A new gradient method with an optimal stepsize property

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.