结构线性方程组Aχ=b和Sylvester矩阵方程的迭代解法
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摘要
本论文主要分为两部分:一部分是考虑了系数矩阵为中心对称矩阵的线性方程组Ax=b的迭代求解;另一部分是研究了控制理论中的Lyapunov矩阵方程和Sylvester矩阵方程的数值求解。
在线性方程组Ax=b的迭代求解中,我们主要考虑来求解一类具有特殊结构的线性方程组,即系数矩阵A是一个中心对称矩阵的情况。在本论文中我们主要针对中心对称矩阵的结构特点构造了几种中心对称分裂格式。与Jacobi迭代方法和Gauss-Seidel迭代方法相比,由这些中心对称分裂格式得到的迭代方法不但收敛速度快而且还有计算和存储上的优势。我们在这里重点考虑中心对称M-阵和中心对称H-阵方程组的迭代求解,对于其它的中心对称矩阵线性方程组的迭代求解还在进一步地研究中。
在控制系统的分析和设计中,矩阵方程和线性矩阵不等式的求解占有十分重要的地位,受到控制学界和数学界的极大关注。在这里我们主要给出了求解矩阵方程的两种迭代解法:
1)利用Kronecker积迭代方法来求解离散的Lyapunov矩阵AXA~T-X+Q=0和连续的Lyapunov矩阵方程AX+XA~T+Q=0.
2)利用基于矩阵分裂的梯度迭代法来求解Sylvester矩阵方程AX+XB=C和AXB+X=C.
The content of this paper consists of two parts:part one is how to solve the linear systems Ax=b iteratively,which coefficient matrices are centrosymmetric matrices; part two pays attention to solving the Lyapunov matrix equations and Sylvester matrix equations in control theory by numcrical methods.
For the iterative solutions of the linear systems Ax=b,wc consider to solve a special linear system which coefficient matrix is a centrosymmetric matrix.In this paper we construct several centrosymmetric splittings according to the special construction of the centrosymmetric matrices.Compared to the Jacobi and Gauss-Seidel methods,the iterative methods based on these centrosymmetric splittings have faster convergence and less cost of computation and store.Here we mainly investigate the iterative methods for the linear systems with a centrosymmetric M-matrix and a centrosymmctric H-matrix, the other cases are also been studied.
In the analyse and design of control systems,the solutions of matrix equations and matrix inequations play an important role,and have caused many attentions in control and mathematical fields.In this paper we give two iterative methods for solving the matrix equations:
1) solving the discrete Lyapunov equation matrix AXA~T-X+Q=0 and continuous Lyapunov matrix equation AX+XA~T+Q=0 by using Kronecker Products iterative method;
2) solving the Sylvester matrix equations AX+XB=C and AXB+X=C by using the gradient iterative methods based on the matrix splittings.
在线性方程组Ax=b的迭代求解中,我们主要考虑来求解一类具有特殊结构的线性方程组,即系数矩阵A是一个中心对称矩阵的情况。在本论文中我们主要针对中心对称矩阵的结构特点构造了几种中心对称分裂格式。与Jacobi迭代方法和Gauss-Seidel迭代方法相比,由这些中心对称分裂格式得到的迭代方法不但收敛速度快而且还有计算和存储上的优势。我们在这里重点考虑中心对称M-阵和中心对称H-阵方程组的迭代求解,对于其它的中心对称矩阵线性方程组的迭代求解还在进一步地研究中。
在控制系统的分析和设计中,矩阵方程和线性矩阵不等式的求解占有十分重要的地位,受到控制学界和数学界的极大关注。在这里我们主要给出了求解矩阵方程的两种迭代解法:
1)利用Kronecker积迭代方法来求解离散的Lyapunov矩阵AXA~T-X+Q=0和连续的Lyapunov矩阵方程AX+XA~T+Q=0.
2)利用基于矩阵分裂的梯度迭代法来求解Sylvester矩阵方程AX+XB=C和AXB+X=C.
The content of this paper consists of two parts:part one is how to solve the linear systems Ax=b iteratively,which coefficient matrices are centrosymmetric matrices; part two pays attention to solving the Lyapunov matrix equations and Sylvester matrix equations in control theory by numcrical methods.
For the iterative solutions of the linear systems Ax=b,wc consider to solve a special linear system which coefficient matrix is a centrosymmetric matrix.In this paper we construct several centrosymmetric splittings according to the special construction of the centrosymmetric matrices.Compared to the Jacobi and Gauss-Seidel methods,the iterative methods based on these centrosymmetric splittings have faster convergence and less cost of computation and store.Here we mainly investigate the iterative methods for the linear systems with a centrosymmetric M-matrix and a centrosymmctric H-matrix, the other cases are also been studied.
In the analyse and design of control systems,the solutions of matrix equations and matrix inequations play an important role,and have caused many attentions in control and mathematical fields.In this paper we give two iterative methods for solving the matrix equations:
1) solving the discrete Lyapunov equation matrix AXA~T-X+Q=0 and continuous Lyapunov matrix equation AX+XA~T+Q=0 by using Kronecker Products iterative method;
2) solving the Sylvester matrix equations AX+XB=C and AXB+X=C by using the gradient iterative methods based on the matrix splittings.
引文
[1]胡家赣,线性代数方程组的迭代解法,北京:科学出版社,1991.
[2]蔡大用,白峰杉,高等数值分析,北京:清华大学出版社,1997.
[3]蔡大用,数值代数,北京:清华大学出版社,1987.
[4]陆金甫,关治译,数值线性代数,北京:人民邮电出版社,2006.
[5]张凯院,徐仲,数值代数,北京:科学出版社,2006.
[6]丁锋,萧德云,多变量系统状态空间模型的递阶辨识.控制与决策,20(2005)848-859.
[7]熊全淹,叶明训,线性代数(第三版),北京:高等教育出版社,1987.
[8]张凯院,Lyapunov型矩阵方程的迭代.校正解法.纯粹数学与应用数学,12(1996)104-108.
[9]Feng Ding,Ming Li,Jiyang Dai,Hierarchical Identification Principle and a Fancily of Iterative Methods,Proceedings of the 25th Chinese Control Conference(2006) 418-422.
[10]R.Chan,M.Ng,Toeplitz Preconditioners for Hermite Toeplitz systems,Linear Algebra Appls.190(1993) 181-208.
[11]Z.-Z.Bai,G.H.Golub,L.-Z.Lu and J.-F.Yin,Block triangular and skew-Hermitian splitting methods for positive definite linear systems,SIAM J.Sci.Comput.,26(2005),p.844-863.
[12]N.Levinson,The Wiener RMS(Root Mean Square) Error Criterion in Filter Design and Prediction,J.Math.Phys.25(1946) 261-278.
[13]Z.-Z.Bai,G.H.Golub,M.K.Ng,Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,SIAM J.Matrix Anal.Appl.,24(2003),p.603-626.
[14]A.Berman,R.J.Plemmons,Nonnegative Matrices in the Mathematical Sciences,Academic Press,New York,1979,reprinted by SIAM,Philadelphia,PA,1994.
[15]C.zhihao,L.zhongyun,Convergence of relaxed parallel multisp litting methods with different weighting schemes,Apply Math.Computation,106(1999) 181-196.
[16]G.Csordas,R.S.Varga,Comparison of regular splittings of matrices,Numer.Math,44(1984)23-35.
[17]G.Baxter,An Asymptotic Result for the Finite Predictor,Math.Scand,10(1962) 137 +144.
[18]L.Elsner,Comparison of weak regular splittings and multisplitting methods,Numer.Math,56(1989)283-289.
[19]L.Elsner,Comparison of weak regular splitting and multisplitting methods,Numer.Math,56(1989) 283-289.
[20]C.Zhihao,L.Zhongyun,Symmetric multisplitting of a symmetric positive defiuite matrix,Linear Algebra Appl.285(1998)309-319.
[21]T.Chen,B.A.Francis,Optimal Sampled-data Control Systems,Springer,London,1995.
[22]G.H.Golub,S.Nash,C.Vanloan,A Hessenberg-Schur Method for the Problem AX+XB=C,IEEE Trans.Automat.Contr.,21(1976) 799-780.
[23] J.Lasalle, S.Lefschetz, Stability of Lyapunov Direct Methods, Academic Press. New York, 1961.
[24] A.Melman, Symmetric centrosymmetric matrix-vector multiplication, Linear Algebra and its Application. 320 (2000) 193-198.
[25] L.Zhongyun, Some properties of centrosymmetric matrices, Applied Mathematics and Computation. 141 (2003) 297-306.
[26] H.Fassbender, K.D.Ikramov, Computing matrix-vector products with centrosymmetric and centrohermitian matrices, Linear Algebra and its Application. 364 (2003) 235-241.
[27] J.W.Demmel, Applied Numerical Linear Algebra, Philadephia: Society for Industrial and Applied Mathematics, 1997.
[28] T. Eirola, O. Nevanlinna, Accelerating with rank-one updates, Linear Algebra Appl. 121 (1989) 511-520.
[29] C. Vuik, H.A. van der Vorst, A comparison of some GMRES-like methods, Linear Algebra Appl. 160 (1992) 131-162.
[30] Zhongyun Liu, HanDong Cao, Huijiong Chen, A note on computing matrix-vector products with generalized centrosymmetric (centrohermitian) matrices, Applied Mathematics and Computation. 169 (2005) 1332-1345.
[31] Zhongyun Liu, Zhaolu Tian and Yanxiang Tan , Computing the least-square solutions for centrohermitian matrix problems , Applied Mathematics and Computation. 174 (2006) 566-577.
[32] A.S. Householder, Theory of Matrices in Numerical Analysis, Blaisdell Publishing Company, Johnson, CO, 1964.
[33] C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Standards 45 (1950) 255-282.
[34] Y. Saad, A exible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Statist. Comput. 14 (1993) 461-469.
[35] M. Engeli, T. Ginsburg, H. Rutishauser, E. Stiefel, Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-Adjoint Boundary Value Problems, Birkhauser, Basel, 1959.
[36] P.N. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM J. Sci. Statist. Comput. 12 (1991) 58-78.
[37] Zhong-Yun Liu , Hui-Jiong Chen , Han-Dong Cao , The computation of the principal square roots of centrosymmetric H-matrices, Applied Mathematics and Computation. 175 (2006) 319-329.
[38] Zhongyun Liu, Heike Faβender, An inverse eigenvalue problem and an associated approximation problem for generalized K—centrohermitian matrices, Journal of Computational and Applied Mathematics. 206 (2007) 578-585.
[39] Zhongyun Liu and Heike Faβender, Some properties of generalized K-centrosymmetric H- matrices,Journal of Computational and Applied Mathematics.Available online 3 April 2007.
[40]M.R.Hestenes,E.L.Stiefel,Methods of conjugate gradients for solving linear systems,J.Res.Nat.Bur.Standards section B 49(1952) 409-436.
[41]J.Cullum,A.Greenbaum,Relations between Galerkin and norm-minimizing iterative meth-ods for solving linear systems,SIAM J.Matrix Anal.Appl.17(1996) 223-247.
[42]Zhongyun Liu,Yulin Zhang and Rui Ralha,Computing the square roots of matrices with central symmetry,Applied Mathematics and Computation.186(2007) 715-726.
[43]J.W.Daniel,The conjugate gradient method for linear and nonlinear operator equations,SIAM J.Numer.Anal.4(1967) 10-26.
[44]A.Greenbaum,V.Ptak,Z.Strakos,Any nonincreasing convergence curve is possible for GMRES,SIAM J.Matrix Anal.Appl.17(1996) 465-469.
[45]J.K.Reid,On the method of conjugate gradients for the solution of large sparse systems of linear equations,in:J.K.Reid(Ed.),Large Sparse Sets of Linear Equations,Academic Press,New York,1971,pp.231-254.
[46]P.Van Dooren,Gramian based model reduction of large-scale dynamical systems,Numerical Analysis,Chapman and Hall,CRC Press,London,(2000) 231-247.
[47]A.C.Antoulas,D.C.Sorensen,Projection methods for balanced model reduction.Technical Report,Rice University,Houston,X,2001.
[48]G.Starke,W.Niethammer,SOR for AX-XB=C,Linear Algebra Appl.,154(1991) 355-375.
[49]Y.Fang,K.A.Loparo,X.Feng,New Estimates for soutions of Lyapunov equations,IEEE Trans.Automat.Control,42(1997) 408-411.
[50]Y.Saad,Iterative Methods for Sparse Linear Systems,PWS Publishing,New York,1996.
[51]Zhaolu Tian,Chuanqing Gu,The iterative methods for centrosymmetric matrices,Appl.Math.Comput.,187(2007) 902-911.
[52]T.Berger,Rate Distortion Theory:A Mathematical Basis for Data Compression,Prentice Hall,Englewood Cliffs,New Jersey,1971.
[53]R.M.Gray,Information Rates of Autoregressive Processes,IEEE Transactions on Information Theory,4(1970) 412-421.
[54]R.M.Gray,On the asymptotic eigenvalue distribution of Toeplitz matrices,IEEE Transac-tions on Information Theory,18(1972) 725-730.
[55]W.E.Arnoldi,The principle of minimized iteration in the solution of the matrix eigenvalue problem,Quart.Appl.Math,9(1951) 17-29.
[56]Z.-Z.Bai,G.H.Golub,M.K.Ng,Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,SIAM J.Matrix Anal.Appl.,24(2003) 603-626.
[57]H.Mukaidani,H.Xu,K.Mizukami,New iterative algorithm for algebra Riccati equation related to H_∞ control of sigularly perturbed systems,IEEE Trans.Automat.Control,46(2001) 1659-1666.
[58] R.H.Bartels, G.W.Stewart, A solution of the equation AX+XB=C, Commun, ACM. 15(1972) 820-826.
[59] B.C.Moore, Principal component analysis in linear systems: controllability, obervability and model reduction, IEEE Trans. Auto. Control AC 26(1981) 17-31.
[60] Y. Saad, M.H. Schultz, GMRES: a generalized minimal residual algorithm for solving a non-symmetric linear systems, SIAM J. Sci. Statist. Comput, 7 (1986) 856-869.
[61] H.X.Chen, Convergent and divergent relationship between MPSD iterative method and Jacobi method, Comm.Appl. Math.Comput.14 (1) (2000) 1-8.
[62] R. Chan, M. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev. 38(1996) 427-482.
[63] W. Trench, An Algorithm for the Inversion of Finite Toeplitz Matrices, SIAM J.Appl. Math. 12 (1964) 515-522.
[64] F.Ding, T.Chen, Gradient Based Iterative Algorithms for Solving a class of Matrix Equations, IEEE Trans. Autom. Control, 50 (2005) 1216-1221.
[65] T.K.Ku, C. J.Kuo, Preconditioned iterative methods for solving Toeplitz-plus-Hankel systems, Tech. Rep, 181, USC, Signal and Image Processing Institute, June 1991.
[66] F.Ding, T.Chen, Iterative least squares solutions of coupled Sylvester matrix equations, Syst.Control Lett., 54 (2005) 95-107.
[67] Ding F, Chen T. Hierarchical gradient-based identification of multivariable discrete-time systems. Automatica,41 (2005) 315-325.
[68] Ding F, Chen T. Hierarchical least squares identification methods for multivariable systems. IEEE Transactions on Automatic Control, 50 (2005)397-402.
[69] F.Weiwei, G.Chuanqing, T.Zhaolu, Jacobi-Gradient Iterative Algorithms for Sylvester Matrix Equations, 14th Conference of the International Linear Algebra Society (2007) 11-14.
[70] A.E. Yagle, New analogs of split algorithms for arbitrary Toeplitz-plus-Hankel matrices, IEEE Trans. Signal Processing. 39 (1991) 2457-2463.
[71] Ding F, Chen T. Hierarchical identification of lifted statespace models for general dual-rate systems. IEEE Transactions on Circuits and Systems-Ⅰ: Regular Papers, 52(2005) 1179-1187.
[72] G.W.Stewart, J.G.Sun, Matrix perturbation Theory, Academic Press, 1990.
[2]蔡大用,白峰杉,高等数值分析,北京:清华大学出版社,1997.
[3]蔡大用,数值代数,北京:清华大学出版社,1987.
[4]陆金甫,关治译,数值线性代数,北京:人民邮电出版社,2006.
[5]张凯院,徐仲,数值代数,北京:科学出版社,2006.
[6]丁锋,萧德云,多变量系统状态空间模型的递阶辨识.控制与决策,20(2005)848-859.
[7]熊全淹,叶明训,线性代数(第三版),北京:高等教育出版社,1987.
[8]张凯院,Lyapunov型矩阵方程的迭代.校正解法.纯粹数学与应用数学,12(1996)104-108.
[9]Feng Ding,Ming Li,Jiyang Dai,Hierarchical Identification Principle and a Fancily of Iterative Methods,Proceedings of the 25th Chinese Control Conference(2006) 418-422.
[10]R.Chan,M.Ng,Toeplitz Preconditioners for Hermite Toeplitz systems,Linear Algebra Appls.190(1993) 181-208.
[11]Z.-Z.Bai,G.H.Golub,L.-Z.Lu and J.-F.Yin,Block triangular and skew-Hermitian splitting methods for positive definite linear systems,SIAM J.Sci.Comput.,26(2005),p.844-863.
[12]N.Levinson,The Wiener RMS(Root Mean Square) Error Criterion in Filter Design and Prediction,J.Math.Phys.25(1946) 261-278.
[13]Z.-Z.Bai,G.H.Golub,M.K.Ng,Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,SIAM J.Matrix Anal.Appl.,24(2003),p.603-626.
[14]A.Berman,R.J.Plemmons,Nonnegative Matrices in the Mathematical Sciences,Academic Press,New York,1979,reprinted by SIAM,Philadelphia,PA,1994.
[15]C.zhihao,L.zhongyun,Convergence of relaxed parallel multisp litting methods with different weighting schemes,Apply Math.Computation,106(1999) 181-196.
[16]G.Csordas,R.S.Varga,Comparison of regular splittings of matrices,Numer.Math,44(1984)23-35.
[17]G.Baxter,An Asymptotic Result for the Finite Predictor,Math.Scand,10(1962) 137 +144.
[18]L.Elsner,Comparison of weak regular splittings and multisplitting methods,Numer.Math,56(1989)283-289.
[19]L.Elsner,Comparison of weak regular splitting and multisplitting methods,Numer.Math,56(1989) 283-289.
[20]C.Zhihao,L.Zhongyun,Symmetric multisplitting of a symmetric positive defiuite matrix,Linear Algebra Appl.285(1998)309-319.
[21]T.Chen,B.A.Francis,Optimal Sampled-data Control Systems,Springer,London,1995.
[22]G.H.Golub,S.Nash,C.Vanloan,A Hessenberg-Schur Method for the Problem AX+XB=C,IEEE Trans.Automat.Contr.,21(1976) 799-780.
[23] J.Lasalle, S.Lefschetz, Stability of Lyapunov Direct Methods, Academic Press. New York, 1961.
[24] A.Melman, Symmetric centrosymmetric matrix-vector multiplication, Linear Algebra and its Application. 320 (2000) 193-198.
[25] L.Zhongyun, Some properties of centrosymmetric matrices, Applied Mathematics and Computation. 141 (2003) 297-306.
[26] H.Fassbender, K.D.Ikramov, Computing matrix-vector products with centrosymmetric and centrohermitian matrices, Linear Algebra and its Application. 364 (2003) 235-241.
[27] J.W.Demmel, Applied Numerical Linear Algebra, Philadephia: Society for Industrial and Applied Mathematics, 1997.
[28] T. Eirola, O. Nevanlinna, Accelerating with rank-one updates, Linear Algebra Appl. 121 (1989) 511-520.
[29] C. Vuik, H.A. van der Vorst, A comparison of some GMRES-like methods, Linear Algebra Appl. 160 (1992) 131-162.
[30] Zhongyun Liu, HanDong Cao, Huijiong Chen, A note on computing matrix-vector products with generalized centrosymmetric (centrohermitian) matrices, Applied Mathematics and Computation. 169 (2005) 1332-1345.
[31] Zhongyun Liu, Zhaolu Tian and Yanxiang Tan , Computing the least-square solutions for centrohermitian matrix problems , Applied Mathematics and Computation. 174 (2006) 566-577.
[32] A.S. Householder, Theory of Matrices in Numerical Analysis, Blaisdell Publishing Company, Johnson, CO, 1964.
[33] C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Res. Nat. Bur. Standards 45 (1950) 255-282.
[34] Y. Saad, A exible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Statist. Comput. 14 (1993) 461-469.
[35] M. Engeli, T. Ginsburg, H. Rutishauser, E. Stiefel, Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-Adjoint Boundary Value Problems, Birkhauser, Basel, 1959.
[36] P.N. Brown, A theoretical comparison of the Arnoldi and GMRES algorithms, SIAM J. Sci. Statist. Comput. 12 (1991) 58-78.
[37] Zhong-Yun Liu , Hui-Jiong Chen , Han-Dong Cao , The computation of the principal square roots of centrosymmetric H-matrices, Applied Mathematics and Computation. 175 (2006) 319-329.
[38] Zhongyun Liu, Heike Faβender, An inverse eigenvalue problem and an associated approximation problem for generalized K—centrohermitian matrices, Journal of Computational and Applied Mathematics. 206 (2007) 578-585.
[39] Zhongyun Liu and Heike Faβender, Some properties of generalized K-centrosymmetric H- matrices,Journal of Computational and Applied Mathematics.Available online 3 April 2007.
[40]M.R.Hestenes,E.L.Stiefel,Methods of conjugate gradients for solving linear systems,J.Res.Nat.Bur.Standards section B 49(1952) 409-436.
[41]J.Cullum,A.Greenbaum,Relations between Galerkin and norm-minimizing iterative meth-ods for solving linear systems,SIAM J.Matrix Anal.Appl.17(1996) 223-247.
[42]Zhongyun Liu,Yulin Zhang and Rui Ralha,Computing the square roots of matrices with central symmetry,Applied Mathematics and Computation.186(2007) 715-726.
[43]J.W.Daniel,The conjugate gradient method for linear and nonlinear operator equations,SIAM J.Numer.Anal.4(1967) 10-26.
[44]A.Greenbaum,V.Ptak,Z.Strakos,Any nonincreasing convergence curve is possible for GMRES,SIAM J.Matrix Anal.Appl.17(1996) 465-469.
[45]J.K.Reid,On the method of conjugate gradients for the solution of large sparse systems of linear equations,in:J.K.Reid(Ed.),Large Sparse Sets of Linear Equations,Academic Press,New York,1971,pp.231-254.
[46]P.Van Dooren,Gramian based model reduction of large-scale dynamical systems,Numerical Analysis,Chapman and Hall,CRC Press,London,(2000) 231-247.
[47]A.C.Antoulas,D.C.Sorensen,Projection methods for balanced model reduction.Technical Report,Rice University,Houston,X,2001.
[48]G.Starke,W.Niethammer,SOR for AX-XB=C,Linear Algebra Appl.,154(1991) 355-375.
[49]Y.Fang,K.A.Loparo,X.Feng,New Estimates for soutions of Lyapunov equations,IEEE Trans.Automat.Control,42(1997) 408-411.
[50]Y.Saad,Iterative Methods for Sparse Linear Systems,PWS Publishing,New York,1996.
[51]Zhaolu Tian,Chuanqing Gu,The iterative methods for centrosymmetric matrices,Appl.Math.Comput.,187(2007) 902-911.
[52]T.Berger,Rate Distortion Theory:A Mathematical Basis for Data Compression,Prentice Hall,Englewood Cliffs,New Jersey,1971.
[53]R.M.Gray,Information Rates of Autoregressive Processes,IEEE Transactions on Information Theory,4(1970) 412-421.
[54]R.M.Gray,On the asymptotic eigenvalue distribution of Toeplitz matrices,IEEE Transac-tions on Information Theory,18(1972) 725-730.
[55]W.E.Arnoldi,The principle of minimized iteration in the solution of the matrix eigenvalue problem,Quart.Appl.Math,9(1951) 17-29.
[56]Z.-Z.Bai,G.H.Golub,M.K.Ng,Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems,SIAM J.Matrix Anal.Appl.,24(2003) 603-626.
[57]H.Mukaidani,H.Xu,K.Mizukami,New iterative algorithm for algebra Riccati equation related to H_∞ control of sigularly perturbed systems,IEEE Trans.Automat.Control,46(2001) 1659-1666.
[58] R.H.Bartels, G.W.Stewart, A solution of the equation AX+XB=C, Commun, ACM. 15(1972) 820-826.
[59] B.C.Moore, Principal component analysis in linear systems: controllability, obervability and model reduction, IEEE Trans. Auto. Control AC 26(1981) 17-31.
[60] Y. Saad, M.H. Schultz, GMRES: a generalized minimal residual algorithm for solving a non-symmetric linear systems, SIAM J. Sci. Statist. Comput, 7 (1986) 856-869.
[61] H.X.Chen, Convergent and divergent relationship between MPSD iterative method and Jacobi method, Comm.Appl. Math.Comput.14 (1) (2000) 1-8.
[62] R. Chan, M. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev. 38(1996) 427-482.
[63] W. Trench, An Algorithm for the Inversion of Finite Toeplitz Matrices, SIAM J.Appl. Math. 12 (1964) 515-522.
[64] F.Ding, T.Chen, Gradient Based Iterative Algorithms for Solving a class of Matrix Equations, IEEE Trans. Autom. Control, 50 (2005) 1216-1221.
[65] T.K.Ku, C. J.Kuo, Preconditioned iterative methods for solving Toeplitz-plus-Hankel systems, Tech. Rep, 181, USC, Signal and Image Processing Institute, June 1991.
[66] F.Ding, T.Chen, Iterative least squares solutions of coupled Sylvester matrix equations, Syst.Control Lett., 54 (2005) 95-107.
[67] Ding F, Chen T. Hierarchical gradient-based identification of multivariable discrete-time systems. Automatica,41 (2005) 315-325.
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