Evaluation of the matrix exponential function using finite elements in time

TL;DR

This paper introduces a novel finite element approach in fictitious time to compute matrix exponentials, aiming for improved accuracy and applicability over existing methods.

Contribution

The paper presents a new finite element method in fictitious time for evaluating matrix exponentials, enhancing accuracy and generality.

Findings

Achieves accurate matrix exponential calculations.

Uses finite elements in fictitious time for improved results.

Provides a set of simultaneous equations for computation.

Abstract

The evaluation of a matrix exponential function is a classic problem of computational linear algebra. Many different methods have been employed for its numerical evaluation [Moler C and van Loan C 1978 SIAM Review 20 4], none of which produce a definitive algorithm which is broadly applicable and sufficiently accurate, as well as being reasonably fast. Herein, we employ a method which evaulates a matrix exponential as the solution to a first-order initial value problem in a fictitious time variable. The new aspect of the present implementation of this method is to use finite elements in the fictitious time variable. [Weatherford C A, Red E, and Wynn A 2002 Journal of Molecular Structure 592 47] Then using an expansion in a properly chosen time basis, we are able to make accurate calculations of the exponential of any given matrix as the solution to a set of simultaneous equations.