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Re: [Help-gsl] A few questions about finding all zeros of Hermite polyno

From: Brian Gough
Subject: Re: [Help-gsl] A few questions about finding all zeros of Hermite polynomial
Date: Tue, 20 Nov 2007 18:59:36 +0000
User-agent: Wanderlust/2.14.0 (Africa) Emacs/22.1 Mule/5.0 (SAKAKI)

At Sun, 18 Nov 2007 14:15:26 +0100,
Jakub Narebski wrote:
> I want to find all zeros of Hermite polynomial up to fairly large order
> (up to H_100). I use poly/gsl_poly_orth.c by Richard J. Mathar (found by
> Google IIRC), which is not present in GSL 1.10, to calculate Hermite
> polynomial using divided differences method.
> Function gsl_poly_complex_solve requires polynomial in generic form;
> would this cause problems wrt. numerical accuracy? 

For orthogonal polynomials there are specialised methods for finding
the roots, e.g. as used for finding the abscissae in Gauss-Hermite
integration -- which are the roots of Hermite polynomials.  These will
work much better than a general polynomial solver as they use the
recurrence relations directly to do Newton-Raphson steps.

> Is there a GSL method to multiply two polynomials in normal form,
> or in divided differences representation?

No, but I would accept some functions for this.

> Is perhaps Jenkins-Traub ethod of finding all zeros of polynomial (RPOLY)
> considered for inclusion in GSL? Is poly/gsl_poly_orth.c?

These require some more feedback and testing from users to get

Brian Gough

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