On the computation of recurrence coefficients for univariate orthogonal polynomials

TL;DR

This paper reviews methods for computing recurrence coefficients of univariate orthogonal polynomials and introduces a new hybrid predictor-corrector algorithm that improves accuracy and efficiency for a broad class of measures.

Contribution

It proposes a novel hybrid predictor-corrector algorithm combined with a stabilized Lanczos procedure for computing recurrence coefficients for various measures.

Findings

The new algorithm achieves higher accuracy than existing methods.

The hybrid approach is more efficient for measures with both continuous and discrete parts.

Numerical experiments demonstrate improved performance in computation.

Abstract

Associated to a finite measure on the real line with finite moments are recurrence coefficients in a three-term formula for orthogonal polynomials with respect to this measure. These recurrence coefficients are frequently inputs to modern computational tools that facilitate evaluation and manipulation of polynomials with respect to the measure, and such tasks are foundational in numerical approximation and quadrature. Although the recurrence coefficients for classical measures are known explicitly, those for nonclassical measures must typically be numerically computed. We survey and review existing approaches for computing these recurrence coefficients for univariate orthogonal polynomial families and propose a novel "predictor-corrector" algorithm for a general class of continuous measures. We combine the predictor-corrector scheme with a stabilized Lanczos procedure for a new hybrid algorithm that computes recurrence coefficients for a fairly wide class of measures that can have both continuous and discrete parts. We evaluate the new algorithms against existing methods in terms of accuracy and efficiency.