Computing the matrix fractional power with the double exponential formula

TL;DR

This paper introduces two novel algorithms based on the double exponential formula for efficiently computing matrix fractional powers, especially for ill-conditioned matrices, with proven convergence and superior performance.

Contribution

The paper develops two new quadrature-based algorithms utilizing the DE formula for matrix fractional powers, including an adaptive method and a truncation error analysis, improving efficiency and accuracy.

Findings

Algorithms achieve desired accuracy efficiently.

DE formula converges faster than Gaussian quadrature for ill-conditioned matrices.

Numerical results demonstrate superior speed and accuracy.

Abstract

Two quadrature-based algorithms for computing the matrix fractional power $A^\alpha$ are presented in this paper. These algorithms are based on the double exponential (DE) formula, which is well-known for its effectiveness in computing improper integrals as well as in treating nearly arbitrary endpoint singularities. The DE formula transforms a given integral into another integral that is suited for the trapezoidal rule; in this process, the integral interval is transformed to the infinite interval. Therefore, it is necessary to truncate the infinite interval into an appropriate finite interval. In this paper, a truncation method, which is based on a truncation error analysis specialized to the computation of $A^\alpha$, is proposed. Then, two algorithms are presented -- one computes $A^\alpha$ with a fixed number of abscissas, and the other computes $A^\alpha$ adaptively. Subsequently, the convergence rate of the DE formula for Hermitian positive definite matrices is analyzed. The convergence rate analysis shows that the DE formula converges faster than the Gaussian quadrature when $A$ is ill-conditioned and $\alpha$ is a non-unit fraction. Numerical results show that our algorithms achieved the required accuracy and were faster than other algorithms in several situations.