TITLE

A New Gradient Method with an Optimal Stepsize Property

AUTHOR(S)
Dai, Y. H.; Yang, X. Q.
PUB. DATE
January 2006
SOURCE
Computational Optimization & Applications;Jan2006, Vol. 33 Issue 1, p73
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
The gradient method for the symmetric positive definite linear system Ax = b is as follows xk+1 = xk - akgk where gk = Axk - b is the residual of the system at xk and ak is the stepsize. The stepsize ak = 2/?1+?n is optimal in the sense that it minimizes the modulus ?I - aA?2, where ?1 and ?n are the minimal and maximal eigenvalues of A respectively. Since ?1 and ?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 ? 8. At the same time, the minimal and maximal eigenvalues, ?1 and ?n, of A and their corresponding eigenvectors can be obtained.
ACCESSION #
20740813

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