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Numerical methods for differential linear matrix equations via Krylov subspace methods. (English) Zbl 07169491
Summary: In the present paper, we present some numerical methods for computing approximate solutions to some large differential linear matrix equations. In the first part of this work, we deal with differential generalized Sylvester matrix equations with full rank right-hand sides using a global Galerkin and a norm-minimization approaches. In the second part, we consider large differential Lyapunov matrix equations with low rank right-hand sides and use the extended global Arnoldi process to produce low rank approximate solutions. We give some theoretical results and present some numerical examples.
MSC:
65F10 Iterative numerical methods for linear systems
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[1] Abou-Kandil, H.; Freiling, G.; Ionescu, V.; Jank, G., Matrix Riccati equations in control and sytems theory, (Systems & Control Foundations & Applications (2003), Birkhauser)
[2] Corless, M. J.; Frazho, A. E., (Linear Systems and Control - An Operator Perspective. Linear Systems and Control - An Operator Perspective, Pure and Applied Mathematics (2003), Marcel Dekker: Marcel Dekker New York-Basel) · Zbl 1050.93001
[3] Bouyouli, R.; Jbilou, K.; Sadaka, R.; Sadok, H., Convergence properties of some block Krylov subspace methods for multiple linear systems, J. Comput. Appl. Math., 196, 498-511 (2006) · Zbl 1100.65024
[4] Jbilou, K.; Messaoudi, A., A computational method for symmetric stein matrix equations, Numer. Linear Algebra Signals Syst. Control, 80, 295-311 (2011) · Zbl 1251.65059
[5] Jbilou A. Messaoudi H. Sadok, K., Global FOM and GMRES algorithms for matrix equations, Appl. Numer. Math., 31, 49-63 (1999) · Zbl 0935.65024
[6] Higham, N. J., The scaling and squaring method for the matrix exponential revisited, SIAM J. Matrix Anal. Appl., 26, 4, 1179-1193 (2005) · Zbl 1081.65037
[7] Moler, C. B.; Van Loan, C. F., Nineteen dubious ways to compute the exponential of a matrix, SIAM Rev., 20, 801-836 (1978), Reprinted and updated as “Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later”, SIAM Review 45(2003), 3-49 · Zbl 0395.65012
[8] Druskin, V.; Knizhnerman, L., Extended Krylov subspaces: approximation of the matrix square root and related functions, SIAM J. Matrix Anal. Appl., 19, 3, 755-771 (1998) · Zbl 0912.65022
[9] Heyouni, M., Extended Arnoldi methods for large Sylvester matrix equations, Appl. Numer. Math., 60, 11, 1171-1182 (2010) · Zbl 1210.65093
[10] Heyouni, M.; Jbilou, K., An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation, Electron. Trans. Numer. Anal., 33, 53-62 (2009) · Zbl 1171.65035
[11] Simoncini, V., A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comput., 29, 3, 1268-1288 (2007) · Zbl 1146.65038
[12] Hached, M.; Jbilou, K., Numerical solutions to large-scale differential Lyapunov matrix equations, Numer. Algorithms, 79, 3, 747-757 (2018) · Zbl 1416.65116
[13] Bartels, R. H.; Stewart, G. W., Algorithm 432: Solution of the matrix equation AX+XB=C, Circ. Syst. Signal Proc., 13, 820-826 (1972) · Zbl 1372.65121
[14] Golub, G. H.; Nash, S.; Van Loan, C., A Hessenberg-Schur method for the problem AX+XB=C, IEEC Trans. Autom. Control, AC-24, 909-913 (1979) · Zbl 0421.65022
[15] Higham, N. J., Functions of Matrices, Theory and Computation (2008), SIAM: SIAM Philadelphia · Zbl 1167.15001
[16] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press · Zbl 0729.15001
[17] Benzi, M.; Razouk, N., Decay rates and O(n) algorithms for approximating functions of sparse matrices, Electron. Trans. Numer. Anal., 28, 16-39 (2007) · Zbl 1171.65034
[18] Gallopoulos, E.; Saad, Y., Efficient solution of parabolic equations by Krylov approximation methods, SIAM J. Sci. Stat. Comput., 13, 1236-1264 (1992) · Zbl 0757.65101
[19] Hochbruck, M.; Lubich, C., On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34, 1911-1925 (1997) · Zbl 0888.65032
[20] Saad, Y., Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 29, 209-228 (1992) · Zbl 0749.65030
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