C. Trefftz, C. C. Huang, P. K. Mckinley, T. Y. Li, and Z. Zeng, "A scalable eigenvalue solver for symmetric tridiagonal matrices," Parallel Computing, Vol. 21, pp. 1213--1240, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....eigenvalue problem can also be transformed into an eigenvalue problem of the form Cx = x; where C = B Gamma1 A: Then we compute the eigenvalues of C. In the science and engineering areas, most applications require the eigenvalues of a large matrix. Especially, symmetric tridiagonal systems [4, 8, 10, 13, 15], banded systems [9] and circulant tridiagonal systems [3, 11] are frequently discussed. For instance, in computer aided design, graphic pattern recognition and image processing, closed curve fitting [2] is important and it can be transformed into a circulant tridiagonal system. In the past, ....

C. Trefftz, C. C. Huang, P. K. Mckinley, T. Y. Li, and Z. Zeng, "A scalable eigenvalue solver for symmetric tridiagonal matrices," Parallel Computing, Vol. 21, pp. 1213--1240, 1995.

.... wide variety of applications, including the dynamic analysis of large scale structures such as aircraft and spacecraft, the prediction of structural responses in solid and soil mechanics, the study of solar convection, the modal analysis of electronic circuits, and the statistical analysis of data [TMLZ93]. Thus the need for faster methods to solve these large eigenvalue problems becomes very important. The problem of finding the eigenvalues of a matrix can be stated as follows: Find the values that satisfy the equation: Ax = x for a vector x, which is called an eigenvector and an eigenvalue. In ....

C. Trefftz, P. K. McKinley, T. Y. Li, and Z. Zeng. A scalable eigenvalue solver for symmetric tridiagonal matrices. In R. Sincovec, D. E. Keyes, M. R. Leuze, L. R. Petzold, and D. A. Reed, editors, Proceedings of the sixth SIAM Conference on Parallel Processing, volume 2, pages 602--609, 1993.

....size 2 i , using the results of stage i Gamma 1 as initial approximations. Due to the independence among computing tasks, this algorithm is inherently parallelizable. We have previously developed a parallel version of the algorithm, which we implemented and tested on both an nCUBE 2 hypercube [15] and on a cluster of workstations [14] Dynamic load balancing was used to compensate for the variance in the number of iterations required to find eigenvalues and, in the cluster case, unequal workloads on processors. In order to compare the split merge version of the singular value algorithm ....

....Both versions were implemented on the two platforms, an nCUBE 2 and on a workstation cluster. Recall that in Li s algorithm, only the eigenvalues of B T B and T that lie in certain ranges, defined by the threshold, are of interest. The parallel implementation of split merge that was reported in [15] calculated all the eigenvalues in the input matrix. The algorithm was modified to find eigenvalues only in a given interval of interest. Consider Fig. 2, where every solid line represents all the eigenvalues of a given submatrix at a particular stage of the algorithm. In the original ....

C. Trefftz, P. K. McKinley, T. Y. Li, and Z. Zeng, A scalable eigenvalue solver for symmetric tridiagonal matrices, in Proceedings of the Sixth SIAM Conference on Parallel Processing, 1993, pp. 602--609.

....algorithm is its natural parallelism, in the sense that each eigenvalue is computed totally independently of the others. Besides, our algorithm has a great capacity for vectorization. These considerations will be discussed in x6 and an intensive experiment will be reported in a separate article [26]. The code of our algorithm is electronically available from Zhonggang Zeng. 2. Laguerre s iteration. 2.1. Evaluation of the determinant and its accuracy. Let T be a symmetric tridiagonal matrix of the form T = fi i Gamma1 ; ff i ; fi i ] 0 B B B B B B B B B B ff 1 fi 1 fi 1 ff 2 fi 2 0 ....

....x 1 ; Delta Delta Delta ; xm . This process can be performed with some do loops described in Fig. 6.1, where the i loops are vector operations. The implementation strategies and experimental results of the parallelization and vectorization of our method will be reported in a separate paper [26]. 6.3. Other possible variations. Our algorithm presented in this paper mainly consists of a split merge process and a carefully implemented Laguerre s iteration. The purpose of the split merge process is to separate the eigenvalues of the target matrix T and obtain initial approximations to ....

C. Trefftz, C. C. Huang, P. McKinley, T. Y. Li, and Z. Zeng, A scalable eigenvalue solver for symmetric tridiagonal matrices. priprint, Michigan State University (1993).

....only need partial eigenvalues of either B B or T . Most importantly, the algorithm is parallel in nature in the sense that each eigenvalue can be computed independently. Preliminary results on parallel implementation show that the split merge algorithm possesses scalability to a large extent [21] for solving symmetric tridiagonal eigenproblems, which is the main building block for our algorithm. The parallel implementation of the algorithm discussed in this article will be reported in a separate paper. In x5, numerical results on substantial variety of matrices describe the comparison of ....

C. Trefftz, P. McKinley, T. Y. Li and Z. Zeng, A scalable eigenvalue solver for symmetric tridiagonal matrices, preprint, Michigan State University, (submitted).

....The speed is sequentially competitive with the QR algorithm and faster than bisection multisection [15] as well as Cuppen s divide and conquer algorithm [6, 7] by a considerable margin. Most importantly, our method is, in contrast to the highly serial QR algorithm, fully parallel and scalable [23]. For a general matrix S, magnitude of the eigenvalues of the pencil (T; S) can be arbitrarily large when S is nearly singular. It is, therefore, inappropriate to apply the algorithm developed in [14] directly to finding zeros of (3) when all eigenvalues of (T; S) are in demand. The modification ....

C. Trefftz, P. McKinley, T. Y. Li and Z. Zeng, A scalable eigenvalue solver for symmetric tridiagonal matrices, Proceedings of The Sixth SIAM Conference on Parallel Processing for Scientific Computing, SIAM publications, pp 602--609, 1993.

....eigenvalues, typically either the smallest or largest. The algorithm has been adapted to find only an arbitrary number of the smallest (or largest) eigenvalues of the matrix. Limited space did not permit discussion of these modifications here; performance results for this algorithm are given in [4]. 5 Reducing Communication Costs In distributed memory systems, efficient communication is critical to performance. Figure 6 shows two traces of the load balancing algorithm using different implementations of broadcast. The shaded areas correspond to the time each node waits for the broadcast ....

C. Trefftz, P. K. McKinley, T. Y. Li, and Z. Zeng, "A scalable eigenvalue solver for symmetric tridiagonal matrices," Tech. Rep. MSU-CPS-ACS-69, Michigan State University, 1992.

....library in solving systems of linear equations is considered. The group table (hashed) 0 1 2 B 1 gid state parent num children[ members[ table entry (DFT source only) Figure 4. Group table design ComPaSS library has also been used in the implementation of a highly parallel eigenvalue solver [19] and implementation of barrier synchronization [18, 5] The following Gaussian elimination example illustrates the use of several data parallel operations. The algorithm is used in solving the equation Ax = b, where A is an n Theta n matrix and x and b are n element vectors. For efficient data ....

C. Trefftz, P. K. McKinley, T.-Y. Li, and Z. Zeng, "A scalable eigenvalue solver for symmetric tridiagonal matrices," accepted to appear in Proc. Sixth SIAM Conference on Parallel Processing for Scientific Computing, Mar. 1993.

....only need partial eigenvalues of either B B or T . Most importantly, the algorithm is parallel in nature in the sense that each eigenvalue can be computed independently. Preliminary results on parallel implementation show that the split merge algorithm possesses scalability to a large extent [21] for solving symmetric tridiagonal eigenproblems, which is the main building block for our algorithm. The parallel implementation of the algorithm discussed in this article will be reported in a separate paper. In Sect. 5, numerical results on substantial variety of matrices describe the ....

Trefftz, C., McKinley, P., Li, T.Y., Zeng, Z.: A Scalable eigenvalue solver for symmetric tridiagonal matrices Parallel Computing (to appear)

....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 type 1 and 2 are discussed here. The following conventions are used in the description of the matrices: ff i ; i = 1; n represent the (main) diagonal entries and fi i ; i = 1; n Gamma 1 denote the offdiagonal entries. For matrices of type 1 (Toeplitz ....

....for types 8 through 12. Clusters of 1, 2, 4 and 8 workstations were used. Figure 8 shows the execution times for matrix types 1 and 2 with 2048 entries. The results for types 3 through 12 are similar to those in Figure 8, and are omitted here for the sake of brevity; those results are available in [13]. The sequential version of the split merge algorithm is significantly faster than the sequential version of bisection. In examining absolute execution times of the parallel algorithm, for matrices with 2048 entries, the execution times of the parallel version of split merge on a cluster with 4 ....

C. Trefftz, P. K. McKinley, T. Y. Li, and Z. Zeng, "A scalable eigenvalue solver for symmetric tridiagonal matrices," Tech. Rep. MSU-CPS-ACS-69, Michigan State University, 1992.

....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 algorithm for MPCs [8]. Using dynamic load balancing and efficient communication routines, we demonstrated good speedup of the algorithm when implemented on a 64 node nCUBE 2 hypercube. In the project described herein, the split merge algorithm was redesigned and implemented atop a cluster of workstations ....

C. Trefftz, P. K. McKinley, T. Y. Li, and Z. Zeng, "A scalable eigenvalue solver for symmetric tridiagonal matrices," in Proceedings of the sixth SIAM conference on Parallel Processing, pp. 602--609, 1993.

....is needed even in cases where the algorithm calls for one process to send a message to all other processes in the application. Multicast is important in many parallel numerical algorithms, including matrix multiplication [3] matrix transpose [2] tridiagonalization [5] eigenvalue computation [19], Gaussian elimination [13] and LU factorization [1] Efficient implementation of multicast is also useful in many other aspects of parallel computing, including support for barrier synchronization [20] memory updates and invalidation in distributed sharedmemory systems [10] and global ....

Trefftz, C., McKinley, P. K., Li, T. Y., and Zeng, Z. A scalable eigenvalue solver for symmetric tridiagonal matrices. In Proceedings of the Sixth SIAM Conference on Parallel Processing (1993), pp. 602--609.

....is needed even in cases where the algorithm calls for one process to send a message to all other processes in the application. Multicast is important in many parallel numerical algorithms, including matrix multiplication [3] matrix transpose [4] tridiagonalization [5] eigenvalue computation [6], Gaussian elimination [7] and LU factorization [8] Efficient implementation of multicast is also useful in many other aspects of parallel computing, including support for barrier synchronization [9] memory updates and invalidation in distributed shared memory systems [10] and global ....

C. Trefftz, P. K. McKinley, T. Y. Li, and Z. Zeng, "A scalable eigenvalue solver for symmetric tridiagonal matrices," in Proceedings of the Sixth SIAM Conference on Parallel Processing, pp. 602--609, 1993.

....operations needed by parallel programs. Much of our earlier and ongoing ComPaSS research has focused on software implementations of collective communication in wormhole routed massively parallel computers [12, 31, 32, 33] and their use to improve the performance of parallel algorithms [34]. In expanding ComPaSS to include cluster platforms, we seek to draw upon and expand earlier contributions in this area. For example, many collective operations rely on efficient point to point communication, so the research projects mentioned above concerning streamlining of protocols and network ....

....on underlying, unreliable hardware multicast functionality. Performance measurements illustrate that both approaches offer substantial improvement over conventional methods. In related projects, we have used the new multicast operations to improve the performance of parallel numerical algorithms [34] and parallel data clustering algorithms [35] ....

C. Trefftz, P. K. McKinley, C. C. Huang, T.-Y. Li, and Z. Zeng, "A scalable eigenvalue solver for symmetric tridiagonal matrices," in Proceedings of the 14th International Conference on Distributed Computing Systems, 1994. accepted to appear.

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