Differential qd Algorithm with Shifts for Rank-Structured Matrices

Although QR iterations dominate in eigenvalue computations, there are several important cases when alternative LR-type algorithms may be preferable, in particular, in the symmetric tridiagonal case where the differential qd algorithm with shifts (dqds) proposed by Fernando and Parlett enjoys often f...

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Vydáno v:SIAM journal on matrix analysis and applications Ročník 33; číslo 4; s. 1153 - 1171
Hlavní autor: Zhlobich, Pavel
Médium: Journal Article
Jazyk:angličtina
Vydáno: Philadelphia Society for Industrial and Applied Mathematics 01.01.2012
Témata:
Accuracy
Algorithms
Eigenvalues
Experiments
Linear algebra
ISSN:0895-4798, 1095-7162
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Shrnutí:Although QR iterations dominate in eigenvalue computations, there are several important cases when alternative LR-type algorithms may be preferable, in particular, in the symmetric tridiagonal case where the differential qd algorithm with shifts (dqds) proposed by Fernando and Parlett enjoys often faster convergence while preserving high relative accuracy. In eigenvalue computations for rank-structured matrices, the QR algorithm is also a popular choice since, in the symmetric case, the rank structure is preserved. In the unsymmetric case, however, the QR algorithm destroys the rank structure and, hence, LR-type algorithms come in to play once again. In the current paper we adapt several variants of qd algorithms to quasi-separable matrices. Remarkably, one of them, when applied to Hessenberg matrices, becomes a direct generalization of the dqds algorithm for tridiagonal matrices. Therefore, it can be applied to such important matrices as companion and confederate and provides an alternative algorithm for finding roots of a polynomial represented in a basis of orthogonal polynomials. Results of preliminary numerical experiments are presented. [PUBLICATION ABSTRACT]
Bibliografie:ObjectType-Article-1
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ISSN:0895-4798
1095-7162
DOI:10.1137/110857155