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""" 

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/ 

 

legadd add two Legendre series. 

legsub subtract one Legendre series from another. 

legmulx multiply a Legendre series in ``P_i(x)`` by ``x``. 

legmul multiply two Legendre series. 

legdiv divide one Legendre series by another. 

legpow raise a Legendre series to a 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 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]) 

 

""" 

[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, 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.]) 

 

""" 

# 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, 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.]) 

 

""" 

# 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. 

 

See Also 

-------- 

legadd, legmul, 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]) 

 

""" 

# 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) 

>>> 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, 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])) 

 

""" 

# 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 

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 

 

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") 

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 ]) 

>>> 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 np.ndim(lbnd) != 0: 

raise ValueError("lbnd must be a scalar.") 

if np.ndim(scl) != 0: 

raise ValueError("scl must be a scalar.") 

if iaxis != axis: 

raise ValueError("The axis must be integer") 

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 

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 Exception: 

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 Exception: 

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.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 

 

""" 

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 or 1-D array_like 

Degree(s) of the fitting polynomials. If `deg` is a single integer 

all terms up to and including the `deg`'th term are included in the 

fit. For NumPy versions >= 1.11.0 a list of integers specifying the 

degrees of the terms to include may be used instead. 

rcond : float, optional 

Relative condition number of the fit. Singular values smaller than 

this relative to the largest singular value will be ignored. The 

default value is len(x)*eps, where eps is the relative precision of 

the float type, about 2e-16 in most cases. 

full : bool, optional 

Switch determining nature of return value. When it is False (the 

default) just the coefficients are returned, when True diagnostic 

information from the singular value decomposition is also returned. 

w : array_like, shape (`M`,), optional 

Weights. If not None, the contribution of each point 

``(x[i],y[i])`` to the fit is weighted by `w[i]`. Ideally the 

weights are chosen so that the errors of the products ``w[i]*y[i]`` 

all have the same variance. The default value is None. 

 

.. versionadded:: 1.5.0 

 

Returns 

------- 

coef : ndarray, shape (M,) or (M, K) 

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 

 

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", 

https://en.wikipedia.org/wiki/Curve_fitting 

 

Examples 

-------- 

 

""" 

x = np.asarray(x) + 0.0 

y = np.asarray(y) + 0.0 

deg = np.asarray(deg) 

 

# check arguments. 

if deg.ndim > 1 or deg.dtype.kind not in 'iu' or deg.size == 0: 

raise TypeError("deg must be an int or non-empty 1-D array of int") 

if deg.min() < 0: 

raise ValueError("expected deg >= 0") 

if x.ndim != 1: 

raise TypeError("expected 1D vector for x") 

if x.size == 0: 

raise TypeError("expected non-empty vector for x") 

if y.ndim < 1 or y.ndim > 2: 

raise TypeError("expected 1D or 2D array for y") 

if len(x) != len(y): 

raise TypeError("expected x and y to have same length") 

 

if deg.ndim == 0: 

lmax = deg 

order = lmax + 1 

van = legvander(x, lmax) 

else: 

deg = np.sort(deg) 

lmax = deg[-1] 

order = len(deg) 

van = legvander(x, lmax)[:, deg] 

 

# set up the least squares matrices in transposed form 

lhs = van.T 

rhs = y.T 

if w is not None: 

w = np.asarray(w) + 0.0 

if w.ndim != 1: 

raise TypeError("expected 1D vector for w") 

if len(x) != len(w): 

raise TypeError("expected x and w to have same length") 

# apply weights. Don't use inplace operations as they 

# can cause problems with NA. 

lhs = lhs * w 

rhs = rhs * w 

 

# set rcond 

if rcond is None: 

rcond = len(x)*np.finfo(x.dtype).eps 

 

# Determine the norms of the design matrix columns. 

if issubclass(lhs.dtype.type, np.complexfloating): 

scl = np.sqrt((np.square(lhs.real) + np.square(lhs.imag)).sum(1)) 

else: 

scl = np.sqrt(np.square(lhs).sum(1)) 

scl[scl == 0] = 1 

 

# Solve the least squares problem. 

c, resids, rank, s = la.lstsq(lhs.T/scl, rhs.T, rcond) 

c = (c.T/scl).T 

 

# Expand c to include non-fitted coefficients which are set to zero 

if deg.ndim > 0: 

if c.ndim == 2: 

cc = np.zeros((lmax+1, c.shape[1]), dtype=c.dtype) 

else: 

cc = np.zeros(lmax+1, dtype=c.dtype) 

cc[deg] = c 

c = cc 

 

# warn on rank reduction 

if rank != order and not full: 

msg = "The fit may be poorly conditioned" 

warnings.warn(msg, pu.RankWarning, stacklevel=2) 

 

if full: 

return c, [resids, rank, s, rcond] 

else: 

return c 

 

 

def 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.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 

 

""" 

# 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) 

basis_name = 'P'