On the best convergence factors of iterative methods of matrix equations based on the gradient and least squares searches
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- 英文论文题名：On the best convergence factors of iterative methods of matrix equations based on the gradient and least squares searches
- 论文作者：Huamin Zhang ; Hongcai Yin
- 英文论文作者：Huamin Zhang ; Hongcai Yin ; School of Science ; Bengbu University ; School of Management Science and Engineering ; Anhui University of Finance & Economics
- 年：2017
- 作者机构：School of Science, Bengbu University;School of Management Science and Engineering, Anhui University of Finance & Economics;
- 英文论文关键词：Gradient-based iterative algorithm ; Least squares based iterative algorithm ; Eigenvalue ; Matrix equation
- 会议召开时间：2017-07-26
- 会议录名称：第36届中国控制会议论文集（A）
- 英文会议录名称：Proceedings of the 36th Chinese Control Conference（A）
- 语种：英文
- 分类号：O241.6
- 学会代码：KZLL
- 会议名称：第36届中国控制会议
- 会议地点：中国辽宁大连
- 主办单位：中国自动化学会控制理论专业委员会
- 学会名称：中国自动化学会控制理论专业委员会
- 页数：5
- 文件大小：270k
- 原文格式：D
- 会议级别：全国
摘要
In this paper, we prove the convergence and determine the best convergence factor of the gradient-based iterative algorithm for the equation Ax = b by analyzing the eigenvalues of the relevant matrices. A new property of eigenvalues related to the symmetric positive definite matrix is established. By using this property, we obtain a family of iterative algorithms for the linear matrix equation and coupled matrix equations and also the best convergence factor is determined. The analysis shows that the least squares based iterative algorithm is one of the special cases of these iterative algorithms. A numerical example illustrates the effectiveness of the proposed algorithm.
In this paper, we prove the convergence and determine the best convergence factor of the gradient-based iterative algorithm for the equation Ax = b by analyzing the eigenvalues of the relevant matrices. A new property of eigenvalues related to the symmetric positive definite matrix is established. By using this property, we obtain a family of iterative algorithms for the linear matrix equation and coupled matrix equations and also the best convergence factor is determined. The analysis shows that the least squares based iterative algorithm is one of the special cases of these iterative algorithms. A numerical example illustrates the effectiveness of the proposed algorithm.
In this paper, we prove the convergence and determine the best convergence factor of the gradient-based iterative algorithm for the equation Ax = b by analyzing the eigenvalues of the relevant matrices. A new property of eigenvalues related to the symmetric positive definite matrix is established. By using this property, we obtain a family of iterative algorithms for the linear matrix equation and coupled matrix equations and also the best convergence factor is determined. The analysis shows that the least squares based iterative algorithm is one of the special cases of these iterative algorithms. A numerical example illustrates the effectiveness of the proposed algorithm.
引文
[1]H.M.Zhang,Reduced-rank gradient-based algorithms for generalized coupled Sylvester matrix equations and its applications,Computers and Mathematics with Applications,70(8):2049–2062,2015.
[2]H.M.Zhang and F.Ding,Iterative algorithms for X+ATX-1A=I by using the hierarchical identification principle,Journal of the Franklin Institute,353(5):1132–1146,2016.
[3]B.Zhou,J.Lam and G.R.Duan,Convergence of gradientbased iterative solution of the coupled Markovian jump Lyapunov equations,Computers and Mathematics with Applications,56(12):3070–3078,2008.
[4]A.G.Wu,G.Feng,G.R.Duan and W.Q.Liu,Iterative solutions to the Kalman-Yakubovich-conjugate matrix equation,Applied Mathematics and Computation,217(9):4427–4438,2011.
[5]A.G.Wu,Y.M.Fu,G.R.Duan,On solutions of matrix equations V-AV F=BW and V-AˉV F=BW,Mathematical and Computer Modelling,47(11-12):1181–1197,2008.
[6]B.Zhou,J.Lam and G.R.Duan,Toward solution of matrix equation,Linear Algebra and Its Applications,435(6):1370–1398,2011.
[7]B.Zhou and G.R.Duan,An explicit solution to the matrix equation AX-X F=BY,Linear Algebra and its Applications,402:345–366,2005.
[8]B.Zhou,Z.Y.Li,G.R.Duan and Y.Wang,Weighted least squares solutions to general coupled Sylvester matrix equations,Journal of Computational and Applied Mathematics,224(2):759–776,2009.
[9]A.G.Wu,B.Li,Y.Yang and G.R.Duan,Finite iterative solutions to coupled Sylvester-conjugate matrix equations,Applied Mathematical Modelling,35(3):1065–1080,2011.
[10]A.G.Wu,L.L.Lv and M.Z.Hou,Finite iterative algorithms for extended Sylvester-conjugate matrix equations,Mathematical and Computer Modelling,54(9-10):2363–2384,2011.
[11]A.G.Wu,L.L.Lv and G.R.Duan,Iterative algorithms for solving a class of complex conjugate and transpose matrix equations,Applied Mathematics and Computation,217(21):8343–8353,2011.
[12]F.Ding,Complexity,convergence and computational efficiency for system identification algorithms,Control and Decision,31(10):1729–1741,2016.
[13]F.Ding and F.F.Wang,Recursive least squares identification algorithms for linear-in-parameter systems with missing data,Control and Decision,31(12):2261–2266,2016.
[14]F.Ding and T.Chen,Gradient based iterative algorithms for solving a class of matrix equations,IEEE Transactions on Automatic Control,50(8):1216–1221,2005.
[15]F.Ding and T.Chen,Iterative least squares solutions of coupled Sylvester matrix equations,Systems&Control Letters,54(2):95–107,2005.
[16]F.Ding and T.Chen,On iterative solutions of general coupled matrix equations,SIAM Journal on Control and Optimization,44(6):2269–2284,2006.
[17]L.Xie,Y.J.Liu and H.Z.Yang,Gradient based and least squares based iterative algorithms for matrix equations AX B+CXTD=F,Applied Mathematics and Computation,217(5):2191–2199,2010.
[18]L.Xie,J.Ding and F.Ding,Gradient based iterative solutions for general linear matrix equations,Computers&Mathematics with Applications,58(7):1441–1448,2009.
[19]J.Ding,Y.J.Liu and F.Ding,Iterative solutions to matrix equations of form Ai X Bi=Fi,Computers&Mathematics with Applications,59(11):3500–3507,2010.
[20]F.Ding and H.M.Zhang,Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems,IET Control Theory and Applications,8(15):1588–1595,2014.
[21]F.Ding,X.P.Liu and J.Ding,Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,Applied Mathematics and Computation,197(1):41–50,2008.
[22]H.M.Zhang and F.Ding,On the Kronecker products and their applications,Journal of Applied Mathematics,2013,Article ID 296185,1-8,2013.
[23]X.D.Zhang,Matrix Analysis and Applications,Beijing:Tsinghua University Press,2004:58–59.
[24]M.H.Lin,Remarks on two recent results of Audenaert,Linear Algebra and its Applcations,489(1):24–29,2016.
[25]H.M.Zhang and F.Ding,A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations,Journal of the Franklin Institute,351(1):340–357,2014.
[2]H.M.Zhang and F.Ding,Iterative algorithms for X+ATX-1A=I by using the hierarchical identification principle,Journal of the Franklin Institute,353(5):1132–1146,2016.
[3]B.Zhou,J.Lam and G.R.Duan,Convergence of gradientbased iterative solution of the coupled Markovian jump Lyapunov equations,Computers and Mathematics with Applications,56(12):3070–3078,2008.
[4]A.G.Wu,G.Feng,G.R.Duan and W.Q.Liu,Iterative solutions to the Kalman-Yakubovich-conjugate matrix equation,Applied Mathematics and Computation,217(9):4427–4438,2011.
[5]A.G.Wu,Y.M.Fu,G.R.Duan,On solutions of matrix equations V-AV F=BW and V-AˉV F=BW,Mathematical and Computer Modelling,47(11-12):1181–1197,2008.
[6]B.Zhou,J.Lam and G.R.Duan,Toward solution of matrix equation,Linear Algebra and Its Applications,435(6):1370–1398,2011.
[7]B.Zhou and G.R.Duan,An explicit solution to the matrix equation AX-X F=BY,Linear Algebra and its Applications,402:345–366,2005.
[8]B.Zhou,Z.Y.Li,G.R.Duan and Y.Wang,Weighted least squares solutions to general coupled Sylvester matrix equations,Journal of Computational and Applied Mathematics,224(2):759–776,2009.
[9]A.G.Wu,B.Li,Y.Yang and G.R.Duan,Finite iterative solutions to coupled Sylvester-conjugate matrix equations,Applied Mathematical Modelling,35(3):1065–1080,2011.
[10]A.G.Wu,L.L.Lv and M.Z.Hou,Finite iterative algorithms for extended Sylvester-conjugate matrix equations,Mathematical and Computer Modelling,54(9-10):2363–2384,2011.
[11]A.G.Wu,L.L.Lv and G.R.Duan,Iterative algorithms for solving a class of complex conjugate and transpose matrix equations,Applied Mathematics and Computation,217(21):8343–8353,2011.
[12]F.Ding,Complexity,convergence and computational efficiency for system identification algorithms,Control and Decision,31(10):1729–1741,2016.
[13]F.Ding and F.F.Wang,Recursive least squares identification algorithms for linear-in-parameter systems with missing data,Control and Decision,31(12):2261–2266,2016.
[14]F.Ding and T.Chen,Gradient based iterative algorithms for solving a class of matrix equations,IEEE Transactions on Automatic Control,50(8):1216–1221,2005.
[15]F.Ding and T.Chen,Iterative least squares solutions of coupled Sylvester matrix equations,Systems&Control Letters,54(2):95–107,2005.
[16]F.Ding and T.Chen,On iterative solutions of general coupled matrix equations,SIAM Journal on Control and Optimization,44(6):2269–2284,2006.
[17]L.Xie,Y.J.Liu and H.Z.Yang,Gradient based and least squares based iterative algorithms for matrix equations AX B+CXTD=F,Applied Mathematics and Computation,217(5):2191–2199,2010.
[18]L.Xie,J.Ding and F.Ding,Gradient based iterative solutions for general linear matrix equations,Computers&Mathematics with Applications,58(7):1441–1448,2009.
[19]J.Ding,Y.J.Liu and F.Ding,Iterative solutions to matrix equations of form Ai X Bi=Fi,Computers&Mathematics with Applications,59(11):3500–3507,2010.
[20]F.Ding and H.M.Zhang,Gradient-based iterative algorithm for a class of the coupled matrix equations related to control systems,IET Control Theory and Applications,8(15):1588–1595,2014.
[21]F.Ding,X.P.Liu and J.Ding,Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,Applied Mathematics and Computation,197(1):41–50,2008.
[22]H.M.Zhang and F.Ding,On the Kronecker products and their applications,Journal of Applied Mathematics,2013,Article ID 296185,1-8,2013.
[23]X.D.Zhang,Matrix Analysis and Applications,Beijing:Tsinghua University Press,2004:58–59.
[24]M.H.Lin,Remarks on two recent results of Audenaert,Linear Algebra and its Applcations,489(1):24–29,2016.
[25]H.M.Zhang and F.Ding,A property of the eigenvalues of the symmetric positive definite matrix and the iterative algorithm for coupled Sylvester matrix equations,Journal of the Franklin Institute,351(1):340–357,2014.