Parallel Computation of functions of matrices and their action on vectors

TL;DR

This paper introduces a new class of parallel algorithms for computing functions of matrices and their action on vectors, enabling high-order approximations with adjustable accuracy and stability, suitable for parallel computing environments.

Contribution

The paper proposes a novel parallel approach using linear systems to approximate matrix functions, with adjustable coefficients for improved accuracy and stability.

Findings

Methods achieve high-order approximations

Coefficients can be tuned for accuracy and stability

Numerical tests demonstrate effectiveness

Abstract

We present a novel class of methods to compute functions of matrices or their action on vectors that are suitable for parallel programming. Solving appropriate simple linear systems of equations in parallel (or computing the inverse of several matrices) and with a proper linear combination of the results, allows us to obtain new high order approximations to the desired functions of matrices. An error analysis to obtain forward and backward error bounds is presented. The coefficients of each method, which depends on the number of processors, can be adjusted to improve the accuracy, the stability or to reduce round off errors of the methods. We illustrate this procedure by explicitly constructing some methods which are then tested on several numerical examples.