基于不完全正交的VRP-IGMRES(m)算法

doi:10.3969/j.issn.1000-1565.2018.05.003

河北大学学报(自然科学版) ›› 2018, Vol. 38 ›› Issue (5): 460-465.DOI: 10.3969/j.issn.1000-1565.2018.05.003

基于不完全正交的VRP-IGMRES(m)算法

郝雪景¹,于春肖¹,任翠环²

收稿日期:2017-12-09 出版日期:2018-09-25 发布日期:2018-09-25
通讯作者: 于春肖(1977—),女,河北平山人,燕山大学教授,博士,主要从事多极边界元法与应用研究.E-mail:chxy@ysu.edu.cn
作者简介:郝雪景(1992—),女,河北邯郸人,燕山大学在读硕士研究生. E-mail:snowscene86@163.com
基金资助:
国家自然科学基金资助项目(11301459);河北省自然科学基金资助项目(A2015203121)

VRP-IGMRES(m)algorithm based on incomplete orthogonalization

HAO Xuejing¹, YU Chunxiao¹, REN Cuihuan²

1. College of Science, Yanshan University, Qinhuangdao 066004, China; 2. Departmentof Mathematics, North China University of Science and Technology, Tangshan 063210, China

Received:2017-12-09 Online:2018-09-25 Published:2018-09-25

摘要/Abstract

摘要： 为提高大型线性方程组的求解效率,在VRP-GMRES(m)算法基础上,利用截断技术,即在构造Krylov子空间的基向量和Hessenberg矩阵时采用不完全正交的Arnoldi过程,提出截断型变参数广义极小残余算法(VRP-IGMRES(m)),并利用连续2次迭代残余向量的夹角余弦与模的关系给出算法的收敛性证明. 最后通过数值算例分析了截断指标对计算精度和计算效率的影响,表明VRP-IGMRES(m)算法在保证计算精度的前提下,可以有效地提高计算效率,并得到了最优截断比的取值大约为0.1,为实际工程问题的求解提供了新的方法.

关键词: VRP-IGMRES(m)算法, 不完全正交, 最优截断比

Abstract: In order to solve large linear equations efficiently, based on the Generalized Minimal Residual with Variable Restart Parameter algorithm(VRP-GMRES(m)), a truncation-pattern Incomplete Generalized Minimal Residual with Variable Restart Parameter algorithm(VRP-IGMRES(m))is proposed using the truncation technology,namely, using incomplete orthogonal Arnoldi process to constructe the base vector of the Krylov subspace and the Hessenberg matrix. The convergence of the algorithm is proved by the relationship between the angle cosine and the modulus of the two successive iterated residual vectors. Finally, through numerical examples to analyze the influence of truncation index on computational accuracy and efficiency, it is found that VRP-IGMRES(m)algorithm can effectively improve the computation efficiency under the premise of guaranteeing the accuracy. The obtained optimal truncation ratio is about 0.1. Our work provides a new method for solving practical engineering problems.

Key words: VRP-IGMRES(m)algorithm, incomplete orthogonalization, optimal truncation ratio

中图分类号:

O242

郝雪景,于春肖,任翠环. 基于不完全正交的VRP-IGMRES(m)算法[J]. 河北大学学报(自然科学版), 2018, 38(5): 460-465.

HAO Xuejing, YU Chunxiao, REN Cuihuan. VRP-IGMRES(m)algorithm based on incomplete orthogonalization[J]. Journal of Hebei University (Natural Science Edition), 2018, 38(5): 460-465.

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基于不完全正交的VRP-IGMRES(m)算法

VRP-IGMRES(m)algorithm based on incomplete orthogonalization

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