Discrete cosine and sine functions of matrices

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Abstract

In this paper, we introduce two methods for constructing matrix-valued cosine and sine functions defined on a discrete domain. The first method develops an algorithm to compute the n × n matrix-valued cosine and sine of a given square matrix A with some restrictions on its eigenvalues. The second method derives a formula based on the Jordan canonical form to compute each term of a Jordan block in the discrete matrix-valued cosine and sine of a given square matrix A. To illustrate the utility of our approach, we present some examples.

Keywords

matrix valued functions / Putzer algorithm / discrete cosine function / Jordan canonical form / difference equations

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References

[1] C. Goodrich, A. C. Peterson, Discrete Fractional Calculus , Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-25562-0

[2] W. Everling, M. I. Liou, On the evaluation of e^AT by power series , Proc. IEEE, 55 (1967), 413. https://doi.org/10.1109/PROC.1967.5503

[3] G. I. Hargreaves, N. J. Higham, Efficient algorithms for the matrix cosine and sine, Numer. Algorithms, 40 (2005), 383-400. https://doi.org/10.1007/s11075-005-8141-0

[4] S. M. Serbin, S. A. Blalock, An algorithm for computing the matrix cosine, SIAM, 1 (1980), 198-204. https://doi.org/10.1137/0901013

[5] A. H. Al-Mohy, N. J. Higham, S. D. Relton, New algorithms for computing the matrix sine and cosine separately or simultaneously, SIAM J. Sci. Comput., 37 (2015), 456-487. http://epubs.siam.org/doi/abs/10.1137/140973979

[6] N. J. Higham, M. I. Smith, Computing the matrix cosine, Numer. Algorithms, 34 (2003), 13-26. https://doi.org/10.1023/A:1026152731904

[7] M. Şeydaoğlu, P. Bader, S. Blanes, F. Casas, Computing the matrix sine and cosine simultaneously with a reduced number of products, Appl. Numer. Math., 163 (2021), 96-107. https://doi.org/10.1016/j.apnum.2021.01.009

[8] A. Shakeri, M. Behroozifar, Approximating the matrix exponential, sine and cosine via the spectral method, Int. J. Nonlinear Anal. Appl., 14 (2023), 2881-2900. https://doi.org/10.22075/ijnaa.2022.26081.3222

[9] H. Qin, X. Lu, An efficient algorithm for basic elementary matrix functions with specified accuracy and application, Appl. Math., 4 (2024), 690-708. https://doi.org/10.3390/appliedmath4020037

[10] N. J. Higham, Functions of Matrices: Theory and Computation , SIAM, Philadelphia, PA, 2008. https://doi.org/10.1137/1.9780898717778

[11] R. A. Horn, C. R. Johnson, Matrix Analysis , Cambridge University Press, Cambridge, MA, USA, 2012. https://doi.org/10.1017/CBO9781139020411

[12] E. J. Putzer, Avoiding the Jordan canonical form in the discussion of linear systems with constant coefficients, Am. Math. Mon., 73 (1966), 2-7. https://doi.org/10.2307/2313914

[13] C. D. Ahlbrand, J. Ridenhour, Floquet theory for time scales and Putzer representations of matrix logarithms, J. Differ. Equations Appl., 9 (2003), 77-92. https://doi.org/10.1080/10236100309487536

[14] S. Elaydi, W. A. Harris Jr, On the computation of A^N , SIAM Rev., 40 (1998), 965-971. https://www.jstor.org/stable/2653046

[15] C. R. Johnson, K. Okubo, R. Reams, Uniqueness of matrix square roots and an application, Linear Algebra Appl., 323 (2001), 51-60. https://doi.org/10.1016/S0024-3795(00)00243-3