一类非线性矩阵方程的正定解
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摘要
讨论非线性矩阵方程X+∑i=1mA*iX-1Ai-∑j=1nB*jX-1Bj=Q的Hermite正定解及其扰动问题。提出了该方程存在唯一正定解的充分条件,给出了迭代解法。讨论了唯一正定解的扰动问题,给出了上界估计,得到了唯一正定解的Rice条件数的显式表达式,并用数值例子对所得结果进行了验证。
The positive definite solutions of a class of nonlinear matrix equation X+∑i=1mA*iX-1Ai-∑j=1nB*jX-1Bj=Q are addressed.Some sufficient conditions for the existence and uniqueness of the positive definite solution to such equations are established.An iterative method for the unique positive definite solution is provided.Perturbation analysis is also conducted.An estimation bound and the explicit expression of Rice condition number of the unique positive definite solution are derived.Several numerical examples are given to illustrate the effectiveness of the above theoretical results.
The positive definite solutions of a class of nonlinear matrix equation X+∑i=1mA*iX-1Ai-∑j=1nB*jX-1Bj=Q are addressed.Some sufficient conditions for the existence and uniqueness of the positive definite solution to such equations are established.An iterative method for the unique positive definite solution is provided.Perturbation analysis is also conducted.An estimation bound and the explicit expression of Rice condition number of the unique positive definite solution are derived.Several numerical examples are given to illustrate the effectiveness of the above theoretical results.
引文
[1]YONG J,ZHOU Z.Stochastic Controls,Hamiltonian Systems and HJB Equations[M].New York:Springer-Verlag,1999.
[2]SAKHNOVICH L A.Interpolation theory and its applications[C]//Mathematicsandits and Applications.Dordrecht:Springer,1997.
[3]BHATIA R.Matrix Analysis,Graduate Texts in Mathematics[M].India:Springer,1997.
[4]BLUZBEE B L,GOLUB B G,NILSON C W.On direct methods for solving Poisson's equations[J].SIAM J Numer Anal,1970,7(4):627-656.
[5]RAN A C M,REURINGS M C B.A nonlinear matrix equation connected to interpolation theory[J].Linear Algebra Appl,2004,379:289-304.DOI:10.1016/S0024-3795(03)00541-X
[6]MEINI B.Efficient computation of the extreme solutions of X+A*X-1A=Q and X-A*X-1A=Q[J].Math Comput,2002,71:1189-1204.
[7]GUO C,KUO Y,LIN W.Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research[J].J Comput Appl Math,2012,236(17):4166-4180.DOI:10.1016/j.cam.2012.05.012
[8]SUN J,XU S.Perturbation analysis of the maximal solution of the matrix equation X+A*X-1A=P(ΙΙ)[J].Linear Algebra Appl,2003,362:211-228.DOI:10.1016/S0024-3795(01)00300-7
[9]HASANOV V I,IVANOV I G.On two perturbation estimates of the extreme solutions to the equations X+A*X-1A=Q[J].Linear Algebra Appl,2006,413:81-92.DOI:10.1016/j.laa.2005.08.013
[10]HE Y,LONG J.On the Hermitian positive definite solution of the nonlinear matrix equation X+∑i=1m Ai*X-1Ai=I[J].Applied Mathematics and Computation,2010,216:3480-3485.
[11]DUAN X,LI C,LIAO A.Solutions and perturbation analysis for the nonlinear matrix equation X+∑i=1m Ai*X-1Ai=I[J].Applied Mathematics and Computation,2011,218:4458-4466.
[12]DUAN X,WANG Q,LIAO A.On the matrix equation X-∑i=1m Ni*X-1Ni=I arising in an interpolation problem[J].Linear and Multilinear Algebra,2013,61(9):1192-1205.
[13]YIN X,FANG L.Perturbation analysis for the positive definite solution of the nonlinear matrix equation X-∑i=1m Ai*X-1Ai=Q[J].Journal of Applied Mathematics and Computing,2013,41:199-211.
[14]BERZIG M.Solving a class of matrix equations via the Bhaskar-Lakshmikantham coupled fixed point theorem[J].Applied Mathematics Letters,2012,25:1638-1643.DOI:10.1016/j.aml.2012.01.028
[15]DUAN X,WANG Q,LI C.Positive definite solution of a class of nonlinear matrix equation[J].Linear and Multilinear Algebra,2014,62(6):839-852.DOI:10.1080/03081087.2013.794230
[16]ZHANG G L.A note on positive definite solution of matrix equation X-M*X-1M+N*X-1N=I[J].Linear and Multilinear Algebra,2016,64(5):951-954.DOI:10.1080/03081087.2015.1068267
[17]BERZIG M,DUAN X,SAMET B.Positive definite solution of the matrix equation X-A*X-1A+B*X-1B=Q via Bhaskar-Lakshmikantham fixed point theorem[J].Mathematical Sciences,2012(6):27-32.
[18]BERINDE V.Generalized coupled fixed point theorems for mixed point monotone mappings in partially ordered metric spaces[J].Nonlinear Analysis,2011,74:7347-7355.
[19]RICE J R.A theory of condition[J].SIAM J Numer Anal,1966(3):287-310.DOI:10.1137/0703023
[2]SAKHNOVICH L A.Interpolation theory and its applications[C]//Mathematicsandits and Applications.Dordrecht:Springer,1997.
[3]BHATIA R.Matrix Analysis,Graduate Texts in Mathematics[M].India:Springer,1997.
[4]BLUZBEE B L,GOLUB B G,NILSON C W.On direct methods for solving Poisson's equations[J].SIAM J Numer Anal,1970,7(4):627-656.
[5]RAN A C M,REURINGS M C B.A nonlinear matrix equation connected to interpolation theory[J].Linear Algebra Appl,2004,379:289-304.DOI:10.1016/S0024-3795(03)00541-X
[6]MEINI B.Efficient computation of the extreme solutions of X+A*X-1A=Q and X-A*X-1A=Q[J].Math Comput,2002,71:1189-1204.
[7]GUO C,KUO Y,LIN W.Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research[J].J Comput Appl Math,2012,236(17):4166-4180.DOI:10.1016/j.cam.2012.05.012
[8]SUN J,XU S.Perturbation analysis of the maximal solution of the matrix equation X+A*X-1A=P(ΙΙ)[J].Linear Algebra Appl,2003,362:211-228.DOI:10.1016/S0024-3795(01)00300-7
[9]HASANOV V I,IVANOV I G.On two perturbation estimates of the extreme solutions to the equations X+A*X-1A=Q[J].Linear Algebra Appl,2006,413:81-92.DOI:10.1016/j.laa.2005.08.013
[10]HE Y,LONG J.On the Hermitian positive definite solution of the nonlinear matrix equation X+∑i=1m Ai*X-1Ai=I[J].Applied Mathematics and Computation,2010,216:3480-3485.
[11]DUAN X,LI C,LIAO A.Solutions and perturbation analysis for the nonlinear matrix equation X+∑i=1m Ai*X-1Ai=I[J].Applied Mathematics and Computation,2011,218:4458-4466.
[12]DUAN X,WANG Q,LIAO A.On the matrix equation X-∑i=1m Ni*X-1Ni=I arising in an interpolation problem[J].Linear and Multilinear Algebra,2013,61(9):1192-1205.
[13]YIN X,FANG L.Perturbation analysis for the positive definite solution of the nonlinear matrix equation X-∑i=1m Ai*X-1Ai=Q[J].Journal of Applied Mathematics and Computing,2013,41:199-211.
[14]BERZIG M.Solving a class of matrix equations via the Bhaskar-Lakshmikantham coupled fixed point theorem[J].Applied Mathematics Letters,2012,25:1638-1643.DOI:10.1016/j.aml.2012.01.028
[15]DUAN X,WANG Q,LI C.Positive definite solution of a class of nonlinear matrix equation[J].Linear and Multilinear Algebra,2014,62(6):839-852.DOI:10.1080/03081087.2013.794230
[16]ZHANG G L.A note on positive definite solution of matrix equation X-M*X-1M+N*X-1N=I[J].Linear and Multilinear Algebra,2016,64(5):951-954.DOI:10.1080/03081087.2015.1068267
[17]BERZIG M,DUAN X,SAMET B.Positive definite solution of the matrix equation X-A*X-1A+B*X-1B=Q via Bhaskar-Lakshmikantham fixed point theorem[J].Mathematical Sciences,2012(6):27-32.
[18]BERINDE V.Generalized coupled fixed point theorems for mixed point monotone mappings in partially ordered metric spaces[J].Nonlinear Analysis,2011,74:7347-7355.
[19]RICE J R.A theory of condition[J].SIAM J Numer Anal,1966(3):287-310.DOI:10.1137/0703023