Resumen de On the computation of symmetric Szeg˝o-type quadrature formulas

A. Bultheel

• 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.