Extracting partial canonical structure for large scale eigenvalue problems |
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Authors: | Bo Kågström Petter Wiberg |
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Institution: | 1. Department of Computing Science, Ume? University, SE-901 87, Ume?, Sweden
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Abstract: | We present methods for computing a nearby partial Jordan-Schur form of a given matrix and a nearby partial Weierstrass-Schur
form of a matrix pencil. The focus is on the use and the interplay of the algorithmic building blocks – the implicitly restarted
Arnoldi method with prescribed restarts for computing an invariant subspace associated with the dominant eigenvalue, the clustering
method for grouping computed eigenvalues into numerically multiple eigenvalues and the staircase algorithm for computing the
structure revealing form of the projected problem. For matrix pencils, we present generalizations of these methods. We introduce
a new and more accurate clustering heuristic for both matrices and matrix pencils. Particular emphasis is placed on reliability
of the partial Jordan-Schur and Weierstrass-Schur methods with respect to the choice of deflation parameters connecting the
steps of the algorithm such that the errors are controlled. Finally, successful results from computational experiments conducted
on problems with known canonical structure and varying ill-conditioning are presented.
This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | partial Jordan canonical stucture eigenvalue problems Jordan-Schur form Weierstrass-Schur form large scale Krylov methods implicitly restarted Arnoldi eigenvalue clustering staircase algorithms 65F15 65F10 15A21 15A22 68N99 |
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