矩阵Mittag-Leffler函数的计算及其在分数阶微积分中的应用
Computing the matrix Mittag-Leffler function with applications to fractional calculus
摘要：
讨论了带矩阵变元的Mittag-Leffler（ML）函数的计算及其在分数阶微积分中的应用。一般来说，在矩阵参数中的标量函数的评估可能需要计算取决于矩阵谱的可能高阶导数。关于ML函数，其任意阶导数的数值计算是一个完全未探索的话题;在本文中，我们解决这个问题，并定制和研究了三种不同的方法。该方法结合在一起，以设计一种算法，能够提供高精度的原始衍生物平衡技术。还研究了矩阵ML函数求值的条件。数值实验表明，该算法具有很高的精度，往往接近机器精度。
英文摘要：
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.
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