Generalized averaged Gauss quadrature rules for the approximation of matrix functionals
Let \(A\) be a large square complex matrix and \(u\), \(v\) complex vectors. Further let \(f\) be a function. In many applications, one has to compute the matrix functional \(u^{\ast}f(A)v\). Usually one reduces \(A\) to a small matrix by a few steps of a Lanczos process and then one evaluates the reduced problem. In this paper, the authors discuss the application of generalized averaged Gauss quadrature rules to the approximation of the matrix functional. First matrix functionals \(u^{\ast}f(A)u\) with a Hermitian matrix \(A\) are approximated by generalized averaged Gauss quadrature rule introduced by \textit{M. M. Spalević} [Math. Comput. 76, No. 259, 1483--1492 (2007; Zbl 1113.65025)]. If \(A\) is non-Hermitian, then \(A\) can be reduced to a small non-Hermitian tridiagonal matrix by a few steps of the non-Hermitian Lanczos process. Then the non-Hermitian tridiagonal matrix defines a Gauss-type quadrature rule, and the authors introduce new generalized averaged Gauss quadrature rules, because this yields higher accuracy and requires almost the same computational effort. In numerical examples, the functions \(f(x) = (1 + x^2)^{-1}\), \(e^x\), and \(\log (x+ c)\) with \(c>0\) are considered. Examples from network analysis are given too.
generalized averaged Gauss quadrature rule
scientific article; zbMATH DE number 6641941
scientific article