Sonderforschungsbereich 393
	Preprint 8 / 2003

Solov'ev, Sergey I. : Preconditioned iterative methods for monotone nonlinear eigenvalue problems

Author(s):

Solov'ev, Sergey I.

Title:

Preconditioned iterative methods for monotone nonlinear eigenvalue problems

Electronic source:

application/pdf
application/postscript

Preprint series:

Technische Universität Chemnitz, SFB 393 (Germany), SFB393-Preprint 8, 2003

Mathematics Subject Classification:

65F15			[ Eigenvalues, eigenvectors ]
65F50			[ Sparse matrices ]
65N25			[ Eigenvalue problems ]

Abstract:

This paper proposes new iterative methods for the efficient computation of the smallest eigenvalue of the symmetric nonlinear matrix eigenvalue problems of large order with a monotone dependence on the spectral parameter. Monotone nonlinear eigenvalue problems for differential equations have important applications in mechanics and physics. The discretization of these eigenvalue problems leads to ill-conditioned nonlinear eigenvalue problems with very large sparse matrices monotone depending on the spectral parameter. To compute the smallest eigenvalue of large matrix nonlinear eigenvalue problem, we suggest preconditioned iterative methods: preconditioned simple iteration method, preconditioned steepest descent method, and preconditioned conjugate gradient method. These methods use only matrix-vector multiplications, preconditioner-vector multiplications, linear operations with vectors and inner products of vectors. We investigate the convergence and derive grid-independent error estimates of these methods for computing eigenvalues. Numerical experiments demonstrate practical effectiveness of the proposed methods for a class of mechanical problems.

Keywords:

eigenvalue, eigenelement, symmetric eigenvalue problem, nonlinear eigenvalue problem, iterative method, preconditioned iterative method, gradient method, steepest descent method, conjugate gradient method

Language:

English

Publication time:

3 / 2003