一类特殊矩阵方程的并行预处理变形共轭梯度算法

曹方颖; 吕全义; 谢公南

doi:10.3879/j.issn.1000-0887.2013.03.004

一类特殊矩阵方程的并行预处理变形共轭梯度算法

doi: 10.3879/j.issn.1000-0887.2013.03.004

¹西北工业大学应用数学系,西安 710129;²西北工业大学机电学院,工程仿真与宇航计算技术联合实验室,西安 710072

基金项目: 国家自然科学基金资助项目（11202164）;陕西省自然科学基金资助项目(2009JM1008)

详细信息

作者简介:
曹方颖(1986—),女,西安人,硕士生(E-mail:caofangying1987@163.com);吕全义(1963—),女,沈阳人,副教授(通讯作者.E-mail:luquan@nwpu.edu.cn).

中图分类号: O246
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出版历程
- 收稿日期: 2013-01-29
- 修回日期: 2013-03-01
- 刊出日期: 2013-03-15

A Parallel Preconditioned Modified Conjugate Gradient Method for a Kind of Matrix Equation

¹Department of Applied Mathematics, Northwestern Polytechnical University,Xi’an 710129, P.R.China;²Engineering Simulation and Aerospace Computing, School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, P.R.China

摘要

摘要: 研究了求解一类矩阵方程AXB=C，提出了一种并行预处理变形共轭梯度法．该方法给出一种迭代法的预处理模式．首先给出的预处理矩阵是严格对角占优矩阵，构造并行迭代求解预处理矩阵方程的迭代格式，进而使用变形共轭梯度法并行求解．通过数值试验，预处理变形共轭梯度法与直接使用变形共轭梯度法相比较，该算法不仅有效提高了收敛速度，而且具有很高的并行性．
- 矩阵方程 /
- 变形共轭梯度法 /
- 预处理矩阵 /
- 并行性
Abstract: In view of a parallel algorithm of preconditioned modified conjugate gradient method for solving a kind of matrix equationAXB=C，a preconditioned model was proposed. Based on this thought, firstly the preconditioned matrix was constructed, which was strictly diagonally dominant matrix, secondly the parallel algorithm for preprocessing matrix equation iterative format was formed, and finally the modified conjugate gradient method was used for parallel solving the preconditioned matrix equation. Through numerical experiments, comparing our algorithm with the modified conjugate gradient method, ours has higher parallel efficiency.
- parallel algorithm /
- preconditioned modified conjugate gradient method /
- preconditioned matrix /
- parallelism

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参考文献(9)

[1]	Penrose R.A generalized inverse for matrices[J]. Mathematical Proceedings of the Cambridge Philosophical Society , 1955, 51（3）: 406-413.
[2]	Lancaster P.Explicit solutions of linear matrix equations[J].SIAM Review, 1970, 72(4): 544-566.
[3]	Mitra S K.The matrix equation AX=C,XB=D[J]. Linear Algebre and Its Application , 1984, 59: 171-181.
[4]	Uhlig F.On the matrix equation AX=B with applications to the generators of controllability matrix[J].Linear Algebre and Its Application , 1987, 85: 203-209.
[5]	XIAO Qing-feng, HU Xi-yan, ZHANG Lei.The symmetric minimal rank solution of the matrix equation AX=B and the optimal approximation[J]. Electronic Journal of Linear Algebra , 2009, 18: 264-273.
[6]	张凯院, 徐仲.数值代数[M].第2版修订本.北京: 科学出版社, 2010.(ZHANG Kai-yuan, XU Zhong. Numerical Algebra [M].Revised 2nd Ed.Beijing: Science Press, 2010.(in Chinese))
[7]	胡家赣.解线性代数方程组的迭代解法[M].北京：科学出版社, 1999: 173-201.(HU Jia-gan.Iterative Solution of Linear Algebraic Equations [M].Beijing: Science Press, 1999: 173-201.(in Chinese))
[8]	陈国良, 安虹, 陈俊, 郑启龙, 单九龙.并行算法实践[M].北京：高等教育出版社, 2004.(CHEN Guo-liang, AN Hong, CHEN Jun, ZHENG Qi-long, SHAN Jiu-long. The Parallel Algorithm [M].Beijing: Higher Education Press, 2004.(in Chinese))
[9]	李晓梅, 吴建平.数值并行算法与软件[M].北京: 科学出版社, 2007.(LI Xiao-mei, WU Jian-ping. Numerical Parallel Algorithm and Software [M].Beijing: Science Press, 2007.(in Chinese))

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一类特殊矩阵方程的并行预处理变形共轭梯度算法

doi: 10.3879/j.issn.1000-0887.2013.03.004

作者简介:
曹方颖(1986—),女,西安人,硕士生(E-mail:caofangying1987@163.com);吕全义(1963—),女,沈阳人,副教授(通讯作者.E-mail:luquan@nwpu.edu.cn).

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A Parallel Preconditioned Modified Conjugate Gradient Method for a Kind of Matrix Equation

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一类特殊矩阵方程的并行预处理变形共轭梯度算法

doi: 10.3879/j.issn.1000-0887.2013.03.004

作者简介: 曹方颖(1986—),女,西安人,硕士生(E-mail:caofangying1987@163.com);吕全义(1963—),女,沈阳人,副教授(通讯作者.E-mail:luquan@nwpu.edu.cn).

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出版历程

A Parallel Preconditioned Modified Conjugate Gradient Method for a Kind of Matrix Equation

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出版历程

目录

作者简介:
曹方颖(1986—),女,西安人,硕士生(E-mail:caofangying1987@163.com);吕全义(1963—),女,沈阳人,副教授(通讯作者.E-mail:luquan@nwpu.edu.cn).