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

TL;DR

This paper introduces a preconditioning technique for Gauss--Legendre quadrature to efficiently compute the logarithm of symmetric positive definite matrices, especially when the matrix has a high condition number.

Contribution

A novel preconditioning method that divides the matrix logarithm to improve computational efficiency for matrices with large condition numbers.

Findings

Effective for matrices with condition numbers between 130 and 3.0×10^5.

Reduces overall computational cost despite requiring two logarithm computations.

Enhances the efficiency of Gauss--Legendre quadrature in high condition number scenarios.

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 $3.0\times 10^5$.