Numerische Mathematik

, Volume 57, Issue 1, pp 695–718 | Cite as

A divide and conquer method for unitary and orthogonal eigenproblems

  • W. B. Gragg
  • L. Reichel


LetH∈ℂn xn be a unitary upper Hessenberg matrix whose eigenvalues, and possibly also eigenvectors, are to be determined. We describe how this eigenproblem can be solved by a divide and conquer method, in which the matrixH is split into two smaller unitary upper Hessenberg matricesH1 andH2 by a rank-one modification ofH. The eigenproblems forH1 andH2 can be solved independently, and the solutions of these smaller eigenproblems define a rational function, whose zeros on the unit circle are the eigenvalues ofH. The eigenvector ofH can be determined from the eigenvalues ofH and the eigenvectors ofH1 andH2. The outlined splitting of unitary upper Hessenberg matrices into smaller such matrices is carried out recursively. This gives rise to a divide and conquer method that is suitable for implementation on a parallel computer.

WhenH∈ℝn xn is orthogonal, the divide and conquer scheme simplifies and is described separately. Our interest in the orthogonal eigenproblem stems from applications in signal processing. Numerical examples for the orthogonal eigenproblem conclude the paper.

Subject Classifications

AMS(MOS): 65F15 CR: G.1.3 


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Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • W. B. Gragg
    • 1
  • L. Reichel
    • 2
  1. 1.Department of MathematicsNaval Postgraduate SchoolMontereyUSA
  2. 2.Bergen Scientific CentreBergenNorway

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