线性方程组分裂迭代法与广义鞍点问题Uzawa算法研究
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摘要
科学与工程的很多重要领域如计算电磁学，高阶微分方程求解，最优化问题，流体力学和油藏模拟等都离不开大型线性代数方程组的求解．大型稀疏线性方程组的求解方法研究已经成为大规模科学与工程计算的核心问题之一，具有重要的理论意义和实际应用价值．本文对求解大型稀疏线性代数方程组的一些迭代解法进行了深入的研究，特别系统研究了非Hermitian线性系统的收敛特性，反对称三角迭代法，交替迭代法的半收敛性理论及广义鞍点问题Uzawa类型算法等．
研究了非Hermitian矩阵线性系统单分裂的收敛性理论．首先，对矩阵的Hermitian和反Hermitian分裂(HSS)添加一个参数α，得到了变形的HSS分裂，利用新的分裂方法建立了非Hermitian正定矩阵单分裂的收敛性理论，给出了取特殊分裂时最优参数的选取方法．同时，将非Hermitian正定矩阵单分裂的收敛定理应用到广义交替迭代法和两步多分裂迭代法中，给出了两种方法的收敛理论．其次，利用与HSS分裂相类似的正规和反Hermitian分裂(NSS)方法，研究了非Hermitian不定矩阵单分裂收敛的等价条件，给出了非Hermitian不定矩阵NSS分裂的相关性质．同时，将所得结论用来判定矩阵是否具有对称占优性．
研究了两类特殊迭代方法—矩阵双分裂迭代法和反对称三角迭代方法．首先，建立了系数矩阵为两类特殊矩阵—H-矩阵和Hermitian正定矩阵时，双分裂迭代法的收敛性理论，并且得到了Hermitian正定矩阵双分裂的比较理论．这些理论为迭代法的选择提供了一些理论依据．其次，给出了反对称三角迭代法中迭代矩阵的两种新的选取方法，对文[65]进行了拓展，得到了反对称三角迭代法收敛的充分条件，对于最优参数的选择也做了相应的介绍．另外，给出了新方法中H_0的一些特殊选取，并得到了迭代法无条件收敛的理论结果．
研究了交替迭代方法．首先对各类交替迭代法，如经典交替迭代法，广义交替迭代法，并行同步迭代法，并行交替同步迭代法的模型一和模型二进行了简单介绍．其次，研究了当系数矩阵为奇异矩阵时各类交替迭代法的半收敛性理论，同时给出了各类交替迭代法的比较理论．
研究了鞍点问题中Uzawa类型的迭代解法．在对各类Uzawa类型算法进行回顾后，提出了三个带松弛因子的非线性Uzawa算法，即非线性Uzawa算法的变形，分析了各算法的收敛性问题，得到三个算法的收敛理论．同时，通过数值实验说明了引入松弛因子的必要性，实验结果表明，前两个带松弛因子的非线性Uzawa算法比原算法所需迭代数少．
Solutions of large-scale sparse linear algebraic systems are deeply involved in vari-ous scientific and engineering fields,such as computational electromagnetics,numericalsolutions of high-order differential equations,optimization problems,fluid mechanics andreservoir modeling.Moreover,research of methods for solving large-scale sparse systemsof linear algebraic equations becomes one of the key issues of large-scale scientific andengineering computing and such research has important theoretic significance and prac-tical applications.This doctoral dissertation deeply studies iteration methods to iterativesolutions of large-scale sparse linear algebraic equations.In particular,convergence char-acteristic of non-Hermitian linear systems,skew-symmetric triangular splitting iterativemethods,semiconvergence theorems of alternating iterative methods and Uzawa-type al-gorithms for generalized saddle point problems are deeply studied in this thesis.
Study convergence theorems of single splittings for linear systems with non-Hermitian matrices.We first present a new iterative methods based on Hermitian andskew-Hermitian splitting(HSS) by appending a parameterαto HSS splitting.Using thisnew method,we establish convergence theorems of single splittings for non-Hermitianpositive definite matrices and give a method for choosing the optimal parameters forthe special splitting.Moreover,we apply convergence theorems of single splittings fornon-Hermitian positive definite to generalized alternating methods and two-stage mul-tisplittings methods and present their convergence theorems.Secondly,with the help ofnormal and skew-Hermitian splitting(NSS),which are similar to HSS splitting,equivalentconditions and related properties for convergence of splitting of non-Hermitian indefinitematrices are presented.Moreover,we use the obtained conclusion to determine whethera matrix has a dominant symmetric part.
Study two classes of special iterative methods:double splitting iterative methods andskew-symmetric triangular iterative methods.We first present convergence theorems ofdouble splittings of Hermitian positive definite matrices and H-matrices.Furthermore,comparison theorems for double splittings of Hermitian positive definite matrices are alsoobtained,which provide theoretical base for the choice of iteration methods.Secondly,wecontribute two new methods for selection of iterative matrices of skew-symmetric triangu- lar iterative methods,extending those in [65] and get sufficient conditions for convergenceof skew-symmetric triangular iterative methods.Choices for the optimal parameters arealso introduced.In addition,we give some special selections of Ho for the new methodsand obtained theoretical results for the unconditional convergence of these methods.
Study alternating iterative methods.We first briefly introduce various types of alter-nating iterative methods,such as the classice alternating iterative methods,the generalizedalternating iterative methods,parallel synchronous iterative methods,parallel alternatingsynchronous iterative methods of model 1 and model 2.Then,we present semiconver-gence theorems of all kinds of alternating iterative methods with singular coefficient ma-trices.Moreover,comparison theorems for alternating iterative methods are also obtained.
Study Uzawa-type iterative methods for generalized saddle point problems,we re-call all types of Uzawa-type algorithms and then present three nonlinear Uzawa-type al-gorithms with relaxation parameters,which the extensions of the original ones.Further-more,convergence of the algorithms is discussed and numerical experiments verify thenecessity of the introduction of the relaxation parameters of these three algorithms,andshow that the nonlinear Uzawa-type algorithms with relaxation parameters requires lessiteration numbers than that of the original algorithms.
研究了非Hermitian矩阵线性系统单分裂的收敛性理论．首先，对矩阵的Hermitian和反Hermitian分裂(HSS)添加一个参数α，得到了变形的HSS分裂，利用新的分裂方法建立了非Hermitian正定矩阵单分裂的收敛性理论，给出了取特殊分裂时最优参数的选取方法．同时，将非Hermitian正定矩阵单分裂的收敛定理应用到广义交替迭代法和两步多分裂迭代法中，给出了两种方法的收敛理论．其次，利用与HSS分裂相类似的正规和反Hermitian分裂(NSS)方法，研究了非Hermitian不定矩阵单分裂收敛的等价条件，给出了非Hermitian不定矩阵NSS分裂的相关性质．同时，将所得结论用来判定矩阵是否具有对称占优性．
研究了两类特殊迭代方法—矩阵双分裂迭代法和反对称三角迭代方法．首先，建立了系数矩阵为两类特殊矩阵—H-矩阵和Hermitian正定矩阵时，双分裂迭代法的收敛性理论，并且得到了Hermitian正定矩阵双分裂的比较理论．这些理论为迭代法的选择提供了一些理论依据．其次，给出了反对称三角迭代法中迭代矩阵的两种新的选取方法，对文[65]进行了拓展，得到了反对称三角迭代法收敛的充分条件，对于最优参数的选择也做了相应的介绍．另外，给出了新方法中H_0的一些特殊选取，并得到了迭代法无条件收敛的理论结果．
研究了交替迭代方法．首先对各类交替迭代法，如经典交替迭代法，广义交替迭代法，并行同步迭代法，并行交替同步迭代法的模型一和模型二进行了简单介绍．其次，研究了当系数矩阵为奇异矩阵时各类交替迭代法的半收敛性理论，同时给出了各类交替迭代法的比较理论．
研究了鞍点问题中Uzawa类型的迭代解法．在对各类Uzawa类型算法进行回顾后，提出了三个带松弛因子的非线性Uzawa算法，即非线性Uzawa算法的变形，分析了各算法的收敛性问题，得到三个算法的收敛理论．同时，通过数值实验说明了引入松弛因子的必要性，实验结果表明，前两个带松弛因子的非线性Uzawa算法比原算法所需迭代数少．
Solutions of large-scale sparse linear algebraic systems are deeply involved in vari-ous scientific and engineering fields,such as computational electromagnetics,numericalsolutions of high-order differential equations,optimization problems,fluid mechanics andreservoir modeling.Moreover,research of methods for solving large-scale sparse systemsof linear algebraic equations becomes one of the key issues of large-scale scientific andengineering computing and such research has important theoretic significance and prac-tical applications.This doctoral dissertation deeply studies iteration methods to iterativesolutions of large-scale sparse linear algebraic equations.In particular,convergence char-acteristic of non-Hermitian linear systems,skew-symmetric triangular splitting iterativemethods,semiconvergence theorems of alternating iterative methods and Uzawa-type al-gorithms for generalized saddle point problems are deeply studied in this thesis.
Study convergence theorems of single splittings for linear systems with non-Hermitian matrices.We first present a new iterative methods based on Hermitian andskew-Hermitian splitting(HSS) by appending a parameterαto HSS splitting.Using thisnew method,we establish convergence theorems of single splittings for non-Hermitianpositive definite matrices and give a method for choosing the optimal parameters forthe special splitting.Moreover,we apply convergence theorems of single splittings fornon-Hermitian positive definite to generalized alternating methods and two-stage mul-tisplittings methods and present their convergence theorems.Secondly,with the help ofnormal and skew-Hermitian splitting(NSS),which are similar to HSS splitting,equivalentconditions and related properties for convergence of splitting of non-Hermitian indefinitematrices are presented.Moreover,we use the obtained conclusion to determine whethera matrix has a dominant symmetric part.
Study two classes of special iterative methods:double splitting iterative methods andskew-symmetric triangular iterative methods.We first present convergence theorems ofdouble splittings of Hermitian positive definite matrices and H-matrices.Furthermore,comparison theorems for double splittings of Hermitian positive definite matrices are alsoobtained,which provide theoretical base for the choice of iteration methods.Secondly,wecontribute two new methods for selection of iterative matrices of skew-symmetric triangu- lar iterative methods,extending those in [65] and get sufficient conditions for convergenceof skew-symmetric triangular iterative methods.Choices for the optimal parameters arealso introduced.In addition,we give some special selections of Ho for the new methodsand obtained theoretical results for the unconditional convergence of these methods.
Study alternating iterative methods.We first briefly introduce various types of alter-nating iterative methods,such as the classice alternating iterative methods,the generalizedalternating iterative methods,parallel synchronous iterative methods,parallel alternatingsynchronous iterative methods of model 1 and model 2.Then,we present semiconver-gence theorems of all kinds of alternating iterative methods with singular coefficient ma-trices.Moreover,comparison theorems for alternating iterative methods are also obtained.
Study Uzawa-type iterative methods for generalized saddle point problems,we re-call all types of Uzawa-type algorithms and then present three nonlinear Uzawa-type al-gorithms with relaxation parameters,which the extensions of the original ones.Further-more,convergence of the algorithms is discussed and numerical experiments verify thenecessity of the introduction of the relaxation parameters of these three algorithms,andshow that the nonlinear Uzawa-type algorithms with relaxation parameters requires lessiteration numbers than that of the original algorithms.
引文
[1] K. Arrow, L. Hurwicz, H. Uzawa. A studies in linear and nonliner programming. Stanford, CA: Stanford University Press, 1958
[2] O. Axelsson. Iterative solution methods. Cambridge University Press, Cambridge., 1996
[3] O. Axelsson, Z.Z. Bai, S.X. Qiu. A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part. Numer. Algorithms., 2004,35:351-372
[4] 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
[5] Z.Z. Bai, S.X. Qiu. Splitting-MINRES methods for linear systems with the coefficient matrix with a dominant indefinite symmetric part. Math. Numer. Sinica., 2002,24:113-128
[6] Z.Z. Bai, G.H. Golub, M.K. Ng. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl., 2003, 24:603-626
[7] Z.Z. Bai, G.H. Golub, J. Y. Pan. Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math., 2004, 98: 1-32
[8] 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 J. Sci. Comput., 2005, 26:844-863
[9] Z.Z. Bai, B.N. Parlett, Z.Q. Wang. On generalized successive overrelaxation methods for augmented linear systems. Numer. Math., 2005,102:1-38
[10] 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 Appl., 2008,428:413-440
[11] G.P. Baker, S.J. Yang. Semi-iterative and iterative methods for singular M-matrices. SIAM J. Matrix Anal. Appl., 1988,9:169-180
[12] M. Benzi, D. B. Szyld. Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods. Numer. Math., 1997,76:309-321
[13] M. Benzi, G.H. Golub. A preconditioner for generalized saddle point problems. SIAM J. Matrix Anal. Appl., 2004, 26:20-41
[14] A. Berman, R.J. Plemmons. Convergent nonnegative matrices and iterative methods for consistent linear systems. Numer. Math., 1978, 31:265-279
[15]A.Berman,R.J.Plemmons.Nonnegative matrices in the mathematical sciences,Academic Press,New York,1979
[16]F.Bertrand,P.A.Tanguy.Krylov-based Uzawa algorithms for the solution of stokes equations using discontinuous-pressure tetrahedral finite elements.J.Comput.Phys.,2002,181:617-638
[17]A.Bjock.Numerical stability of methods for sloving augmentde systems.Jerusalem:Proceedings of Recent Developments in Optimization Theory and Nonlinear Analysis.,1995
[18]M.A.Botchev,L.A.Krukier.About iterative solution of strongly nonsymmetric linear equation systems.Zh.Vychisl.Mat.i.Mat.Fiz.,1997,37(11):1283-1293(in Russian)
[19]J.H.Bramble,J.E.Pasciak and A.T.Vassilev.Analysis of the inexact Uzawa algorithm for saddle point problems.SIAM.J.Numer.Anal.,1997,34:1072-1092
[20]J.H.Bramble,I.E.Pasciak,A.T.Vassilev.Uzawa type algorithms for non-symmetric saddle point problems.Math.Comput.,2000,69:667-689
[21]R.Bru,L.Elsner,M.Neumann.Models of parallel chaotic iteration methods.Linear Algebra Appl.,1988,103:175-192
[22]R.Bru,V.Migallon,J.Pinades.Chaotic methods for the parallel solution of linear systems.Compu System Engin.,1995,5:385-390
[23]曹志浩．数值线性代数．复旦大学出版社，上海．1996，95-107
[24]Z.H.Cao.Fast Uzawa algorithm for generalized saddle point problems.Appl.Numer.Math.,2003,46:157-171
[25]Z.H.Cao.Fast Uzawa algorithm for sloving non-symmetric stabilized saddle point problems.Numer.Linear Algebra Appl.,Vol 11,No.1 2004:1-24
[26]R.H.Chan,M.K.Ng.Conjugate gradient methods for Toepliz systems.SIAM Review.,1996,38:427-482
[27]X.Chen.On preconditioned Uzawa methods and SOR methods for saddle point problems.J.Comput.Appl.Math.,1998,100:207-224
[28]J.H.Chen,W.G.Li.Inexact Newton-splitting methods for non-symmetric nonlinear problems with a dominant symmetric part.Chinese J.Numer.Math.Appl.,2005,27:32-47
[29]J.H.Chen,W.G.Li.Equivalent Conditions for convergence of splittings of non-Hermitian indefinite matrices.J.Sci.Comput.,2006,30:117-130
[30]Y.L.Chen,X.Y.Tan.Semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices.Appl.Mathe.Comput.,2005,167:930-956
[31] J.J. Climent, C. Perea. Some comparison theorems for weak nonnegative splittings of bounded operators. Linear Algebra Appl., 1998,275/276: 77-106
[32] J.J. Climent, C. Perea. Convergence and comparison theorems for multisplittings, Numeri. Linear Algebra Appl., 1999,6: 93-107
[33] J.J. Climent, C. Perea. Convergence and compariso theorems for a generalized alternating iterative method. Appl. Math. Comp., 2003,143:1-14
[34] J.J. Climent, C. Perea, L. Tortosa. A. Zamora. Convergence theorems for parallel alternating iterative methods. Appl. Math. Comput., 2004,148:497-517
[35] V. Conrad, Y. Wallach. Alterating methods for sets of linear equations. Numer. Mathe., 1979,32:105-108
[36] M.-R. Cui. Analysis of iterative algorithms of Uzawa type for saddle point problems. Appl. Numer. Math., 2004,50:133-146
[37] H. Elman, G.H. Golub. Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal., 1994,31:1645-1661
[38] H.C. Elman, D.J. Silvester. Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations. SIAM J. Sci. Comput., 1996,17:33-46
[39] H.C. Elman, D.J. Silvester. A.J. Wathen. Finite elements and fast iterative solvers: with application in incompressible fluid dynamics. Oxford: Oxford University Press., 2005.
[40] L. Eisner. Comparisons of weak regular splittings and multisplitting methods. Numer. Math.,1989,56:283-289
[41] L. Eisner, Kh.D. Ikramov. Normal matrices: An update. Numer. Math., Linear Algebra Appl.,1998,285:291-303
[42] M. Fortin, R. Golwninski. Augmentde lagrangian methods: Applications to the numerical solution of boundary-value problems. Amsterdam: Elsevier Science Publishers B.V., 1983:47-95
[43] A. Frommer, G. Mayer. Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl., 1989, 119:141-152
[44] A. Frommer, G. Mayer. On the theory and practice of multisplitting methods in parallel computation. Computing., 1992,49:63-74
[45] A. Frommer, B. Pohla. A comparison result for multisplittings and waveform relaxation methods.Numer. Linear Algebra Appl., 1995, 2: 335-346
[46] R.E. Funderlic, J.B. Mankin. Solution of homogeneous systems of linear equational model. SIAM J. Sci. Statist. Comput., 1981, 2:375-383
[47]P.E.Gill,W.Murray,M.H.Wright.Practical optimization[M].Academic Press,New York.,1981
[48]G.H.Golub,R.S.Varga.Chebyshev semi-iterative methods,successive overrrelaxation iterative methods,and second order Richardson iterative methods.I.Numer.Math.,1961,3:147-168
[49]G.H.Golub,X.Wu,J.Y.Yuan.SOR-Like methods for augmented systems.BIT,2001,41:71-85
[50]R.Grone,C.R.Johnson,E.M.Sa,H.Wolkowicz.Normal matrices.Linear Algebra Appl.,1987,87:213-225
[51]W.Hackbusch.Iterative solutions of sparse linear systems.Academic Press,New York.1994
[52]A.Hadjidimos.On the optimization of the classical iterative schemes for the solution of complex singular linear systems.SIAM J.Algebra Discrete Meth.,1985,6:555-566
[53]L.Hemmingsson,K.Otto.Analysis of semi-Toepliz preconditioners for first-order PDEs.SIAM J.Sci.Comput.,1996,17:47-64
[54]Q.Hu,J.Zou.Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems.Numer.Math.,2002,93:333-359
[55]Y.Huang,Y.Song.Semiconvergence of block AOR method for singular p-cyclic matrices.Appl.Mathe.Comput.,2007,189:1637-1647
[56]M.Huhtanen,O.Nevanlinna.Minimal decompositions and iterative methods.Numer.Math.,2000,86:257-281
[57]W.Joubert.A robust GMRES-based adaptive polynomial preconditioned algorithm for nonsymmetric linear systems.SIAM J.Sci.Comput.,1994,15:427-439
[58]L.Kaufman.Matrix methods for queuing problems.SIAM J.Sci.Statist.Comput.,1983,4:525-552
[59]A.Klawonn.Block-triangular preconditioners for saddle point problems with a penalty term.SIAM J.Sci.Comput.,1998,19:172-184
[60]L.A.Krukier.Iterative method solution of implicit difference schemes approximated for one class of quasilinear equation systems.Izv.Vyssh.Uchebn.Zaved.Mat.,1979,7:41-52
[61]L.A.Kru(?)kier.Construction of the operator B in implicit two-layer iterative schemes for providing convergence in the case of dissipative operator A.Izv.Vyssh.Uchebn.Zaved.Mat.,1983,5:41-47
[62]L.A.Krukier.Mathematical modelling of the Azov sea hydrodynamics in realizing the projects of reconstructing its ecosystem.,Math.Modelling.,1991,3:3-20
[63]L.A.Krukier,L.G.Chikina.The use of the iterative methods for solution of the steady convection-diffusion equations in the incompressible medium with predominating convection.Proc.International Conference on the Use of the Mathematical Modelling for the Solution of Problems in Science and Technic,Izevsk.,1996:246-257
[64]L.A.Krukier.Skew-symmetric preconditioners for strongly nonsymmetric linear equation systems.Proc.of 15th IMACS World Congress of Scientific Computation,Berlin.,1997,Vol 2:533-538
[65]L.A.Krukier.Convergence acceleration of triangular iterative methods based on the skewsymmetric part of the matrix.Appl.Numer.Math.,1999,30:281-290
[66]L.A.Krnkier,L.G.Chikina and T.V.Belokon.Triangular skew-symmetric iterative solvers for strongly nonsymmetric positive real linear system of equations.Appl.Numer.Math.,2002,41:89-105
[67]Y.Q.Lin,Y.H.Cao.A new nonlinear Uzawa algorithm for generalized saddle point problems.Appl.Mathe.Comput.,2006,175:1432-1454
[68]W.M.Lin,S.Xu.A new improved Uzawa method for finite element solution of stokes problem.Comput.Mech.,2001,27:305-310
[69]刘国新，王卫国．关于结构KKT方程组的扰动分析．计算数学．，2004，26(2)：179-188
[70]W.Li,W.Sun.Comparison results for parallel multisplitting methods with applications to AOR methods.Linear Algebra Appl.,2001,331:131-144
[71]W.Li,L.Elsner,L.Z.Lu.Comparisons of spectral radii and the theorem of Stein-Rosenberg.Linear Algebra Appl.,2002,348:283-287
[72]T.A.Manteuffel,G.Starke.On hybrid iterative methods for nonsymmetric systems of linear equations.Numer.Math.,1996,73:489-506
[73]I.Marek,D.B.Szyld.Comparison theorems for weak splittings of bounded operators.Numer.Mathe.,1990,58:389-397
[74]C.D.Meyer.R.J.Plemmons.Convergent powers of a matrix with applications to iterative methods for singular linear systems.SIAM J.Numer.Anal.,1977,14:699-705
[75]V.A.Miller,M.Neumann.A note on comparison theorems for nonnegative matrices.Numer.Math.,1985,47:427-434
[76]M.F.Murphy,G.H.Golub,A.J.Wathen.A note on preconditioning for indefinite linear systems.Tech.Report,SCCM-99-03,Stanford University,1999
[77]M.F.Murphy,G.H.Golub,AJ.Wathen.A note on preconditioning for indefinite linear systems.SIAM J.Sci.Comput.,2000,21:1969-1972
[78]R.Nabben.A note on comparison theorems for splittings and multisplittings of Hermitian positive definite matrices.Linear Algebra Appl.,1996,233:67-80
[79]N.M.Nachtigal,L.Reichel and L.N.Trefethen.A hybrid GMRES algorithm for nonsymmetric matrix iterations.SIAM J Matrix Anal.Appl.,1992,13:796-825
[80]M.Neumann,R.].Plemmons.Convergent nonnegative matrices and iterative methods for consisitent linear systems.Numer,Math.,1978,31:265-279
[81]M.Neumann,R.J.Plemmons.Convergence of parallel multisplitting iterative methods for Mmatrices.Linear Algebra Appl.,1987,88/89:559-573
[82]H.Nochetto,J.H.Pyo.Optimal relaxation parameter for the Uzawa method.Numer.Math.,2004,98:695-702
[83]D.P.O'Leary,R.E.White.Multisplittings of matrices and parallel solution of linear systems.SIAM J.Algebraic Discrete Methods,1985,6:630-640
[84]J.M.Ortega,W.C.Rheinboldt.Monotone iterations for nonlinear equations with application to Gauss-Seidel methods.SIAM J.Numer.Anal.,1967,4:171-190
[85]J.M.Ortega,and R.J.Plemmons.Extensions of the Ostrowski-Reich theorem for SOR iterations.Linear Algebra Appl.,1979,28:177-191
[86]J.M.Ortega.Numerical analysis.A Sencond Course,Academic Press,New York,Reprinted by SIAM,Philadelphia,1990
[87]D.Peaceman,H.Rachford.The numerical solution of elliptic and parabolic differential equations.J.SIAM,1955,3:28-41
[88]R.J.Plemmons.M-matrices leading to semiconvergent splittings.Linear Algebra Appl.,1976,15:243-252
[89]T.Rees,C.Greif.A preconditioner for linear systems arising from interior point optimization methods.SIAM J.Sci.Comput.,2007,29(5):1992-2007
[90]A.Ruhe.Closest normal matrix finally found.BIT.,1987,27:585-598
[91]T.Rusten,T.Winther.A preconditioned iterative method for saddle point problems.SIAM J.Matrix Anal.Appl.,1992,13:887-904
[92]Y.Saad,M.H.Schultz.GMRES:A generalized minimal residual algorithm for solving nonsymmetric linear systems.SIAM J.Sci.Star.Comput.,1986,7:856-869
[93] Y. Saad. Iterative methods for sparse linear systems. Boston: PWS Publishing Company.,1996:240-243
[94] S.Q. Shen, T.Z. Huang. Convergence and comparison theorems for double splittings of matrices.Comp. Math. Appl., 2006,51:1751-1760
[95] S.Q. Shen, T.Z. Huang, J.L. Shao. Convergence and comparison results for double splittings of Hermitian positive definite matrices. CALCLOL., 2007, 44:127-135
[96] S.Q. Shen, T.Z. Huang. New comparison results for parallel multisplitting iterative methods. Appl. Math. Comput., 2008,206:738-747
[97] A.A. Samarskij, E.S. Nikolaev. Numerical methods for grid equations. Vol. 2:Iterative Meth-ods(Birkhauser, Basel)., 1989
[98] C. Siefert, E.D. Sturler. Preconditioners for generalized saddle-point problems. SLAM J. Numer.Anal., 2006,44:1275-1296
[99] V. Simoncini, M. Benzi. Spectral properties of the Hermitian and skew-Hermitian splitting pre-conditioner for saddle point problems. SLAM J. Matrix Anal. Appl., 2004,6:377-389
[100] Y. Song. Some comparison theorems for nonnegative splittings of matrices. Numer. Math.,1993,65:245 - 252
[101] Y. Song. P-cyclic matrices and block accelerated overrelaxation method., Math. Appl., 1997,10(2):32-36
[102] Y. Song. Semiconvergence of extrapolated iterative methods for singular linear systems. J. Comput. Appl. Math., 1999,106:117-129
[103] Y. Song. Semiconvergence of nonnegative splittings for singular matrices. Numer. Math., 2000,85:109-127
[104] Y. Song. Semiconvergence of block SOR method for singular linear systems with p-cyclic matrices. J. Comput. Appl. Math., 2001,130:217-229
[105] Y. Song, L. Wang. On the semiconvergence of extrapolated iterative methods for singular linear systems. Appl. Numer. Math., 2003,44:401-413
[106] D.J. Silvester, H.C. Elman, A. Ramage. LFISS: Incompressible flow iterative solution sofeware. http://www.manchester.ac.uk/ifss
[107] E.D. Sturler, J. Liesen. Block-diagonal and constraint preconditioners for nonsymmertic indefinite linear systems. Part I:theory. SLAM J. Sci. Comput., 2005, 26:1598-1619
[108] D. B. Szyld, M. T. Jones. Two-stage and multisplitting methods for the parallel solution of linear systems. SLAM J. Matrix Anal. Appl., 1992,13: 671-679
[109]R.S.Varga.Matrix iterative analysis.Prentice-Hall:Englewood Cliffs,NJ,1962
[110]E.L.Wachspress.Iterative solution of elliptic systems and applications to the neutron equations of reactor physics.Prentice-Hall,Englewood Cliffs,NJ,1966
[111]D.Wang.On the convergence of the parallel multisplitting AOR algorithm.Linear Algebra Appl.,1991,154/156:473-486
[112]C.L.Wang,Z.Z.Bai.Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices.Linear Algebra Appl.,2001,330:215-218
[113]C.L.Wang.T.Z.Huang.New convergence results for alternating methods.J.Comput.Appl.Math.,2001,135:325-333
[114]C.L.Wang.Nonstationary multisplittings with general weighting matrices for non-Hermitian positive definite systems.Appl.Mathe.Lett.,2003,16:919-924
[115]C.L.Wang,J.H.Zhao.Further results on regular splittings.and multisplittings.Internat.J.Comput.Math.,2005,82:421-431
[116]王国荣．矩阵与算子广义逆．科学出版社，北京．1998
[117]L.Wang,Z.Z.Bai.Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts.BIT.Numer.Mathe.,2004,11:363-386
[118]L.Wang.Semiconvergence of two-stage iterative methods for singular linear systems.Linear Algebra Appl.,2007,422:824-838
[119]L.Wang,Z.Z.Bal.Convergence conditions for splitting iteration methods for non-Hermitian linear systems.Linear Algebra Appl.,2008,428:453-468
[120]A.Wathen,D.Silvester.Fast iterative solution of stabilised Stokes systems part Ⅰ:using simple diagonal preconditioners.SIAM J.Numer.Anal.,1993,30:630-649
[121]A.Wathen,D.Silvester.Fast iterative solution of stabilised Stokes systems part Ⅱ:using general block diagonal preconditioners.SIAM J.Numer.Anal.,1994,31:1352-1367
[122]Y.M.Wei.A characterization and representation of the Drazin inverse.SIAM J.Matrix Anal.Appl.,1996,17:744-747
[123]Y.M.Wei.Index splitting for the Drazin inverse and the singular linear system.Appl.Math.Comput.,1998,95:115-124
[124]R.E.White.Multisplitting of a symmetric positive definite matrix.SIAM J.Matrix Anal.Appl.,1990,11:69-82
[125]Z.I.Woznicki.Estimation of the optimum relaxation factors in partial factorization iterative methods.SIAM.J.Matrix Anal.Appl.,1993,14:59-73
[126]Z.I.Wo(?)nicki.Nonnegative splitting theory.Japan J.Indust.Appl.Math.,1994,11:289-342
[127]Z.I.Wo(?)nicki.Comparison theorems for splittings of monotone matrices.Nonlinear analysis,1997,30:1251-1262
[128]Z.I.Wo(?)nicki.Basic comparison theorems for weak and weaker matrix splittings.Electron.J.Linear Algebra,2001,8:53-59
[129]D.M.Young.Iterative solution of large linear systems.Academic Press,New York,1971
[130]J.Y.Yuan.The Ostrowski-Reich theorem for SOR iterations:extensions to the rank deficient case.Linear Algebra Appl.,2000,315:189-196
[131]A.Zsaki,D.Rixen,M.Paraschivoiu.A substructure based iterative inner solver coupled with Uzawa's algorithm for the stokes problem.Internat.J.Numer.Meth.Fluids.,2003,43:215-230
[2] O. Axelsson. Iterative solution methods. Cambridge University Press, Cambridge., 1996
[3] O. Axelsson, Z.Z. Bai, S.X. Qiu. A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part. Numer. Algorithms., 2004,35:351-372
[4] 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
[5] Z.Z. Bai, S.X. Qiu. Splitting-MINRES methods for linear systems with the coefficient matrix with a dominant indefinite symmetric part. Math. Numer. Sinica., 2002,24:113-128
[6] Z.Z. Bai, G.H. Golub, M.K. Ng. Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems. SIAM J. Matrix Anal. Appl., 2003, 24:603-626
[7] Z.Z. Bai, G.H. Golub, J. Y. Pan. Preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite linear systems. Numer. Math., 2004, 98: 1-32
[8] 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 J. Sci. Comput., 2005, 26:844-863
[9] Z.Z. Bai, B.N. Parlett, Z.Q. Wang. On generalized successive overrelaxation methods for augmented linear systems. Numer. Math., 2005,102:1-38
[10] 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 Appl., 2008,428:413-440
[11] G.P. Baker, S.J. Yang. Semi-iterative and iterative methods for singular M-matrices. SIAM J. Matrix Anal. Appl., 1988,9:169-180
[12] M. Benzi, D. B. Szyld. Existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods. Numer. Math., 1997,76:309-321
[13] M. Benzi, G.H. Golub. A preconditioner for generalized saddle point problems. SIAM J. Matrix Anal. Appl., 2004, 26:20-41
[14] A. Berman, R.J. Plemmons. Convergent nonnegative matrices and iterative methods for consistent linear systems. Numer. Math., 1978, 31:265-279
[15]A.Berman,R.J.Plemmons.Nonnegative matrices in the mathematical sciences,Academic Press,New York,1979
[16]F.Bertrand,P.A.Tanguy.Krylov-based Uzawa algorithms for the solution of stokes equations using discontinuous-pressure tetrahedral finite elements.J.Comput.Phys.,2002,181:617-638
[17]A.Bjock.Numerical stability of methods for sloving augmentde systems.Jerusalem:Proceedings of Recent Developments in Optimization Theory and Nonlinear Analysis.,1995
[18]M.A.Botchev,L.A.Krukier.About iterative solution of strongly nonsymmetric linear equation systems.Zh.Vychisl.Mat.i.Mat.Fiz.,1997,37(11):1283-1293(in Russian)
[19]J.H.Bramble,J.E.Pasciak and A.T.Vassilev.Analysis of the inexact Uzawa algorithm for saddle point problems.SIAM.J.Numer.Anal.,1997,34:1072-1092
[20]J.H.Bramble,I.E.Pasciak,A.T.Vassilev.Uzawa type algorithms for non-symmetric saddle point problems.Math.Comput.,2000,69:667-689
[21]R.Bru,L.Elsner,M.Neumann.Models of parallel chaotic iteration methods.Linear Algebra Appl.,1988,103:175-192
[22]R.Bru,V.Migallon,J.Pinades.Chaotic methods for the parallel solution of linear systems.Compu System Engin.,1995,5:385-390
[23]曹志浩．数值线性代数．复旦大学出版社，上海．1996，95-107
[24]Z.H.Cao.Fast Uzawa algorithm for generalized saddle point problems.Appl.Numer.Math.,2003,46:157-171
[25]Z.H.Cao.Fast Uzawa algorithm for sloving non-symmetric stabilized saddle point problems.Numer.Linear Algebra Appl.,Vol 11,No.1 2004:1-24
[26]R.H.Chan,M.K.Ng.Conjugate gradient methods for Toepliz systems.SIAM Review.,1996,38:427-482
[27]X.Chen.On preconditioned Uzawa methods and SOR methods for saddle point problems.J.Comput.Appl.Math.,1998,100:207-224
[28]J.H.Chen,W.G.Li.Inexact Newton-splitting methods for non-symmetric nonlinear problems with a dominant symmetric part.Chinese J.Numer.Math.Appl.,2005,27:32-47
[29]J.H.Chen,W.G.Li.Equivalent Conditions for convergence of splittings of non-Hermitian indefinite matrices.J.Sci.Comput.,2006,30:117-130
[30]Y.L.Chen,X.Y.Tan.Semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices.Appl.Mathe.Comput.,2005,167:930-956
[31] J.J. Climent, C. Perea. Some comparison theorems for weak nonnegative splittings of bounded operators. Linear Algebra Appl., 1998,275/276: 77-106
[32] J.J. Climent, C. Perea. Convergence and comparison theorems for multisplittings, Numeri. Linear Algebra Appl., 1999,6: 93-107
[33] J.J. Climent, C. Perea. Convergence and compariso theorems for a generalized alternating iterative method. Appl. Math. Comp., 2003,143:1-14
[34] J.J. Climent, C. Perea, L. Tortosa. A. Zamora. Convergence theorems for parallel alternating iterative methods. Appl. Math. Comput., 2004,148:497-517
[35] V. Conrad, Y. Wallach. Alterating methods for sets of linear equations. Numer. Mathe., 1979,32:105-108
[36] M.-R. Cui. Analysis of iterative algorithms of Uzawa type for saddle point problems. Appl. Numer. Math., 2004,50:133-146
[37] H. Elman, G.H. Golub. Inexact and preconditioned Uzawa algorithms for saddle point problems. SIAM J. Numer. Anal., 1994,31:1645-1661
[38] H.C. Elman, D.J. Silvester. Fast nonsymmetric iterations and preconditioning for Navier-Stokes equations. SIAM J. Sci. Comput., 1996,17:33-46
[39] H.C. Elman, D.J. Silvester. A.J. Wathen. Finite elements and fast iterative solvers: with application in incompressible fluid dynamics. Oxford: Oxford University Press., 2005.
[40] L. Eisner. Comparisons of weak regular splittings and multisplitting methods. Numer. Math.,1989,56:283-289
[41] L. Eisner, Kh.D. Ikramov. Normal matrices: An update. Numer. Math., Linear Algebra Appl.,1998,285:291-303
[42] M. Fortin, R. Golwninski. Augmentde lagrangian methods: Applications to the numerical solution of boundary-value problems. Amsterdam: Elsevier Science Publishers B.V., 1983:47-95
[43] A. Frommer, G. Mayer. Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl., 1989, 119:141-152
[44] A. Frommer, G. Mayer. On the theory and practice of multisplitting methods in parallel computation. Computing., 1992,49:63-74
[45] A. Frommer, B. Pohla. A comparison result for multisplittings and waveform relaxation methods.Numer. Linear Algebra Appl., 1995, 2: 335-346
[46] R.E. Funderlic, J.B. Mankin. Solution of homogeneous systems of linear equational model. SIAM J. Sci. Statist. Comput., 1981, 2:375-383
[47]P.E.Gill,W.Murray,M.H.Wright.Practical optimization[M].Academic Press,New York.,1981
[48]G.H.Golub,R.S.Varga.Chebyshev semi-iterative methods,successive overrrelaxation iterative methods,and second order Richardson iterative methods.I.Numer.Math.,1961,3:147-168
[49]G.H.Golub,X.Wu,J.Y.Yuan.SOR-Like methods for augmented systems.BIT,2001,41:71-85
[50]R.Grone,C.R.Johnson,E.M.Sa,H.Wolkowicz.Normal matrices.Linear Algebra Appl.,1987,87:213-225
[51]W.Hackbusch.Iterative solutions of sparse linear systems.Academic Press,New York.1994
[52]A.Hadjidimos.On the optimization of the classical iterative schemes for the solution of complex singular linear systems.SIAM J.Algebra Discrete Meth.,1985,6:555-566
[53]L.Hemmingsson,K.Otto.Analysis of semi-Toepliz preconditioners for first-order PDEs.SIAM J.Sci.Comput.,1996,17:47-64
[54]Q.Hu,J.Zou.Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems.Numer.Math.,2002,93:333-359
[55]Y.Huang,Y.Song.Semiconvergence of block AOR method for singular p-cyclic matrices.Appl.Mathe.Comput.,2007,189:1637-1647
[56]M.Huhtanen,O.Nevanlinna.Minimal decompositions and iterative methods.Numer.Math.,2000,86:257-281
[57]W.Joubert.A robust GMRES-based adaptive polynomial preconditioned algorithm for nonsymmetric linear systems.SIAM J.Sci.Comput.,1994,15:427-439
[58]L.Kaufman.Matrix methods for queuing problems.SIAM J.Sci.Statist.Comput.,1983,4:525-552
[59]A.Klawonn.Block-triangular preconditioners for saddle point problems with a penalty term.SIAM J.Sci.Comput.,1998,19:172-184
[60]L.A.Krukier.Iterative method solution of implicit difference schemes approximated for one class of quasilinear equation systems.Izv.Vyssh.Uchebn.Zaved.Mat.,1979,7:41-52
[61]L.A.Kru(?)kier.Construction of the operator B in implicit two-layer iterative schemes for providing convergence in the case of dissipative operator A.Izv.Vyssh.Uchebn.Zaved.Mat.,1983,5:41-47
[62]L.A.Krukier.Mathematical modelling of the Azov sea hydrodynamics in realizing the projects of reconstructing its ecosystem.,Math.Modelling.,1991,3:3-20
[63]L.A.Krukier,L.G.Chikina.The use of the iterative methods for solution of the steady convection-diffusion equations in the incompressible medium with predominating convection.Proc.International Conference on the Use of the Mathematical Modelling for the Solution of Problems in Science and Technic,Izevsk.,1996:246-257
[64]L.A.Krukier.Skew-symmetric preconditioners for strongly nonsymmetric linear equation systems.Proc.of 15th IMACS World Congress of Scientific Computation,Berlin.,1997,Vol 2:533-538
[65]L.A.Krukier.Convergence acceleration of triangular iterative methods based on the skewsymmetric part of the matrix.Appl.Numer.Math.,1999,30:281-290
[66]L.A.Krnkier,L.G.Chikina and T.V.Belokon.Triangular skew-symmetric iterative solvers for strongly nonsymmetric positive real linear system of equations.Appl.Numer.Math.,2002,41:89-105
[67]Y.Q.Lin,Y.H.Cao.A new nonlinear Uzawa algorithm for generalized saddle point problems.Appl.Mathe.Comput.,2006,175:1432-1454
[68]W.M.Lin,S.Xu.A new improved Uzawa method for finite element solution of stokes problem.Comput.Mech.,2001,27:305-310
[69]刘国新，王卫国．关于结构KKT方程组的扰动分析．计算数学．，2004，26(2)：179-188
[70]W.Li,W.Sun.Comparison results for parallel multisplitting methods with applications to AOR methods.Linear Algebra Appl.,2001,331:131-144
[71]W.Li,L.Elsner,L.Z.Lu.Comparisons of spectral radii and the theorem of Stein-Rosenberg.Linear Algebra Appl.,2002,348:283-287
[72]T.A.Manteuffel,G.Starke.On hybrid iterative methods for nonsymmetric systems of linear equations.Numer.Math.,1996,73:489-506
[73]I.Marek,D.B.Szyld.Comparison theorems for weak splittings of bounded operators.Numer.Mathe.,1990,58:389-397
[74]C.D.Meyer.R.J.Plemmons.Convergent powers of a matrix with applications to iterative methods for singular linear systems.SIAM J.Numer.Anal.,1977,14:699-705
[75]V.A.Miller,M.Neumann.A note on comparison theorems for nonnegative matrices.Numer.Math.,1985,47:427-434
[76]M.F.Murphy,G.H.Golub,A.J.Wathen.A note on preconditioning for indefinite linear systems.Tech.Report,SCCM-99-03,Stanford University,1999
[77]M.F.Murphy,G.H.Golub,AJ.Wathen.A note on preconditioning for indefinite linear systems.SIAM J.Sci.Comput.,2000,21:1969-1972
[78]R.Nabben.A note on comparison theorems for splittings and multisplittings of Hermitian positive definite matrices.Linear Algebra Appl.,1996,233:67-80
[79]N.M.Nachtigal,L.Reichel and L.N.Trefethen.A hybrid GMRES algorithm for nonsymmetric matrix iterations.SIAM J Matrix Anal.Appl.,1992,13:796-825
[80]M.Neumann,R.].Plemmons.Convergent nonnegative matrices and iterative methods for consisitent linear systems.Numer,Math.,1978,31:265-279
[81]M.Neumann,R.J.Plemmons.Convergence of parallel multisplitting iterative methods for Mmatrices.Linear Algebra Appl.,1987,88/89:559-573
[82]H.Nochetto,J.H.Pyo.Optimal relaxation parameter for the Uzawa method.Numer.Math.,2004,98:695-702
[83]D.P.O'Leary,R.E.White.Multisplittings of matrices and parallel solution of linear systems.SIAM J.Algebraic Discrete Methods,1985,6:630-640
[84]J.M.Ortega,W.C.Rheinboldt.Monotone iterations for nonlinear equations with application to Gauss-Seidel methods.SIAM J.Numer.Anal.,1967,4:171-190
[85]J.M.Ortega,and R.J.Plemmons.Extensions of the Ostrowski-Reich theorem for SOR iterations.Linear Algebra Appl.,1979,28:177-191
[86]J.M.Ortega.Numerical analysis.A Sencond Course,Academic Press,New York,Reprinted by SIAM,Philadelphia,1990
[87]D.Peaceman,H.Rachford.The numerical solution of elliptic and parabolic differential equations.J.SIAM,1955,3:28-41
[88]R.J.Plemmons.M-matrices leading to semiconvergent splittings.Linear Algebra Appl.,1976,15:243-252
[89]T.Rees,C.Greif.A preconditioner for linear systems arising from interior point optimization methods.SIAM J.Sci.Comput.,2007,29(5):1992-2007
[90]A.Ruhe.Closest normal matrix finally found.BIT.,1987,27:585-598
[91]T.Rusten,T.Winther.A preconditioned iterative method for saddle point problems.SIAM J.Matrix Anal.Appl.,1992,13:887-904
[92]Y.Saad,M.H.Schultz.GMRES:A generalized minimal residual algorithm for solving nonsymmetric linear systems.SIAM J.Sci.Star.Comput.,1986,7:856-869
[93] Y. Saad. Iterative methods for sparse linear systems. Boston: PWS Publishing Company.,1996:240-243
[94] S.Q. Shen, T.Z. Huang. Convergence and comparison theorems for double splittings of matrices.Comp. Math. Appl., 2006,51:1751-1760
[95] S.Q. Shen, T.Z. Huang, J.L. Shao. Convergence and comparison results for double splittings of Hermitian positive definite matrices. CALCLOL., 2007, 44:127-135
[96] S.Q. Shen, T.Z. Huang. New comparison results for parallel multisplitting iterative methods. Appl. Math. Comput., 2008,206:738-747
[97] A.A. Samarskij, E.S. Nikolaev. Numerical methods for grid equations. Vol. 2:Iterative Meth-ods(Birkhauser, Basel)., 1989
[98] C. Siefert, E.D. Sturler. Preconditioners for generalized saddle-point problems. SLAM J. Numer.Anal., 2006,44:1275-1296
[99] V. Simoncini, M. Benzi. Spectral properties of the Hermitian and skew-Hermitian splitting pre-conditioner for saddle point problems. SLAM J. Matrix Anal. Appl., 2004,6:377-389
[100] Y. Song. Some comparison theorems for nonnegative splittings of matrices. Numer. Math.,1993,65:245 - 252
[101] Y. Song. P-cyclic matrices and block accelerated overrelaxation method., Math. Appl., 1997,10(2):32-36
[102] Y. Song. Semiconvergence of extrapolated iterative methods for singular linear systems. J. Comput. Appl. Math., 1999,106:117-129
[103] Y. Song. Semiconvergence of nonnegative splittings for singular matrices. Numer. Math., 2000,85:109-127
[104] Y. Song. Semiconvergence of block SOR method for singular linear systems with p-cyclic matrices. J. Comput. Appl. Math., 2001,130:217-229
[105] Y. Song, L. Wang. On the semiconvergence of extrapolated iterative methods for singular linear systems. Appl. Numer. Math., 2003,44:401-413
[106] D.J. Silvester, H.C. Elman, A. Ramage. LFISS: Incompressible flow iterative solution sofeware. http://www.manchester.ac.uk/ifss
[107] E.D. Sturler, J. Liesen. Block-diagonal and constraint preconditioners for nonsymmertic indefinite linear systems. Part I:theory. SLAM J. Sci. Comput., 2005, 26:1598-1619
[108] D. B. Szyld, M. T. Jones. Two-stage and multisplitting methods for the parallel solution of linear systems. SLAM J. Matrix Anal. Appl., 1992,13: 671-679
[109]R.S.Varga.Matrix iterative analysis.Prentice-Hall:Englewood Cliffs,NJ,1962
[110]E.L.Wachspress.Iterative solution of elliptic systems and applications to the neutron equations of reactor physics.Prentice-Hall,Englewood Cliffs,NJ,1966
[111]D.Wang.On the convergence of the parallel multisplitting AOR algorithm.Linear Algebra Appl.,1991,154/156:473-486
[112]C.L.Wang,Z.Z.Bai.Sufficient conditions for the convergent splittings of non-Hermitian positive definite matrices.Linear Algebra Appl.,2001,330:215-218
[113]C.L.Wang.T.Z.Huang.New convergence results for alternating methods.J.Comput.Appl.Math.,2001,135:325-333
[114]C.L.Wang.Nonstationary multisplittings with general weighting matrices for non-Hermitian positive definite systems.Appl.Mathe.Lett.,2003,16:919-924
[115]C.L.Wang,J.H.Zhao.Further results on regular splittings.and multisplittings.Internat.J.Comput.Math.,2005,82:421-431
[116]王国荣．矩阵与算子广义逆．科学出版社，北京．1998
[117]L.Wang,Z.Z.Bai.Skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems of strong skew-Hermitian parts.BIT.Numer.Mathe.,2004,11:363-386
[118]L.Wang.Semiconvergence of two-stage iterative methods for singular linear systems.Linear Algebra Appl.,2007,422:824-838
[119]L.Wang,Z.Z.Bal.Convergence conditions for splitting iteration methods for non-Hermitian linear systems.Linear Algebra Appl.,2008,428:453-468
[120]A.Wathen,D.Silvester.Fast iterative solution of stabilised Stokes systems part Ⅰ:using simple diagonal preconditioners.SIAM J.Numer.Anal.,1993,30:630-649
[121]A.Wathen,D.Silvester.Fast iterative solution of stabilised Stokes systems part Ⅱ:using general block diagonal preconditioners.SIAM J.Numer.Anal.,1994,31:1352-1367
[122]Y.M.Wei.A characterization and representation of the Drazin inverse.SIAM J.Matrix Anal.Appl.,1996,17:744-747
[123]Y.M.Wei.Index splitting for the Drazin inverse and the singular linear system.Appl.Math.Comput.,1998,95:115-124
[124]R.E.White.Multisplitting of a symmetric positive definite matrix.SIAM J.Matrix Anal.Appl.,1990,11:69-82
[125]Z.I.Woznicki.Estimation of the optimum relaxation factors in partial factorization iterative methods.SIAM.J.Matrix Anal.Appl.,1993,14:59-73
[126]Z.I.Wo(?)nicki.Nonnegative splitting theory.Japan J.Indust.Appl.Math.,1994,11:289-342
[127]Z.I.Wo(?)nicki.Comparison theorems for splittings of monotone matrices.Nonlinear analysis,1997,30:1251-1262
[128]Z.I.Wo(?)nicki.Basic comparison theorems for weak and weaker matrix splittings.Electron.J.Linear Algebra,2001,8:53-59
[129]D.M.Young.Iterative solution of large linear systems.Academic Press,New York,1971
[130]J.Y.Yuan.The Ostrowski-Reich theorem for SOR iterations:extensions to the rank deficient case.Linear Algebra Appl.,2000,315:189-196
[131]A.Zsaki,D.Rixen,M.Paraschivoiu.A substructure based iterative inner solver coupled with Uzawa's algorithm for the stokes problem.Internat.J.Numer.Meth.Fluids.,2003,43:215-230