# Algorithms for the computation of the matrix logarithm based on the double exponential formula

## TL;DR

This paper introduces a novel approach for computing the matrix logarithm using the double exponential numerical quadrature, improving efficiency and accuracy especially for ill-conditioned matrices.

## Contribution

It develops a method to select finite integration intervals and presents two algorithms, including one with error control, for matrix logarithm computation using the double exponential formula.

## Key findings

- Enhanced convergence for ill-conditioned matrices
- Effective finite interval selection method
- Algorithms with error control for improved accuracy

## Abstract

We consider the computation of the matrix logarithm by using numerical quadrature. The efficiency of numerical quadrature depends on the integrand and the choice of quadrature formula. The Gauss--Legendre quadrature has been conventionally employed; however, the convergence could be slow for ill-conditioned matrices. This effect may stem from the rapid change of the integrand values. To avoid such situations, we focus on the double exponential formula, which has been developed to address integrands with endpoint singularity. In order to utilize the double exponential formula, we must determine a suitable finite integration interval, which provides the required accuracy and efficiency. In this paper, we present a method for selecting a suitable finite interval based on an error analysis as well as two algorithms, and one of these algorithms addresses error control.