\input texinfo   @c -*-texinfo-*-
@c %**start of header
@setfilename gsl_integration_hermite.info
@settitle Gauss-Hermite Quadrature
@c %**end of header

@setchapternewpage odd

@node Hermite Quadrature
@chapter Hermite Quadrature

@tex
\gdef\beforedisplay{}
\gdef\afterdisplay{}
@end tex

@c begin GSL-manual

@c Version 1: Konrad Griessinger (konradg(at)gmx.net), 04/2014

@cindex Hermite quadrature


@node Hermite quadrature
@subsection Hermite quadrature


To evaluate integrals of the form 

@tex
\beforedisplay
$$
\eqalign{
\int_{-\infty}^{\infty} e^{-x^2/2} f(x) dx & \quad \rm{(probabilist)}  \cr
\int_{-\infty}^{\infty} e^{-x^2} f(x) dx & \quad \rm{(physicist)}
}
$$
\afterdisplay
@end tex
@ifinfo

@example
\int_{-\infty}^{\infty} e^{-x^2/2} f(x) dx 
\int_{-\infty}^{\infty} e^{-x^2} f(x) dx 
@end example

@end ifinfo
@noindent

Gauss-Hermite quadrature can be used. For polynomials, this method gives the exact result when using @var{n} sample points to integrate a polynomial of order @math{2n-1} or less. For any function @math{f(x)} that can be approximated by a polynomial, the result of the quadrature will give a reasonably accurate approximation of the above integral for sufficiently large @var{n}. (See e.g. Introduction to Numerical Analysis (1992) by J. Stoer, R. Bulirsch.)  

 Unlike most other numerical integration routines within the library, these routines do not accept absolute or relative error bounds. 

@deftypefun double gsl_sf_hermite_prob_quad (const int @var{n}, const gsl_function * @var{func})
This routine computes the @var{n}-point probabilists' Gauss-Hermite quadrature of the function @var{func}.
@end deftypefun

@deftypefun double gsl_sf_hermite_phys_quad (const int @var{n}, const gsl_function * @var{func})
This routine computes the @var{n}-point physicists' Gauss-Hermite quadrature of the function @var{func}.
@end deftypefun


@c end GSL-manual

@page
@printindex cp

@contents
@bye