Abstract
In this paper, a homotopy algorithm for finding all eigenpairs of a real symmetric matrix pencil (A, B) is presented, whereA andB are realn×n symmetric matrices andB is a positive semidefinite matrix. In the algorithm, pencil (A, B) is first reduced to a pencil\((\tilde A,\tilde B)\), where\(\tilde A\) is a symmetric tridiagonal matrix and\(\tilde B\) is a positive definite and diagonal matrix. Then, the “Divide and Conquer” strategy with homotopy continuation approach is used to find all eigenpairs of pencil\((\tilde A,\tilde B)\). One can easily form the eigenpair (x,λ) of pencil (A, B) from the eigenpair (y, λ) of pencil\((\tilde A,\tilde B)\) with a few computations. Numerical comparisons of our algorithm with the QZ algorithm in the widely used EISPACK library are presented. Numerical results show that our algorithm is strongly competitive in terms of speed, accuracy and orthogonality. The performance of the parallel version of our algorithm is also presented.
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Communicated by C. Brezinski
Research supported in part by NSF under Grant CCR-9024840.
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Li, K., Li, TY. A homotopy algorithm for a symmetric generalized eigenproblem. Numer Algor 4, 167–195 (1993). https://doi.org/10.1007/BF02142745
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DOI: https://doi.org/10.1007/BF02142745