Roundoff error analysis of the double exponential formula-based method for the matrix sign function

TL;DR

This paper analyzes the numerical stability of a double-exponential formula-based method for computing the matrix sign function, revealing how ill-conditioned matrices affect accuracy.

Contribution

It provides a detailed roundoff error analysis explaining the accuracy deterioration for ill-conditioned or nonnormal matrices in the method.

Findings

Accuracy decreases for ill-conditioned matrices

Error analysis explains stability issues

Method has large parallelism potential

Abstract

In this paper, we perform a roundoff error analysis of an integration-based method for computing the matrix sign function recently proposed by Nakaya and Tanaka. The method expresses the matrix sign function using an integral representation and computes the integral numerically by the double-exponential formula. While the method has large-grain parallelism and works well for well-conditioned matrices, its accuracy deteriorates when the input matrix is ill-conditioned or highly nonnormal. We investigate the reason for this phenomenon by a detailed roundoff error analysis.