numpy.polynomial.legendre.legfromroots()

numpy.polynomial.legendre.legfromroots(roots) [source]

Generate a Legendre series with given roots.

The function returns the coefficients of the polynomial

$p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),$

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

$p(x) = c_0 + c_1 * L_1(x) + ... + c_n * L_n(x)$

The coefficient of the last term is not generally 1 for monic polynomials in Legendre form.

Parameters:

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).

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).

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])

Links:

https://docs.scipy.org/doc/numpy-1.11.0/reference/generated/numpy.polynomial.legendre.legfromroots.html

doc_NumPy

2025-01-10 15:47:30

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