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Computing the Coefficients of a Recurrence Formula for Numerical Integration by Moments and Modified Moments

M. Cecchi
M. N. Zaglia
Journal / Anthology

Journal of Computational and Applied Mathematics
Year: 1993
Volume: 49
Page range: 207-216

To evaluate the class of integrals f1-1e-axf(x) dx, where alpha is an element of R+ and the function f(x) is known only approximately in a tabular form, we wish to use a Gaussian quadrature formula. Nodes and weights have to be computed using the family of nomic orthogonal polynomials, with respect to the weight function e-ax, obtained through the three-term recurrence relation Pk+1 (x) = (x+Bk+1) Pk(x) - Ck+1Pk-1(x. To guarantee a good precision, we must evaluate carefully the values for the coefficients Bk+1 and Ck+1. Such evaluations are made completely formally through a Mathematica program to obtain great precision. A comparison between various methods, starting from moments and modified moments, is shown. Numerical results are also presented.

*Applied Mathematics > Numerical Methods
*Mathematics > Calculus and Analysis > Calculus
*Mathematics > Calculus and Analysis > Special Functions

Orthogonal polynomials, recurrence relation for orthogonal polynomials, Gaussian quadrature, moments, modified moments, symbolic computation