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).

翻译：众所周知，双参数Mittag-Leffler（ML）函数在分数阶微积分中扮演着关键角色。本文研究当该函数的参数为方阵时的计算问题。解决该问题的有效方法需计算高阶导数或使用混合精度算术。本文提出一种无需导数且完全基于IEEE标准双精度算术的替代方法。在满足特定条件时，该方法采用ML函数的泰勒级数表示；否则，切换至结合柯西积分公式的Schur-Parlett技术。文中详细讨论了合适围道选取的理论与数值问题，并分析了所提算法性能的相关理论与数值问题。一系列数值实验表明，在IEEE双精度算术条件下，本文创新方法在精度和CPU时间方面均与现有最先进方法具有竞争力。对于舒尔分解包含大块聚集特征值的矩阵，本文方法性能远优于其他方法。由于无需高效计算高阶导数，该方法还具有易于扩展至其他矩阵函数（如特殊函数）的额外优势。