Certain modifications of the $AB$-algorithm
Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part V, Tome 111 (1981), pp. 117-136
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One considers various modifications of the $AB$-algorithm for the solution of the complete (partial) eigenvalue problem of a regular pencil $A-\lambda B$ 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 $A-\lambda B$ 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.
@article{ZNSL_1981_111_a8,
author = {V. N. Kublanovskaya and V. N. Simonova},
title = {Certain modifications of the $AB$-algorithm},
journal = {Zapiski Nauchnykh Seminarov POMI},
pages = {117--136},
publisher = {mathdoc},
volume = {111},
year = {1981},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_111_a8/}
}
V. N. Kublanovskaya; V. N. Simonova. Certain modifications of the $AB$-algorithm. Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms. Part V, Tome 111 (1981), pp. 117-136. http://geodesic.mathdoc.fr/item/ZNSL_1981_111_a8/