García, I.

Garzón, E. M.

Garzón E. M.

García I.

University of Szeged

Parallel Implementation of the Large Sparse and Symmetric EigenproblemE. M. Garzón and I. GarcíaThe aim of this work is to solve the eigenproblem for large symmetric and sparse matrices througha divide–and–conquer strategy put forward by Cuppen [1, 2]. This method has been implementedon a Cray T3E multiprocessor system, using PVM parallel interface and a detailed analysis of theobtained performance has been carried out.The solution of the eigenproblem has been decomposed in the next phases:	 Obtaining a structured matrix from the input matrix  . In this phase, both a tridiagonal ma-trix    and an orthonormal matrix  are obtained from , where  is similar to . Thesematrices hold the relation    . The Lanczos Method with Complete Reorhtogonal-ization has been implemented to be carried out at this stage [3, 4, 5].	 Solving the eigenproblem of  . The goal of this stage is to compute the diagonal  andorthonormal  matrices. These matrices hold the relation     . The divide-and-conquer method (dc) has been implemented to solve this problem. This method is decomposedin the following stages:1. Decomposition in  subproblems (with   ), through the Rank-One Modificationof  .2. Solving eigenproblem of each subproblem through the  algorithm.3. Reconstructing. The solution of the eigenproblem for each link of subproblems is com-puted through the solution of each. Once it has been determined the solution of all thelinks of the same dimension, other stage of the reconstruction begins in which pairs ofprevious stage subproblems are linked. As the process develops, the number of subprob-lems is halved, until just one link of is solved. In this way, the graph of tasks connectedwith the development of this stage can be represented through a binary tree.	 Computing the eigenvectors of . If  denotes the orthonormal matrix whose columns are theeigenvectors of , then    . On the other hand, the output matrices generated in theprevious stages hold the relation      , so    . Thus, this stageis carried out with a matrices product.The eigenproblem of a sparse matrix is decomposed in several procedures whose performance issequential, therefore it has been established a parallel implementation for each procedure in an inde-pendent way. Nevertheless, in order to link all the procedures in parallel, the output data distributionof a procedure establishes the parallel implementation of the procedure that receives these data asinput.The parallel implementation in phase 1 (Lanczos) is based on a decomposition in domains ofinput matrix. The input data are irregular, since  is sparse. Thus, a stage of preprocessing hasbeen designed, namely Pivoting-Block, that is quick and guarantees that the input data partitions arebalanced and also the operations associated to them [6, 7].Then the parallel implementation of the Cuppen’s method is based on decomposition in domainsin both, the input and output data. The binary tree of tasks is distributed so that the same number ofbranches of the tree is allocated to each PE. If  refers to the number of PEs of the multiprocessorsystem and    then initially each PE starts the reconstruction process independently, until   . Then groups of PEs that collaborate in the solution of a pair of subproblems are defined.As the reconstruction stages develop the subproblems are of a greater dimension and the number ofPEs that collaborate is larger too.The product    is carried out starting with a partition of  and  by rows among PEs.This partition causes a remarkable penalization in the communications that should be established. Itwill be proved that the phases with the largest computational cost are the first and the last. This imple-mentation will be evaluated for matrices of dimension between 782 and 7168, obtaining superlinealspeed-up in most cases. It will be checked how the superlineal speed-up is due to low efficiencymanagement of memory hierarchy with few PEs. It wil be proved that the management improvessubstantially as the number of PEs is increased.34References[1] Cuppen, J.J.M, "A Divide and Conquer Method for the Symmetric Tridiagonal Eigenproblem",Numer. Math, vol. 36, pag. 177-195, 1981.[2] Dongarra, J.J. and Sorensen, D.C " Fully Parallel Algorithm for the Symmetric Eigenvalue Prob-lem", SIAM J. Sci. Stat. Comput., vol 8, num. 2, pag. 139-154, 1987.[3] Paige, C.C. "Practical Use of the Symmetric Lanczos Process with Re-Orthogonalization.", BIT,10, pag. 183-195. 1970.[4] Paige, C.C. "Accuracy and Effectiveness of the Lanczos Algorithm for the Symmetric Eigen-problem.", Linear Algebra and its Applications, 34, pag. 235-258, 1980.[5] Parlett, B.N. and Nour-Omid, B. "Towards a Black Box Lanczos Program.", Computers PhysicsCommunications. Vol 53, pag 169-179, 1989.[6] Garzón, E.M. and García, I. "Parallel Implementation of the Lanczos Method for Sparse Matri-ces: Analysis of the Data Distributions." Proceedings ICS’96. 1996.[7] Garzón, E.M. and García, I. "Evaluation of the Work Load Balance in Irregular Problems UsingValue Based Data Distributions." Proceedings of Euro-PDS’97, pag 137-143, 1997.35

Parallel implementation of the large sparse and symmetric eigenproblem :  [abstract]

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Parallel implementation of the large sparse and symmetric eigenproblem

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