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

Abstract

This paper aims to develop efficient numerical methods for computing the inverse of matrix $φ$-functions, $ψ_\ell(A) := (φ_\ell(A))^{-1}$, for $\ell =1,2,\ldots,$ when $A$ is a large and sparse matrix with eigenvalues in the open left half-plane. While $φ$-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 $ψ_\ell(A)$, 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 Padé approximants for approximating $ψ_1(A/2^s)$, where $s$ is a suitably chosen integer, necessary at the root of the squaring process. Numerical experiments demonstrate the effectiveness of the proposed approach.