Computation of generalized matrix functions

TL;DR

This paper introduces efficient numerical algorithms based on Gaussian quadrature and Golub--Kahan bidiagonalization for computing generalized matrix functions, with applications demonstrated in network analysis.

Contribution

The paper presents novel algorithms for generalized matrix functions using Gaussian quadrature and Golub--Kahan bidiagonalization, including block variants.

Findings

Algorithms are effective and efficient in numerical experiments.

Block variants improve computational performance.

Applications in network analysis demonstrate practical utility.

Abstract

We develop numerical algorithms for the efficient evaluation of quantities associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel, Linear and Multilinear Algebra 1(2), 1973, pp. 163-171]. Our algorithms are based on Gaussian quadrature and Golub--Kahan bidiagonalization. Block variants are also investigated. Numerical experiments are performed to illustrate the effectiveness and efficiency of our techniques in computing generalized matrix functions arising in the analysis of networks.