An implementation of a divide and conquer algorithm for the unitary eigen problem

  • Authors:

  • G. S. Ammar;L. Reichel;D. C. Sorensen

  • Affiliations:

  • -;-;-

  • Venue:
  • ACM Transactions on Mathematical Software (TOMS)
  • Year:
  • 1992

Quantified Score

Hi-index 0.00

Visualization

Abstract

We present a FORTRAN implementation of a divide-and-conquer methodfor computing the spectral resolution of a unitary upper Hessenbergmatrix H. Any such matrixH of ordern, normalized so that its subdiagonalelements are nonnegative, can be written as a product ofn−1 Givensmatrices and a diagonal matrix. This representation, which we refer toas the Schur parametric form of H,arises naturally in applications such as in signal processing and in thecomputation of Gauss-Szego quadrature rules. Our programs utilize theSchur parametrization to compute the spectral decomposition ofH without explicitly forming theelements of H. If only theeigenvalues and first components of the eigenvectors are desired, as inthe applications mentioned above, the algorithm requires onlyO(n2) arithmeticoperations. Experimental results presented indicate that the algorithmis reliable and competitive with the general QR algorithm applied tothis problem. Moreover, the algorithm can be easily adapted for parallelimplementation.—Authors' Abstract