A new relaxation method for obtaining the lowest eigenvalue and eigenvector of a matrix equation
Yoshiaki Muda · International Journal for Numerical Methods in Engineering · 1973
Abstract The matrix eigenvalue problem Hui = λi ui is considered. It is shown that when a new approximate vector v(n+1) to u1 (the eigenvector of the lowest eigenvalue) is computed from the present one v(n) by the relation v(n+1) = (1− αH + βH2) v(n) or v(n+1) = (1− αH + βH2 – γH3) v(n), the convergence rate is at least double that of the gradient method which corresponds to set β = γ = 0. Moreover, by choosing parameters α, β, or γ properly, one can get about three to five times faster convergence rate than that of the latter method, for H having very small γ2–γ1 and very large λN (the largest eigenvalue), further modifications are suggested. The relation with the Richardson method is also discussed.