Computing the matrix Mittag-Leffler function with applications to fractional calculus

TL;DR

This paper develops and investigates three methods for accurately computing the matrix Mittag-Leffler function, addressing the challenge of derivatives of arbitrary order, with applications in fractional calculus.

Contribution

It introduces a novel algorithm combining three methods and a derivatives balancing technique for high-accuracy computation of matrix ML functions.

Findings

The proposed algorithm achieves near machine precision accuracy.

The methods effectively handle derivatives of high order.

The study includes analysis of the conditioning of matrix ML function evaluation.

Abstract

The computation of the Mittag-Leffler (ML) function with matrix arguments, and some applications in fractional calculus, are discussed. In general the evaluation of a scalar function in matrix arguments may require the computation of derivatives of possible high order depending on the matrix spectrum. Regarding the ML function, the numerical computation of its derivatives of arbitrary order is a completely unexplored topic; in this paper we address this issue and three different methods are tailored and investigated. The methods are combined together with an original derivatives balancing technique in order to devise an algorithm capable of providing high accuracy. The conditioning of the evaluation of matrix ML functions is also studied. The numerical experiments presented in the paper show that the proposed algorithm provides high accuracy, very often close to the machine precision.