Computing Enclosures for the Matrix Mittag–Leffler Function

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...

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Bibliographic Details
Published inJournal of scientific computing Vol. 87; no. 2; p. 62
Main Author Miyajima, Shinya
Format Journal Article
LanguageEnglish
Published New York Springer US 01.05.2021
Springer Nature B.V
Subjects
Algorithms
Approximation
Computation
Computational Mathematics and Numerical Analysis
Differential equations
Eigenvalues
Enclosures
Fractional calculus
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Parameters
Partial differential equations
Theoretical
Fractional differential equation
33E12
65F60
65G20
26A33
Verified block diagonalization
Self-validating numerical algorithm
15A15
Mittag–Leffler function
33F05
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Summary: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.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-021-01447-6