Linking the TPR1, DPR1 and Arrow-Head Matrix Structures

Some recent polynomial root-finders rely on effective solution of the eigenproblem for special matrices such as DPR1 (that is, diagonal plus rank-one) and arrow-head matrices. We examine the correlation between these two classes and their links to the Frobenius companion matrix, and we show a Gauss...

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Bibliographic Details
Published in:Computers & mathematics with applications (1987) 2006-11, Vol.52 (10), p.1603-1608
Main Authors: Pan, V.Y., Kunin, M., Murphy, B., Rosholt, R.E., Tang, Y., Yan, X., Cao, W.
Format: Article
Language:English
Subjects:
Blocking
Computer simulation
DPR1
Mathematical analysis
Mathematical models
Matrices
Matrix methods
Similarity
Similarity transforms
The algebraic eigenproblem
TPR1 and arrow-head matrices
Transforms
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Summary:Some recent polynomial root-finders rely on effective solution of the eigenproblem for special matrices such as DPR1 (that is, diagonal plus rank-one) and arrow-head matrices. We examine the correlation between these two classes and their links to the Frobenius companion matrix, and we show a Gauss similarity transform of a TPR1 (that is, triangular plus rank-one) matrix into DPR1 and arrow-head matrices. Theoretically, the known unitary similarity transforms of a general matrix into a block triangular matrix with TPR1 diagonal blocks enable the extension of the cited effective eigen-solvers from DPR1 and arrow-head matrices to general matrices. Practically, however, the numerical stability problems with these transforms may limit their value to some special classes of input matrices.
ISSN:0898-1221
1873-7668
DOI:10.1016/j.camwa.2005.04.020