Abstract
One considers various modifications of the AB-algorithm for the solution of the complete (partial) eigenvalue problem of a regular pencil A-λB of square matrices. A modification of the AB-algorithm is suggested which allows to eliminate in a finite number of steps the zero and the infinite eigenvalues of the pencil A-λB and to lower its dimensions. For regular pencils with real eigenvalues a modification ot the AB-algorithm with a shift is presented. For a well-defined choice of the shifts one proves the quadratic convergence of the algorithm, successively to each eigenvalue of the pencil, starting with the smallest one. For a pencil whose eigenvalues can be divided into the groups of “large” and “small” eigenvalues, one considers a modification of the AB-algorithm, allowing to obtain approximations to the indicated groups of eigenvalues as solutions of a problem for pencils of lower dimensions.
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Literature cited
V. N. Kublanovskaya, “The AB -algorithm and its properties,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,102, 42–60 (1980).
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 117–136, 1981
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Kublanovskaya, V.N., Simonova, V.N. Certain modifications of the AB-algorithm. J Math Sci 24, 75–89 (1984). https://doi.org/10.1007/BF01230268
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DOI: https://doi.org/10.1007/BF01230268