A preconditioning technique of Gauss--Legendre quadrature for the logarithm of symmetric positive definite matrices

2024/10/29 by Fuminori Tatsuoka, Tatsuoka, Fuminori, Tomohiro Sogabe +5
Computer Science · Physics and Astronomy · #Matrix Theory and Algorithms #Electromagnetic Scattering and Analysis #Advanced Mathematical Theories and Applications

paper · pdf · doi:10.48550/arxiv.2410.22014

Abstract

This note considers the computation of the logarithm of symmetric positive definite matrices using the Gauss--Legendre (GL) quadrature. The GL quadrature becomes slow when the condition number of the given matrix is large. In this note, we propose a technique dividing the matrix logarithm into two matrix logarithms, where the condition numbers of the divided logarithm arguments are smaller than that of the original matrix. Although the matrix logarithm needs to be computed twice, each computation can be performed more efficiently, and it potentially reduces the overall computational cost. It is shown that the proposed technique is effective when the condition number of the given matrix is approximately between 130 and 3.0× 10⁵.

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