On a conjecture of G. Forsythe

A conjecture of Forsythe on the asymptotic behavior of the $s$-step method of steepest descent for a quadratic functional is confirmed for the two-step method, and the essential range of the asymptotic rate of convergence is found. Conditions are determined for the eigenvalues of the matrix to be in the asymptotic spectrum of the method. Devices for increasing the efficiency of the $s$-step method are proposed and justified on the basis of the results obtained.
Bibliography: 20 titles.