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(i) To develop and extend parallel algorithms for the computation of partial eigensolutions;(ii) For each of a collection of machines with different architectures to determine which of the algorithms is (are) the most efficient.Progress:The project is being undertaken in conjunction with an EPSRC supported project (GR/J41857) led by Dr J. S. Weston at the University of Ulster at Coleraine. Work in the first ten months of the project has concentrated on parallel versions of two methods for the partial eigensolution of real symmetric matrices: the Lanczos method and the simultaneous iteration method. The methods rely heavily on matrix-matrix products, matrix-vector products and vector orthogonalisation. A new convergence monitoring procedure [1] has been developed for the Lanczos method which may be efficiently implemented on parallel machines.A comparison of the relative merits of the Lanczos and simultaneous iteration methods when executed on a dual-processor Convex C3840 has been made. This work is reported in [2].An efficient parallel method for the orthogonalisation of sets of vectors, based on Choleski decomposition and implemented in a SIMD environment, has been investigated [3]A general paper [4] was presented at the Para94 Conference in Copenhagen, June 21-23, 1994, and paper [5] has been accepted for presentation at ICIAM 95. Papers [6] and [7] have been submitted to ParCo95 and paper [8] is in preparation.[1] Szularz M, Weston J, Murphy K and Clint M, Monitoring the Convergence of the Lanczos Algorithm in Parallel Computing Environments, to be published in Journal of Parallel Algorithms and Applications.[2] Weston J, Szularz M, Clint M and Murphy K, The Parallel Computation of Partial Eigensolutions on a Shared Memory Multiprocessor, submitted to Euro-Par95.[3] Clint M, Weston J S and Flannagan J B, Efficient Gram-Schmidt Orthogonalisation on an Array Processor, Parallel Processing: Conpar 94 - VAPP VI ( B Buchberger, J Volkert, eds) LNCS 854, Springer-Verlag, 218-228 (1994).[4] Szularz M, Weston J, Murphy K and Clint M, Parallel algorithms for the partial eigensolution of large sparse matrices on novel architecture computers, in Parallel Scientific Computing, LNCS No 879, J Dongarra and J Wasniewski (Eds), Springer-Verlag, (1994) 469-482.[5] Weston J, Szularz M, Clint M, and Murphy K, Parallel Lanczos algorithms for novel architecture computers: a comparative evaluation, to appear in special volumes of the Zeitschrift fur Angewandte Mathematik und Mechanik, ZAMM.[6] Szularz M, Weston J, Murphy K and Clint M, An Improved Parallel Lanczos Algorithm, submitted to ParCo95.[7] Murphy K, Szularz M, Clint M and J S Weston, Simultaneous Iteration in Parallel Computing Environments: A Comparative Evaluation, submitted to ParCo95.[8] Clint M, Murphy K, Weston J S and Szularz M, The parallel computation of partial eigensolutions using an enhanced Lanczos method, in preparation.
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