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http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600657

Kurzfassung in Englisch

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.

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