A parallel algorithm for computing the extremal eigenvalues of very large sparse matrices

This paper proposed the several real life applications for big data analytic using parallel computing software. Some parallel computing software under consideration are Parallel Virtual Machine, MATLAB Distributed Computing Server and Compute Unified Device Architecture to simulate the big data problems. The parallel computing is able to overcome the poor performance at the runtime, speedup and efficiency of programming in sequential computing. The mathematical models for the big data analytic are based on partial differential equations and obtained the large sparse matrices from discretization and development of the linear equation system. Iterative numerical schemes are used to solve the problems. Thus, the process of computational problems are summarized in parallel algorithm. Therefore, the parallel algorithm development is based on domain decomposition of problems and the architecture of difference parallel computing software. The parallel performance evaluations for distributed and shared memory architecture are investigated in terms of speedup, efficiency, effectiveness and temporal performance.

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An Efficient Parallel Algorithm for Extreme Eigenvalues of Sparse Nonsymmetric Matrices

The International Journal of Supercomputing Applications ◽

10.1177/109434209200600106 ◽

1992 ◽

Vol 6 (1) ◽

pp. 98-111 ◽

Cited By ~ 2

Author(s):

S. K. Kim ◽

A. T. Chrortopoulos

Keyword(s):

Shared Memory ◽

Message Passing ◽

Sparse Matrices ◽

Data Locality ◽

Main Memory ◽

Global Memory ◽

Global Communication ◽

Step Method ◽

Arnoldi Algorithm ◽

Large Sparse Matrices

Main memory accesses for shared-memory systems or global communications (synchronizations) in message passing systems decrease the computation speed. In this paper, the standard Arnoldi algorithm for approximating a small number of eigenvalues, with largest (or smallest) real parts for nonsymmetric large sparse matrices, is restructured so that only one synchronization point is required; that is, one global communication in a message passing distributed-memory machine or one global memory sweep in a shared-memory machine per each iteration is required. We also introduce an s-step Arnoldi method for finding a few eigenvalues of nonsymmetric large sparse matrices. This method generates reduction matrices that are similar to those generated by the standard method. One iteration of the s-step Arnoldi algorithm corresponds to s iterations of the standard Arnoldi algorithm. The s-step method has improved data locality, minimized global communication, and superior parallel properties. These algorithms are implemented on a 64-node NCUBE/7 Hypercube and a CRAY-2, and performance results are presented.

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