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Computing Enclosures for the Matrix Mittag–Leffler Function

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Abstract

We propose two algorithms for numerically calculating interval matrices including two-parameter matrix Mittag–Leffler (ML) functions. We first present an algorithm for computing enclosures for scalar ML functions. Then, the two proposed algorithms are developed by exploiting the scalar algorithm and verified block diagonalization. The first algorithm relies on a numerical spectral decomposition. The cost of this algorithm is only cubic plus that of the scalar algorithm if the second parameter is not too small. The second algorithm is based on a numerical Jordan decomposition, and can also be applied to defective matrices. The cost of this algorithm is quartic plus that of the scalar algorithm. A numerical experiment illustrates an application to a fractional differential equation.

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Acknowledgements

The author is grateful to Prof. Roberto Garrappa (Universit à degli Studi di Bari) and the anonymous referees for their advice and comments.

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  1. Faculty of Science and Engineering, Iwate University, 4-3-5 Ueda, Morioka-shi, Iwate, 020-8551, Japan

    Shinya Miyajima

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  1. Shinya Miyajima
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Correspondence to Shinya Miyajima.

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This work was partially supported by JSPS KAKENHI Grant Numbers JP16K05270, JP21K03363.

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Miyajima, S. Computing Enclosures for the Matrix Mittag–Leffler Function. J Sci Comput 87, 62 (2021). https://doi.org/10.1007/s10915-021-01447-6

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  • DOI: https://doi.org/10.1007/s10915-021-01447-6

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