ON THE CONVERGENCE OF THE SELF-CONSISTENT FIELDITERATION FOR A CLASS OF NONLINEAR EIGENVALUEPROBLEMS

摘要

We investigate the convergence of the self-consistent field (SCF) iteration used to solvea class of nonlinear eigenvalue problems. We show that for the class of problems considered, the SCFiteration 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 tothe nonlinear eigenvalue problem. We identify the condition under which the SCF iteration becomesa contractive fixed point iteration that guarantees its convergence. This condition is characterized byan upper bound placed on a parameter that weighs the contribution from the nonlinear componentof the eigenvalue problem. We derive such a bound for the general case as well as for a special casein which the dimension of the problem is 2.