Extended block Hessenberg method for large-scale Sylvester differential matrix equations

Document Type : Research Paper

Author

Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran

Abstract

In this paper, we consider large-scale low-rank Sylvester differential matrix equations. We present two iterative methods for the approximate solution of such differential matrix equations. In the first method, exploiting the extended block Krylov method, we approximate the exponential matrix in the exact solution. In the second method, we first project the initial value problem onto an extended block Krylov subspace and acquire a low-dimensional low-rank Sylvester differential matrix equation. Then the reduced Sylvester differential matrix equation is solved by the backward differentiation formula method (BDF) and the derived solution is used to construct the low-rank approximate solution of the original initial value problem. The iterative approaches are followed until some certain accuracy is obtained. We give some theoretical results and some numerical examples to show the efficiency of the proposed methods.

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References

[1] Abdaoui, I., Elbouyahyaoui, L., & Heyouni, H. (2020). The simpler block CMRH method for linear systems. Numerical Algorithms, 84, 1265-1293. https://doi.org/10.1007/s11075-019-00814-7
[2] Abidi, O., Heyouni, M., & Jbilou, K. (2017). On some properties of the extended block and global Arnoldi methods with applications to model reduction. Numerical Algorithms, 75, 285-304. https://doi.org/10.1007/s11075-016-0207-7
[3] Abou-Kandil, H., Freiling, G., Ionescu, V. & Jank, G. (2003). Matrix Riccati equations in control and systems theory. Birkhauser.
[4] Addam, M., Heyouni, M., & Sadok, H. (2017). The block Hessenberg process for matrix equations. Electronic Transactions on Numerical Analysis, 46, 460-473.
[5] Agoujil, S., Bentbib, A. H., Jbilou, K. & Sadek, El M. (2014). A minimal residual norm method for large-scale Sylvester matrix equations. Electronic Transactions on Numerical Analysis, 43, 45-59. https://doi.org/10.1016/j.amc.2006.10.011
[6] Amato, F., Ambrosino, R., Ariola, M., Cosentino, C., & De Tommasi, G. (2014). Finite-Time stability and control. Springer.
[7] Antoulas, A. C. (2005). Approximation of Large-Scale Dynamical Systems. SIAM.
[8] Azizizadeh, N., Tajaddini, A., & Ra ei, R.(2023). Implicitly restarted global Krylov subspace methods for matrix equations AXB = C. Mathematical Sciences, https://doi.org/10.1007/s40096-023-00516-1
[9] Azizizadeh, N., Tajaddini, A., & Wu, G.(2018). Weighted and de ated global GMRES algorithms for solving large Sylvester matrix equations. Numerical Algorithms, 82, 155-181. https://doi.org/10.1007/s11075-018-0597-9
[10] Behr, M., Benner, P., & Heiland, J. (2019). Solution formulas for di erential Sylvester and Lyapunov equations. Calcolo, 56(4), 51-84. https://doi.org/10.1007/s10092-019-0348-x
[11] Benner, P., & and Mena, H. (2004). BDF methods for large-scale di erential Riccati equations. Proceedings of Mathematical Theory of Network and Systems, MTNS.
[12] Benner, P. & Mena, H. (2013). Rosenbrock methods for solving Riccati di erential equations. IEEE Transactions on Automatic Control, 58(11), 2950-2957. https://doi.org/10.1109/TAC.2013.2258495
[13] Blanquer, I., Claramunt, H., Hernandez, V., & Vidal, A. M. (1998). Solving the Generalized Lyapunov Equation by the Bartels-Stewart Method using Standard Software Libraries for Linear Algebra Computations. IFAC Proceedings Volumes, 31(18), 387-392.
[14] Boisvert, R., Pozo, R., Remington, K., Barrett, R., & Dongarra, J. (1997). The Matrix Market: A web resource for test matrix collections, in Quality of Numerical Software, Assessment and Enhancement, R. Boisvert, ed., Chapman & Hall, London, 125-137.
[15] Bouhamidi, A., Elbouyahyaoui, L., & Heyouni, M. (2024). The constant solution method for solving large-scale di erential Sylvester matrix equations with time invariant coecients. Numerical Algorithms, 96, 449-488. https://doi.org/10.1007/s11075-023-01653-3
[16] Butcher, J. C. (2008). Numerical methods for ordinary di erential equations. John Wiley & Sons.
[17] Golub, G. H., Nash, S., & Van Loan, C. (1979). A Hessenberg-Schur method for the problem AX + XB = C. IEEE Transactions on Automatic Control, 24, 909-913. https://doi.org/10.1109/tac.1979.1102170
[18] Hached, M., & Jbilou, K. (2018). Computational Krylov-based methods for large-scale di erential Sylvester matrix problems. Numerical Linear Algebra with Applications, 25(5), e2187. https://doi.org/10.1002/nla.2187
[19] Hached, M., & Jbilou, K. (2018). Numerical solutions to large-scale di erential Lyapunov matrix equations. Numerical Algorithms, 79, 741-757. https://doi.org/10.1007/s11075-017-0458-y
[20] Hached, M., & Jbilou, K. (2020). Numerical methods for di erential linear matrix equations via Krylov subspace methods. Journal of Computational and Applied Mathematics, 370, 112-647. https://doi.org/10.1016/j.cam.2019.112674
[21] Heyouni, M., & Jbilou, K. (2009). An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation. Electronic Transactions on Numerical Analysis, 33, 53-62. http://eudml.org/doc/130626
[22] Heyouni, M., Saberi-Movahed, F., & Tajaddini, A.(2019). On global Hessenberg based methods for solving Sylvester matrix equations. Computers & Mathematics with Applications, 77(1), 77-92. https://doi.org/10.1016/j.camwa.2018.09.015
[23] Higham, N. J. (2005). The scaling and squaring method for the matrix exponential revised. SIAM Journal on Matrix Analysis and Applications, 26(4), 1179-1193. https://doi.org/10.1137/04061101X
[24] Penzl, T. (2000). LYAPACK. A MATLAB Toolbox for large Lyapunov and Riccati equations, Model Reduction Problems, and Linear-Quadratic Optimal Control Problems.
[25] Rosenbrock, H. H., (1963). Some general implicit processes for the numerical solution of di erential equations. The Computer Journal, 5(4), 329-330. https://doi.org/10.1093/comjnl/5.4.329
[26] Saad, Y. (1989). Numerical solution of large Lyapunov equations. Research Institute for Advanced Computer Science, NASA Ames Research Center.
[27] Saad, Y. (1992). Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM Journal on Numerical Analysis, 29, 209-228. https://doi.org/10.1137/0729014
[28] Sadek, E. M., Bentbib, A., Sadek, L., & Alaoui, H. (2020). Global extended Krylov subspace methods for large-scale di erential Sylvester matrix equations. Journal of Applied Mathematics and Computing, 62, 157-177. https://doi.org/10.1007/s12190-019-01278-7
[29] Sadek, L., Sadek, El M. & Alaoui, H. T. (2022). On Some Numerical Methods for Solving Large Di erential Nonsymmetric Stein Matrix Equations, Mathematical and Computational Applications, 27(4: 69), https://doi.org/10.3390/mca27040069
[30] Sadok, H. (1999). CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm. Numerical Algorithms 20(4), 303-321. https://doi.org/10.1023/A:1019164119887
[31] Simoncini, V. (2007). A new iterative method for solving large-scale Lyapunov matrix equations. SIAM Journal on Scienti c Computing, 29(3), 1268-1288. https://doi.org/10.1137/06066120X
[32] Tajaddini, A., Wu, G., Saberi-Movahed, F., & Azizizadeh, N.(2021). Two New Variants of the Simpler Block GMRES Method with Vector De ation and Eigenvalue De ation for Multiple Linear Systems. Journal of Scienti c Computing, 86(9), 1-33. https://doi.org/10.1007/s10915-020-01376-w

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History

  • Receive Date: 07 December 2023
  • Revise Date: 05 April 2024
  • Accept Date: 22 May 2024

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