线性方程组和鞍点问题的迭代法与预处理技术研究
|
摘要
大规模科学计算和工程技术中许多问题的解决，最终归结为大型稀疏线性方程组的求解，其求解时间在整个问题求解时间中占有很大的比重，有的甚至达到80％．由于现今科学研究和大型项目中各种复杂的课题对计算精度和计算速度的要求越来越高．因此，作为大规模科学计算基础的线性代数方程组的高效数值求解引起了人们的普遍关注．这种方程组的求解一般采用迭代法，所以，迭代法的收敛性和收敛速度就成为人们关注的焦点，为许多专家和学者所研究．本文对与大型稀疏线性方程组迭代求解有关的特殊矩阵迭代法进行了深入和系统的研究，特别研究了松弛型矩阵多分裂迭代法的收敛性，两种Krylov子空间方法的性能和鞍点问题的预处理技术．全文共六章，分四个部分：
研究了H-矩阵的松弛型矩阵多分裂法，并给出详细的理论分析和敛散速度的比较．一方面，给出了松弛型矩阵多分裂TOR法，研究了方法的收敛性，比较了他们的敛散速度，分别进行了串行和并行试验，验证了所提方法的优越性．另一方面，给出了松弛型矩阵多分裂USAOR法，研究了方法的收敛性，给出了实现算法的例子，并用数值试验与存在的方法进行了比较．
进一步研究了一些H-矩阵松弛型矩阵多分裂法新的收敛性结果．分别为非线性方程组的非定常矩阵多分裂法，线性互补问题的矩阵多分裂法，松弛型矩阵多分裂SSOR法和松弛型矩阵多分裂TOR法，构建了相应方法的收敛性理论，得到了新的更弱的收敛性条件，进行了数值试验的比较．基于多分裂法的并行性，矩阵多分裂的研究和理论分析对于多分裂预处理子的构造有一定的理论和应用价值．我们的方法选取参数的余地更大，当选取近似最优参数时，能实现更快的收敛速度和构造出更有效的预处理子．
基于BiCR算法设计了求解非对称线性方程组Krylov子空间平方共轭残差(CRS)算法和适合分布式并行计算的改进的平方共轭残差(ICRS)算法，并对两种算法进行了理论分析和算法比较，串行和并行数值试验表明所提方法具有较好的收敛速度和并行性能．
研究了鞍点问题特殊矩阵迭代求解预处理技术．首先，对内点优化问题产生的一类鞍点问题给出了一种预处理技术，进行了相应的理论分析和数值试验．接着，基于离散化混合型时谐Maxwell方程的块三角鞍点问题和特殊矩阵的结构，提出了带多个参数的预处理技术，并进行了理论分析和参数的理论选取．理论分析表明提出的预处理子有更好的特征值聚集性．数值试验也表明本章提出的预处理子性能大大优于免增广和免Schur余块对角预处理子．
Solutions of large-scale sparse linear systems arise widely in large-scale scientific computing, and engineering disciplines. Moreover, solving large-scale sparse linear systems plays a key role in scientific and engineering computing and the corresponding time of computing always arrives at 80%, so more and more people pay attention to find effective numerical solutions of linear algebraic systems based on large-scale scientific and engineering computing. Solutions of large-scale sparse linear systems are always iterative methods, so convergence and convergence rate of iterative methods are deeply studied by many experts and scholars. The author mainly studies some iterative methods of special matrices related to iteration solutions of large-scale sparse linear systems, especially convergence of relaxed matrix multisplitting iterative methods, performance of two Krylov subspace methods, and preconditioning techniques of saddle point problems. This paper consists of four parts with six chapters.
Research on relaxed matrix multisplitting iterative methods for H-matrices, and gives detailedly theoretical analysis and comparisons of convergence and divergence rates. On the one hand, we present relaxed matrix multisplitting TOR iterative method, study convergence of our method, compare the corresponding convergence and divergence rates, and carry out sequential and parallel experiments which show the validity of our method. On the other hand, we give relaxed matrix multisplitting USAOR iterative method, study convergence of our method, show the examples to put algorithms in practice, and carry out numerical experiments to compare with the corresponding existing methods.
Further study new convergence of some relaxed matrix multisplitting iterative methods for H-matrices. We analyze non-stationary matrix multisplitting iterative methods for almost linear systems, matrix multisplitting iterative methods for the linear complementarity problems, relaxed matrix multisplitting SSOR iterative method, and relaxed matrix multisplitting TOR iterative method for linear systems, respectively. Moreover, this chapter studies convergence theory of the corresponding methods, finds new weaker convergence conditions, and carries out some comparisons of numerical experiments. Based on the parallelity of multisplitting, the study and theory of matrix multisplitting contributing to the structure of multisplitting preconditioners have many theoretical and applied values. Our methods have more optional parameters, so we can obtain more quick convergence rate and structure more effective preconditioners when choosing the approximately optimal relaxed parameters.
Design Krylov subspace conjugate residual squared (CRS) algorithm for nonsym-metric linear systems and the improved conjugate residual squared (ICRS) algorithm for distributed parallel computing based on biconjugate residual method (BiCR) algorithm, and give theoretical analysis and comparisons of algorithms for two algorithms. In the end, sequential and parallel numerical experiments show that CRS and ICRS methods have more quick convergence rate than BiCR method and that ICRS method has better parallel performance than CRS method and so on.
Research on generalized block preconditioning techniques for iteration solutions of saddle point problems which arise from the discretized time-harmonic Maxwell equations and interior point optimization method. On the one hand, new block triangular precondi-tioner for linear systems arising from interior point optimization method is obtained, and the corresponding theoretical analysis and numerical experiments are given to show the validity of our preconditioner. On the other hand, we deeply study block triangular preconditioners with multi-parameters based on block triangular saddle point linear systems and structure of special matrices arising from time-harmonic Maxwell equations, and give theoretical analysis and theoretical choice of optimal parameters. In particular, theoretical analysis shows that all eigenvalues of two generalized block triangular preconditioners are strongly clustered. Finally, numerical experiments show that the quality of the presented block triangular preconditioners is better than that of the corresponding block diagonal augmentation-free and Schur complement-free preconditioners.
研究了H-矩阵的松弛型矩阵多分裂法，并给出详细的理论分析和敛散速度的比较．一方面，给出了松弛型矩阵多分裂TOR法，研究了方法的收敛性，比较了他们的敛散速度，分别进行了串行和并行试验，验证了所提方法的优越性．另一方面，给出了松弛型矩阵多分裂USAOR法，研究了方法的收敛性，给出了实现算法的例子，并用数值试验与存在的方法进行了比较．
进一步研究了一些H-矩阵松弛型矩阵多分裂法新的收敛性结果．分别为非线性方程组的非定常矩阵多分裂法，线性互补问题的矩阵多分裂法，松弛型矩阵多分裂SSOR法和松弛型矩阵多分裂TOR法，构建了相应方法的收敛性理论，得到了新的更弱的收敛性条件，进行了数值试验的比较．基于多分裂法的并行性，矩阵多分裂的研究和理论分析对于多分裂预处理子的构造有一定的理论和应用价值．我们的方法选取参数的余地更大，当选取近似最优参数时，能实现更快的收敛速度和构造出更有效的预处理子．
基于BiCR算法设计了求解非对称线性方程组Krylov子空间平方共轭残差(CRS)算法和适合分布式并行计算的改进的平方共轭残差(ICRS)算法，并对两种算法进行了理论分析和算法比较，串行和并行数值试验表明所提方法具有较好的收敛速度和并行性能．
研究了鞍点问题特殊矩阵迭代求解预处理技术．首先，对内点优化问题产生的一类鞍点问题给出了一种预处理技术，进行了相应的理论分析和数值试验．接着，基于离散化混合型时谐Maxwell方程的块三角鞍点问题和特殊矩阵的结构，提出了带多个参数的预处理技术，并进行了理论分析和参数的理论选取．理论分析表明提出的预处理子有更好的特征值聚集性．数值试验也表明本章提出的预处理子性能大大优于免增广和免Schur余块对角预处理子．
Solutions of large-scale sparse linear systems arise widely in large-scale scientific computing, and engineering disciplines. Moreover, solving large-scale sparse linear systems plays a key role in scientific and engineering computing and the corresponding time of computing always arrives at 80%, so more and more people pay attention to find effective numerical solutions of linear algebraic systems based on large-scale scientific and engineering computing. Solutions of large-scale sparse linear systems are always iterative methods, so convergence and convergence rate of iterative methods are deeply studied by many experts and scholars. The author mainly studies some iterative methods of special matrices related to iteration solutions of large-scale sparse linear systems, especially convergence of relaxed matrix multisplitting iterative methods, performance of two Krylov subspace methods, and preconditioning techniques of saddle point problems. This paper consists of four parts with six chapters.
Research on relaxed matrix multisplitting iterative methods for H-matrices, and gives detailedly theoretical analysis and comparisons of convergence and divergence rates. On the one hand, we present relaxed matrix multisplitting TOR iterative method, study convergence of our method, compare the corresponding convergence and divergence rates, and carry out sequential and parallel experiments which show the validity of our method. On the other hand, we give relaxed matrix multisplitting USAOR iterative method, study convergence of our method, show the examples to put algorithms in practice, and carry out numerical experiments to compare with the corresponding existing methods.
Further study new convergence of some relaxed matrix multisplitting iterative methods for H-matrices. We analyze non-stationary matrix multisplitting iterative methods for almost linear systems, matrix multisplitting iterative methods for the linear complementarity problems, relaxed matrix multisplitting SSOR iterative method, and relaxed matrix multisplitting TOR iterative method for linear systems, respectively. Moreover, this chapter studies convergence theory of the corresponding methods, finds new weaker convergence conditions, and carries out some comparisons of numerical experiments. Based on the parallelity of multisplitting, the study and theory of matrix multisplitting contributing to the structure of multisplitting preconditioners have many theoretical and applied values. Our methods have more optional parameters, so we can obtain more quick convergence rate and structure more effective preconditioners when choosing the approximately optimal relaxed parameters.
Design Krylov subspace conjugate residual squared (CRS) algorithm for nonsym-metric linear systems and the improved conjugate residual squared (ICRS) algorithm for distributed parallel computing based on biconjugate residual method (BiCR) algorithm, and give theoretical analysis and comparisons of algorithms for two algorithms. In the end, sequential and parallel numerical experiments show that CRS and ICRS methods have more quick convergence rate than BiCR method and that ICRS method has better parallel performance than CRS method and so on.
Research on generalized block preconditioning techniques for iteration solutions of saddle point problems which arise from the discretized time-harmonic Maxwell equations and interior point optimization method. On the one hand, new block triangular precondi-tioner for linear systems arising from interior point optimization method is obtained, and the corresponding theoretical analysis and numerical experiments are given to show the validity of our preconditioner. On the other hand, we deeply study block triangular preconditioners with multi-parameters based on block triangular saddle point linear systems and structure of special matrices arising from time-harmonic Maxwell equations, and give theoretical analysis and theoretical choice of optimal parameters. In particular, theoretical analysis shows that all eigenvalues of two generalized block triangular preconditioners are strongly clustered. Finally, numerical experiments show that the quality of the presented block triangular preconditioners is better than that of the corresponding block diagonal augmentation-free and Schur complement-free preconditioners.
引文
[1] H. B. An, Z. Z. Bai. A globally convergent Newton-GMRES method for large sparse systems of nonlinear equations. Applied Numerical Mathematics, 2007,57:235-252
[2] H. B. An, Z.-Z. Bai. NGLM:A globally convergent Newton-GMRES method. Mathematica Nu-merica Sinica, 2005, 27(2):151-174
[3] P. Arbenz, R. Geus. Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems. Applied Numerical Mathematics, 2005, 54:07-121
[4] J. Arnal, V. Migall(?)n, Jos(?) Penad(?). Non-stationary parallel multisplitting algorithms for almost linear systems. Numerical linear algebra and its applications, 1999,6:79-92
[5] Z. Z. Bai. On the convergence domains of the matrix multisplitting relaxed methods for linear systems. Applied Mathematics Journal of Chinese University, 1998,13b(l):45-52
[6] Z. Z. Bai. On the convergence of the multisplitting methods for the linear complementarity problem. SIAM Journal on Matrix Analysis and Application, 1999, 21:67-78
[7] Z. Z. Bai, D. J. Evans. Matrix multisplitting relaxation methods for linear complementarity problems. International Journal of Computer Mathematics, 1997,63:309-326
[8] Z. Z. Bai. Comparisons of the convergence and divergence rates of the parallel matrix multisplitting iteration methods. Chinese Journal of Engineering Mathematics, 1994,1:99-102
[9] Z. Z. Bai. Parallel nonlinear AOR method and its convergence. Computers and Mathematics with Applications, 1996, 31(2):21-31
[10] Z. Z. Bai, G. H. Golub, L. Z. Lu, J. F. Yin. Block-Triangular and skew-Hermitian splitting methods for positive definite linear systems. SIAM Journal on Scientific Computing, 2005, 26:844-863
[11] Z. Z. Bai, G. H. Golub, M. K. Ng. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM Journal on Matrix Analysis and Application,2003, 24:603-626
[12] Z. Z. Bai, G. H. Golub, M. K. Ng. On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra and its Application, 2008,428:413-440
[13] Z. Z. Bai, G. H. Golub, M. K. Ng. On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iteration. available online at http://wwwsccm. stanford.edu/wrap/pub-tech.html
[14] Z. Z. Bai, B. N. Parlett, Z. Q. Wang. On generalized successive overrelaxation methods for augmented linear systems. Numerische Mathematik, 2005,102:1-38
[15] R. E. Bank, B. D. Welfert, H. Yserentant. A class of iterative methods for sloving saddle point problems. Numerische Mathematik, 1990, 56:645-666
[16] M. Benzi, M. J. Gander, G. H. Golub. Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems. BIT, 2003,43:881-900
[17] M. Benzi, G. H. Golub. A preconditioner for generalized saddle point problems. SIAM Journal on Matrix Analysis and Application, 2004,26:20-41
[26] M. Benzi, G. H. Golub, J. Liesen. Numerical solution of saddle point problems. Acta Numerica,2005,14:1-137
[19] M. Benzi, J. Liu. An efficient solver for the incompressible Navier-Stokes equation in rotation form. SIAM Journal on Scientific Computing, 2007,29:1959-1981
[25] M. Benzi, J. Liu. Block preconditioning for saddle point systems with indefinite (1,1) block. International Journal of Computer Mathematics, 2007,84:1117-1129
[21] M. Benzi, M. K. Ng. Preconditioned iterative methods for weighted Toeplitz least squares problems. SIAM Journal on Matrix Analysis and Application, 2006, 27:1106-1124
[27] M. Benzi, M. A. Olshanskii. An augmented lagrangian-based approach to the Oseen problem. SIAM Journal on Scientific Computing, 2006, 28:2095-2113
[23] M. Benzi, V. Simoncini. On the eigenvalues of a class of saddle point matrices. Numerische Mathematik, 2006, 103:173-196
[24] M. Benzi, D. B. Szyld. Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods. Numerische Mathematik, 1997,76:309-321
[25] M. Benzi J. Liu. Block preconditioning for saddle point systems with indefinite (1,1) block. International Journal of Computer Mathematics, 2007, 84(8): 1117-1129
[26] M. Benzi, G. H. Golub, J. Liesen. Numerical solution of saddle point problems. Acta Numerica,2005,14:1-137
[27] M. Benzi, M. A. Olshanskii. An augmented Lagrangian-based approach to the Oseen problem. SIAM Journal on Scientific Computing, 2006, 28:2095-2113
[28] L. Bergamaschi, J. Gondzio, G. Zilli. Preconditioning indefinite systems in interior point methods for optimization. Computational Optimization and Applications, 2004,28:149-171
[29] A. Berman, R. J. Plemmons. Nonnegative Matrices in the Mathematical Science, Academic Press, New York, third edition, 1979, Reprinted by SIAM, Philadelphia, 1994
[30] G. Birkhoff. Numerical Solution of Elliptic Equation. volume 1 of CBMS Regional Conference Series in Applied Mathematics, Society for industrial and Applied Mathematics, Philadelphia,1970
[31] P. N. Brown, Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SLAM Journal on Scientific and Statistical Computing, 1990,11:450-481
[32] R. Bru, V. Migallon, J. Penades, D.B. Szyld. Parallel, synchronous and asynchronous two-stage multisplitting methods. Electronic Transactions on Numerical Analysis, 1995,3:24-38
[33] R. Bru, L. Eisner, M. Neumman. Models of parallel chaotic iterative methods. Linear Algebra and Its Applications, 1988,103:175-192
[34] A. Berman, R. J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York, 1979
[35] H. M. Bucker, M. Sauren. A parallel version of the quasi-minimal residual method based on coupled two-term recurrences. In Proceedings of Workshop on Applied Parallel computing in Industrial Problems and Optimization (Para96), Technical University of Denmark, Lyngby, Denmark, Springer-Verlab, August 1996
[36] Z. H. Cao. Fast Uzawa algorithm for generalized saddle point problems. Applied Numerical Mathematics, 2003,46:157 - 171
[37] Z. H. Cao. Positive stable block triangular preconditioners for symmetric saddle point problems. Applied Numerical Mathematics, 2007, 57:899-910
[38] Z. H. Cao, Z. Y. Lin. Convergence of relaxed parallel multisplitting methods with different weighting schemes. Applied Mathematics and Computation, 1996, 106:181-196
[39] Z. H. Cao. On the convergence of nested stationary iterative methods. Linear Algebra and Its Applications, 1995,221:159-170
[40] R. Chan, X. Jin, M. Yeung. The circulant operator in the Banach algebra of matrices. Linear Algebra and its Applications, 1991, 149:41-53
[41] T. Chan. An optimal circulant preconditioner for Toeplitz systems. SIAM Journal on Scientific Computing, 1988,9:766-771
[42] D. W. Chang. Convergence analysis of the parallel multisplitting TOR methods. Journal of Computational and Applied Mathematics, 1996,72:169-177
[43] D. W. Chang. The parallel multisplitting TOR(MTOR) method for linear systems. Computers and Mathematics with Applications, 2001,41:215-227
[44] D. Chazan, W. Miranker. Chaotic relaxation. Linear Algebra and Its Applications, 1969, 2:199-222
[45] 陈国良著。并行计算-结构、算法、编程。高等教育出版社,北京,2003
[46] R. S. Chen, X. W. Ping, Edward K. N. Yung, C. H. Chan, Z. P. Nie, J. Hu. Application of diagonally perturbed incomplete factorization preconditioned conjugate gradient algorithms for edge finite-element analysis of helmholtz equations. IEEE Transactions on Antennas and Propagation,2006,54:1064-1068
[47] Z. Chen, Q. Du, J. Zou. Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SLAM Journal on Numerical Analysis, 2000,37:1542-1570
[48] C. M. Cheng, X. Q. Jin, S. W. Vong, W. Wang. A note on spectra of optimal and superoptimal preconditioned matrices. Linear Algebra and its Application, 2007, 422:482-485
[49] L. H. Chi, J. Liu, X. P. Liu, Q. F. Hu, X. M. Li. An improved conjugate residual algorithm for large symmetric linear systems. In Computational Physics, Proceedings of the Joint Conference of ICCP6 and CCP2003, Rinton Press, New Jersey, USA, 2005, 325-328
[50] K. A. Cliffe, T. J. Garratt, A. Spence. Eigenvalues of block matrices arising from problems in fluid mechanics. SIAM Journal on Matrix Analysis and Applications, 1994,15:1310-1318
[51] J. J. Climent, C. Perea. Some comparison theorems for weak nonnegative splittings of bounded operators. Linear Algebra and its Applications, 1998, 275/276:77-106
[52] J. J. Climent, C. Perea, L. Tortosa, A. Zamora. Convergence theorems for parallel alternating iterative methods. Applied Mathematics and Computation, 2004, 148:497-517
[53] J. J. Climent, C. Perea. Convergence and comparison theorems for multisplittings. Numerical Linear Algebra and Its applications, 1999, 6:93-107
[54] R. W. Cottle, G. B. Dantzig. Complementarity pivot theorey of mathematical programming. Linear Algebra and Its Applications, 1968, 1:103-125
[55] R. W. Cottle, G. H. Golub, R. S. Sacher. On the solution of large, structured linear complementarity problems: The block partitioned case. Applied Mathematics and Optimization, 1978,4:347-363
[56] C. W. Cryer. The solution of a quadratic programming problem using systematic overrelaxation.SIAM Journal of Control, 1971, 9:385-393
[57] P. Davis. Circulant Matrices. second ed., Chelsea Publishing, New York, 1994
[58] R. S. Dembo, S.C. Eisenstat, T. Steihaug. Inexact Newton methods. SIAM Journal on Numerical Analysis, 1982, 19:400-408
[59] L. Demkowicz, L. Vardapetyan. Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements. Computer Methods in Applied Mechanics and Engineering,1998,152:103-124
[60] J. W. Demmel. Applied Numerical Linear Algebra. SIAM, Philadelphia, 1997
[61] H. S. Dollar. Constraint-style preconditioners for regularized saddle point problems. SIAM Journal on Matrix Analysis and Application, 2007,29:672-684
[62] H. S. Dollar, A. J. Wathen. Approximate factorization constraint preconditioners for saddle-point matrices. SIAM Journal on Scientific Computing, 2006,27:1555-1572
[63] S. C. Eisenstat, H.F. Walker. Globally convergent inexact Newton methods. SIAM Journal on Optimization, 1994,4:393-422
[64] L. Eisner. Comparisons of weak regular splittings and multisplitting methods. Numerische Math-ematik, 1989,56:283-289
[65] L. Eisner, A. Frommer, R. Nabben, H. Schneider, D. B. Szyld. Conditions for strict inequality in comparisons of spectral radii of splittings of different matrices. Linear Algebra and its Applications, 2003, 263:65-80
[66] L. Eisner, M. Neumman, B. Vemmer. The effect the number of processors on the convergence of the parallel block Jacobi method. Linear Algebra and its Applications, 1991,154-156:311-330
[67] K. Fan. Topological proffs for certain theorems on matrices with nonnegative elements. Monat-shefte fur Mathematik, 1958,62:219-237
[68] 冯果忱。非线性方程组迭代解法。上海：上海科学技术出版社,1989
[69] R. Fletcher. Practical Methods of Optimization. 2nd edn (Chichester: John Wiley), 1987
[70] A. Forsgren, P. E. Gill, M. H. Wright. Interior methods for nonlinear optimization. SIAM Review,2002,44:525-597
[71] M. Fortin, R. Glowinski. Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problems (Amsterdam: North-Holland), 1983
[72] R. W. Freund, N. M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numerische Mathematik, 1991, 60:315-339
[73]A.Frommer,G.Mayer.Convergence of relaxed parallel multisplitting methods.Linear Algebra and its Applications,1989,119:141-152
[74]A.Frommer,G.Mayer.On the theory and practice of multisplitting methods in parallel computation.Computing,1992,49:62-74
[75]R.Fuster,V.Migall(?)n,J.Penad(?)s.Non-stationary parallel multisplitting AOR methods.Electronic Transaction On Numerical Analysis,1996,4:1-13
[76]T.B.Gan,T.Z.Huang.Simple criteria for nonsingular H-matrices.Linear Algebra and its Applications,2003,374:317-326
[77]干泰彬，黄廷祝．非奇H-矩阵的实用充分条件，计算数学，2004，1：109-116
[78]A.Gantner,R.H.W.Hoppe,D.K(?)ster,K.Siebert,A.Wixforth.Numerical simulation of piezo-electrically agitated surface acoustic waves on microfluidic biochips.Computing and Visualization in Science,2007,10:145-161
[79]R.Glowinski.Finite element methods for incompressible viscous flow,Handbook of Numerical Analysis.Volume IX,Numerical Methods for Fluids (Part 3),2003
[80]G.H.Golub,C.Greif.On solving block-structured indefinite linear systems.SIAM Joumal on Scientific Computing,2003,24:2076-2092
[81]G.H.Golub,C.Greif,J.M.Varah.An algebraic analysis of a block diagonal preconditioner for sadie point systems.SIAM Journal on Matrix Analysis and Applications,2006,27:779-792
[82]G.H.Golub,X.Wu,J.Y.Yuan.SOR-Like methods for augmented systems.BIT,2001,41:71-85
[83]J.Gopalakrishnan,J.E.Pasciak,L.F.Demkowicz.Analysis of a multigrid algo-rithm for time harmonic Maxwell equations.SIAM Journal on Numerical Analysis,2004,42:90-108
[84]C.Greif,D.Sch(?)tzau.Preconditioners for the discretized time-harmonic Maxwell equations in mixed form.Numerical Linear Algebra and Its Applications,2007,14:281-297
[85]谷同祥．大型稀疏线性代数方程组的并行非定常迭代方法．博士论文，中国工程物理研究院，2001
[86]T.X.Gu,J.Z.Li.Relaxed Parallel Multispliting Iterative Methods with Arbitrary Weighting Schemes.Journal of Xinyang Teachers College,1994,7(2):126-131
[87]T.X.Gu,X.P.Liu,L.J.Shen.Relaxed parallel two-stage multisplitting methods.International Journal of Computer Mathematics,2000,75:351-363
[88]T.X.Gu,X.P.Liu.Parallel two-stage multisplitting iterative methods.International Journal of Computer Mathematics,1998,20(2):153-166
[89]T.X.Gu,X.Y.Zuo,L.T.Zhang,W.Q.Zhang,Z.Q.Sheng.An improved Bi-Conjugate residual algorithm suitable for distributed parallel computing.Applied Mathematics and Computation,2007,186:1243-1253
[90]T.X.Gu,X.P.Liu,Z.Y.Mo.Multiple search direction conjugate gradient method I:methods and their propositions.International Journal of Computer Mathematics,2004,81(9):1133-1143
[91]R.A.Horn,C.A.Johnson.Matrix Analysis.Cambridge University Press,Cambridge,1985
[92]R.A.Horn,C.R.Johnson.Topics in matrix analysis.Cambridge University Press,Cambridge,England,1991
[93]P.Houston,A.Schneebeli,D.Sch(?)tzau.Mixed discontinuous Galerkin approximation of the Maxwell operator:the definite case.Mod(?)lisation Mathematique et Analyse Num(?)rique,2005,39:727-754
[94]胡家赣．线性代数方程组迭代解法．北京：科学出版社，1997
[95]T.Z.Huang,S.M.Zhong,Z.L.Li,Matrix Theory.Higher Education Press,Bei Jing,2003
[96]黄廷祝．非奇H-矩阵的简捷判据．计算数学，1993，3：318-327
[97]黄廷祝．Ostrowski定理的推广与非奇H矩阵的条件．计算数学，1994，1：19-24
[98]T.Z.Huang,J.Leng,E.Wachspress,Y.Tang.Characterization of H-matrices.Computers and Mathematics with Applications,2004,48:1587-1601
[99]A.N.Iusem.On the convergence of iterative methods for symmetric linear complementarity problems.Mathematical Programming,1993,59:33-48
[100]A.N.Iusem.On the convergence of iterative methods for nonsymmetric linear complementarity problems.Computational and Applied Mathematics,1991,10:27-41
[101]K.C.Jea,D.M.Young.Generalized conjugate gradient acceleration of nonsymmetrizable iterative methods.Linear Algebra and its Applications,1980,34:159-194
[102]C.Keller,N.I.M.Gould,A.J.Wathen.Constraint preconditioning for indefinite linear systems.SIAM Journal on Matrix Analysis and Application,2000,21:1300-1317
[103]C.T.Kelley.Iterative methods for linear and nonlinear equations.SIAM,Philadelphia,1995
[104]A.Klawonn.Block-triangular preconditioners for saddle point problems with a penalty term.SIAM Journal on Scientific Computing,1998,19:172-184
[105]P.E.Kloeden,D.Yuan.Comvergence of relaxed chaotic parallel iterative methods.Bulletin of the Australian Mathematical Society,1994,50:167-176
[106]J.Kuang,J.Ji.A survey of AOR and TOR methods.Journal of Computational and Applied Mathematics,1998,24:3-12
[107]V.Kumar,V.N.Rao V.Parallel Depth-First Search,Part II:Analysis.In Int' 1 J.of Parallel programming,1987,16(6):501-519
[108]S.J.Lai,B.Z.Wang,Y.Duan.Application of the RBF-based meshless method to solve 2-D time domain Maxwell's equations.Microwave and Millimeter Wave Technology,2008,2:749-751
[109]S.J.Lai,B.Z.Wang.Simulation of bounces on power distribution with finite-difference timedomain.Journal of Electromagnetic Waves and Applications,2008,22:1223-1231
[110]C.E.Lemke.Bimatrix equilibrium points and mathematical programming.Management Science,1965,11:681-689
[111]李建宇．解非线性方程组的牛顿-并行矩阵多分裂算法．四川师范大学学报(自然科学报)，1995，18(4)：51-55
[112]X.P.Liu,T.X.Gu,X.D.Hang,Z.Q.Sheng.A parallel version of QMRCGSTAB method for large linear systems in distributed parallel environments.Applied Mathematics and Computation,2006,172(2):744-752
[113]Z.Y.Liu,H.B.Wu,C.C.Shi.Convergence of nonstationary parallel multisplitting AOR method.Journal of Fudan University (Natural Science),1999,38(2):228-233
[114]Z.Q.Luo,P.Tseng.On the convergence of a matrix splitting algorithm for the symmetric monotone linear complementarity problem.SIAM Journal on Control and Optimization,1991,29:1037-1060
[115]O.L.Mangasarian.Solution of symmetric linear complementarity problems by iterative methods.Journal of Optimization Theory and Application,1977,22:465-485
[116]M.M.Martins.On the convergence of the modified overrelaxation method.Linear Algebra and its Applications,1986,81:55-73
[117]M.M.Martins.Some new results on the convergence of the SSOR and USSOR mehtods.Linear Algebra and its Applications,1988,106:185-193
[118]J.Mas,V.Migallon,J.Penad (?) s,D.B.Szyld.Nonstationary parallel relaxed multisplitting methods.Linear Algebra and its Applications,1992,241-243:733-747
[119]J.Mas,V.Migall(?)n,J.Penad(?)s,D.B.Szyld.Non-stationary parallel relaxed multisplitting methods.Linear Algebra and Its Applications,1996,241-243:733-748
[120] P. Monk. Finite element methods for Maxwell's equations. Oxford University Press, New York,2003
[121] M. F. Murphy, G. H. Golub, A. J. Wathen. A note on preconditioning for indefinite linear systems. SIAM Journal on Scientific Computing, 2000,21:1969-1972
[122] R. Nabben. A note on comparison theorems for splittings and multisplittings of Hermitian positive definite matrices. Linear Algebra and its Applications, 1996,233:67-80
[123] A. Neumaier. The extremal case of some matrix inequalities. Archiv der Mathematik, 1984,43:137-141
[124] M. Neumann, R. J. Plemmons. Convergence of parallel multisplitting iterative methods for M-matrices. Linear Algebra and its Applications, 1987,88-89:559-573
[125] J. Nocedal, S. J. Wright. Numerical Optimization. Springer, New York, 1999
[126] D. P. O'Leary, R. E. White. Multi-splittings of matrices and parallel solution of linear systems.SIAM Journal on Algebraic and Discrete Mathematics, 1985,6:630-640
[127] J. M. Ortega, W.C. Rheinboldt. Iterative solution of nonlinear equations in several variables. Academic Press, New York, 1970
[128] J. Y. Pan, M. K. Ng, Z. Z. Bai. New preconditioners for saddle point problems. Applied Mathematics and Computation, 2006, 172:762-771
[129] J. S. Pang. Necessary and sufficient conditions for the convergence of iterative methods for the linear complementarity problem. Journal of Optimization Theory and Application, 1984,42:1-17
[130] I. Perugia, D. Schotzau, P. Monk. Stabilized interior penalty methods for the time-harmonic Maxwell equations. Computer Methods in Applied Mechanics and Engineering, 2002,191:4675-4697
[131] T. Rees, C. Greif. A preconditioner for linear systems arising from interior point optimization methods. SIAM Journal on Scientific Computing, 2007, 29:1992-2007
[132] W. C. Rheinboldt. Methods for solving systems of nonlinear equations. SIAM, Philadelphia,1998
[133] W. C. Rheinholdt, J. S. Vandergrafy. A simple approach to the Perron-Frobenius theory for positive Iperators on general partially-ordered finite-dimension linear spaces. Mathematics of Computation, 1973, 27:139-145
[134] F. Robert, M. Charnay, F. Musy. Iterations chaotiques serie-parallel pour des equations non-lineaires de point fixe. Aplikace Matematiky, 1975, 20:1-38
[135] P. L. Rui, R. S. Chen. An efficient sparse approximate inverse preconditioning for FMM implementation. Microwave and Optical Technology Letters, 2007,49:1746-1750
[136] P. L. Rui, R. S. Chen, Edward K. N. Yung, C. H. Chan. Application of a two-step preconditionning strategy to the finite element analysis for electromagnetic problems. Microwave and Optical Technology Letters, 2006,48:1623-1627
[137] P. L. Rui, H. Yong, R. S. Chen. Multipreconditioned GMRES method for electromagnetic wave scattering problems. Microwave and Optical Technology Letters, 2008,50:150-152
[138] Y. Saad. Iterative methods for sparse linear systems. PWS Publishing Company, Boston, 1996
[139] Y. Saad. Krylov subspace method for solving large unsymmetric linear systems. Mathematics of Computation, 1981, 37:105-126
[140] C. Siefert, E.D. Sturler. Preconditioners for generalized saddle-point problems. SIAM Journal on Numerical Analysis, 2006,44:1275-1296
[141] D. J. Silvester, H .C. Elman, A. Ramage. IFISS:IncompressibIe Flow Iterative Solution Software. http://www.manchester.ac.uk/ifiss
[142] D. Silvester, A. Wathen. Fast iterative solution of stabilised Stokes systems part II: using general block preconditioners. SIAM Journal on Numerical Analysis, 1994, 31:1352-1367
[143] V. Simoncini. Block triangular preconditioners for symmetric saddle-point problems. Applied Numerical Mathematics, 2004,49:63-80
[144] V. Simoncini, M. Benzi. Spectral properties of the Hermitian and skew-Hermitian splitting pre-conditioner for saddle point problems. SIAM Journal on Matrix Analysis and Applications, 2004, 6:377-389
[145] T. Sogabe, S. L. Zhang. Extended conjugate residual methods for solving nonsymmetric linear systems. In Numerical Linear Algebra and Optimization, Edited by Ya-Xiang Yuan, Science Press, Beijing/New York, 2003, 88-99
[149] J. M. Song, C. C. Lu, W. C. Chew. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Trans Antennas Propag, 1997,45:1488-1493
[147] Y. Z. Song. Comparison theorems for splittings of matrices. Numerical Mathematics, 2002,92(3):563-591
[150] Y. Z. Song. Convergence of parallel multisplitting methods. Linear Algebra and its Applications, 1994,50:213-232
[149] Y. Song. On the convergence of the generalized AOR method. Linear Algebra and its Applications, 1997,256:199-218
[150]Y.Song,D.Yuan.On the convergence of relaxed parallel chaotic iterations for H-matrix.International Journal of Computer Mathematics,1994,52:195-209
[151]P.Sonneveld.CGS:a fast lanczos-type solver for nonsymmetric linear systems.SIAM Journal on Scientific and Statistical Computing,1989,10(1):36-52
[152]E.D.Sturler,H.A.van der Vorst.Reducing the effect of the global communication in GMRES(m) and CG on parallel distributed memory computers.Applied Numerical Mathematics,1985,18:441-459
[153]E.D.Sturler.A performance model for Krylov subspace methods on mesh-based parallel computers.Parallel Computing,1996,22:57-74
[154]E.D.Sturler,J.Liesen.Block-diagonal and constraint preconditioners for nonsymmertic indefinite linear systems.Part I:theory,SIAM Journal on Scientific Computing,2005,26:1598-1619
[155]D.B.Szyld,M.T.Jones.Two-stage and multisplitting methods for the parallel solution of linear systems.SIAM Journal on Matrix Analysis and Applications,1992,13:671-679
[156]E.Tyrtyshnikov.Optimal and superoptimal circulant preconditioners.SIAM Journal on Matrix Analysis and Applications,1992,13:459-473
[157]R.S.Varga.Matrix Iterative Analysis.Springer-Vedag,New York,2000
[158]H.A.van der Vorst.Bi-CGSTAB:a fast and smoothly converging variant of bi-CG for the solution of nonsymmetric linear systems.SIAM Journal on Scientific and Statistical Computing,1992,13:631-644
[159]H.A.Van der Vorst.Iterative Krylov Methods for Large Linear Systems.Cambridge Monographs on Applied and Computational Mathematics,Cambridge University Press,Cambridge,UK,2003
[160]C.L.Wang,Z.Z.Bai.Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices.Linear Algebra and its Application,2001,330:215-218
[161]C.L.Wang,T.Z.Huang.New convergence results for alternating methods.Journal of Computational and Applied Mathematics,2001,135:325-333
[162]C.L.Wang,J.H.Zhao.Further results on regular splittings and multisplittings.International Journal of Computer Mathematics,2005,82:421-431
[163]D.R.Wang.On the convergence of parallel multisplitting AOR algorithm.Linear Algebra and its Applications,1991,154/156:473-486
[164]王德人，白中治．大型线性最小二乘问题的并行矩阵多分裂松弛算法．高校计算数学学报，1993，15(1)：72-85
[165]L.Wang,Z.Z.Bai.Convergence conditions for splitting iteration methods for non-Hermitian linear systems.Linear Algebra and its Application,2008,428:453-468
[166]A.Wathen,D.Silvester.Fast iterative solution of stabilised Stokes systems part Ⅰ:using simple diagonal preconditioners.SIAM Journal on Numerical Analysis,1993,30:630-649
[167]A.Wathen,D.Silvester.Fast iterative solution of stabilized stokes systems Part Ⅱ:Using general block diagonal preconditioners.SIAM Journal on Numerical Analysis,1994,31:1352-1367
[168]R.E.White.Parallel algorithms for nonlinear problems.SIAM Journal on Algebraic and Discrete Methods,1986,7:137-149
[169]Z.I.Wo(?)nicki.AGA two-sweep iterative methods and their application for the solution of linear equation systems.Linear Algebra and its Application,1989,121:702-710
[170]Z.I.Wo(?)nicki.Nonnegative splitting theory.Japan Journal of Industrial and Applied Mathematics,1994,11:289-342
[171]Z.I.Wo(?)nicki.Comparison theorems for splittings of monotone matrices.Nonlinear analysis,1997,30:1251-1262
[172]Z.I.Wo(?)nicki.Basic comparison theorems for weak and weaker matrix splittings.Electronic Journal of Linear Algebra,2001,8:53-59
[173]吴健平，王正华，李晓梅．稀疏线性方程组的高效求解与并行计算．湖南科学技术出版社，湖南，2004
[174]S.X.Xu.Convergence of nonstationary parallel multisplitting SOR methods.Mathematical Theory and Application,2000,20(1):116-118
[175]T.R.Yang,H.X.Lin.The improved quasi-minimal residual method on massively distributed memory computers.In Proceedings of The International Conference on High Performance Computing and Networking (HPCN-97),April 1997
[176]T.R.Yang.The improved CGS method for large and sparse linear systems on bulk synchronous parallel architectures.2002 5th Intern.Conf.Algorithms and Architectures for Parallel Processing,IEEE Computer Society,2002,232-237
[177]T.R.Yang,R.P.Brent.The improved BiCGSTAB method for large and sparse nonsymmetric linear systems on parallel distributed memory architectures.2001 5th Intern.Conf.Algorithms and Architectures for Parallel Processing,IEEE Computer Society,2002,324-328
[178]T.R.Yang,R.P.Brent.The improved BiCG method for large and sparse linear systems on parallel distributed memory architectures.Information Journal,2003,6:349-360
[179]D.M.Young.Iterative Solution of Large Linear Systems.Academic Press,New York,1972
[180]J.H.Yun.Performance of ILU factorization perconditioners based on multisplitting.Numerical Mathematics,2003,95:761-779
[181]J.H.Yun.preconditioners for a block-tridiagonal matrix.The Journal of Supercomputing,2003,24:69-89
[182]J.H.Yun.Convergence of nonstationary multisplitting methods using ILU factorizations.Journal of Computational and Applied Mathematics,2005,180:245-263
[183]J.H.Yun.Convergence of SSOR multisplitting method for an H-matrix.Journal of Computational and Applied Mathematics,2008,217:252-258
[184]D.Yuan.On the convergence of parallel multisplitting asynchronous GAOR method for Hmatrix.Applied Mathematics and Computation,2005,160:477-485
[185]Y.Zhang.The USAOR iterative method for linear systems (in Chinese).Numerical Mathematics,1987,9(4):354-365
[186]张宝琳，谷同祥，莫则尧．数值并行计算原理与方法．北京：国防工业出版社，1999
[187]L.T.Zhang,T.Z.Huang,T.X.Gu.Global relaxed non-stationary multisplitting multiparameters methods.International Journal of Computer Mathematics,2008,85(2):211-224
[188]L.T.Zhang,T.Z.Huang,T.X.Gu,X.L.Guo.Convergence of relaxed multisplitting USAOR method for an H-matrix.Applied Mathematics and Computation,2008,202(1):121-132
[189]左宪禹．分布式并行环境下的Krylov子空间方法．硕士论文，河南师范大学，2003
[2] H. B. An, Z.-Z. Bai. NGLM:A globally convergent Newton-GMRES method. Mathematica Nu-merica Sinica, 2005, 27(2):151-174
[3] P. Arbenz, R. Geus. Multilevel preconditioned iterative eigensolvers for Maxwell eigenvalue problems. Applied Numerical Mathematics, 2005, 54:07-121
[4] J. Arnal, V. Migall(?)n, Jos(?) Penad(?). Non-stationary parallel multisplitting algorithms for almost linear systems. Numerical linear algebra and its applications, 1999,6:79-92
[5] Z. Z. Bai. On the convergence domains of the matrix multisplitting relaxed methods for linear systems. Applied Mathematics Journal of Chinese University, 1998,13b(l):45-52
[6] Z. Z. Bai. On the convergence of the multisplitting methods for the linear complementarity problem. SIAM Journal on Matrix Analysis and Application, 1999, 21:67-78
[7] Z. Z. Bai, D. J. Evans. Matrix multisplitting relaxation methods for linear complementarity problems. International Journal of Computer Mathematics, 1997,63:309-326
[8] Z. Z. Bai. Comparisons of the convergence and divergence rates of the parallel matrix multisplitting iteration methods. Chinese Journal of Engineering Mathematics, 1994,1:99-102
[9] Z. Z. Bai. Parallel nonlinear AOR method and its convergence. Computers and Mathematics with Applications, 1996, 31(2):21-31
[10] Z. Z. Bai, G. H. Golub, L. Z. Lu, J. F. Yin. Block-Triangular and skew-Hermitian splitting methods for positive definite linear systems. SIAM Journal on Scientific Computing, 2005, 26:844-863
[11] Z. Z. Bai, G. H. Golub, M. K. Ng. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM Journal on Matrix Analysis and Application,2003, 24:603-626
[12] Z. Z. Bai, G. H. Golub, M. K. Ng. On inexact Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. Linear Algebra and its Application, 2008,428:413-440
[13] Z. Z. Bai, G. H. Golub, M. K. Ng. On successive-overrelaxation acceleration of the Hermitian and skew-Hermitian splitting iteration. available online at http://wwwsccm. stanford.edu/wrap/pub-tech.html
[14] Z. Z. Bai, B. N. Parlett, Z. Q. Wang. On generalized successive overrelaxation methods for augmented linear systems. Numerische Mathematik, 2005,102:1-38
[15] R. E. Bank, B. D. Welfert, H. Yserentant. A class of iterative methods for sloving saddle point problems. Numerische Mathematik, 1990, 56:645-666
[16] M. Benzi, M. J. Gander, G. H. Golub. Optimization of the Hermitian and skew-Hermitian splitting iteration for saddle-point problems. BIT, 2003,43:881-900
[17] M. Benzi, G. H. Golub. A preconditioner for generalized saddle point problems. SIAM Journal on Matrix Analysis and Application, 2004,26:20-41
[26] M. Benzi, G. H. Golub, J. Liesen. Numerical solution of saddle point problems. Acta Numerica,2005,14:1-137
[19] M. Benzi, J. Liu. An efficient solver for the incompressible Navier-Stokes equation in rotation form. SIAM Journal on Scientific Computing, 2007,29:1959-1981
[25] M. Benzi, J. Liu. Block preconditioning for saddle point systems with indefinite (1,1) block. International Journal of Computer Mathematics, 2007,84:1117-1129
[21] M. Benzi, M. K. Ng. Preconditioned iterative methods for weighted Toeplitz least squares problems. SIAM Journal on Matrix Analysis and Application, 2006, 27:1106-1124
[27] M. Benzi, M. A. Olshanskii. An augmented lagrangian-based approach to the Oseen problem. SIAM Journal on Scientific Computing, 2006, 28:2095-2113
[23] M. Benzi, V. Simoncini. On the eigenvalues of a class of saddle point matrices. Numerische Mathematik, 2006, 103:173-196
[24] M. Benzi, D. B. Szyld. Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods. Numerische Mathematik, 1997,76:309-321
[25] M. Benzi J. Liu. Block preconditioning for saddle point systems with indefinite (1,1) block. International Journal of Computer Mathematics, 2007, 84(8): 1117-1129
[26] M. Benzi, G. H. Golub, J. Liesen. Numerical solution of saddle point problems. Acta Numerica,2005,14:1-137
[27] M. Benzi, M. A. Olshanskii. An augmented Lagrangian-based approach to the Oseen problem. SIAM Journal on Scientific Computing, 2006, 28:2095-2113
[28] L. Bergamaschi, J. Gondzio, G. Zilli. Preconditioning indefinite systems in interior point methods for optimization. Computational Optimization and Applications, 2004,28:149-171
[29] A. Berman, R. J. Plemmons. Nonnegative Matrices in the Mathematical Science, Academic Press, New York, third edition, 1979, Reprinted by SIAM, Philadelphia, 1994
[30] G. Birkhoff. Numerical Solution of Elliptic Equation. volume 1 of CBMS Regional Conference Series in Applied Mathematics, Society for industrial and Applied Mathematics, Philadelphia,1970
[31] P. N. Brown, Y. Saad. Hybrid Krylov methods for nonlinear systems of equations. SLAM Journal on Scientific and Statistical Computing, 1990,11:450-481
[32] R. Bru, V. Migallon, J. Penades, D.B. Szyld. Parallel, synchronous and asynchronous two-stage multisplitting methods. Electronic Transactions on Numerical Analysis, 1995,3:24-38
[33] R. Bru, L. Eisner, M. Neumman. Models of parallel chaotic iterative methods. Linear Algebra and Its Applications, 1988,103:175-192
[34] A. Berman, R. J. Plemmons. Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York, 1979
[35] H. M. Bucker, M. Sauren. A parallel version of the quasi-minimal residual method based on coupled two-term recurrences. In Proceedings of Workshop on Applied Parallel computing in Industrial Problems and Optimization (Para96), Technical University of Denmark, Lyngby, Denmark, Springer-Verlab, August 1996
[36] Z. H. Cao. Fast Uzawa algorithm for generalized saddle point problems. Applied Numerical Mathematics, 2003,46:157 - 171
[37] Z. H. Cao. Positive stable block triangular preconditioners for symmetric saddle point problems. Applied Numerical Mathematics, 2007, 57:899-910
[38] Z. H. Cao, Z. Y. Lin. Convergence of relaxed parallel multisplitting methods with different weighting schemes. Applied Mathematics and Computation, 1996, 106:181-196
[39] Z. H. Cao. On the convergence of nested stationary iterative methods. Linear Algebra and Its Applications, 1995,221:159-170
[40] R. Chan, X. Jin, M. Yeung. The circulant operator in the Banach algebra of matrices. Linear Algebra and its Applications, 1991, 149:41-53
[41] T. Chan. An optimal circulant preconditioner for Toeplitz systems. SIAM Journal on Scientific Computing, 1988,9:766-771
[42] D. W. Chang. Convergence analysis of the parallel multisplitting TOR methods. Journal of Computational and Applied Mathematics, 1996,72:169-177
[43] D. W. Chang. The parallel multisplitting TOR(MTOR) method for linear systems. Computers and Mathematics with Applications, 2001,41:215-227
[44] D. Chazan, W. Miranker. Chaotic relaxation. Linear Algebra and Its Applications, 1969, 2:199-222
[45] 陈国良著。并行计算-结构、算法、编程。高等教育出版社,北京,2003
[46] R. S. Chen, X. W. Ping, Edward K. N. Yung, C. H. Chan, Z. P. Nie, J. Hu. Application of diagonally perturbed incomplete factorization preconditioned conjugate gradient algorithms for edge finite-element analysis of helmholtz equations. IEEE Transactions on Antennas and Propagation,2006,54:1064-1068
[47] Z. Chen, Q. Du, J. Zou. Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SLAM Journal on Numerical Analysis, 2000,37:1542-1570
[48] C. M. Cheng, X. Q. Jin, S. W. Vong, W. Wang. A note on spectra of optimal and superoptimal preconditioned matrices. Linear Algebra and its Application, 2007, 422:482-485
[49] L. H. Chi, J. Liu, X. P. Liu, Q. F. Hu, X. M. Li. An improved conjugate residual algorithm for large symmetric linear systems. In Computational Physics, Proceedings of the Joint Conference of ICCP6 and CCP2003, Rinton Press, New Jersey, USA, 2005, 325-328
[50] K. A. Cliffe, T. J. Garratt, A. Spence. Eigenvalues of block matrices arising from problems in fluid mechanics. SIAM Journal on Matrix Analysis and Applications, 1994,15:1310-1318
[51] J. J. Climent, C. Perea. Some comparison theorems for weak nonnegative splittings of bounded operators. Linear Algebra and its Applications, 1998, 275/276:77-106
[52] J. J. Climent, C. Perea, L. Tortosa, A. Zamora. Convergence theorems for parallel alternating iterative methods. Applied Mathematics and Computation, 2004, 148:497-517
[53] J. J. Climent, C. Perea. Convergence and comparison theorems for multisplittings. Numerical Linear Algebra and Its applications, 1999, 6:93-107
[54] R. W. Cottle, G. B. Dantzig. Complementarity pivot theorey of mathematical programming. Linear Algebra and Its Applications, 1968, 1:103-125
[55] R. W. Cottle, G. H. Golub, R. S. Sacher. On the solution of large, structured linear complementarity problems: The block partitioned case. Applied Mathematics and Optimization, 1978,4:347-363
[56] C. W. Cryer. The solution of a quadratic programming problem using systematic overrelaxation.SIAM Journal of Control, 1971, 9:385-393
[57] P. Davis. Circulant Matrices. second ed., Chelsea Publishing, New York, 1994
[58] R. S. Dembo, S.C. Eisenstat, T. Steihaug. Inexact Newton methods. SIAM Journal on Numerical Analysis, 1982, 19:400-408
[59] L. Demkowicz, L. Vardapetyan. Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements. Computer Methods in Applied Mechanics and Engineering,1998,152:103-124
[60] J. W. Demmel. Applied Numerical Linear Algebra. SIAM, Philadelphia, 1997
[61] H. S. Dollar. Constraint-style preconditioners for regularized saddle point problems. SIAM Journal on Matrix Analysis and Application, 2007,29:672-684
[62] H. S. Dollar, A. J. Wathen. Approximate factorization constraint preconditioners for saddle-point matrices. SIAM Journal on Scientific Computing, 2006,27:1555-1572
[63] S. C. Eisenstat, H.F. Walker. Globally convergent inexact Newton methods. SIAM Journal on Optimization, 1994,4:393-422
[64] L. Eisner. Comparisons of weak regular splittings and multisplitting methods. Numerische Math-ematik, 1989,56:283-289
[65] L. Eisner, A. Frommer, R. Nabben, H. Schneider, D. B. Szyld. Conditions for strict inequality in comparisons of spectral radii of splittings of different matrices. Linear Algebra and its Applications, 2003, 263:65-80
[66] L. Eisner, M. Neumman, B. Vemmer. The effect the number of processors on the convergence of the parallel block Jacobi method. Linear Algebra and its Applications, 1991,154-156:311-330
[67] K. Fan. Topological proffs for certain theorems on matrices with nonnegative elements. Monat-shefte fur Mathematik, 1958,62:219-237
[68] 冯果忱。非线性方程组迭代解法。上海：上海科学技术出版社,1989
[69] R. Fletcher. Practical Methods of Optimization. 2nd edn (Chichester: John Wiley), 1987
[70] A. Forsgren, P. E. Gill, M. H. Wright. Interior methods for nonlinear optimization. SIAM Review,2002,44:525-597
[71] M. Fortin, R. Glowinski. Augmented Lagrangian Methods: Application to the Numerical Solution of Boundary-Value Problems (Amsterdam: North-Holland), 1983
[72] R. W. Freund, N. M. Nachtigal. QMR: a quasi-minimal residual method for non-Hermitian linear systems. Numerische Mathematik, 1991, 60:315-339
[73]A.Frommer,G.Mayer.Convergence of relaxed parallel multisplitting methods.Linear Algebra and its Applications,1989,119:141-152
[74]A.Frommer,G.Mayer.On the theory and practice of multisplitting methods in parallel computation.Computing,1992,49:62-74
[75]R.Fuster,V.Migall(?)n,J.Penad(?)s.Non-stationary parallel multisplitting AOR methods.Electronic Transaction On Numerical Analysis,1996,4:1-13
[76]T.B.Gan,T.Z.Huang.Simple criteria for nonsingular H-matrices.Linear Algebra and its Applications,2003,374:317-326
[77]干泰彬，黄廷祝．非奇H-矩阵的实用充分条件，计算数学，2004，1：109-116
[78]A.Gantner,R.H.W.Hoppe,D.K(?)ster,K.Siebert,A.Wixforth.Numerical simulation of piezo-electrically agitated surface acoustic waves on microfluidic biochips.Computing and Visualization in Science,2007,10:145-161
[79]R.Glowinski.Finite element methods for incompressible viscous flow,Handbook of Numerical Analysis.Volume IX,Numerical Methods for Fluids (Part 3),2003
[80]G.H.Golub,C.Greif.On solving block-structured indefinite linear systems.SIAM Joumal on Scientific Computing,2003,24:2076-2092
[81]G.H.Golub,C.Greif,J.M.Varah.An algebraic analysis of a block diagonal preconditioner for sadie point systems.SIAM Journal on Matrix Analysis and Applications,2006,27:779-792
[82]G.H.Golub,X.Wu,J.Y.Yuan.SOR-Like methods for augmented systems.BIT,2001,41:71-85
[83]J.Gopalakrishnan,J.E.Pasciak,L.F.Demkowicz.Analysis of a multigrid algo-rithm for time harmonic Maxwell equations.SIAM Journal on Numerical Analysis,2004,42:90-108
[84]C.Greif,D.Sch(?)tzau.Preconditioners for the discretized time-harmonic Maxwell equations in mixed form.Numerical Linear Algebra and Its Applications,2007,14:281-297
[85]谷同祥．大型稀疏线性代数方程组的并行非定常迭代方法．博士论文，中国工程物理研究院，2001
[86]T.X.Gu,J.Z.Li.Relaxed Parallel Multispliting Iterative Methods with Arbitrary Weighting Schemes.Journal of Xinyang Teachers College,1994,7(2):126-131
[87]T.X.Gu,X.P.Liu,L.J.Shen.Relaxed parallel two-stage multisplitting methods.International Journal of Computer Mathematics,2000,75:351-363
[88]T.X.Gu,X.P.Liu.Parallel two-stage multisplitting iterative methods.International Journal of Computer Mathematics,1998,20(2):153-166
[89]T.X.Gu,X.Y.Zuo,L.T.Zhang,W.Q.Zhang,Z.Q.Sheng.An improved Bi-Conjugate residual algorithm suitable for distributed parallel computing.Applied Mathematics and Computation,2007,186:1243-1253
[90]T.X.Gu,X.P.Liu,Z.Y.Mo.Multiple search direction conjugate gradient method I:methods and their propositions.International Journal of Computer Mathematics,2004,81(9):1133-1143
[91]R.A.Horn,C.A.Johnson.Matrix Analysis.Cambridge University Press,Cambridge,1985
[92]R.A.Horn,C.R.Johnson.Topics in matrix analysis.Cambridge University Press,Cambridge,England,1991
[93]P.Houston,A.Schneebeli,D.Sch(?)tzau.Mixed discontinuous Galerkin approximation of the Maxwell operator:the definite case.Mod(?)lisation Mathematique et Analyse Num(?)rique,2005,39:727-754
[94]胡家赣．线性代数方程组迭代解法．北京：科学出版社，1997
[95]T.Z.Huang,S.M.Zhong,Z.L.Li,Matrix Theory.Higher Education Press,Bei Jing,2003
[96]黄廷祝．非奇H-矩阵的简捷判据．计算数学，1993，3：318-327
[97]黄廷祝．Ostrowski定理的推广与非奇H矩阵的条件．计算数学，1994，1：19-24
[98]T.Z.Huang,J.Leng,E.Wachspress,Y.Tang.Characterization of H-matrices.Computers and Mathematics with Applications,2004,48:1587-1601
[99]A.N.Iusem.On the convergence of iterative methods for symmetric linear complementarity problems.Mathematical Programming,1993,59:33-48
[100]A.N.Iusem.On the convergence of iterative methods for nonsymmetric linear complementarity problems.Computational and Applied Mathematics,1991,10:27-41
[101]K.C.Jea,D.M.Young.Generalized conjugate gradient acceleration of nonsymmetrizable iterative methods.Linear Algebra and its Applications,1980,34:159-194
[102]C.Keller,N.I.M.Gould,A.J.Wathen.Constraint preconditioning for indefinite linear systems.SIAM Journal on Matrix Analysis and Application,2000,21:1300-1317
[103]C.T.Kelley.Iterative methods for linear and nonlinear equations.SIAM,Philadelphia,1995
[104]A.Klawonn.Block-triangular preconditioners for saddle point problems with a penalty term.SIAM Journal on Scientific Computing,1998,19:172-184
[105]P.E.Kloeden,D.Yuan.Comvergence of relaxed chaotic parallel iterative methods.Bulletin of the Australian Mathematical Society,1994,50:167-176
[106]J.Kuang,J.Ji.A survey of AOR and TOR methods.Journal of Computational and Applied Mathematics,1998,24:3-12
[107]V.Kumar,V.N.Rao V.Parallel Depth-First Search,Part II:Analysis.In Int' 1 J.of Parallel programming,1987,16(6):501-519
[108]S.J.Lai,B.Z.Wang,Y.Duan.Application of the RBF-based meshless method to solve 2-D time domain Maxwell's equations.Microwave and Millimeter Wave Technology,2008,2:749-751
[109]S.J.Lai,B.Z.Wang.Simulation of bounces on power distribution with finite-difference timedomain.Journal of Electromagnetic Waves and Applications,2008,22:1223-1231
[110]C.E.Lemke.Bimatrix equilibrium points and mathematical programming.Management Science,1965,11:681-689
[111]李建宇．解非线性方程组的牛顿-并行矩阵多分裂算法．四川师范大学学报(自然科学报)，1995，18(4)：51-55
[112]X.P.Liu,T.X.Gu,X.D.Hang,Z.Q.Sheng.A parallel version of QMRCGSTAB method for large linear systems in distributed parallel environments.Applied Mathematics and Computation,2006,172(2):744-752
[113]Z.Y.Liu,H.B.Wu,C.C.Shi.Convergence of nonstationary parallel multisplitting AOR method.Journal of Fudan University (Natural Science),1999,38(2):228-233
[114]Z.Q.Luo,P.Tseng.On the convergence of a matrix splitting algorithm for the symmetric monotone linear complementarity problem.SIAM Journal on Control and Optimization,1991,29:1037-1060
[115]O.L.Mangasarian.Solution of symmetric linear complementarity problems by iterative methods.Journal of Optimization Theory and Application,1977,22:465-485
[116]M.M.Martins.On the convergence of the modified overrelaxation method.Linear Algebra and its Applications,1986,81:55-73
[117]M.M.Martins.Some new results on the convergence of the SSOR and USSOR mehtods.Linear Algebra and its Applications,1988,106:185-193
[118]J.Mas,V.Migallon,J.Penad (?) s,D.B.Szyld.Nonstationary parallel relaxed multisplitting methods.Linear Algebra and its Applications,1992,241-243:733-747
[119]J.Mas,V.Migall(?)n,J.Penad(?)s,D.B.Szyld.Non-stationary parallel relaxed multisplitting methods.Linear Algebra and Its Applications,1996,241-243:733-748
[120] P. Monk. Finite element methods for Maxwell's equations. Oxford University Press, New York,2003
[121] M. F. Murphy, G. H. Golub, A. J. Wathen. A note on preconditioning for indefinite linear systems. SIAM Journal on Scientific Computing, 2000,21:1969-1972
[122] R. Nabben. A note on comparison theorems for splittings and multisplittings of Hermitian positive definite matrices. Linear Algebra and its Applications, 1996,233:67-80
[123] A. Neumaier. The extremal case of some matrix inequalities. Archiv der Mathematik, 1984,43:137-141
[124] M. Neumann, R. J. Plemmons. Convergence of parallel multisplitting iterative methods for M-matrices. Linear Algebra and its Applications, 1987,88-89:559-573
[125] J. Nocedal, S. J. Wright. Numerical Optimization. Springer, New York, 1999
[126] D. P. O'Leary, R. E. White. Multi-splittings of matrices and parallel solution of linear systems.SIAM Journal on Algebraic and Discrete Mathematics, 1985,6:630-640
[127] J. M. Ortega, W.C. Rheinboldt. Iterative solution of nonlinear equations in several variables. Academic Press, New York, 1970
[128] J. Y. Pan, M. K. Ng, Z. Z. Bai. New preconditioners for saddle point problems. Applied Mathematics and Computation, 2006, 172:762-771
[129] J. S. Pang. Necessary and sufficient conditions for the convergence of iterative methods for the linear complementarity problem. Journal of Optimization Theory and Application, 1984,42:1-17
[130] I. Perugia, D. Schotzau, P. Monk. Stabilized interior penalty methods for the time-harmonic Maxwell equations. Computer Methods in Applied Mechanics and Engineering, 2002,191:4675-4697
[131] T. Rees, C. Greif. A preconditioner for linear systems arising from interior point optimization methods. SIAM Journal on Scientific Computing, 2007, 29:1992-2007
[132] W. C. Rheinboldt. Methods for solving systems of nonlinear equations. SIAM, Philadelphia,1998
[133] W. C. Rheinholdt, J. S. Vandergrafy. A simple approach to the Perron-Frobenius theory for positive Iperators on general partially-ordered finite-dimension linear spaces. Mathematics of Computation, 1973, 27:139-145
[134] F. Robert, M. Charnay, F. Musy. Iterations chaotiques serie-parallel pour des equations non-lineaires de point fixe. Aplikace Matematiky, 1975, 20:1-38
[135] P. L. Rui, R. S. Chen. An efficient sparse approximate inverse preconditioning for FMM implementation. Microwave and Optical Technology Letters, 2007,49:1746-1750
[136] P. L. Rui, R. S. Chen, Edward K. N. Yung, C. H. Chan. Application of a two-step preconditionning strategy to the finite element analysis for electromagnetic problems. Microwave and Optical Technology Letters, 2006,48:1623-1627
[137] P. L. Rui, H. Yong, R. S. Chen. Multipreconditioned GMRES method for electromagnetic wave scattering problems. Microwave and Optical Technology Letters, 2008,50:150-152
[138] Y. Saad. Iterative methods for sparse linear systems. PWS Publishing Company, Boston, 1996
[139] Y. Saad. Krylov subspace method for solving large unsymmetric linear systems. Mathematics of Computation, 1981, 37:105-126
[140] C. Siefert, E.D. Sturler. Preconditioners for generalized saddle-point problems. SIAM Journal on Numerical Analysis, 2006,44:1275-1296
[141] D. J. Silvester, H .C. Elman, A. Ramage. IFISS:IncompressibIe Flow Iterative Solution Software. http://www.manchester.ac.uk/ifiss
[142] D. Silvester, A. Wathen. Fast iterative solution of stabilised Stokes systems part II: using general block preconditioners. SIAM Journal on Numerical Analysis, 1994, 31:1352-1367
[143] V. Simoncini. Block triangular preconditioners for symmetric saddle-point problems. Applied Numerical Mathematics, 2004,49:63-80
[144] V. Simoncini, M. Benzi. Spectral properties of the Hermitian and skew-Hermitian splitting pre-conditioner for saddle point problems. SIAM Journal on Matrix Analysis and Applications, 2004, 6:377-389
[145] T. Sogabe, S. L. Zhang. Extended conjugate residual methods for solving nonsymmetric linear systems. In Numerical Linear Algebra and Optimization, Edited by Ya-Xiang Yuan, Science Press, Beijing/New York, 2003, 88-99
[149] J. M. Song, C. C. Lu, W. C. Chew. Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects. IEEE Trans Antennas Propag, 1997,45:1488-1493
[147] Y. Z. Song. Comparison theorems for splittings of matrices. Numerical Mathematics, 2002,92(3):563-591
[150] Y. Z. Song. Convergence of parallel multisplitting methods. Linear Algebra and its Applications, 1994,50:213-232
[149] Y. Song. On the convergence of the generalized AOR method. Linear Algebra and its Applications, 1997,256:199-218
[150]Y.Song,D.Yuan.On the convergence of relaxed parallel chaotic iterations for H-matrix.International Journal of Computer Mathematics,1994,52:195-209
[151]P.Sonneveld.CGS:a fast lanczos-type solver for nonsymmetric linear systems.SIAM Journal on Scientific and Statistical Computing,1989,10(1):36-52
[152]E.D.Sturler,H.A.van der Vorst.Reducing the effect of the global communication in GMRES(m) and CG on parallel distributed memory computers.Applied Numerical Mathematics,1985,18:441-459
[153]E.D.Sturler.A performance model for Krylov subspace methods on mesh-based parallel computers.Parallel Computing,1996,22:57-74
[154]E.D.Sturler,J.Liesen.Block-diagonal and constraint preconditioners for nonsymmertic indefinite linear systems.Part I:theory,SIAM Journal on Scientific Computing,2005,26:1598-1619
[155]D.B.Szyld,M.T.Jones.Two-stage and multisplitting methods for the parallel solution of linear systems.SIAM Journal on Matrix Analysis and Applications,1992,13:671-679
[156]E.Tyrtyshnikov.Optimal and superoptimal circulant preconditioners.SIAM Journal on Matrix Analysis and Applications,1992,13:459-473
[157]R.S.Varga.Matrix Iterative Analysis.Springer-Vedag,New York,2000
[158]H.A.van der Vorst.Bi-CGSTAB:a fast and smoothly converging variant of bi-CG for the solution of nonsymmetric linear systems.SIAM Journal on Scientific and Statistical Computing,1992,13:631-644
[159]H.A.Van der Vorst.Iterative Krylov Methods for Large Linear Systems.Cambridge Monographs on Applied and Computational Mathematics,Cambridge University Press,Cambridge,UK,2003
[160]C.L.Wang,Z.Z.Bai.Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices.Linear Algebra and its Application,2001,330:215-218
[161]C.L.Wang,T.Z.Huang.New convergence results for alternating methods.Journal of Computational and Applied Mathematics,2001,135:325-333
[162]C.L.Wang,J.H.Zhao.Further results on regular splittings and multisplittings.International Journal of Computer Mathematics,2005,82:421-431
[163]D.R.Wang.On the convergence of parallel multisplitting AOR algorithm.Linear Algebra and its Applications,1991,154/156:473-486
[164]王德人，白中治．大型线性最小二乘问题的并行矩阵多分裂松弛算法．高校计算数学学报，1993，15(1)：72-85
[165]L.Wang,Z.Z.Bai.Convergence conditions for splitting iteration methods for non-Hermitian linear systems.Linear Algebra and its Application,2008,428:453-468
[166]A.Wathen,D.Silvester.Fast iterative solution of stabilised Stokes systems part Ⅰ:using simple diagonal preconditioners.SIAM Journal on Numerical Analysis,1993,30:630-649
[167]A.Wathen,D.Silvester.Fast iterative solution of stabilized stokes systems Part Ⅱ:Using general block diagonal preconditioners.SIAM Journal on Numerical Analysis,1994,31:1352-1367
[168]R.E.White.Parallel algorithms for nonlinear problems.SIAM Journal on Algebraic and Discrete Methods,1986,7:137-149
[169]Z.I.Wo(?)nicki.AGA two-sweep iterative methods and their application for the solution of linear equation systems.Linear Algebra and its Application,1989,121:702-710
[170]Z.I.Wo(?)nicki.Nonnegative splitting theory.Japan Journal of Industrial and Applied Mathematics,1994,11:289-342
[171]Z.I.Wo(?)nicki.Comparison theorems for splittings of monotone matrices.Nonlinear analysis,1997,30:1251-1262
[172]Z.I.Wo(?)nicki.Basic comparison theorems for weak and weaker matrix splittings.Electronic Journal of Linear Algebra,2001,8:53-59
[173]吴健平，王正华，李晓梅．稀疏线性方程组的高效求解与并行计算．湖南科学技术出版社，湖南，2004
[174]S.X.Xu.Convergence of nonstationary parallel multisplitting SOR methods.Mathematical Theory and Application,2000,20(1):116-118
[175]T.R.Yang,H.X.Lin.The improved quasi-minimal residual method on massively distributed memory computers.In Proceedings of The International Conference on High Performance Computing and Networking (HPCN-97),April 1997
[176]T.R.Yang.The improved CGS method for large and sparse linear systems on bulk synchronous parallel architectures.2002 5th Intern.Conf.Algorithms and Architectures for Parallel Processing,IEEE Computer Society,2002,232-237
[177]T.R.Yang,R.P.Brent.The improved BiCGSTAB method for large and sparse nonsymmetric linear systems on parallel distributed memory architectures.2001 5th Intern.Conf.Algorithms and Architectures for Parallel Processing,IEEE Computer Society,2002,324-328
[178]T.R.Yang,R.P.Brent.The improved BiCG method for large and sparse linear systems on parallel distributed memory architectures.Information Journal,2003,6:349-360
[179]D.M.Young.Iterative Solution of Large Linear Systems.Academic Press,New York,1972
[180]J.H.Yun.Performance of ILU factorization perconditioners based on multisplitting.Numerical Mathematics,2003,95:761-779
[181]J.H.Yun.preconditioners for a block-tridiagonal matrix.The Journal of Supercomputing,2003,24:69-89
[182]J.H.Yun.Convergence of nonstationary multisplitting methods using ILU factorizations.Journal of Computational and Applied Mathematics,2005,180:245-263
[183]J.H.Yun.Convergence of SSOR multisplitting method for an H-matrix.Journal of Computational and Applied Mathematics,2008,217:252-258
[184]D.Yuan.On the convergence of parallel multisplitting asynchronous GAOR method for Hmatrix.Applied Mathematics and Computation,2005,160:477-485
[185]Y.Zhang.The USAOR iterative method for linear systems (in Chinese).Numerical Mathematics,1987,9(4):354-365
[186]张宝琳，谷同祥，莫则尧．数值并行计算原理与方法．北京：国防工业出版社，1999
[187]L.T.Zhang,T.Z.Huang,T.X.Gu.Global relaxed non-stationary multisplitting multiparameters methods.International Journal of Computer Mathematics,2008,85(2):211-224
[188]L.T.Zhang,T.Z.Huang,T.X.Gu,X.L.Guo.Convergence of relaxed multisplitting USAOR method for an H-matrix.Applied Mathematics and Computation,2008,202(1):121-132
[189]左宪禹．分布式并行环境下的Krylov子空间方法．硕士论文，河南师范大学，2003