numpy.polynomial.legendre.legfromroots — NumPy v2.2 Manual (original) (raw)

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:

rootsarray_like

Sequence containing the roots.

Returns:

outndarray

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]) # may vary