A matrix approach to the computation of quadrature formulas on the unit circle

• Published in: Applied numerical mathematics Vol. 58; no. 3; pp. 296 - 318
• Main Authors: Cantero, María José, Cruz-Barroso, Ruymán, González-Vera, Pablo
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.03.2008; Elsevier
• Subjects: Approximations and expansions; Exact sciences and technology; Five-diagonal matrices; Fourier analysis; Hessenberg matrices; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical approximation; Orthogonal Laurent polynomials; Recurrence relations; Rogers–Szegö q-polynomials; Sciences and techniques of general use; Special functions; Szegö polynomials; Szegö quadrature formulas; Szegö polynomials; Five-diagonal matrices; Orthogonal Laurent polynomials; Recurrence relations; Szegö quadrature formulas; Rogers–Szegö q-polynomials; Hessenberg matrices; Szego polynomials; Multiplication; Recurrence relation; Orthogonal polynomial; Rogers-Szego q-polynomials; Numerical approximation; Numerical analysis; Cubature; Applied mathematics; Matrix calculus; Quadrature formula
• ISSN: 0168-9274; 1873-5460
• DOI: 10.1016/j.apnum.2006.11.009

In this paper we consider a general sequence of orthogonal Laurent polynomials on the unit circle and we first study the equivalences between recurrences for such families and Szegö's recursion and the structure of the matrix representation for the multiplication operator in Λ when a general sequence of orthogonal Laurent polynomials on the unit circle is considered. Secondly, we analyze the computation of the nodes of the Szegö quadrature formulas by using Hessenberg and five-diagonal matrices. Numerical examples concerning the family of Rogers–Szegö q-polynomials are also analyzed.