对称正定线性方程组具有任意权矩阵的多分裂迭代求解

doi:10.12286/jssx.2014.1.27

计算数学 ›› 2014, Vol. 36 ›› Issue (1): 27-34.DOI: 10.12286/jssx.2014.1.27

对称正定线性方程组具有任意权矩阵的多分裂迭代求解

温瑞萍¹, 孟国艳², 王川龙¹

1. 太原师范学院数学系, 太原 030012;
2. 忻州师范学院计算机科学系, 山西忻州 034000

收稿日期:2013-01-10 出版日期:2014-02-15 发布日期:2014-02-12
基金资助:
国家自然科学基金（11071184）项目，山西省自然科学基金（2010011006，2012011015-6）项目资助.

PDF ( 1234 ) 引用被引次数 ( 7 )

温瑞萍, 孟国艳, 王川龙. 对称正定线性方程组具有任意权矩阵的多分裂迭代求解[J]. 计算数学, 2014, 36(1): 27-34.

Wen Ruiping, Meng Guoyan, Wang Chuanlong. MULTISPLITTING ITERATIVE METHODS WITH GENERAL WEIGHTING MATRICES FOR SOLVING SYMMETRIC POSITIVE DEFINITE LINEAR SYSTEMS[J]. Mathematica Numerica Sinica, 2014, 36(1): 27-34.

MULTISPLITTING ITERATIVE METHODS WITH GENERAL WEIGHTING MATRICES FOR SOLVING SYMMETRIC POSITIVE DEFINITE LINEAR SYSTEMS

Wen Ruiping¹, Meng Guoyan², Wang Chuanlong¹

1. Department of Mathematics, Taiyuan Normal University, Taiyuan 030012, China;
2. Department of computer Science, Xinzhou Normal University, Xinzhou 034000, Shanxi, China

Received:2013-01-10 Online:2014-02-15 Published:2014-02-12

本文利用优化模型研究求解对称正定线性方程组Ax=b的多分裂并行算法的权矩阵. 在我们的多分裂并行算法中，m个分裂仅要求其中之一为P-正则分裂而其余的则可以任意构造，这不仅大大降低了构造多分裂的难度，而且也放宽了对权矩阵的限制（不像标准的多分裂迭代方法中要求权矩阵为预先给定的非负数量矩阵）. 并且证明了新的多分裂迭代法是收敛的. 最后，通过数值例子展示了新算法的有效性.

关键词： 对称正定矩阵, 权矩阵, 多分裂, 收敛性

By making use of optimal models, we study the weighting matrices of the multisplitting parallel methods for solving the symmetric positive definite linear system Ax=b. In our multisplitting there is only one that is required to be P-regular splitting and all the others can be constructed arbitrarily, which not only decreases the difficulty of constructing the multisplitting of the coefficient matrix A, but also relaxes the constraints to the weighting matrices (unlike the standard methods, they are not necessarily nonnegative diagonal scalar matrices or given in advance). We then prove the convergence of this new method. Finally, numerical experiments show that the method is efficient.

Key words: symmetric positive definite matrix, weighting matrices, multisplitting, convergence

MR(2010)主题分类:

65F10
15A06

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[1] Migallón V, Penadés J. Convergence of two-stage iterative methods for hermitian positive definite matrices[J]. Appl. Math. Lett., 1997, 10: 79-83.

[2] Bai Z Z. A new generalized asynchronous parallel multisplitting iteration method[J]. J. Comput. Math., 1999, 17: 449-456.

[3] Bai Z Z. On the convergence of parallel nonstationary multisplitting iteration methods[J]. J. Comput. Appl. Math., 2003, 159: 1-11.

[4] Bai Z Z, Benzi M, Chen F. Modified HSS iteration methods for a class of complex symmetric linear systems[J]. Computing, 2010, 87: 93-111.

[5] Bai Z Z , Sun J C, Wang D R. A unified framework for the construction of various matrix multislpitting iterative methods for large sparse system of linear equations[J]. Comput. Math. Appl., 1996, 32: 51-76.

[6] Bai Z Z, Evans D J. Blockwise matrix multi-splitting multi-parameter block relaxation methods[J]. Int. J. Comput. Math., 1997, 64: 103-118.

[7] Bru R, Elsner L, Neumann M. Models of parallel chaotic iteration methods[J]. Linear Algebra Appl., 1988, 103: 175-192.

[8] Bru R, Migallón V, Penadés J, Szyld D.-B. Parallel synchronous and asynchronous two-stage multisplitting methods[J]. Electron. Trans. Numer. Anal., 1995, 3: 24-38.

[9] Cao Z H. Nonstationary two-stage multisplitting methods with overlapping blocks[J]. Linear Algebra Appl., 1998, 285: 153-163.

[10] Cao Z H, Liu Z Y. Symmetric multisplitting methods of a symmetric positive definite matrix[J]. Linear Algebra Appl., 1998, 285: 309-319.

[11] Castel M J, Migallón V, Penadés J. Convergence of non-stationary multisplitting methods for Hermitian positive definite matrices[J]. Math. Comput., 1998, 67: 209-220.

[12] Frommer A, Mayer G. Convergence of relaxed parallel multisplitting methods[J]. Linear Algebra Appl., 1989, 119: 141-152.

[13] Frommer A, Syzld D B. Weighted max norms, splitttngs, and overlapping additive Schwarz iterations[ J]. Numer. Math., 1997, 5: 48-62.

[14] Jones M T, Szyld D B. Two-stage multisplitting methods with overlapping blocks[J]. Numer. Linear Algebra Appl., 1996, 3: 113-124.

[15] Migallón V., Penadés J, Szyld D B. Nonstationary multisplttings with general weighting matrices[ J]. SIAM J. Matrix Anal. Appl., 2001, 22: 1089-1094.

[16] Nabben R. A note on comparison theorems for splittings and multisplittings of Hermitian positive definite matrices[J]. Linear Algebra Appl., 1996, 233: 67-83.

[17] Neuman M, Plemmons R J. Convergence of parallel multisplitting iterative methods for Mmatrices[ J]. Linear Algebra Appl., 1987, 88: 559-573.

[18] O'Leary D P, White R. Multi-splittings of matrices and parallel solutions of linear systems[J]. SIAM J. Alg. Discr. Meth., 1985, 6: 630-640.

[19] Szyld D B, Jones M T. Two-stage and multisplitting methods for the parallel solution of linear systems[J]. SIAM J. Matrix Anal. Appl., 1992, 13: 671-679.

[20] Wang D R. On the convergence of the parallel multisplitting AOR algorithm[J]. Linear Algebra Appl., 1991, 154/156: 473-486.

[21] Wang D R, Bai Z Z, Evans D J. A class of asynchronous parallel matrix multisplitting relaxation method[J]. Parallel Algorithms Appl., 1994, 2: 173-192.

[22] Wen R P, Wang C L, Meng G Y. Convergence theorems for block splitting iterative methods for linear systems[J]. J. Comput. Appl. Math., 2007, 202: 540-547.

[23] White R E. Multisplitting with different weighting schemes[J]. SIAM J. Matrix Anal. Appl., 1998, 10: 481-493.

[24] White R E. Multisplitting of a symmetric positive definite matrix[J]. SIAM J. Matrix Anal. Appl., 1990, 11: 69-82.

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摘要

对称正定线性方程组具有任意权矩阵的多分裂迭代求解

MULTISPLITTING ITERATIVE METHODS WITH GENERAL WEIGHTING MATRICES FOR SOLVING SYMMETRIC POSITIVE DEFINITE LINEAR SYSTEMS

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