Eigen-solving via reduction to DPR1 matrices

Authors:
V. Y. Pan;B. Murphy;R. E. Rosholt;Y. Tang;X. Wang;A. Zheng
Affiliations:
Department of Mathematics and Computer Science, Lehman College, City University of New York, Bronx, NY 10468, USA;Department of Mathematics and Computer Science, Lehman College, City University of New York, Bronx, NY 10468, USA;Department of Mathematics and Computer Science, Lehman College, City University of New York, Bronx, NY 10468, USA;Ph.D. Program in Computer Science, The City University of New York, New York, NY 10036, USA;Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China;Ph.D. Program in Mathematics, The City University of New York, New York, NY 10036, USA
Venue:
Computers & Mathematics with Applications
Year:
2008

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Abstract

Highly effective polynomial root-finders have been recently designed based on eigen-solving for DPR1 (that is diagonal + rank-one) matrices. We extend these algorithms to eigen-solving for the general matrix by reducing the problem to the case of the DPR1 input via intermediate transition to a TPR1 (that is triangular + rank-one) matrix. Our transforms use substantially fewer arithmetic operations than the QR classical algorithms but employ non-unitary similarity transforms of a TPR1 matrix, whose representation tends to be numerically unstable. We, however, operate with TPR1 matrices implicitly, as with the inverses of Hessenberg matrices. In this way our transform of an input matrix into a similar DPR1 matrix partly avoids numerical stability problems and still substantially decreases arithmetic cost versus the QR algorithm.