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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References
Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.: Mittag–Leffler Functions. Related Topics and Applications. Springer, Berlin (2014)
Haubold, H.J., Mathai, A.M., Saxena, R.K.: Mittag–Leffler functions and their applications. J. Appl. Math. 2011, 298628 (2011)
Gorenflo, R., Loutchko, J., Luchko, Y.: Computation of the Mittag–Leffler function \(E_{\alpha ,\beta }(z)\) and its derivative. Fract. Calc. Appl. Anal. 5, 491–518 (2002)
Seybold, H., Hilfer, R.: Numerical algorithm for calculating the generalized Mittag–Leffler function. SIAM J. Numer. Anal. 47(1), 69–88 (2008)
Garrappa, R.: Numerical evaluation of two and three parameter Mittag–Leffler functions. SIAM J. Numer. Anal. 53(3), 1350–1369 (2015)
Higham, N.J.: Functions of Matrices: Theory and Computation. SIAM Publications, Philadelphia (2008)
Garrappa, R., Popolizio, M.: Computing the matrix Mittag–Leffler function with applications to fractional calculus. J. Sci. Comput. 77, 129–153 (2018)
Duan, J., Chen, L.: Solution of fractional differential equation systems and computation of matrix Mittag–Leffler functions. Symmetry 10(10), 503 (2018)
Popolizio, M.: On the matrix Mittag–Leffler function: theoretical properties and numerical computation. Mathematics 7(12), 1140 (2019)
Higham, N.J., Liu, X.: A multiprecision derivative-free Schur–Parlett algorithm for computing matrix functions. MIMS EPrint 2020.19, (2020) http://eprints.maths.manchester.ac.uk/2781/
Bochev, P., Markov, S.: A self-validating numerical method for the matrix exponential. Computing 43, 59–72 (1989)
Bochev, P.: Simultaneous self-verified computation of \(\exp (A)\) and \(\int _0^1\exp (As)ds\). Computing 45, 183–191 (1990)
Frommer, A., Hashemi, B.: Verified computation of square roots of a matrix. SIAM J. Matrix Anal. Appl. 31, 1279–1302 (2009)
Frommer, A., Hashemi, B., Sablik, T.: Computing enclosures for the inverse square root and the sign function of a matrix. Linear Algebra Appl. 456, 199–213 (2014)
Miyajima, S.: Fast enclosure for a matrix inverse square root. Linear Algebra Appl. 467, 116–135 (2015)
Miyajima, S.: Fast verified computation for the matrix principal \(p\)th root. J. Comput. Appl. Math. 330, 276–288 (2018)
Miyajima, S.: Verified computation of the matrix exponential. Adv. Comput. Math. 45, 137–152 (2019)
Miyajima, S.: Verified computation for the matrix principal logarithm. Linear Algebra Appl. 569, 38–61 (2019)
Miyajima, S.: Verified computation for the matrix Lambert \(W\) function. Appl. Math. Comput. 362, 124555 (2019)
Frommer, A., Hashemi, B.: Computing enclosures for the matrix exponential. SIAM J. Matrix Anal. Appl. 41(4), 1674–1703 (2020)
Miyajima, S.: Verified computation for the geometric mean of two matrices. Jpn. J. Ind. Appl. Math. 38, 211–232 (2021)
Miyajima, S.: Verified computation of real powers of matrices. J. Comput. Appl. Math. 391, 113431 (2021)
Miyajima, S.: Verified computation of matrix gamma function. Linear Multilinear Algebra https://doi.org/10.1080/03081087.2020.1757602
Rohn, J.: VERSOFT: Verification Software in MATLAB/INTLAB. http://uivtx.cs.cas.cz/~rohn/matlab
Rump, S.M.: INTLAB-INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–107. Kluwer, Dordrecht (1999)
Miyajima, S.: Fast enclosure for all eigenvalues and invariant subspaces in generalized eigenvalue problems. SIAM J. Matrix Anal. Appl. 35, 1205–1225 (2014)
Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1991)
de Figueiredo, L.H., Stolfi, J.: Affine arithmetic: concepts and applications. Numer. Algorithms 37, 147–158 (2004)
Rump, S.M.: Verification methods: Rigorous results using floating-point arithmetic. Acta Numer. 19, 287–449 (2010)
Rump, S.M., Zemke, J.-P.M.: On eigenvector bounds. BIT Numer. Math. 43, 823–837 (2003)
Ogita, T., Rump, S.M., Oishi, S.: Accurate sum and dot product. SIAM J. Sci. Comput. 26(6), 1955–1988 (2005)
Zeng, Z., Li, T.-Y.: NAClab: A Matlab toolbox for numerical algebraic computation. ACM Commun. Comput. Algebra 47, 170–173 (2013)
Fasi, M., Higham, N.J., Iannazzo, B.: An algorithm for the matrix Lambert \(W\) function. SIAM J. Matrix Anal. Appl. 36, 669–685 (2015)
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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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
Keywords
- Fractional differential equation
- Mittag–Leffler function
- Verified block diagonalization
- Self-validating numerical algorithm