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Parallel eigenvalue reordering in real Schur forms

by R. Granat and B. Kagstroem and D. Kressner

(Report number 2008-16)

Abstract
A parallel algorithm for reordering the eigenvalues in the real Schur form of a matrix is presented and discussed. Our novel approach adopts computational windows and delays multiple outside-window updates until each window has been completely reordered locally. By using multiple concurrent windows the parallel algorithm has a high level of concurrency, and most work is level 3 BLAS operations. The presented algorithm is also extended to the generalized real Schur form. Experimental results for ScaLAPACK-style Fortran 77 implementations on a Linux cluster confirm the efficiency and scalability of our algorithms in terms of more than 16 times of parallel speedup using 64 processor for large scale problems. Even on a single processor our implementation is demonstrated to perform significantly better compared to the state-of-the-art serial implementation.

Keywords: Parallel algorithms, eigenvalue problems, invariant subspaces, direct reordering, Sylvester matrix equations, condition number estimates

@Techreport{GKK08_380,
  author = {R. Granat and B. Kagstroem and D. Kressner},
  title = {Parallel eigenvalue reordering in real Schur forms},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2008-16},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2008/2008-16.pdf },
  year = {2008}
}

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