Construction of Gaussian quadrature formulas for even weight functions

Mohammad Masjed-Jamei,Gradimir Milovanovic

doi:10.2298/aadm1701177m

Abstract

Instead of a quadrature rule of Gaussian type with respect to an even weight function on (?a, a) with n nodes, we construct the corresponding Gaussian formula on (0, a2) with only [(n+1)/2] nodes. Especially, such a procedure is important in the cases of nonclassical weight functions, when the elements of the corresponding three-diagonal Jacobi matrix must be constructed numerically. In this manner, the influence of numerical instabilities in the process of construction can be significantly reduced, because the dimension of the Jacobi matrix is halved. We apply this approach to Pollaczek?s type weight functions on (?1, 1), to the weight functions on R which appear in the Abel-Plana summation processes, as well as to a class of weight functions with four free parameters, which covers the generalized ultraspherical and Hermite weights. Some numerical examples are also included.

Highlights

Let P be the set of all algebraic polynomials and Pn be its subset of degree at most n
The corresponding polynomials πk are known as strong non–classical orthogonal polynomials, and their recursion coefficients must be constructed numerically from the moment information
In the eighties of the last century, Walter Gautschi developed the so-called constructive theory of orthogonal polynomials on R, with effective algorithms for numerically generating the first n recursion coefficients (the method of moments, the discretized Stieltjes–Gautschi procedure, and the Lanczos algorithm), which allow us to compute all orthogonal polynomials of degree ≤ n by a straightforward application of the three-term recurrence relation

Summary

INTRODUCTION

Let P be the set of all algebraic polynomials and Pn be its subset of degree at most n. For many weight functions the coefficients βk in (2) are not explicitly known In such cases, the corresponding polynomials πk are known as strong non–classical orthogonal polynomials, and their recursion coefficients must be constructed numerically from the moment information. The corresponding polynomials πk are known as strong non–classical orthogonal polynomials, and their recursion coefficients must be constructed numerically from the moment information Such problems are very sensitive with respect to small perturbations in the input data. In the eighties of the last century, Walter Gautschi developed the so-called constructive theory of orthogonal polynomials on R, with effective algorithms for numerically generating the first n recursion coefficients (the method of (modified) moments, the discretized Stieltjes–Gautschi procedure, and the Lanczos algorithm), which allow us to compute all orthogonal polynomials of degree ≤ n by a straightforward application of the three-term recurrence relation.

TRANSFORMATION AND PRESERVATION OF ORTHOGONALITY

CONSTRUCTION OF TWO RULES OF GAUSSIAN TYPE

GAUSSIAN RULES RELATED TO THE POLLACZEK WEIGHT

A CLASS OF SYMMETRIC WEIGHTS ON R

A CLASS OF SYMMETRIC WEIGHTS WITH FOUR FREE PARAMETERS