A Distributed Memory Implementation of the Nonsymmetric QR Algorithm

The QR algorithm is the crux of the serial nonsymmetric eigenvalue problem. Recent eeorts to parallelize this algorithm have made signiicant advances towards solving the parallel nonsym-metric eigenvalue problem. Most methods to date suuer a scalability problem. In this talk we discuss an approach for parallelizing QR which overcomes many of the disadvantages to date. We also give insights into what is necessary for a parallel algorithm to work using these strategies. Performance of a parallel implementation on the Intel Paragon TM system is reported.

[1]  Al Geist,et al.  Finding eigenvalues and eigenvectors of unsymmetric matrices using a distributed-memory multiprocessor , 1990, Parallel Comput..

[2]  Robert A. van de Geijn,et al.  Deferred Shifting Schemes for Parallel QR Methods , 1993, SIAM J. Matrix Anal. Appl..

[3]  D. Sorensen,et al.  Block reduction of matrices to condensed forms for eigenvalue computations , 1990 .

[4]  Daniel Boley,et al.  A parallel QR algorithm for the nonsymmetric eigenvalue problem , 1989 .

[5]  James Demmel,et al.  Modeling the benefits of mixed data and task parallelism , 1995, SPAA '95.

[6]  Greg Henry Improving the Unsymmetric Parallel QR Algorithm on Vector Machines , 1993, PPSC.

[7]  James Demmel,et al.  The Spectral Decomposition of Nonsymmetric Matrices on Distributed Memory Parallel Computers , 1997, SIAM J. Sci. Comput..

[8]  Jack J. Dongarra,et al.  A Parallel Algorithm for the Reduction of a Nonsymmetric Matrix to Block Upper-Hessenberg Form , 1995, Parallel Comput..

[9]  Gregory Mark Henry Improving data re-use in eigenvalue-related computations , 1994 .

[10]  Robert A. van de Geijn Implementing the qr-algorithm on an array of processors , 1987 .

[11]  Bruce Hendrickson,et al.  The Torus-Wrap Mapping for Dense Matrix Calculations on Massively Parallel Computers , 1994, SIAM J. Sci. Comput..

[12]  David S. Watkins,et al.  Shifting Strategies for the Parallel QR Algorithm , 1994, SIAM J. Sci. Comput..

[13]  G. W. Stewart,et al.  A parallel implementation of the QR-algorithm , 1987, Parallel Comput..

[14]  G. A. Geist,et al.  Finding eigenvalues and eigenvectors of unsymmetric matrices using a hypercube multiprocessor , 1989, C3P.

[15]  J. Demmel,et al.  An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems , 1997 .

[16]  E. Jessup A case against a divide and conquer approach to the nonsymmetric eigenvalue problem , 1993 .

[17]  David S. Watkins,et al.  The transmission of shifts and shift blurring in the QR algorithm , 1996 .

[18]  Robert A. van de Geijn,et al.  Storage Schemes for Parallel Eigenvalue Algorithms , 1988 .

[19]  Bo Kågström,et al.  GEMM-Based Level-3 BLAS , 1991 .

[20]  James Demmel,et al.  ScaLAPACK: A Portable Linear Algebra Library for Distributed Memory Computers - Design Issues and Performance , 1995, Proceedings of the 1996 ACM/IEEE Conference on Supercomputing.

[21]  L. Kaufman,et al.  Squeezing the most out of eigenvalue solvers on high-performance computers , 1986 .

[22]  Jack J. Dongarra,et al.  A set of level 3 basic linear algebra subprograms , 1990, TOMS.

[23]  Steven Huss-Lederman,et al.  A Parallelizable Eigensolver for Real Diagonalizable Matrices with Real Eigenvalues , 1997, SIAM J. Sci. Comput..

[24]  J. H. Wilkinson The algebraic eigenvalue problem , 1966 .

[25]  R. Byers Numerical Stability and Instability in Matrix Sign Function Based Algorithms , 1986 .

[26]  James Demmel,et al.  On a Block Implementation of Hessenberg Multishift QR Iteration , 1989, Int. J. High Speed Comput..

[27]  Jack J. Dongarra,et al.  Algorithm 710: FORTRAN subroutines for computing the eigenvalues and eigenvectors of a general matrix by reduction to general tridiagonal form , 1990, TOMS.

[28]  Vipin Kumar,et al.  The Scalability of FFT on Parallel Computers , 1993, IEEE Trans. Parallel Distributed Syst..

[29]  Eleanor Chu,et al.  New Distributed-Memory Parallel Algorithms for Solving Nonsymmetric Eigenvalue Problems , 1995, PPSC.

[30]  L. Auslander,et al.  On parallelizable eigensolvers , 1992 .

[31]  Anshul Gupta,et al.  On the scalability of FFT on parallel computers , 1990, [1990 Proceedings] The Third Symposium on the Frontiers of Massively Parallel Computation.

[32]  James Demmel,et al.  Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I , 1993, PPSC.

[33]  David S. Watkins,et al.  Fundamentals of matrix computations , 1991 .

[34]  Jack J. Dongarra,et al.  A Parallel Algorithm for the Nonsymmetric Eigenvalue Problem , 1993, SIAM J. Sci. Comput..

[35]  Robert A. van de Geijn,et al.  Parallelizing the QR Algorithm for the Unsymmetric Algebraic Eigenvalue Problem: Myths and Reality , 1996, SIAM J. Sci. Comput..