Lanzcos algorithm and the complex Gauss quadrature

• ID: 2744, RIV: 10360929
• ISSN: neuvedeno, ISBN: neuvedeno
• zdroj: neuvedeno
• klíčová slova: Quasi-definite linear functionals; Gauss quadrature; orthogonal polynomials; complex Jacobi matrices; matching moments; Lanczos algorithm
• autoři: Stefano Pozza, Miroslav Pranić, Zdeněk Strakoš
• autoři z KNM: Strakoš Zdeněk

Abstrakt

Gauss quadrature can be naturally generalized to approximate quasi-definite linear functionals, where the interconnections with (formal) orthogonal polynomials, Padé approximants, (complex) Jacobi matrices and Lanczos algorithm are analogous to those in the positive definite case. In this paper we show that existence of the n-weight (complex) Gauss quadrature corresponds to performing successfully the first n steps of the Lanczos algorithm for generating the biorthogonal bases of the two associated Krylov subspaces. We also prove that the Jordan decomposition of the (complex) Jacobi matrix can be explicitly expressed in terms of the Gauss quadrature nodes and weights and the associated orthogonal polynomials. Since the output of the Lanczos algorithm can be made real whenever the input is real, it can be shown that the value of the Gauss quadrature is a real number whenever all relevant moments of the quasi-definite linear functional are real.