A new gradient method with an optimal stepsize property

doi:10.1007/s10589-005-5959-2

	A new gradient method with an optimal stepsize property
	Dai, YH; Yang, XQ
	2006
发表期刊	COMPUTATIONAL OPTIMIZATION AND APPLICATIONS
ISSN	0926-6003
卷号	33 期号:1 页码:73-88
摘要	The gradient method for the symmetric positive definite linear system Ax = b is as follows x(k+1) = x(k) - alpha(k)g(k) where g(k) = Ax(k) - b is the residual of the system at x(k) and alpha(k) is the stepsize. The stepsize alpha(k) = 2/lambda(1)+lambda(n) is optimal in the sense that it minimizes the modulus parallel to I - alpha A parallel to(2), where.1 and.n are the minimal and maximal eigenvalues of A respectively. Since lambda(1) and lambda(n) are unknown to users, it is usual that the gradient method with the optimal stepsize is only mentioned in theory. In this paper, we will propose a new stepsize formula which tends to the optimal stepsize as k -> infinity. At the same time, the minimal and maximal eigenvalues,.1 and.n, of A and their corresponding eigenvectors can be obtained.
关键词	linear system gradient method steepest descent method (shifted) power method
DOI	10.1007/s10589-005-5959-2
语种	英语
WOS研究方向	Operations Research & Management Science ; Mathematics
WOS类目	Operations Research & Management Science ; Mathematics, Applied
WOS记录号	WOS:000235317800004
出版者	SPRINGER
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/3679
专题	计算数学与科学工程计算研究所
通讯作者	Dai, YH
作者单位	1.Chinese Acad Sci, Acad Math& Syst Sci, Inst Computat Math & Sci Engn Comp, State Key Lab Sci & Engn Comp, Beijing 100080, Peoples R China 2.Hong Kong Polytech Univ, Dept Appl Math, Kowloon, Hong Kong, Peoples R China
推荐引用方式 GB/T 7714	Dai, YH,Yang, XQ. A new gradient method with an optimal stepsize property[J]. COMPUTATIONAL OPTIMIZATION AND APPLICATIONS,2006,33(1):73-88.
APA	Dai, YH,&Yang, XQ.(2006).A new gradient method with an optimal stepsize property.COMPUTATIONAL OPTIMIZATION AND APPLICATIONS,33(1),73-88.
MLA	Dai, YH,et al."A new gradient method with an optimal stepsize property".COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 33.1(2006):73-88.