt < 0: + raise ValueError("The order of integration must be non-negative") + if len(k) > cnt: + raise ValueError("Too many integration constants") + if np.ndim(lbnd) != 0: + raise ValueError("lbnd must be a scalar.") + if np.ndim(scl) != 0: + raise ValueError("scl must be a scalar.") + iaxis = normalize_axis_index(iaxis, c.ndim) + + if cnt == 0: + return c + + c = np.moveaxis(c, iaxis, 0) + 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.moveaxis(c, 0, iaxis) + 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 + 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. + + """ + c = np.array(c, ndmin=1, copy=False) + 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 + + """ + return pu._valnd(legval, c, x, y) + + +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 + + """ + return pu._gridnd(legval, c, x, y) + + +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 + + """ + return pu._valnd(legval, c, x, y, z) + + +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 + + """ + return pu._gridnd(legval, c, x, y, z) + + +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 = pu._deprecate_as_int(deg, "deg") + if ideg < 0: + raise ValueError("deg must be non-negative") + + x = np.array(x, copy=False, 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.moveaxis(v, 0, -1) + + +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] + 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 + + """ + return pu._vander_nd_flat((legvander, legvander), (x, y), deg) + + +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 + + """ + return pu._vander_nd_flat((legvander, legvander, legvander), (x, y, z), deg) + + +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 or 1-D array_like + Degree(s) of the fitting polynomials. If `deg` is a single integer + all terms up to and including the `deg`'th term are included in the + fit. For NumPy versions >= 1.11.0 a list of integers specifying the + degrees of the terms to include may be used instead. + rcond : float, optional + Relative condition number of the fit. Singular values smaller than + this relative to the largest singular value will be ignored. The + default value is len(x)*eps, where eps is the relative precision of + the float type, about 2e-16 in most cases. + full : bool, optional + Switch determining nature of return value. When it is False (the + default) just the coefficients are returned, when True diagnostic + information from the singular value decomposition is also returned. + w : array_like, shape (`M`,), optional + Weights. If not None, the weight ``w[i]`` applies to the unsquared + residual ``y[i] - y_hat[i]`` at ``x[i]``. Ideally the weights are + chosen so that the errors of the products ``w[i]*y[i]`` all have the + same variance. When using inverse-variance weighting, use + ``w[i] = 1/sigma(y[i])``. The default value is None. + + .. 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`. If `deg` is specified as a list, coefficients for + terms not included in the fit are set equal to zero in the + returned `coef`. + + [residuals, rank, singular_values, rcond] : list + These values are only returned if ``full == True`` + + - residuals -- sum of squared residuals of the least squares fit + - rank -- the numerical rank of the scaled Vandermonde matrix + - singular_values -- singular values of the scaled Vandermonde matrix + - rcond -- value of `rcond`. + + For more details, see `numpy.linalg.lstsq`. + + Warns + ----- + RankWarning + The rank of the coefficient matrix in the least-squares fit is + deficient. The warning is only raised if ``full == False``. The + warnings can be turned off by + + >>> import warnings + >>> warnings.simplefilter('ignore', np.RankWarning) + + See Also + -------- + numpy.polynomial.polynomial.polyfit + numpy.polynomial.chebyshev.chebfit + numpy.polynomial.laguerre.lagfit + numpy.polynomial.hermite.hermfit + numpy.polynomial.hermite_e.hermefit + legval : Evaluates a Legendre series. + legvander : Vandermonde matrix of Legendre series. + legweight : Legendre weight function (= 1). + numpy.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", + https://en.wikipedia.org/wiki/Curve_fitting + + Examples + -------- + + """ + return pu._fit(legvander, x, y, deg, rcond, full, w) + + +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 + -------- + numpy.polynomial.polynomial.polyroots + numpy.polynomial.chebyshev.chebroots + numpy.polynomial.laguerre.lagroots + numpy.polynomial.hermite.hermroots + numpy.polynomial.hermite_e.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]) # may vary + + """ + # 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]]) + + # rotated companion matrix reduces error + m = legcompanion(c)[::-1,::-1] + 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 = pu._deprecate_as_int(deg, "deg") + if ideg <= 0: + raise ValueError("deg must be a positive 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.eigvalsh(m) + + # 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 + symbol : str, optional + Symbol used to represent the independent variable in string + representations of the polynomial expression, e.g. for printing. + The symbol must be a valid Python identifier. Default value is 'x'. + + .. versionadded:: 1.24 + + """ + # 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 + domain = np.array(legdomain) + window = np.array(legdomain) + basis_name = 'P' |