Computing the Matrix Mittag-Leffler Function with Applications to Fractional Calculus

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. , avialable at https://doi.org/ 10.1007/s10915-018-0699-5. This work is supported under the GNCS-INdAM 2017 project "Analisi numerica per modelli descritti da operatori frazionari". arXiv:1804.04883v2 [math.NA] 1 Dec 2019where |δ 1 |, |δ 2 |, . . . , |δ J | < and terms proportional to O( 2 ) have been discarded. It is thus immediate to derive the following bound for the round-off errorThe order by which the terms c j are summed is relevant especially for the reliability of the above estimator; clearly, an ascending sorting of |c j | makes the estimate (16) more conservative and hence more useful for practical use. It is therefore advisable, especially in the more compelling cases (namely, for arguments z with large modulus and | arg(z)| > απ/2) to perform a preliminary sorting of the terms in (14) with respect to their modulus.An alternative estimate of the round-off error can be obtained after reformulating (15) asŜ J = S J + J j=1 S j δ j from which one can easily derive S J −Ŝ J ≤ J j=1

Numerical Solution of Multiterm Fractional Differential Equations Using the Matrix Mittag–Leffler Functions

“…However, for a long time, this has been considered only as a theoretical tool because of the lack of effective methods to numerically approximate this function. Only recently have many advances been made for the numerical evaluation of the scalar ML function [26][27][28][29]; the case of matrix arguments has since been analyzed [30,31], and finally a numerical algorithm has been accomplished, which reaches very high accuracies [32]. In this paper, we show the effectiveness of the matrix approach when solving MTFDEs, both in terms of execution time and in terms of accuracy, and also in comparison with some well-established numerical methods.…”

“…The numerical computation of matrix functions is an extensively studied topic that has deserved great attention during the last decades (we refer to Higham [37] for a complete treatise and a full list of references). Only recent studies have considered matrix arguments for the ML function (see, e.g., [30][31][32]38,39]). To be precise, even the numerical scalar case has received poor attention, and only recently has Garrappa [29] developed a powerful Matlab routine (ml.m, available on Matlab website) that gives very accurate results for arguments all over the complex plane.…”

“…To be precise, even the numerical scalar case has received poor attention, and only recently has Garrappa [29] developed a powerful Matlab routine (ml.m, available on Matlab website) that gives very accurate results for arguments all over the complex plane. Thereafter, an effective numerical procedure for the matrix case has been proposed [32], and we apply this for our computations. Interestingly, the computation of the matrix ML function turns out to be very accurate, practically close to the machine precision.…”

“…In Section 3, we address the numerical solution of this equivalent system. This essentially grounds on ML functions with matrix arguments, and we introduce the numerical method proposed by Garrappa and Popolizio [32] to compute matrix ML functions. The performance is tested in Section 4, where we give a comparison with PI methods for several tests; in the same section, a brief description of these methods is presented.…”

A Practical Guide to Prabhakar Fractional Calculus

“…. This last expression turns out to be particularly useful when combined with (4.4) or (4.5), since it allows to express derivatives of the two-parameter ML function as combination of different instances of the same function, a result widely exploited in [40] for numerical purposes.…”