奇异线性系统Drazin逆解的DQMR算法
|
摘要
近年来,关于奇异线性系统Drazin逆解的算法引起了众多学者的广泛关注,并获得了依赖于Krylov子空间的大量研究成果.然而,Krylov子空间法十分繁琐且解决奇异线性不相容系统十分困难.基于此,利用投影法给出了相容或非相容奇异线性系统Ax=b的Drazin逆解的DQMR算法,其中A∈?~(n×n)是一个具有任意指标的奇异Hermitian矩阵. DQMR算法"类似"于非奇异系统的QMR算法.
In recent years, the algorithm of Drazin inverse solution for singular linear systems has caused wide concern by many scholars, and many research results that depend on the Krylov subspace are obtained. However, the Krylov subspace method is very cumbersome and it is very difficult to solve the singular linear incompatible system. Based on this, in this paper, the DQMR algorithm for Drazin inverse solution of compatible or non-compatible singular linear systems Ax=b is given by using projection method, where A∈?~(n×n) is a singular Hermitian matrix with arbitrary index. The DQMR algorithm of singular systems is "similar" to the QMR algorithm of non-singular systems.
In recent years, the algorithm of Drazin inverse solution for singular linear systems has caused wide concern by many scholars, and many research results that depend on the Krylov subspace are obtained. However, the Krylov subspace method is very cumbersome and it is very difficult to solve the singular linear incompatible system. Based on this, in this paper, the DQMR algorithm for Drazin inverse solution of compatible or non-compatible singular linear systems Ax=b is given by using projection method, where A∈?~(n×n) is a singular Hermitian matrix with arbitrary index. The DQMR algorithm of singular systems is "similar" to the QMR algorithm of non-singular systems.
引文
[1] 王松桂.广义逆矩阵及其应用[M].北京:北京工业大学出版社,2006.WANG S G.Generalized Inverse Matrix and its Application[M].Beijing:Beijing University of Technology Press,2016.(Ch).
[2] 翁业早.广义逆矩阵理论的研究与应用[D].南京:南京邮电大学,2016.WENG Y Z.Research and Application of the Theory of Generalized Inverse Matrix[D].Nanjing:Nanjing University of Posts and Telecommunications,2016.(Ch).
[3] ISRAEL B,GREVILE T N E.Generalized Inverses:Theory and Applications[M].2nd Ed.New York:Springer-Verlag,2003.
[4] 王芳,程俊荣.求解奇异线性方程组的两种预条件QMR算法[J].温州大学学报(自然科学版),2013,34(1):24-30.WANG F,CHENG J R.The two preconditioned QMR algorithms for solving [J].Journal of Wenzhou University (Natural Sciences),2013,34(1):24-30.(Ch).
[5] 刘华磊.解线性方程组的简单GMRES算法研究[D].南京:南京航空航天大学,2007.LIU H L.Research on the Simpler GMRES for Linear Systems[D].Nanjing:Nanjing University of Aeronautics and Astronautics,2007.(Ch).
[6] 戴华.求解大规模矩阵问题的Krylov子空间方法[J].南京航空航天大学学报,2001,33(2):139-146.DAI H.Krylov subspace methods for solving large scale matrix problems[J].Journal of Nanjing University of Aeronautics& Astronautics,2001,33(2):139-146.(Ch).
[7] WEI Y,WU H.Convergence properties of Krylov subspace methods for singular system with arbitrary index[J].Journal of Computational and Applied Mathematics,2000,114:305-318.
[8] SIDI A.A unified approach to Krylov subspace methods for the Drazin-inverse solution of singular non-symmetric linear systems[J].Linear Algebra and its Applications,1999,298:99-113.
[9] SMOCH L.Some result about GMRES in the singular case [J].Numerical Algorithms,1999,22:193-212.
[10] SIDI A,KLUZNER V.A Bi-CG type iterative method for Drazin inverse solution of singular inconsistent non-symmetric linear systems of arbitrary index[J].The Electronic Journal of Linear Algebra,1999,6:72-94.
[11] SIDI A.DGMRES:A GMRES-Type algorithm for Drazin-inverse solution of singular non-symmetric linear systems[J].Linear Algebra and its Applications,2001,335:189-204.
[12] FAEZEH T,REZA B.New methods for computing the Drazin-inverse solution of singular linear systems[J].Applied Mathematics & Computation,2017,294:343-352.
[13] ZHOU J Y,WEI Y M.DFOM algorithm and error analysis for projection methods for solving singular linear system [J].Applied Mathematics and Computation,2004,157:313-329.
[14] 周富照,田时宇,袁艳杰.一类矩阵方程组的正交投影迭代解法[J].吉首大学学报(自然科学版),2015,36(3):1-6.ZHOU F Z,TIAN S Y,YUAN Y J.An orthogonal projection iteration method for a matrix equations[J].Journal of Jishou University (Natural Science),2015,36(3):1-6.(Ch).
[15] FAN H Y,ZHANG L P.Numerical solution to a linear equation with tensor product structure[J].Numerical Lin Alg with Applic.2017,24(6):1-22.
[2] 翁业早.广义逆矩阵理论的研究与应用[D].南京:南京邮电大学,2016.WENG Y Z.Research and Application of the Theory of Generalized Inverse Matrix[D].Nanjing:Nanjing University of Posts and Telecommunications,2016.(Ch).
[3] ISRAEL B,GREVILE T N E.Generalized Inverses:Theory and Applications[M].2nd Ed.New York:Springer-Verlag,2003.
[4] 王芳,程俊荣.求解奇异线性方程组的两种预条件QMR算法[J].温州大学学报(自然科学版),2013,34(1):24-30.WANG F,CHENG J R.The two preconditioned QMR algorithms for solving [J].Journal of Wenzhou University (Natural Sciences),2013,34(1):24-30.(Ch).
[5] 刘华磊.解线性方程组的简单GMRES算法研究[D].南京:南京航空航天大学,2007.LIU H L.Research on the Simpler GMRES for Linear Systems[D].Nanjing:Nanjing University of Aeronautics and Astronautics,2007.(Ch).
[6] 戴华.求解大规模矩阵问题的Krylov子空间方法[J].南京航空航天大学学报,2001,33(2):139-146.DAI H.Krylov subspace methods for solving large scale matrix problems[J].Journal of Nanjing University of Aeronautics& Astronautics,2001,33(2):139-146.(Ch).
[7] WEI Y,WU H.Convergence properties of Krylov subspace methods for singular system with arbitrary index[J].Journal of Computational and Applied Mathematics,2000,114:305-318.
[8] SIDI A.A unified approach to Krylov subspace methods for the Drazin-inverse solution of singular non-symmetric linear systems[J].Linear Algebra and its Applications,1999,298:99-113.
[9] SMOCH L.Some result about GMRES in the singular case [J].Numerical Algorithms,1999,22:193-212.
[10] SIDI A,KLUZNER V.A Bi-CG type iterative method for Drazin inverse solution of singular inconsistent non-symmetric linear systems of arbitrary index[J].The Electronic Journal of Linear Algebra,1999,6:72-94.
[11] SIDI A.DGMRES:A GMRES-Type algorithm for Drazin-inverse solution of singular non-symmetric linear systems[J].Linear Algebra and its Applications,2001,335:189-204.
[12] FAEZEH T,REZA B.New methods for computing the Drazin-inverse solution of singular linear systems[J].Applied Mathematics & Computation,2017,294:343-352.
[13] ZHOU J Y,WEI Y M.DFOM algorithm and error analysis for projection methods for solving singular linear system [J].Applied Mathematics and Computation,2004,157:313-329.
[14] 周富照,田时宇,袁艳杰.一类矩阵方程组的正交投影迭代解法[J].吉首大学学报(自然科学版),2015,36(3):1-6.ZHOU F Z,TIAN S Y,YUAN Y J.An orthogonal projection iteration method for a matrix equations[J].Journal of Jishou University (Natural Science),2015,36(3):1-6.(Ch).
[15] FAN H Y,ZHANG L P.Numerical solution to a linear equation with tensor product structure[J].Numerical Lin Alg with Applic.2017,24(6):1-22.