Scaling-and-squaring method for computing the inverses of matrix $\varphi$-functions

TL;DR

This paper introduces an efficient scaling-and-squaring method, based on Newton-Schulz iteration, for computing inverses of matrix φ-functions, crucial in exponential integrators and boundary value problems, especially for large sparse matrices.

Contribution

It adapts the scaling-and-squaring technique with Newton-Schulz iteration to compute matrix φ-function inverses, including analysis and Padé approximants for improved accuracy.

Findings

Method converges rapidly for large sparse matrices.

Numerical experiments confirm the efficiency and accuracy of the approach.

Padé approximants enhance the approximation of inverse φ-functions.

Abstract

This paper aims to develop efficient numerical methods for computing the inverse of matrix $\varphi$-functions, $\psi_\ell(A) := (\varphi_\ell(A))^{-1}$, for $\ell =1,2,\ldots,$ when $A$ is a large and sparse matrix with eigenvalues in the open left half-plane. While $\varphi$-functions play a crucial role in the analysis and implementation of exponential integrators, their inverses arise in solving certain direct and inverse differential problems with non-local boundary conditions. We propose an adaptation of the standard scaling-and-squaring technique for computing $\psi_\ell(A)$, based on the Newton-Schulz iteration for matrix inversion. The convergence of this method is analyzed both theoretically and numerically. In addition, we derive and analyze Pad\'e approximants for approximating $\psi_1(A/2^s)$, where $s$ is a suitably chosen integer, necessary at the root of the squaring process. Numerical experiments demonstrate the effectiveness of the proposed approach.