We survey general techniques and open problems in numerical linear algebra on parallel architectures. We first discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing efficient algorithms. We illustrate these principles using current architectures and software systems, and by showing how one would implement matrix multiplication. Then, we present direct and iterative algorithms for solving linear systems of equations, linear least squares problems, the symmetric eigenvalue problem, the nonsymmetric eigenvalue problem, the singular value decomposition, and generalizations of these to two matrices. We consider dense, band and sparse matrices.

Abstract In this paper we present an O(nk) procedure, Algorithm MR 3, for computing k eigenvectors of an n \Theta n symmetric tridiagonal matrix T. A salient feature of the algorithm is that a number of different LDL t products (L unit lower triangular, D diagonal) are computed. In exact arithmetic each LDL t is a factorization of a translate of T. We call the various LDL t

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. In this paper, preliminary research results on a new algorithm for finding all the eigenvalues and eigenvectors of a real diagonalizable matrix with real eigenvalues are presented. The basic mathematical theory behind this approach is reviewed and is followed by a discussion of the numerical considerations of the actual implementation. The numerical algorithm has been tested on thousands of matrices on both a Cray-2 and an IBM RS/6000 Model 580 workstation. The results of these tests are presented. Finally, issues concerning the parallel implementation of the algorithm are discussed. The algorithm's heavy reliance on matrix--matrix multiplication, coupled with the divide and conquer nature of this algorithm, should yield a highly parallelizable algorithm. Key words. eigenvalues, divide and conquer algorithm, invariant subspaces, parallel algorithm AMS subject classification. 65F15 PII. S1064827592228833 1. Introduction. Computation of all the eigenvalues and eigenvectors of a dens...

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This report discusses a serial implementation of Cuppen's divide and conquer algorithm for computing all eigenvalues and eigenvectors of a real symmetric matrix T = Q Q T. This method is compared with the LAPACK implementations of QR, bisection/inverse iteration, and root-free QR/inverse iteration to nd all of the eigenvalues and eigenvectors. On a DEC Alpha using optimized Basic Linear Algebra Subroutines (BLAS), divide and conquer was uniformly the fastest algorithm by a large margin for large tridiagonal eigenproblems. When Fortran BLAS were used, bisection/inverse iteration was somewhat faster (up to a factor of 2) for very large matrices (n 500) without clustered eigenvalues. When eigenvalues were clustered, divide and conquer was up to 80 times faster. The speedups over QR were so large in the tridiagonal case that the overall problem, including reduction to tridiagonal form, sped up by a factor of 2.5 over QR for n 500. Nearly universally, the matrix of eigenvectors generated by divide and con-

. We investigate direct algorithms to solve linear banded systems of equations on MIMD multiprocessor computers with distributed memory. We show that it is hard to beat ordinary one-processor Gaussian elimination. Numerical computation results from the Intel Paragon are given. 1. Introduction In a project on divide and conquer algorithms in numerical linear algebra, the authors studied parallel algorithms to solve systems of linear equations and eigenvalue problems. The latter consisted in a study of the divide and conquer algorithm proposed by Cuppen [4] and stabilized by Sorensen and Tang [11]. This algorithm is evolving as the standard algorithm for solving the symmetric tridiagonal eigenvalue problem on sequential as on parallel computers. In [7], Gates and Arbenz report on the first successful parallel implementation of the algorithm. They observed almost optimal speedups on the Intel Paragon. The accuracy observed is as good as with any other known (fast) algorithm. The...

. In this paper we present an algorithm for the eigenvalue problem of symmetric tridiagonal matrices. Our algorithm employs the determinant evaluation, split-and-merge strategy and Laguerre's iteration. The method directly evaluates eigenvalues and uses inverse iteration as an option when eigenvectors are needed. This algorithm combines the advantages of existing algorithms such as QR, bisection/multisection and Cuppen's divide-and-conquer method. It is fully parallel, and competitive in speed with the most efficient QR algorithm in serial mode. On the other hand, our algorithm is as accurate as any standard algorithm for the symmetric tridiagonal eigenproblem and enjoys the flexibility in evaluating partial spectrum. Key words. eigenvalue, Laguerre's iteration, symmetric tridiagonal matrix 1. Introduction. For a symmetric tridiagonal matrix T with nonzero subdiagonal entries, the eigenvalues of T or the zeros of its characteristic polynomial f() = det[T \Gamma I ] (1.1) are all rea...

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