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During the last years nonlinear eigenvalue problems of the type T (λ)x = 0 became more and more important in many applications with the rapid development of computing performance. In parallel the minmax theory for symmetric nonlinear eigenvalue problems was developed which reveals a strong connection to corresponding linear problems. We want to review the current theory for symmetric nonlinear ...

In linear stability analysis of a large-scale dynamical system, we need to compute the rightmost eigenvalue(s) for a series of large generalized eigenvalue problems. Existing iterative eigenvalue solvers are not robust when no estimate of the rightmost eigenvalue(s) is available. In this study, we show that such an estimate can be obtained from Lyapunov inverse iteration applied to a special ei...

Recently the homotopy method has been applied to solve linear algebraic eigenvalue problems. On the basis of theoretical advantages and practical experiments, the method has been suggested as a serious alternative to EISPACK for finding all isolated eigenpairs of large, sparse linear algebraic eigenvalue problems on SIMD machines. This note offers a simpler proof than Li and Sauer’s of the exis...

The trust-region subproblem of minimizing a quadratic function subject to a norm constraint arises in the context of trust-region methods in optimization and in the regularization of discrete forms of ill-posed problems, including non-negative regularization by means of interior-point methods. A class of efficient methods and software for solving large-scale trust-region subproblems is based on...

Large-scale eigenvalue problems arise in a number of DOE applications. This paper provides an ovcrvicw of thc recent dcvclopriicrit of cigciivaluc computation in the coritcxt of two SciDAC applications. We emphasize the importance of Krylov suhspace methods, and point out its limitations. M7e discuss the value of alternative approaches that are more amenable to the use of prcconditioncrs, arid ...