Volume 19, pp. 113-134, 2005.

Orthogonal Laurent polynomials and quadratures on the unit circle and the real half-line

Ruymán Cruz-Barroso and Pablo González-Vera

Abstract

The purpose of this paper is the computation of quadrature formulas based on Laurent polynomials in two particular situations: the Real Half-Line and the Unit Circle. Comparative results and a connection with the split Levinson algorithm are established. Illustrative numerical examples are approximate integrals of the form $${\int }_{-1}^1\frac{f(x)}{(x+\lambda {)}^r}\omega (x)dx,r=1,2,3,\dots$$ with $f(x)$ a continuous function on $[-1,1]$, $\omega (x)\ge 0$ a weight function on this interval and $\lambda \in \mathbf{R}$ such that $|\lambda |>1$ is required. Here the classical Gaussian quadrature is an extremely slow procedure.

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Key words

orthogonal Laurent polynomials, L-Gaussian quadrature, Szegő quadrature, three-term recurrence relations, split Levinson algorithm, numerical quadrature.

AMS subject classifications

41A55, 33C45, 65D30.