Journal ArticleDOI
A fully parallel algorithm for the symmetric eigenvalue problem
Jack Dongarra,Danny C. Sorensen +1 more
TLDR
In this article, the authors present a parallel algorithm for the symmetric algebraic eigenvalue problem, which is based upon a divide and conquer scheme suggested by Cuppen for computing the eigensystem of a symmetric tridiagonal matrix.Abstract:
In this paper we present a parallel algorithm for the symmetric algebraic eigenvalue problem. The algorithm is based upon a divide and conquer scheme suggested by Cuppen for computing the eigensystem of a symmetric tridiagonal matrix. We extend this idea to obtain a parallel algorithm that retains a number of active parallel processes that is greater than or equal to the initial number throughout the course of the computation. We give a new deflation technique which together with a robust root finding technique will assure computation of an eigensystem to full accuracy in the residuals and in the orthogonality of eigenvectors. A brief analysis of the numerical properties and sensitivity to round off error is presented to indicate where numerical difficulties may occur. The algorithm is able to exploit parallelism at all levels of the computation and is well suited to a variety of architectures.Computational results are presented for several machines. These results are very encouraging with respect to both...read more
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Proceedings ArticleDOI
LAPACK: a portable linear algebra library for high-performance computers
E. Anderson,Zhaojun Bai,Jack Dongarra,Anne Greenbaum,A. McKenney,J. Du Croz,S. Hammarling,James Demmel,Christian Bischof,Danny C. Sorensen +9 more
TL;DR: The goal of the LAPACK project is to design and implement a portable linear algebra library for efficient use on a variety of high-performance computers, based on the widely used LINPACK and EISPACK packages, but extends their functionality in a number of ways.
Journal ArticleDOI
Large-Scale Sparse Singular Value Computations
TL;DR: Four numerical methods for computing the singular value decomposition (SVD) of large sparse matrices on a multiprocessor architecture are presented and may help advance the development of future out-of-core sparse SVD methods, which can be used to handle extremely large sparsematrices associated with extremely large databases in query-based information-retrieval applications.
Journal ArticleDOI
Eigenvalue computation in the 20th century
TL;DR: The intention of this contribution is to sketch the main developments of this century, especially as they relate to one another, and to give an impression of the state of the art at the turn of the authors' century.
Journal ArticleDOI
Solving a Polynomial Equation: Some History and Recent Progress
TL;DR: The history of the algorithmic approach to this problem is recalled, some successful solution algorithms are reviewed, and some algorithms of 1995 are outlined that solve this problem at a surprisingly low computational cost.
References
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Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Journal ArticleDOI
Smoothing by spline functions. II
TL;DR: In this paper, the authors generalize the results of [4] and modify the algorithm presented there to obtain a better rate of convergence, which is the same as in this paper.
Book
Introduction to matrix computations
TL;DR: Rounding-Error Analysis of Solution of Triangular Systems and of Gaussian Elimination.
Book
Matrix Eigensystem Routines - Eispack Guide
TL;DR: Eispack as discussed by the authors is an Eispack subroutine that uses handbook algol procedures to validate and validate EISPACKs and is used for EisPacks.
Journal ArticleDOI
Rank-one modification of the symmetric eigenproblem
TL;DR: An algorithm is presented for computing the eigensystem of the rank-one modification of a symmetric matrix with known eIGensystem and the explicit computation of the updated eigenvectors and the treatment of multiple eigenvalues.