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A New Gradient Method with an Optimal Stepsize Property

Authors:

Affiliation:

(1) State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, P.R. China;(2) Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, Hong Kong

Abstract:

The gradient method for the symmetric positive definite linear system $$Ax=b$$

is as follows

$x_{k + 1}=x_{k}-alpha_{k} g_{k}$

(1)

where $g_{k}=Ax_{k}-b$ is the residual of the system at x_k and α_k is the stepsize. The stepsize $alpha_{k} = frac{2}{{lambda_{1}+lambda_{n}}}$ is optimal in the sense that it minimizes the modulus $||I - alpha A||_{2}$ , where λ₁ and λ_n are the minimal and maximal eigenvalues of A respectively. Since λ₁ and λ_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 $$k to infty$$

. At the same time, the minimal and maximal eigenvalues, λ₁ and λ_n, of A and their corresponding eigenvectors can be obtained. This research was initiated while the first author was visiting The Hong Kong Polytechnic University. This author was supported by the Chinese NSF grants (No. 40233029 and 101071104) and an innovation fund of Chinese Academy of Sciences. This author was supported by a grant from the Research Committee of the Hong Kong Polytechnic University (A-PC36).

Keywords:

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

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