Results 1  10
of
52
Parallel Numerical Linear Algebra
, 1993
"... 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 illust ..."
Abstract

Cited by 575 (26 self)
 Add to MetaCart
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, and the singular value decomposition. We consider dense, band and sparse matrices.
Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators
, 2005
"... ..."
(Show Context)
Design of a Parallel Nonsymmetric Eigenroutine Toolbox, Part I
, 1993
"... The dense nonsymmetric eigenproblem is one of the hardest linear algebra problems to solve effectively on massively parallel machines. Rather than trying to design a "black box" eigenroutine in the spirit of EISPACK or LAPACK, we propose building a toolbox for this problem. The tools ar ..."
Abstract

Cited by 65 (13 self)
 Add to MetaCart
The dense nonsymmetric eigenproblem is one of the hardest linear algebra problems to solve effectively on massively parallel machines. Rather than trying to design a "black box" eigenroutine in the spirit of EISPACK or LAPACK, we propose building a toolbox for this problem. The tools are meant to be used in different combinations on different problems and architectures. In this paper, we will describe these tools which include basic block matrix computations, the matrix sign function, 2dimensional bisection, and spectral divide and conquer using the matrix sign function to find selected eigenvalues. We also outline how we deal with illconditioning and potential instability. Numerical examples are included. A future paper will discuss error analysis in detail and extensions to the generalized eigenproblem.
Convergence of Algorithms of Decomposition Type for the Eigenvalue Problem
 Linear Algebra Appl
, 1995
"... We develop the theory of convergence of a generic GR algorithm for the matrix eigenvalue problem that includes the QR, LR, SR, and other algorithms as special cases. Our formulation allows for shifts of origin and multiple GR steps. The convergence theory is based on the idea that the GR algorithm p ..."
Abstract

Cited by 56 (14 self)
 Add to MetaCart
(Show Context)
We develop the theory of convergence of a generic GR algorithm for the matrix eigenvalue problem that includes the QR, LR, SR, and other algorithms as special cases. Our formulation allows for shifts of origin and multiple GR steps. The convergence theory is based on the idea that the GR algorithm performs nested subspace iteration with a change of coordinate system at each step. Thus the convergence of the GR algorithm depends on the convergence of certain sequences of subspaces. It also depends on the quality of the coordinate transformation matrices, as measured by their condition numbers. We show that with a certain obvious shifting strategy the GR algorithm typically has a quadratic asymptotic convergence rate. For matrices possessing certain special types of structure, cubic convergence can be achieved. Key words. eigenvalue, QR algorithm, GR algorithm, subspace iteration, convergence AMS(MOS) subject classifications. 65F15, 15A18 Running head: Convergence of Eigenvalue Algori...
A Parallel Implementation of the Nonsymmetric QR Algorithm for Distributed Memory Architectures
 SIAM J. SCI. COMPUT
, 2002
"... One approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the QR algorithm. Not long ago, this was widely considered to be a hopeless task. Recent efforts have led to significant advances, although the methods proposed up to now have suffered from scalability problems ..."
Abstract

Cited by 37 (3 self)
 Add to MetaCart
One approach to solving the nonsymmetric eigenvalue problem in parallel is to parallelize the QR algorithm. Not long ago, this was widely considered to be a hopeless task. Recent efforts have led to significant advances, although the methods proposed up to now have suffered from scalability problems. This paper discusses an approach to parallelizingthe QR algorithm that greatly improves scalability. A theoretical analysis indicates that the algorithm is ultimately not scalable, but the nonscalability does not become evident until the matrix dimension is enormous. Experiments on the Intel Paragon system, the IBM SP2 supercomputer, the SGI Origin 2000, and the Intel ASCI Option Red supercomputer are reported.
The spectral decomposition of nonsymmetric matrices on distributed memory parallel computers
 SIAM J. Sci. Comput
, 1997
"... Abstract. The implementation and performance of a class of divideandconquer algorithms for computing the spectral decomposition of nonsymmetric matrices on distributed memory parallel computers are studied in this paper. After presenting a general framework, we focus on a spectral divideandconqu ..."
Abstract

Cited by 32 (11 self)
 Add to MetaCart
Abstract. The implementation and performance of a class of divideandconquer algorithms for computing the spectral decomposition of nonsymmetric matrices on distributed memory parallel computers are studied in this paper. After presenting a general framework, we focus on a spectral divideandconquer (SDC) algorithm with Newton iteration. Although the algorithm requires several times as many floating point operations as the best serial QR algorithm, it can be simply constructed from a small set of highly parallelizable matrix building blocks within Level 3 basic linear algebra subroutines (BLAS). Efficient implementations of these building blocks are available on a wide range of machines. In some illconditioned cases, the algorithm may lose numerical stability, but this can easily be detected and compensated for. The algorithm reached 31 % efficiency with respect to the underlying PUMMA matrix multiplication and 82 % efficiency with respect to the underlying ScaLAPACK matrix inversion on a 256 processor Intel Touchstone Delta system, and 41 % efficiency with respect to the matrix multiplication in CMSSL on a 32 node Thinking Machines CM5 with vector units. Our performance model predicts the performance reasonably accurately. To take advantage of the geometric nature of SDC algorithms, we have designed a graphical user interface to let the user choose the spectral decomposition according to specified regions in the complex plane.
A Parallelizable Eigensolver for Real Diagonalizable Matrices with Real Eigenvalues
, 1991
"... . 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 consi ..."
Abstract

Cited by 27 (6 self)
 Add to MetaCart
(Show Context)
. 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 Cray2 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 matrixmatrix multiplication, coupled with the divide and conquer nature of this algorithm, should yield a highly parallelizable algorithm. 1. Introduction. Computation of all the eigenvalues and eigenvectors of a dense matrix is essential for solving problems in many fields. The everincreasing computational power available from modern supercomputers offers the potenti...
QRlike Algorithms for Eigenvalue Problems
 SIAM J. Matrix Anal. Appl
, 2000
"... . In the year 2000 the dominant method for solving matrix eigenvalue problems is still the QR algorithm. This paper discusses the family of GR algorithms, with emphasis on the QR algorithm. Included are historical remarks, an outline of what GR algorithms are and why they work, and descriptions ..."
Abstract

Cited by 27 (11 self)
 Add to MetaCart
. In the year 2000 the dominant method for solving matrix eigenvalue problems is still the QR algorithm. This paper discusses the family of GR algorithms, with emphasis on the QR algorithm. Included are historical remarks, an outline of what GR algorithms are and why they work, and descriptions of the latest, highly parallelizable, versions of the QR algorithm. Now that we know how to parallelize it, the QR algorithm seems likely to retain its dominance for many years to come. 1. Introduction Since the early 1960's the standard algorithms for calculating the eigenvalues and (optionally) eigenvectors of "small" matrices have been the QR algorithm [29] and its variants. This is still the case in the year 2000 and is likely to remain so for many years to come. For us a small matrix is one that can be stored in the conventional way in a computer's main memory and whose complete eigenstructure can be calculated in a matter of minutes without exploiting whatever sparsity the matrix m...
The Transmission of Shifts and Shift Blurring in the QR Algorithm
, 1992
"... The QR algorithm is one of the most widely used algorithms for calculating the eigenvalues of matrices. The multishift QR algorithm with multiplicity m is a version that effects m iterations of the QR algorithm at a time. It is known that roundoff errors cause the multishift QR algorithm to perform ..."
Abstract

Cited by 21 (7 self)
 Add to MetaCart
The QR algorithm is one of the most widely used algorithms for calculating the eigenvalues of matrices. The multishift QR algorithm with multiplicity m is a version that effects m iterations of the QR algorithm at a time. It is known that roundoff errors cause the multishift QR algorithm to perform poorly when m is large. In this paper the mechanism by which the shifts are transmitted through the matrix in the course of a multishift QR iteration is identified. Numerical evidence showing that the mechanism works well when m is small and poorly when m is large is presented. When the mechanism works poorly, the convergence of the algorithm is degraded proportionately. 1. Introduction The QR algorithm is one of the most widely used algorithms for calculating the eigenvalues of matrices [7], [9], [16]. It is therefore worrisome that attempts to parallelize the QR algorithm have been mostly unsatisfactory. (However, the work of Henry and van de Geijn [10], [11] is recent good news.) One atte...