Computation of the exponential function of matrices by a formula without oscillatory integrals on infinite intervals

TL;DR

This paper introduces a new quadrature-based method for computing matrix exponentials that avoids oscillatory integrals on infinite intervals, improving efficiency especially for matrices with large imaginary eigenvalues.

Contribution

It presents a novel integral formula using a simple path and standard quadrature rules, reducing computational cost compared to existing methods based on Laplace or Fourier transforms.

Findings

Outperforms existing formulas for matrices with large imaginary eigenvalues

Uses double-exponential and Gauss-Legendre quadrature with rigorous error bounds

Provides a non-oscillatory integral approach for matrix exponential computation

Abstract

We propose a quadrature-based formula for computing the exponential function of matrices with a non-oscillatory integral on an infinite interval and an oscillatory integral on a finite interval. In the literature, existing quadrature-based formulas are based on the inverse Laplace transform or the Fourier transform. We show these expressions are essentially equivalent in terms of complex integrals and choose the former as a starting point to reduce computational cost. By choosing a simple integral path, we derive an integral expression mentioned above. Then, we can easily apply the double-exponential formula and the Gauss-Legendre formula, which have rigorous error bounds. As numerical experiments show, the proposed formula outperforms the existing formulas when the imaginary parts of the eigenvalues of matrices have large absolute values.