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Manuscript Title: Abscissae and weights for the Gauss-Hermite quadrature formula.
Authors: T. Takemasa
Program title: GAUSSH
Catalogue identifier: ABBL_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 48(1988)265
Programming language: Fortran.
Computer: FACOM M-780/20.
Operating system: IV/F4.
RAM: 264K words
Word size: 8
Keywords: General purpose, Numerical quadrature, Orthogonal, Hermite polynomials, Newton-raphson's method, Lagrange, Gauss-hermite Quadrature formula.
Classification: 4.11.

Nature of problem:
In the study of physical problems, the need often arises for evaluating definite integrals numerically. The Gaussian quadrature is known as one of the most efficient quadrature schemes. The present program generates the abscissae and weights for the Gauss-Hermite quadrature formulas for integrals of the form integral (-infinity, +infinity) e**-x**2 f(x)dx very rapidly and with high accuracy even in the case of a great many abscissae.

Solution method:
The Gauss-Hermite integration may be based on the orthogonal property of the Hermite polynomials. The zeros of the Hermite polynomial are found by the Newton-Raphson method with suitable initial approximations. Then the corresponding weights are evaluated. This method is very fast in computational times compared with the one of Rysavy.

In principle, there are no restrictions, but the number of zeros must not exceed two thousands in the present version. This restriction can readily be removed by just increasing the size of dimensions of arrays.

Running time:
The running time depends on the number of zeros n. The test case (n= 100) requires 0.08s to run on the FACOM M-780/20 computer at Kyushu University. The case for n=2000 takes 16 s.