Scaling-and-squaring method for computing the inverses of matrix $\ddot{I}†$-functions

Abstract

This paper aims to develop efficient numerical methods for computing the inverse of matrix $\ddot{I}†$-functions, $\ddot{I}{ˆ}_{\ell }(A):=(\ddot{I}{†}_{\ell }(A){)}^{-1}$, for $\ell =1,2,\dots ,$ when $A$ is a large and sparse matrix with eigenvalues in the open left half-plane. While $\ddot{I}†$-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 $\ddot{I}{ˆ}_{\ell }(A)$, 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 $\ddot{I}{ˆ}_1(A/2^s)$, where $s$ is a suitably chosen integer, necessary at the root of the squaring process. Numerical experiments demonstrate the effectiveness of the proposed approach.