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Continuous methods for symmetric generalized eigenvalue problems. (English) Zbl 1140.65029

The authors consider generalized eigenvalue problems $Ax=\lambda Bx$, where $A$, $B$ are symmetric real $n×n$ matrices and $B$ is positive semi-definite. At first the existence of solutions of the generalized eigenvalue problem is discussed. Then some conditions are formulated under which the generalized eigenvalue problem can be reduced to a standard eigenvalue problem.

The computation of the minimal eigenvalue and the corresponding eigenvector is discussed. Optimization problems are given which are equivalent to the eigenvalue problems. For finding the eigenvector a continuous method is presented. This method includes a merit function and a system of ordinary differential equations (ODE system). It is shown that the solution of the ODE system converges to an eigenvector. Conditions are given under which this is the eigenvector corresponding to the minimal eigenvalue of the generalized eigenvalue problem.

Then, the presented approach is extended to the computation of the $k$th generalized eigenvalue and the corresponding eigenvector. The numerical experiments presented confirm the theoretical results.

##### MSC:
 65F15 Eigenvalues, eigenvectors (numerical linear algebra) 65L15 Eigenvalue problems for ODE (numerical methods)