S. Huss-Lederman, A. Tsao and G. Zhang. A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrices. In Proceedings Sixth SIAM Conf. on Parallel Processing for Scientific Computing. SIAM, 1993. 37  Home/Search   Document Not in Database   Summary   Related Articles   Check

....exist for computing the eigenvalues of a symmetric matrix, including bisection methods, the QR algorithm, divide and conquer (Cuppen s algorithm) and Jacobi s method. More recently, novel algorithms based on invariant subspace decomposition that use matrix matrix multiplication have been proposed [2, 16]. For a complete survey and references, we suggest the reader turn to [12, 14] Jacobi s method is the oldest, dating back to the mid 1800 s. It has fallen out of favor and been resurrected on many occasions. Its most recent resurgence is mainly due to better stability properties [5] and ....

S. Huss-Lederman, A. Tsao and G. Zhang. A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrices. In Proceedings Sixth SIAM Conf. on Parallel Processing for Scientific Computing. SIAM, 1993. 37

.... [33, 32] All these methods su#er from the use of fine grain parallelism, instability, slow or misconvergence in the presence of clustered eigenvalues of the original problem or some constructed subproblems [16] The other algorithms most closely related to the approach used here may be found in [2, 6, 24], where symmetric matrices or, more generally, matrices with real spectra are treated. One of the notable features of the SDC algorithm is that it can calculate just those eigenvalues (and the corresponding invariant subspace) in a user specified region of the complex plane. To help the user ....

S. HUSS-LEDERMAN, A. TSAO, AND G. ZHANG, A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrices, in Proc. of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, SIAM, Philadelphia, PA, 1993.

....reduction of width nb necessitates the update of a n Theta nb strip of Q from the right. This is the same kernel as in the POST step. 3 A Parallel Implementation To distribute the matrix across a distributed memory machine, we chose a blocked twodimensional torus wrapping (see, for example [17, 11, 10]) A scalar wrap mapping and onedimensional (i.e. row or column oriented distributions) are special cases of this mapping. This mapping also has been selected in other efforts to develop linear algebra basis software for massively parallel machines, for example the ScaLAPACK project [9, 11] We ....

....routines in the Chamelon programming system that our implementation builds on. Lastly we mention that, in the end, we plan to exploit the divide and conquer nature of the tridiagonalization of matrices with only 0 and 1 eigenvalues [6] as it arises in the ISDA eigenvalue solver framework [17]. With the 2D torus wrapped data mapping, this approach would not significantly reduce the communication requirements of the code, but would approximately halve the number of arithmetic operations required. Acknowledgements The authors would like to thank Bill Gropp for his help and support with ....

S. Huss-Lederman, A. Tsao, and G. Zhang, A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrices, in Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientfic Computing, R. Sincovec, ed., Philadelphia, 1993, SIAM.

....reduction of width nb necessitates the update of a n Theta nb strip of Q from the right. This is the same kernel as in the POST step. 3 A Parallel Implementation To distribute the matrix across a distributed memory machine, we chose a blocked twodimensional torus wrapping (see, for example [17, 11, 10]) A scalar wrap mapping and onedimensional (i.e. row or column oriented distributions) are special cases of this mapping. This mapping also has been selected in other efforts to develop linear algebra basis software for massively parallel machines, for example the ScaLAPACK project [9, 11] We ....

....routines in the Chamelon programming system that our implementation builds on. Lastly we mention that, in the end, we plan to exploit the divide and conquer nature of the tridiagonalization of matrices with only 0 and 1 eigenvalues [6] as it arises in the ISDA eigenvalue solver framework [17]. With the 2D torus wrapped data mapping, this approach would not significantly reduce the communication requirements of the code, but would approximately halve the number of arithmetic operations required. Acknowledgements The authors would like to thank Bill Gropp for his help and support with ....

S. Huss-Lederman, A. Tsao, and G. Zhang, A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrices, in Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientfic Computing, R. Sincovec, ed., Philadelphia, 1993, SIAM.

....ffl We propose a realistic and inexpensive stopping criterion for the inner loop iteration. Many simplifications in these algorithms are possible in case the matrix A is symmetric. The PRISM project, with which this work is associated, is also producing algorithms for the symmetric case; see [10, 9, 37, 5, 40] for more details. The rest of this paper is organized as follows. In section 2 we present our two algorithms for the ordinary and generalized spectral divide and conquer problems, discuss some implementation details and options, and show how to divide the spectrum along arbitrary circles and ....

S. Huss-Lederman, A. Tsao, and G. Zhang. A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrics. In Proceedings of the Sixth SIAM Conference on Parallel Proceesing for Scientific Computing. SIAM, 1993.

....in a slightly different location. Compared with the other approaches mentioned above, we believe the algorithms discussed in this paper offer an effective tradeoff between parallelizability and stability. The other algorithms most closely related to the approaches used here may be found in [3, 9, 36], where symmetric matrices, or more generally matrices with real spectra, are treated. Another advantage of the algorithms described in this paper is that they can compute just those eigenvalues (and the corresponding invariant subspace) in a user specified region of the complex plane. To help the ....

S. Huss-Lederman, A. Tsao, and G. Zhang. A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrices. In Proceedings of the Sixth SIAM Conference on Parallel Proceesing for Scientific Computing. SIAM, 1993.

.... [33, 32] All these methods suffer from the use of fine grain parallelism, instability, slow or misconvergence in the presence of clustered eigenvalues of the original problem or some constructed subproblems [16] The other algorithms most closely related to the approach used here may be found in [2, 6, 24], where symmetric matrices, or more generally matrices with real spectra, are treated. One of the notable features of the SDC algorithm is that it can calculate just those eigenvalues (and the corresponding invariant subspace) in a user specified region of the complex plane. To help the user ....

S. Huss-Lederman, A. Tsao, and G. Zhang. A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrices. In Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing. SIAM, 1993.

....are available, for Sun and Cray machines. Contact: Steve Lederman, Supercomputing Research Center, USA Email: lederman super.org FTP: Implementations and technical reports are available for anonymous ftp at ftp: ftp.super.org pub prism Comments: Tested and available at Daresbury [3] References: [19, 10, 15, 11, 12, 115, 79, 18, 80, 16, 77, 17, 76, 78] 3 GRID BASED TOOLS 16 3 Grid based tools 3.1 AMR Name: AMR , Adaptive Mesh Refinement Class Library Description: A C class library for building self adaptive mesh refinement applications. Parallelisation and array handling are inherited from P (see entry for P ) Systems: Built on P ....

....are available, for Sun and Cray machines. Contact: Steve Lederman, Supercomputing Research Center, USA Email: lederman super.org FTP: Implementations and technical reports are available for anonymous ftp at ftp: ftp.super.org pub prism Comments: Tested and available at Daresbury [3] References: [19, 10, 15, 11, 12, 115, 79, 18, 80, 16, 77, 17, 76, 78] 7.2 BLACS Name: BLACS: Basic Linear Algebra Communication Subroutines Description: Package of communication skeletons for use in parallel linear algebra codes on message passing machines. Designed for efficient communication operations on 2D arrays and sub arrays on a rectangular mesh of ....

S E Huss-Lederman, A Tsao, and G Zhang. A parallel implementation of the invariant subspace decomposition algorithm for dense symmetric matrices. In Proceedings of the sixth SIAM conference on Parallel Processing for Scientific Computation, March 1993. Available by anonymous ftp from ftp.super.org in file pub/prism/wn9.ps.

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