The Scaling, Splitting and Squaring Method for the Exponential of Perturbed Matrices

TL;DR

This paper introduces splitting methods for efficiently computing the exponential of perturbed matrices, leveraging matrix structure and small perturbations to improve performance over existing techniques.

Contribution

It presents a novel splitting approach that combines scaling, Padé or Taylor approximations, and optimized squaring to reduce computational cost for perturbed matrix exponentials.

Findings

Improved efficiency in medium-precision matrix exponential computations.

Theoretical analysis of local error and error propagation.

Numerical experiments demonstrating superior performance over existing methods.

Abstract

We propose splitting methods for the computation of the exponential of perturbed matrices which can be written as the sum $A=D+\varepsilon B$ of a sparse and efficiently exponentiable matrix $D$ with sparse exponential $e^D$ and a dense matrix $\varepsilon B$ which is of small norm in comparison with $D$. The predominant algorithm is based on scaling the large matrix $A$ by a small number $2^{-s}$, which is then exponentiated by efficient Pad\'e or Taylor methods and finally squared in order to obtain an approximation for the full exponential. In this setting, the main portion of the computational cost arises from dense-matrix multiplications and we present a modified squaring which takes advantage of the smallness of the perturbed matrix $B$ in order to reduce the number of squarings necessary. Theoretical results on local error and error propagation for splitting methods are complemented with numerical experiments and show a clear improvement over existing methods when medium precision is sought.