On a self-consistent-field-like iteration for maximizing the sum of the Rayleigh quotients
Journal of Computational and Applied Mathematics
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We investigate the convergence of the self-consistent field (SCF) iteration used to solve a class of nonlinear eigenvalue problems. We show that for the class of problems considered, the SCF iteration produces a sequence of approximate solutions that contain two convergent subsequences. These subsequences may converge to two different limit points, neither of which is the solution to the nonlinear eigenvalue problem. We identify the condition under which the SCF iteration becomes a contractive fixed point iteration that guarantees its convergence. This condition is characterized by an upper bound placed on a parameter that weighs the contribution from the nonlinear component of the eigenvalue problem. We derive such a bound for the general case as well as for a special case in which the dimension of the problem is $2$.