A New Iterative Scheme for Solving the Semi Sylvester Equation

ABSTRACT

In this paper, the Galerkin projection method is used for solving the semi Sylvester equation. Firstly the semi Sylvester equation is reduced to the multiple linear systems. To apply the Galerkin projection method, some propositions are presented. The presented scheme is compared with the L-GL-LSQR algorithm in point of view CPU-time and iteration number. Finally, some numerical experiments are presented to show that the efficiency of the new scheme.

Cite this paper

S. Karimi and F. Attarzadeh, "A New Iterative Scheme for Solving the Semi Sylvester Equation,"*Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 6-10. doi: 10.4236/am.2013.41002.

S. Karimi and F. Attarzadeh, "A New Iterative Scheme for Solving the Semi Sylvester Equation,"

References

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[2] G. H. Golub and C. F. Van Loan, “Matrix Computations,” 3rd Edition, Johns Hopkins University Press, Baltimore, 1996.

[3] V. Sima, “Algorithms for Linear-Quadratic Optimization,” Vol. 200, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1996.

[4] B. Datta, “Numerical Methods for Linear Control Systems,” Elsevier Academic Press, New York, 2004.

[5] A. Lu and E. Wachspress, “Solution of Lyapunov Equations by ADI Iteration,” Computers & Mathematics with Applications, Vol. 21, 1991, pp. 43-58. doi:10.1016/0898-1221(91)90124-M

[6] A. C. Antoulas, “Approximation of Large-Scale Dynamical Systems,” Advances in Design and Control, SIAM, Philadelphia, 2005.

[7] U. Baur and P. Benner, “Cross-Gramian Based Model Reduction for Data-Sparse Systems,” Technical Report, Fakultat fur Mathematik, TU Chemnitz, Chemnitz, 2007.

[8] D. Sorensen and A. Antoulas, “The Sylvester Equation and Approximate Balanced Reduction,” Linear Algebra and Its Applications, Vol. 351-352, 2002, pp. 671-700. doi:10.1016/S0024-3795(02)00283-5

[9] G. H. Golub, S. Nash and C. Van Loan, “A Hessenberg Schur Method for the Problem AX + XB = C,” IEEE Transactions on Automatic Control, Vol. 24, No. 6, 1979, pp. 909-913. doi:10.1109/TAC.1979.1102170

[10] P. Benner, “Factorized Solution of Sylvester Equations with Applications in Control,” In: B. De Moor, B. Motmans, J. Willems, P. Van Dooren and V. Blondel, Eds., Proceedings of the 16th International Symposium on Mathematical Theory of Network and Systems, Leuven, 5-9 July 2004.

[11] W. D. Hoskins, D. S. Meek and D. J. Walton, “The Numerical Solution of the Matrix Equation XA + AY = F,” BIT, Vol. 17, 1977, pp. 184-190.

[12] D. Y. Hu and L. Reichel, “Application of ADI Iterative Methods to the Restoration of Noisy Images,” Linear Algebra and Its Applications, Vol. 172, No. 15, 1992, pp. 283-313. doi:10.1016/0024-3795(92)90031-5

[13] A. El Guennouni, K. Jbilou and A. J. Riquet, “Block Krylov Subspace Methods for Solving Large Sylvester Equations,” Numerical Algorithms, Vol. 29, No. 1, 2002, pp. 75-96. doi:10.1023/A:1014807923223

[14] K. Jbilou, “Low Rank Approximate Solutions to Large Sylvester Matrix Equations,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, pp. 365-376. doi:10.1016/j.amc.2005.11.014

[15] M. Robb and M. Sadkane, “Use of Near-Breakdowns in the Block Arnoldi Method for Solving Large Sylvester Equations,” Applied Numerical Mathematics, Vol. 58, No. 4, 2008, pp. 486-498. doi:10.1016/j.apnum.2007.01.025

[16] G. H. Golub, S. Nash and C. Van Loan, “A Hessenberg Schur Method for the Problem AX + XB = C,” IEEE Transactions on Automatic Control, No. 24, No. 6, 1979, pp. 909-913.

[17] T. Chan and M. Ng, “Galerkin Projection Methods for Solving Multiple Linear Systems,” Department of Mathematics, University of California, Los Angeles, 1996.

[18] S. Karimi, “A New Iterative Solution Method for Solving Multiple Linear Systeams,” Advances in Linear Algebra and Matrix Theory, Vol. 1, 2012, pp. 25-30. doi:10.4236/alamt.2012.23004

[1] R. Bhatia and P. Rosenthal, “How and Why to Solve the Operator Equation AX - XB = Y,” Bulletin London Mathematical Society, Vol. 29, No. 1, 1997, pp. 1-21. doi:10.1112/S0024609396001828

[2] G. H. Golub and C. F. Van Loan, “Matrix Computations,” 3rd Edition, Johns Hopkins University Press, Baltimore, 1996.

[3] V. Sima, “Algorithms for Linear-Quadratic Optimization,” Vol. 200, Pure and Applied Mathematics, Marcel Dekker, Inc., New York, 1996.

[4] B. Datta, “Numerical Methods for Linear Control Systems,” Elsevier Academic Press, New York, 2004.

[5] A. Lu and E. Wachspress, “Solution of Lyapunov Equations by ADI Iteration,” Computers & Mathematics with Applications, Vol. 21, 1991, pp. 43-58. doi:10.1016/0898-1221(91)90124-M

[6] A. C. Antoulas, “Approximation of Large-Scale Dynamical Systems,” Advances in Design and Control, SIAM, Philadelphia, 2005.

[7] U. Baur and P. Benner, “Cross-Gramian Based Model Reduction for Data-Sparse Systems,” Technical Report, Fakultat fur Mathematik, TU Chemnitz, Chemnitz, 2007.

[8] D. Sorensen and A. Antoulas, “The Sylvester Equation and Approximate Balanced Reduction,” Linear Algebra and Its Applications, Vol. 351-352, 2002, pp. 671-700. doi:10.1016/S0024-3795(02)00283-5

[9] G. H. Golub, S. Nash and C. Van Loan, “A Hessenberg Schur Method for the Problem AX + XB = C,” IEEE Transactions on Automatic Control, Vol. 24, No. 6, 1979, pp. 909-913. doi:10.1109/TAC.1979.1102170

[10] P. Benner, “Factorized Solution of Sylvester Equations with Applications in Control,” In: B. De Moor, B. Motmans, J. Willems, P. Van Dooren and V. Blondel, Eds., Proceedings of the 16th International Symposium on Mathematical Theory of Network and Systems, Leuven, 5-9 July 2004.

[11] W. D. Hoskins, D. S. Meek and D. J. Walton, “The Numerical Solution of the Matrix Equation XA + AY = F,” BIT, Vol. 17, 1977, pp. 184-190.

[12] D. Y. Hu and L. Reichel, “Application of ADI Iterative Methods to the Restoration of Noisy Images,” Linear Algebra and Its Applications, Vol. 172, No. 15, 1992, pp. 283-313. doi:10.1016/0024-3795(92)90031-5

[13] A. El Guennouni, K. Jbilou and A. J. Riquet, “Block Krylov Subspace Methods for Solving Large Sylvester Equations,” Numerical Algorithms, Vol. 29, No. 1, 2002, pp. 75-96. doi:10.1023/A:1014807923223

[14] K. Jbilou, “Low Rank Approximate Solutions to Large Sylvester Matrix Equations,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, pp. 365-376. doi:10.1016/j.amc.2005.11.014

[15] M. Robb and M. Sadkane, “Use of Near-Breakdowns in the Block Arnoldi Method for Solving Large Sylvester Equations,” Applied Numerical Mathematics, Vol. 58, No. 4, 2008, pp. 486-498. doi:10.1016/j.apnum.2007.01.025

[16] G. H. Golub, S. Nash and C. Van Loan, “A Hessenberg Schur Method for the Problem AX + XB = C,” IEEE Transactions on Automatic Control, No. 24, No. 6, 1979, pp. 909-913.

[17] T. Chan and M. Ng, “Galerkin Projection Methods for Solving Multiple Linear Systems,” Department of Mathematics, University of California, Los Angeles, 1996.

[18] S. Karimi, “A New Iterative Solution Method for Solving Multiple Linear Systeams,” Advances in Linear Algebra and Matrix Theory, Vol. 1, 2012, pp. 25-30. doi:10.4236/alamt.2012.23004