×

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.

MSC:

15A24 Matrix equations and identities

Software:

mctoolbox
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bhattacharyya, S. P.; de Souza, E., Pole assignment via Sylvester’s equation, Systems & Control Letters, 1, 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, 2, 361-383 (2004) · Zbl 1056.93036
[3] Lam, J.; Yan, W.; Hu, T., Pole assignment with eigenvalue and stability robustness, International Journal of Control, 72, 13, 1165-1174 (1999) · Zbl 1047.93519
[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; 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, 4, 1401-1418 (2000) · Zbl 0958.65052
[6] Gugercin, S.; Sorensen, D. C.; Antoulas, A. C., A modified low-rank Smith method for large-scale Lyapunov equations, Journal Numerical Algorithms, 32, 1, 27-55 (2003) · Zbl 1034.93020
[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
[8] Miller, D. F., The iterative solution of the matrix equation \(X A + B X + C = 0\), Linear Algebra and Its Applications, 105, 131-137 (1988) · Zbl 0663.65032
[9] Duan, G. R., The solution to the matrix equation \(A V + B W = E V J + R\), Applied Mathematics Letters, 17, 10, 1197-1204 (2004)
[10] Zhou, B.; Duan, G. R., A new solution to the generalized Sylvester matrix equation \(A V - E V F = B W\), Systems & Control Letters, 55, 3, 193-198 (2006)
[11] Hu, Q.; Cheng, D., The polynomial solution to the Sylvester matrix equation, Applied Mathematics Letters, 19, 9, 859-864 (2006) · Zbl 1117.15011
[12] Smith, R. A., Matrix equation \(X A + B X = C\), SIAM Journal on Applied Mathematics, 16, 1, 198-201 (1968) · Zbl 0157.22603
[13] Wachspress, E. L., Iterative solution of the Lyapunov matrix equation, Applied Mathematics Letters, 1, 1, 87-90 (1988) · Zbl 0631.65037
[14] Davison, E. J.; Man, F. T., The numerical solution of \(A^\prime Q + Q A = C\), IEEE Transactions on Automatic Control, 13, 4, 448-449 (1968)
[15] Sadkane, M., Estimates from the discrete-time Lyapunov equation, Applied Mathematics Letters, 16, 3, 313-316 (2003) · Zbl 1068.93047
[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
[17] Wang, Q.; Lam, J.; Wei, Y.; Chen, T., Iterative solutions of coupled discrete Markovian jump Lyapunov equations, Computers and Mathematics with Applications, 55, 4, 843-850 (2008) · Zbl 1139.60334
[18] Varga, R. S., Matrix Iterative Analysis (2000), Springer · Zbl 0133.08602
[19] Higham, N. J., Accuracy and Stability of Numerical Algorithms (1996), SIAM Philadelphia · Zbl 0847.65010
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.