Efficient scaling and squaring method for the matrix exponential

TL;DR

This paper introduces an efficient algorithm for computing the matrix exponential that outperforms existing methods by integrating various approximation techniques within a scaling and squaring framework, adaptable to Lie algebra matrices.

Contribution

The paper presents a novel algorithm combining Taylor, partitioned, and Padé methods with scaling and squaring, improving performance and flexibility over current software implementations.

Findings

Superior computational performance demonstrated in numerical experiments

Flexible implementation avoiding matrix inverses for certain problems

Preserves Lie group properties when using diagonal Padé approximants

Abstract

This work presents a new algorithm to compute the matrix exponential within a given tolerance. Combined with the scaling and squaring procedure, the algorithm incorporates Taylor, partitioned and classical Pad\'e methods shown to be superior in performance to the approximants used in state-of-the-art software. The algorithm computes matrix--matrix products and also matrix inverses, but it can be implemented to avoid the computation of inverses, making it convenient for some problems. If the matrix A belongs to a Lie algebra, then exp(A) belongs to its associated Lie group, being a property which is preserved by diagonal Pad\'e approximants, and the algorithm has another option to use only these. Numerical experiments show the superior performance with respect to state-of-the-art implementations.