Computing the Mittag-Leffler Function of a Matrix Argument

TL;DR

This paper introduces a derivative-free, efficient method for computing the two-parameter Mittag-Leffler function of a matrix, applicable in fractional calculus, that outperforms existing methods especially for matrices with clustered eigenvalues.

Contribution

The paper presents a novel, derivative-free algorithm for matrix Mittag-Leffler functions that combines Taylor series and Schur-Parlett techniques, improving accuracy and efficiency.

Findings

Competitive with state-of-the-art in accuracy and CPU time

Outperforms existing methods for matrices with clustered eigenvalues

Easily extendable to other matrix functions

Abstract

It is well-known that the two-parameter Mittag-Leffler (ML) function plays a key role in Fractional Calculus. In this paper, we address the problem of computing this function, when its argument is a square matrix. Effective methods for solving this problem involve the computation of higher order derivatives or require the use of mixed precision arithmetic. In this paper, we provide an alternative method that is derivative-free and works entirely using IEEE standard double precision arithmetic. If certain conditions are satisfied, our method uses a Taylor series representation for the ML function; if not, it switches to a Schur-Parlett technique that will be combined with the Cauchy integral formula. A detailed discussion on the choice of a convenient contour is included. Theoretical and numerical issues regarding the performance of the proposed algorithm are discussed. A set of numerical experiments shows that our novel approach is competitive with the state-of-the-art method for IEEE double precision arithmetic, in terms of accuracy and CPU time. For matrices whose Schur decomposition has large blocks with clustered eigenvalues, our method far outperforms the other. Since our method does not require the efficient computation of higher order derivatives, it has the additional advantage of being easily extended to other matrix functions (e.g., special functions).