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On Smith-type iterative algorithms for the Stein matrix equation. (English) Zbl 1179.15016
Summary: This note studies the iterative solution to the Stein matrix equation. Firstly, it is shown that the recently developed Smith$(l)$ iteration converges to the exact solution for arbitrary initial condition whereas a special initial condition is required in the literature. Secondly, by presenting a new accelerative Smith iteration named the $r$-Smith iteration that includes the well-known ordinary Smith accelerative iteration as a special case, we have shown that the $r$-Smith accelerative iteration requires less computation than the Smith iteration and the Smith$(l)$ iteration, and the ordinary Smith accelerative iteration requires the least computations comparing with other Smith-type iterations.

15A24Matrix equations and identities
Full Text: DOI
[1] Bhattacharyya, S. P.; De Souza, E.: Pole assignment via Sylvester’s equation, Systems control letters 1, No. 4, 261-263 (1981) · Zbl 0473.93037
[2] Hu, T.; Lin, Z.; Lam, J.: Unified gradient approach to performance optimization under a pole assignment constrain, Journal of optimization theory and applications 121, No. 2, 361-383 (2004) · Zbl 1056.93036 · doi:10.1023/B:JOTA.0000037409.86566.f9
[3] Lam, J.; Yan, W.; Hu, T.: Pole assignment with eigenvalue and stability robustness, International journal of control 72, No. 13, 1165-1174 (1999) · Zbl 1047.93519 · doi:10.1080/002071799220326
[4] B. Zhou, G.R. Duan, Parametric approach for the normal Luenberger function observer design in second-order linear systems, in: Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 1423--1428
[5] Penzl, T.: A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM journal on scientific computing 21, No. 4, 1401-1418 (2000) · Zbl 0958.65052 · doi:10.1137/S1064827598347666
[6] Gugercin, S.; Sorensen, D. C.; Antoulas, A. C.: A modified low-rank Smith method for large-scale Lyapunov equations, Journal numerical algorithms 32, No. 1, 27-55 (2003) · Zbl 1034.93020 · doi:10.1023/A:1022205420182
[7] Chu, D.; Van Dooren, P.: A novel numerical method for exact model matching problem with stability, Automatica 42, 1697-1704 (2006) · Zbl 1130.93332 · doi:10.1016/j.automatica.2006.04.024
[8] Miller, D. F.: The iterative solution of the matrix equation XA+BX+C=0, Linear algebra and its applications 105, 131-137 (1988) · Zbl 0663.65032 · doi:10.1016/0024-3795(88)90008-0
[9] Duan, G. R.: The solution to the matrix equation AV+BW=EVJ+R, Applied mathematics letters 17, No. 10, 1197-1204 (2004) · Zbl 1065.15015 · doi:10.1016/j.aml.2003.05.012
[10] Zhou, B.; Duan, G. R.: A new solution to the generalized Sylvester matrix equation AV-EVF=BW, Systems control letters 55, No. 3, 193-198 (2006) · Zbl 1129.15300 · doi:10.1016/j.sysconle.2005.07.002
[11] Hu, Q.; Cheng, D.: The polynomial solution to the Sylvester matrix equation, Applied mathematics letters 19, No. 9, 859-864 (2006) · Zbl 1117.15011 · doi:10.1016/j.aml.2005.09.005
[12] Smith, R. A.: Matrix equation XA+BX=C, SIAM journal on applied mathematics 16, No. 1, 198-201 (1968) · Zbl 0157.22603 · doi:10.1137/0116017
[13] Wachspress, E. L.: Iterative solution of the Lyapunov matrix equation, Applied mathematics letters 1, No. 1, 87-90 (1988) · Zbl 0631.65037 · doi:10.1016/0893-9659(88)90183-8
[14] Davison, E. J.; Man, F. T.: The numerical solution of A’Q+QA=C, IEEE transactions on automatic control 13, No. 4, 448-449 (1968)
[15] Sadkane, M.: Estimates from the discrete-time Lyapunov equation, Applied mathematics letters 16, No. 3, 313-316 (2003) · Zbl 1068.93047 · doi:10.1016/S0893-9659(03)80050-2
[16] Hammarling, S. J.: Numerical solution of the stable, non-negative definite Lyapunov equation, IMA journal of numerical analysis 2, 303-323 (1982) · Zbl 0492.65017 · doi:10.1093/imanum/2.3.303
[17] Wang, Q.; Lam, J.; Wei, Y.; Chen, T.: Iterative solutions of coupled discrete Markovian jump Lyapunov equations, Computers and mathematics with applications 55, No. 4, 843-850 (2008) · Zbl 1139.60334 · doi:10.1016/j.camwa.2007.04.031
[18] Varga, R. S.: Matrix iterative analysis, (2000) · Zbl 0998.65505
[19] Higham, N. J.: Accuracy and stability of numerical algorithms, (1996) · Zbl 0847.65010