orthax.orthfromroots

orthax.orthfromroots(roots, rec)Source

Generate an orthogonal series with given roots.

The function returns the coefficients of the polynomial

\[p(x) = (x - r_0) * (x - r_1) * ... * (x - r_n),\]

in the form of an orthogonal series with recurrence relation rec, 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

\[p(x) = c_0 + c_1 * P_1(x) + ... + c_n * P_n(x)\]

Parameters:

• roots (array_like) – Sequence containing the roots.

• rec (AbstractRecurrenceRelation) – Recurrence relation for the family of orthogonal polynomials.

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).