预条件平方Smith法求解连续Lyapunov方程
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摘要
探讨了如何数值求解连续时间的Lyapunov矩阵方程AX+XA~T+BB~T=0,给出了一种预条件的平方Smith算法,该算法首先利用交替方向隐式法即ADI法处理连续Lyapunov方程,构造出含ADI参数的对称Stein方程;然后利用平方Smith法迭代产生Krylov子空间中的低秩逼近形式。得到一些数值实验,这些例子表明预条件平方Smith法是非常有效的。
This paper proposes a preconditioned squared Smith algorithm to solve the continuous-time Lyapunov matrix equations AX+XA~T+BB~T=0 numerically.The method first uses the alternating directional implicit(ADI)method and transforms the original equations to the equivalent symmetric Stein matrix equations with some ADI parameters.Then we adopt the squared Smith algorithm to seek solutions of the Stein equations by generating the squared Smith iterations in some low-rank forms with the Krylov subspaces.And we give some numerical experiments to show the efficiency of this algorithm finally.
This paper proposes a preconditioned squared Smith algorithm to solve the continuous-time Lyapunov matrix equations AX+XA~T+BB~T=0 numerically.The method first uses the alternating directional implicit(ADI)method and transforms the original equations to the equivalent symmetric Stein matrix equations with some ADI parameters.Then we adopt the squared Smith algorithm to seek solutions of the Stein equations by generating the squared Smith iterations in some low-rank forms with the Krylov subspaces.And we give some numerical experiments to show the efficiency of this algorithm finally.
引文
[1]LASALLE J P,LEFSCHETZ S.Stability by Lyapunov’s Direct Method:With Applications[M].New York:Academic Press,1961.
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[3]SAFONOV M G,CHIANG R Y.A Schur method for balanced-truncation model reduction[J].IEEE Transactions on Automatic Control,1989,34(7):729-733.
[4]TOMBS M S,POSTLETHWAITE I.Truncated balanced realization of a stable non-minimal state-space system[J].International Journal of Control,1987,46(4):1319-1330.
[5]BARTELS R H,STEWART G W.Solution of the matrix equation AX+XB=C[J].Communications of the ACM,1972,15(9):820-826.
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[7]HU D Y,REICHEL L.Krylov-subspace methods for the Sylvester equation[J].Linear Algebra and Its Applications,1992,172:283-313.
[8]JAIMOUKHA I M,KASENALLY E M.Krylov subspace methods for solving large Lyapunov equations[J].SIAM Journal on Numerical Analysis,1994,31(1):227-251.
[9]JBILOU K,RIQUET A J.Projection methods for large Lyapunov matrix equations[J].Linear Algebra and Its Applications,2006,415(2):344-358.
[10]SAAD Y.Numerical solution of large Lyapunov equations[J].Signal Processing,Scattering&Operator Theory&Numerical Methods,Proc Mtns,2010,47(2):503-511.
[11]SIMONCINI V.A new iterative method for solving largescale Lyapunov matrix equations[J].SIAM Journal on Scientific Computing,2007,29(3):1268-1288.
[12]LU A,Wachspress E L.Solution of Lyapunov equations by alternating direction implicit iteration[J].Computers&Mathematics with Applications,1991,21(9):43-58.
[13]PENZL T.A cyclic low-rank Smith method for large sparse Lyapunov equations[J].SIAM Journal on Scientific Computing,1999,21(4):1401-1418.
[14]YOUNG D.The numerical solution of elliptic and parabolic partial differential equations[J].Survey of Numerical Analysis,1961:380-438.
[15]WACHSPRESS E L.Iterative solution of the Lyapunov matrix equation[J].Applied Mathematics Letters,1988,1(1):87-90.
[16]YOUSEF S.Iterative Methods for Sparse Linear Systems[M].Philadelphia:SIAM,2003.
[17]HOM R A,JOHNSON C R.Topics in Matrix Analysis[M].New York:Cambridge UP,1991.
[18]BENNER P,KHOURY G E,SADKANE M.On the squared Smith method for large-scale Stein equations[J].Numerical Linear Algebra with Applications,2014,21(5):645-665.
[19]LANCASTER P,RODMAN L.Algebraic Riccati Equations[M].Oxford:Clarendon Press,1995.
[20]GOLUB G H,VAN LOAN C F.Matrix Computations[M].Baltimore:JHU Press,2012.
[21]JBILOU K.ADI preconditioned Krylov methods for large Lyapunov matrix equations[J].Linear Algebra and Its Applications,2010,432(10):2473-2485.
[22]SADKANE M.A low-rank Krylov squared Smith method for large-scale discrete-time Lyapunov equations[J].Linear Algebra and Its Applications,2012,436(8):
[2]MOORE B C.Principal component analysis in linear systems:Controllability,observability,and model reduction[J].IEEE Transactions on Automatic Control,1981,26(1):17-32.
[3]SAFONOV M G,CHIANG R Y.A Schur method for balanced-truncation model reduction[J].IEEE Transactions on Automatic Control,1989,34(7):729-733.
[4]TOMBS M S,POSTLETHWAITE I.Truncated balanced realization of a stable non-minimal state-space system[J].International Journal of Control,1987,46(4):1319-1330.
[5]BARTELS R H,STEWART G W.Solution of the matrix equation AX+XB=C[J].Communications of the ACM,1972,15(9):820-826.
[6]GOLUB G H,NASH S,VAN LOAN C.A Hessenberg-Schur method for the problemAX+XB=C[J].IEEE Transactions on Automatic Control,1979,24(6):909-913.
[7]HU D Y,REICHEL L.Krylov-subspace methods for the Sylvester equation[J].Linear Algebra and Its Applications,1992,172:283-313.
[8]JAIMOUKHA I M,KASENALLY E M.Krylov subspace methods for solving large Lyapunov equations[J].SIAM Journal on Numerical Analysis,1994,31(1):227-251.
[9]JBILOU K,RIQUET A J.Projection methods for large Lyapunov matrix equations[J].Linear Algebra and Its Applications,2006,415(2):344-358.
[10]SAAD Y.Numerical solution of large Lyapunov equations[J].Signal Processing,Scattering&Operator Theory&Numerical Methods,Proc Mtns,2010,47(2):503-511.
[11]SIMONCINI V.A new iterative method for solving largescale Lyapunov matrix equations[J].SIAM Journal on Scientific Computing,2007,29(3):1268-1288.
[12]LU A,Wachspress E L.Solution of Lyapunov equations by alternating direction implicit iteration[J].Computers&Mathematics with Applications,1991,21(9):43-58.
[13]PENZL T.A cyclic low-rank Smith method for large sparse Lyapunov equations[J].SIAM Journal on Scientific Computing,1999,21(4):1401-1418.
[14]YOUNG D.The numerical solution of elliptic and parabolic partial differential equations[J].Survey of Numerical Analysis,1961:380-438.
[15]WACHSPRESS E L.Iterative solution of the Lyapunov matrix equation[J].Applied Mathematics Letters,1988,1(1):87-90.
[16]YOUSEF S.Iterative Methods for Sparse Linear Systems[M].Philadelphia:SIAM,2003.
[17]HOM R A,JOHNSON C R.Topics in Matrix Analysis[M].New York:Cambridge UP,1991.
[18]BENNER P,KHOURY G E,SADKANE M.On the squared Smith method for large-scale Stein equations[J].Numerical Linear Algebra with Applications,2014,21(5):645-665.
[19]LANCASTER P,RODMAN L.Algebraic Riccati Equations[M].Oxford:Clarendon Press,1995.
[20]GOLUB G H,VAN LOAN C F.Matrix Computations[M].Baltimore:JHU Press,2012.
[21]JBILOU K.ADI preconditioned Krylov methods for large Lyapunov matrix equations[J].Linear Algebra and Its Applications,2010,432(10):2473-2485.
[22]SADKANE M.A low-rank Krylov squared Smith method for large-scale discrete-time Lyapunov equations[J].Linear Algebra and Its Applications,2012,436(8):