Re: A few questions about finding all zeros of Hermite polynomial

On Tue, 20 Nov 2007, Brian Gough wrote:
> 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? 

I found it does; I think I hit numerical accuracy when calculating
coefficients of Hermite polynomial using gsl_poly_dd_taylor.

> 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.

Thanks for help. Can you recommend some starting point or literature?


Most probably I try to solve generic problem to solve easier
specialized one. What I really want to do is to calculate integrals
between wavefunctions of quantum harmonic oscillator, 

  <n|f(x)|m> = \int_{-inf}^{inf} dx psi_m(x) f(x) psi_n(x)

where

  psi_n(x) = N_n exp(-x^2) H_n(x)

where N_n is normalization so \int dx [psi_n(x)]^2 = 1, and H_n(x)
is Hermite polynomial of degree n.

Could you recommend how to calculate this?

>> Is perhaps Jenkins-Traub method 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
> incorporated.

What is needed in terms of feedback and testing from users for
poly/gsl_poly_orth.c? I think I can write texinfo documentation
for it, but I don't know what is required of testing and feedback.

-- 
Jakub Narebski
Poland