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Manuscript Title: Abscissae and weights for the Gauss-Laguerre quadrature formula.
Authors: T. Takemasa
Program title: GAUSSLA
Catalogue identifier: ABFJ_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 52(1988)133
Programming language: Fortran.
Computer: FACOM M-780/20.
Operating system: IV/F4 MSP.
RAM: 216K words
Word size: 8
Keywords: General purpose, Numerical quadrature, Orthogonal, Laguerre polynomials, Newton-raphson's method, Lagrange, Gauss-laguerre Quadrature formula.
Classification: 4.11.

Nature of problem:
In the study of physical problems, the need often arises for evaluating definite intergrals 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-Laguerre quadrature formula for integrals very rapidly and with high accuracy even in the case of a great many abscissae.

Solution method:
The Gauss-Laguerre quadrature formula may be based on the orthogonal property of the Laguerre polynomials. The abscissae are given by the zeros of the Laguerre polynomials, which 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.

Restrictions:
In principle, there are no restrictions except for alpha > -1.0, but the number of zeros must not exceed one thousand 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 and alpha = 0.0 requires 0.29s to run on the FACOM M-780/20 computer at Kysushu University. The case for n = 1000 and alpha = 5.0 takes 22 s.