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

CAO Fang-ying; Lü Quan-yi; XIE Gong-nan

doi:10.3879/j.issn.1000-0887.2013.03.004

Volume 34 Issue 3

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2013 > 34(3): 240-251

CAO Fang-ying, Lü Quan-yi, XIE Gong-nan. A Parallel Preconditioned Modified Conjugate Gradient Method for a Kind of Matrix Equation[J]. Applied Mathematics and Mechanics, 2013, 34(3): 240-251. doi: 10.3879/j.issn.1000-0887.2013.03.004

Citation:

PDF( 401 KB)

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

doi: 10.3879/j.issn.1000-0887.2013.03.004

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

Received Date: 2013-01-29
Rev Recd Date: 2013-03-01
Publish Date: 2013-03-15

Abstract

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

FullText(HTML)

References(9)

References

[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))