On iterative methods for solving a semi-linear eigenvalue problem

Abstract

For the semi-linear eigenvalue problem of the form Ax = lF{x), where F is a nonlinear map-ping, we present some methods for numerical solution. For this problem, we first describe a practical SOR method: in this method, the overrelaxation parameter automatically estimated is used instead of the optimum value, since the eigenvalue is not known a priori. We discuss the convergence of this Newton-like method. We then present a conjugate gradient (CG) method for the eigenvalue problem. We also discuss some preconditioning techniques for the present CG method. Finally, a comparison of the convergence rates for these methods is made with a numerical example.

Keywords:

• Nonlinear SOR
• Newton method
• CG method
• preconditioned CG method
• eigenvalue problem

C.R. Categories:

• G.1.2