A new gradient method with an optimal stepsize property

被引：21

作者：

Dai, YH ^{[1
]}

Yang, XQ

机构：

[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

来源：

COMPUTATIONAL OPTIMIZATION AND APPLICATIONS | 2006年 / 33卷 / 01期

关键词：

linear system; gradient method; steepest descent method; (shifted) power method;

D O I：

10.1007/s10589-005-5959-2

中图分类号：

C93 [管理学]; O22 [运筹学];

学科分类号：

070105 ; 12 ; 1201 ; 1202 ; 120202 ;

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.

引用

页码：73 / 88

页数：16