Computational Aspects of Orthogonal Polynomials

Abstract

Our concern here is with computational methods for generating orthogonal poly- nomials and related quantities. We focus on the case where the underlying measure of integra- tion is nonclassical. The main problem, then, is that of computing the coefficients in the basic recurrence relation satisfied by orthogonal polynomials. Two principal methods are considered, one based on modified moments, the other on inner product representations of the coefficients. The first method is the more economical one, but may be subject to ill-conditioning. A study is made of the underlying reasons for instability. The second method, suitably implemented, is more widely applicable, but less economical. A number of problem areas in the physical sci- ences and in applied mathematics are described where these methods find useful applications.