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Comparisons of weak regular splittings and multisplitting methods

Ludwig Elsner
1989 Numerische Mathematik  
Comparison results for weak regular splittings of monotone matrices are derived.  ...  As an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings.  ...  Multisplittings and Weak Regular Splittings Comparison Results for Weak Regular Splittings We try here to generalize comparison results for regular splittings to the case of weak regular splittings.  ... 
doi:10.1007/bf01409790 fatcat:fmcfl7mlibfz3hlsrqwcjzwpba

Symmetric multisplitting of a symmetric positive definite matrix

Zhi-Hao Cao, Zhong-Yun Liu
1998 Linear Algebra and its Applications  
A parallel symmetric multisplitting method for solving a symmetric positive system Ax = b is presented. Here the s.p.d.  ...  The convergence of the presented parallel symmetric multisplitting method is also discussed by using this tool. 0 1998 Elsevier Science. Inc. All rights reserved.  ...  Acknowledgements We wish to thank the referee for suggestions and comments which helped improve some results of the paper, especially Theorem 3 and Corollary 1.  ... 
doi:10.1016/s0024-3795(98)10151-9 fatcat:zncbfhelqnc7vl2hrmyglm7euy

Nonstationary two-stage multisplitting methods for symmetric positive definite matrices

Zhong-Yun Liu, Lu Lin, Chun-Chao Shi
2000 Applied Mathematics Letters  
The main tool for the construction of the two-stage multisplitting and related theoretical investigation is the diagonally compensated reduction (cf. [l]).  ...  Nonstationary synchronous two-stage multisplitting methods for the solution of the symmetric positive definite linear system of equations are considered.  ...  developed and analyzed by Axelsson and Kolotilina in [l] , are different from the traditional iterative methods based on P-regular splitting and P-regular splitting theorem (cf  ... 
doi:10.1016/s0893-9659(00)00095-1 fatcat:wfbgdxvw3jeq3bmwozh6qt3tyq

On parallel multisplitting methods for symmetric positive semidefinite linear systems

Guangxi Cao, Yongzhong Song
2009 Numerical Linear Algebra with Applications  
The semiconvergence of the parallel multisplitting method is discussed. The results here generalize some known results for the nonsingular linear systems to the singular systems.  ...  It is well known that if B is nonsingular, then a splitting B = M − N is convergent if and only if (M −1 N )<1.  ...  ACKNOWLEDGEMENTS The authors would like to thank the anonymous reviewers for their helpful comments and advice in improving the manuscript.  ... 
doi:10.1002/nla.619 fatcat:o6sj37uvv5fnhfrure6f42f32e

Convergence and comparison theorems for multisplittings

Joan‐Josep Climent, Carmen Perea
1999 Numerical Linear Algebra with Applications  
We then derive convergence and comparison results for proper weak regular multisplittings.  ...  In this paper, we first prove a few comparison results between two proper weak regular splittings which are useful in getting the iterative solution of a large class of rectangular (square singular) linear  ...  Sivakumar, Professor, Indian Institute of Technology Madras for his valuable comments on an earlier version of the manuscript.  ... 
doi:10.1002/(sici)1099-1506(199903)6:2<93::aid-nla149>3.0.co;2-8 fatcat:zmghsqcsr5eevba2b3k57osnlq

A multisplitting method for symmetric linear complementarity problems

Naoki Machida, Masao Fukushima, Toshihide Ibaraki
1995 Journal of Computational and Applied Mathematics  
The results obtained from those splitting iterations are combined to define the multisplitting iterates. Thus, the method may be effectively implemented on multiprocessors.  ...  In particular, we establish some convergence results for the multisplitting method, which generalize the corresponding convergence results for the splitting method for LCP.  ...  The results obtained from those splitting iterations are combined to define the multisplitting iterates. Thus, the method may be effectively implemented on multiprocessors.  ... 
doi:10.1016/0377-0427(94)00103-2 fatcat:5avsr53tabarhahlzii6pghd5e

Convergence theory of iterative methods based on proper splittings and proper multisplittings for rectangular linear systems

Vaibhav Shekhar, Chinmay Giri, Debasisha Mishra
2020 Filomat  
With the aim to extend the convergence theory of proper multisplittings, this paper further adds a few results.  ...  Math. 158 (2003), 43-48: MR2013603] introduced the notion of proper multisplittings to solve the system Ax = b on parallel and vector machines, and established convergence theory for a subclass of proper  ...  made on an earlier draft of this paper.  ... 
doi:10.2298/fil2006835s fatcat:sx7jci2uiff7vhyfybde2af73i

Convergence of parallel multisplitting iterative methods for M-matrices

M. Neumann, R.J. Plemmons
1987 Linear Algebra and its Applications  
One such is the multisplitting iterative algorithm suggested by O'Leary and White.  ...  Comparison results between multisplitting methods are established in terms of monotonic norms and, for the case where A is irreducible, in terms of the asymptotic convergence rate.  ...  We shall establish in Section 2 a heuristic principle that indicates that the rate of convergence of the multisplitting method depends on the splittings and not on the way in which the load is distributed  ... 
doi:10.1016/0024-3795(87)90125-x fatcat:eowfsj255na2vgj5ryksys5iha

Comparison results for proper multisplittings of rectangular matrices [article]

Chinmay Kumar Giri, Debasisha Mishra
2016 arXiv   pre-print
We then derive convergence and comparison theorems for proper multisplittings with the help of the theory of proper weak regular splittings.  ...  To this end, we propose a few comparison theorems for proper weak regular splittings and proper nonnegative splittings first.  ...  Sivakumar for his valuable comments on an earlier version of the manuscript.  ... 
arXiv:1610.01051v1 fatcat:fsi6u5iys5goda3zxqay3gx6sy

Convergence of SSOR multisplitting method for an H-matrix

Jae Heon Yun
2008 Journal of Computational and Applied Mathematics  
A splitting A = M − N is called regular if M −1 0 and N 0, and weak regular if M −1 0 and M −1 N 0. * Tel.  ...  We also introduce an application of the SSOR multisplitting method. is called a splitting of A when M is nonsingular.  ...  Acknowledgements The author would like to thank the anonymous referee for useful comments and constructive suggestions which substantially improved the quality of this paper.  ... 
doi:10.1016/j.cam.2007.06.030 fatcat:r43iraonkbebddzu3fe7hsh4qi

A note on comparison theorems for splittings and multisplittings of Hermitian positive definite matrices

Reinhard Nabben
1996 Linear Algebra and its Applications  
For M-matrices A and weak regular splittings there exist wellknown comparison theorems. Here, we give a comparison theorem for splittings of Sermitian positive definite matrices.  ...  We discuss iterative methods for the solution of the linear system Az = b, which are based on a single splitting or a multisplitting of A.  ...  Let A + 0, and let (Mk, Nk, Ek) pTT1 be a multisplitting of A, where for each k, (Mk,Nk) as a P-regular splitting and El, = QkI.  ... 
doi:10.1016/0024-3795(94)00050-6 fatcat:xeef3yktdbakdjyurxhgo3m76e

On parallel multisplitting methods for non-Hermitian positive definite linear systems [article]

Cheng-yi Zhang, Shuanghua Luo, Yan Zhu
2014 arXiv   pre-print
To solve non-Hermitian linear system Ax=b on parallel and vector machines, some paralell multisplitting methods are considered.  ...  results for the Hermitian positive definite case; ii) We extend the positive-definite and skew-Hermitian splitting (PSS) method methods in [ SIAM J.  ...  Many thanks also to Professor Michele Benzi for suggesting the topic of this paper and for helpful suggestions.  ... 
arXiv:1410.3197v1 fatcat:o6ee3r3xwnet7f4t5ephzr4wim

Sequential and parallel synchronous alternating iterative methods

Joan-Josep Climent, Carmen Perea, Leandro Tortosa, Antonio Zamora
2003 Mathematics of Computation  
In this paper new results are introduced when A is a monotone matrix using a weak nonnegative multisplitting of the second type and when A is a symmetric positive definite matrix using a P -regular multisplitting  ...  Also, when matrix A is symmetric positive definite and the multisplittings are P -regular, the schemes are also convergent.  ...  Bru, Elsner, and Neumann [3] also introduce the following convergence result for the parallel iterative method (22) that we quote for further references.  ... 
doi:10.1090/s0025-5718-03-01607-7 fatcat:co7yammjofhd5o2knsjqsz5ybm

On the convergence of subproper (multi)-splitting methods for solving rectangular linear systems

Lijing Lin, Yimin Wei
2008 Calcolo  
We give a convergence criterion for stationary iterative schemes based on subproper splittings for solving rectangular systems and show that, for special splittings, convergence and quotient convergence  ...  We also analyze the convergence of multisplitting algorithms for the solution of rectangular systems when the coefficient matrices have special properties and the linear systems are consistent.  ...  Bini and the referee for their very detailed comments, which greatly improved the paper. They also thank Prof. Zhi-Hao Cao for his inspired guidance in studying singular linear equations.  ... 
doi:10.1007/s10092-008-0141-8 fatcat:q2vaqsqul5calp5m3dwjkocdfq

Convergence of non-stationary parallel multisplitting methods for hermitian positive definite matrices

M. Jesús Castel, Violeta Migallón, José Penadés
1998 Mathematics of Computation  
Non-stationary multisplitting algorithms for the solution of linear systems are studied.  ...  Asynchronous versions of these algorithms are considered and their convergence investigated.  ...  Let A = F j −G j , 1 ≤ j ≤ r, be P-regular splittings. Assume further that lim l→∞ q(l, j) = ∞, 1 ≤ j ≤ r.  ... 
doi:10.1090/s0025-5718-98-00893-x fatcat:q3ir52ieuveshgufu3647whtom
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