P. Arbenz, K. Gates, and Ch. Sprenger. A parallel implementation of the symmetric tridiagonal qr algorithm. In Frontier's 92, McLEan, Virginia, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to find the singular values of a bidiagonal matrix is an important part of the overall process of finding the singular values of a real matrix A. Parallel processing can be used to reduce the execution time. Different parallel methods have been used to solve the SVD problem of bidiagonal matrices [2, 3, 7, 9], and space limitations do not allow us to review them here. Many algorithms use an approach that first finds the eigenvalues of a symmetric tridiagonal (ST) matrix that is related to the bidiagonal matrix B. Li et al. [10] recently proposed a new SVD algorithm that combines two eigenvalue ....

P. Arbenz, K. Gates, and C. Sprenger, A parallel implementation of the symmetric tridiagonal QR algorithm, in Proceedings of the Fourth Symposium on the Frontiers of Massively Paralllel Computation, IEEE CS Press, 1992.

....is presented in Section 7, and Section 8 concludes the paper. 2 Related Work Several different approaches have been used for finding the eigenvalues of real, symmetric, tridiagonal matrices on parallel computers. These approaches can be divided into four broad classes: parallel QR methods [7, 8], Cuppen s divide and conquer method [9] Sturm s sequence evaluations [10] used as the basis of bisection and multisection) and homotopy methods [11] Parallelization of QR was first discussed by Sameh and Kuck [7] Cuppen s algorithm has been parallelized in shared memory machines by Dongarra ....

....that parallelizes the evaluation of Sturm s sequence to improve the bisection process; their algorithm was implemented on a shared memory system. A homotopy method for hypercubes was described by Li, Zhang, and Sun [11] With the advent of workstation clusters, Arbenz, Gates, and Sprenger [8] conducted a study comparing the performance of an eigenproblem solver (both eigenvalues and eigenvectors) on a nCUBE 2 and a (Sun IPC) workstation cluster (as well as other platforms) That study focused on a modified version of QR for finding both eigenvalues and eigenvectors. The finding of the ....

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P. Arbenz, K. Gates, and C. Sprenger, "A parallel implementation of the symmetric tridiagonal qr algorithm," in Proceedings of the Fourth Symposium on the Frontiers of Massively Paralllel Computation, IEEE CS Press, 1992.

....once the initial data has been broadcast to the participating nodes, each node can work on its part of the problem without any communication with other nodes, except for reporting the final results to the initiating node. QR methods are difficult to parallelize when only the eigenvalues are sought [14]. Ipsen and Jessup [6] report that parallel bisection is faster than divide and conquer [15] hence the decision to use bisection in comparisons reported here. Twelve types of standard input matrices were used in the experiments and are described in Table 1. The following conventions are used in ....

....for matrices of type 1 through 12 on a cluster of workstations 25 6 Related Work Several different approaches have been used for finding the eigenvalues of real, symmetric, tridiagonal matrices on parallel computers. These approaches can be divided into four broad classes: parallel QR methods [19, 14], Cuppen s divide and conquer method [15] Sturm s sequence evaluations [17] used as the basis of bisection and multisection) and homotopy methods [20] Parallelization of QR was first discussed by Sameh and Kuck [19] Cuppen s algorithm has been parallelized in shared memory machines by ....

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P. Arbenz, K. Gates, and C. Sprenger, "A parallel implementation of the symmetric tridiagonal qr algorithm," in Proceedings of the Fourth Symposium on the Frontiers of Massively Paralllel Computation, IEEE CS Press, 1992.

.... to find all the eigenvalues is sequential in nature and is not easily parallelized on modern parallel machines despite the attempts in [96, 132, 93] However, the O(n 3 ) computation in accumulating the Givens rotations into the eigenvector matrix is trivially and efficiently parallelized, see [3] for more details. But the higher operation count and inability to exploit fast matrix multiply based operations make the QR algorithm much slower than divide and conquer and also, slower on average than bisection followed by inverse iteration. Prior to beginning this work, we were hopeful of ....

P. Arbenz, K. Gates, and Ch. Sprenger. A parallel implementation of the symmetric tridiagonal QR algorithm. In Frontier's 92. McLean, 1992.

....approaches have been used for finding the eigenvalues of real, symmetric, tridiagonal matrices on parallel computers. These approaches can be divided into four broad classes: Cuppen s divide and conquer method [5] Sturm s sequence evaluations [6] homotopy methods [7] and parallelization of QR [8]. Cuppen s algorithm has been parallelized in shared memory machines [9] The routines BISECT and INVIT, from EISPACK, were parallelized, again for shared memory machines [10] A parallel version of Cuppen s algorithm for the hypercube was presented in [11] where static task assignment was used. ....

....again for shared memory machines [10] A parallel version of Cuppen s algorithm for the hypercube was presented in [11] where static task assignment was used. A homotopy method for hypercubes was described in [7] QR was first parallelized by Kuck and Sameh [12] a new parallel version of QR [8] has been implemented in a Cray Y MP, an Alliant FX 80, a Sequent Symmetry, a nCUBE 2, a Thinking Machines CM200, and a cluster of Sun SPARCstations. 7 Conclusions and Future Work The general structure of the sequential split merge algorithm lends itself very well for a very efficient ....

P. Arbenz, K. Gates, and C. Sprenger, "A parallel implementation of the symmetric tridiagonal qr algorithm," in Proceedings of the Fourth Symposium on the Frontiers of Massively Paralllel Computation, IEEE CS Press, 1992.

....symmetric tridiagonal matrices. As one of the most fundamental problems of computational mathematics, this problem continues to receive considerable attention in the literature due to its wide applicability. Several parallel and distributed algorithms have been developed to address this problem [2, 3, 4, 5, 6]. We study the split merge algorithm, designed originally for shared memory parallel architectures by Li and Zeng [7] This algorithm is inherently parallel and takes advantage of a fast iteration technique, namely, Laguerre s method. We have previously designed and implemented a version of this ....

....once the initial data has been broadcast to the participating nodes, each node can work on its part of the problem without any communication with other nodes, except for reporting the final results to the initiating node. QR methods are difficult to parallelize when only the eigenvalues are sought [3]. Ipsen and Jessup [6] report that parallel bisection is faster than divide and conquer, hence the decision to use bisection in comparisons reported here. Twelve standard types of input matrices were used in the experiments and are described in the technical report [13] The results for matrices ....

P. Arbenz, K. Gates, and C. Sprenger, "A parallel implementation of the symmetric tridiagonal QR algorithm, " in Proceedings of the Fourth Symposium on the Frontiers of Massively Paralllel Computation, IEEE CS Press, 1992.

....thesis on network computing [25] He used Sun s XDR (eXternal Data Representation) library to support distributed computing in heterogeneous networks. The resulting Sciddle 2. 0 software 2 was used, among other things, for a parallel implementation of the symmetric tridiagonal QR algorithm [5]. In August 1993, Christoph Sprenger released Sciddle 3.0 3 . Now the stub generator was based on the UNIX tools lex and yacc. More computer architectures were supported, and the software was documented [7, 26] A complete rewrite of the Sciddle monitor lead to Sciddle 3.1 4 which was released ....

Peter Arbenz, Kevin Gates, and Christoph Sprenger, A Parallel Implementation of the Symmetric Tridiagonal QR Algorithm, in Frontiers '92, Proceedings of the Fourth Symposium on the Frontiers of Massively Parallel Computation, ed. H. J. Siegel, IEEE Computer Society Press, Los Alamitos, Californa, 1992, pp. 382--388.

....QR iteration for finding the eigenvalues of a symmetric tridiagonal matrix must be done sequentially with usually just one eigenvalue converging at a time. Thus, it is not suitable for parallel computers. However, when computing the eigenvector matrix Q too, it is especially easy to parallelize [7, 26]. Each processor redundantly runs the entire algorithm updating the tridiagonal matrix by forming PAP , but only computes n=p of the columns of PQ, where p is the number of processors and n is the dimension of A. At the end each processor has all eigenvalues and n=p components of each ....

P. Arbenz, K. Gates, and Ch. Sprenger. A parallel implementation of the symmetric tridiagonal QR algorithm. In Frontier's 92, McLean, Virginia, 1992.

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P. Arbenz, K. Gates, and Ch. Sprenger. A parallel implementation of the symmetric tridiagonal qr algorithm. In Frontier's 92, McLEan, Virginia, 1992.

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