Linking the TPR1, DPR1 and Arrow-Head Matrix Structures

  • Authors:

  • V. Y. Pan;M. Kunin;B. Murphy;R. E. Rosholt;Y. Tang;X. Yan;W. Cao

  • Affiliations:

  • Department of Mathematics and Computer Science, Lehman College, City University of New York, Bronx, NY 10468, U.S.A. and Ph.D. Program in Computer Science, Graduate School and University Center, C ...;Ph.D. Program in Computer Science, Graduate School and University Center, City University of New York, New York, NY 10016, U.S.A.;Department of Mathematics and Computer Science, Lehman College, City University of New York, Bronx, NY 10468, U.S.A. and Ph.D. Program in Computer Science, Graduate School and University Center, C ...;Department of Mathematics and Computer Science, Lehman College, City University of New York, Bronx, NY 10468, U.S.A.;Ph.D. Program in Computer Science, Graduate School and University Center, City University of New York, New York, NY 10016, U.S.A.;Ph.D. Program in Computer Science, Graduate School and University Center, City University of New York, New York, NY 10016, U.S.A.;Ph.D. Program in Computer Science, Graduate School and University Center, City University of New York, New York, NY 10016, U.S.A.

  • Venue:
  • Computers & Mathematics with Applications
  • Year:
  • 2006

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Abstract

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.