On the computation of symmetric Szeg˝o-type quadrature formulas
- Autores: A. Bultheel
- Localización: Monografías de la Real Academia de Ciencias Exactas, Físicas, Químicas y Naturales de Zaragoza, ISSN 1132-6360, Nº. 33, 2010, págs. 177-196
- Idioma: inglés
- Enlaces
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Resumen
By z = ei and x = cos , one may relate x ∈ I = (−1, 1], with ∈ (−, ] and a point z on the complex unit circle T. Hence there is a connection between the integrals of 2-periodic functions, integrals of functions over I and over T. The well known Gauss quadratures approximate the integrals over I and their circle counterparts are the Szeg˝o quadratures. When none, one or both endpoints of I are added to the usual Gauss nodes, one obtains the Gauss-type (Radau and Lobatto) quadratures. The circular counterparts are called Szeg˝o-type quadratures.
If the integrand and the weight function are symmetric for upper and lower half of T, the choice of complex conjugate Szeg˝o nodes with equal weights seems to be natural, and in that case, the Gauss nodes in I are just the projections of the Szeg˝o nodes. Also the weights are related, and it becomes numerically interesting to compute the Szeg˝o quadrature from the corresponding Gauss quadrature which reduces the computational cost considerably. Especially when the weights and nodes are computed via an eigenvalue problem, which for Gauss works with a tri-diagonal Jacobi matrix, but requires an upper Hessenberg matrix in the Szeg˝o case.
