Abstract
Many situations, as for example within the context of Fractional Calculus theory, require computing the Mittag–Leffler (ML) function with matrix arguments. In this paper, we collect theoretical properties of the matrix ML function. Moreover, we describe the available numerical methods aimed at this purpose by stressing advantages and weaknesses.
Highlights
The Mittag–Leffler (ML) function has earned the title of “Queen function of fractional calculus” [1,2,3] for the fundamental role it plays within this subject
It becomes clear that the difficult goal of settling the best numerical method for the exponential function becomes even more tough when treating the matrix ML function. In this case, we can affirm that a top-ranked method exists; it was recently proposed [18] and is based on the combination of the Schur–Parlett method [19] and the Optimal Parabolic Contour (OPC) method
[4] for the scalar ML function and its derivative. This method starts from a Schur decomposition, with reordering and blocking, of the matrix argument and applies the Parlett’s recurrence to compute the function in the triangular factor
Summary
The Mittag–Leffler (ML) function has earned the title of “Queen function of fractional calculus” [1,2,3] for the fundamental role it plays within this subject. An analog of this strategy for the ML function presents more difficulties since it can be regarded as a solution of the more involved FDEs. It becomes clear that the difficult goal of settling the best numerical method for the exponential function becomes even more tough when treating the matrix ML function. In this case, we can affirm that a top-ranked method exists; it was recently proposed [18] and is based on the combination of the Schur–Parlett method [19] and the Optimal Parabolic Contour (OPC) method [4] for the scalar ML function and its derivative Speaking, this method starts from a Schur decomposition, with reordering and blocking, of the matrix argument and applies the Parlett’s recurrence to compute the function in the triangular factor.
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