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16
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 ..."
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Cited by 575 (26 self)
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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.
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 ..."
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Cited by 65 (13 self)
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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.
An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems
, 1997
"... We discuss an inversefree, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB. This algorithm is based on earlier ones of Bulgakov, Godunov ..."
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Cited by 64 (11 self)
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We discuss an inversefree, highly parallel, spectral divide and conquer algorithm. It can compute either an invariant subspace of a nonsymmetric matrix A, or a pair of left and right deflating subspaces of a regular matrix pencil A − λB. This algorithm is based on earlier ones of Bulgakov, Godunov and Malyshev, but improves on them in several ways. This algorithm only uses easily parallelizable linear algebra building blocks: matrix multiplication and QR decomposition, but not matrix inversion. Similar parallel algorithms for the nonsymmetric eigenproblem use the matrix sign function, which requires matrix inversion and is faster but can be less stable than the new algorithm.
The Matrix Sign Decomposition and its Relation to the Polar Decomposition
, 1994
"... The sign function of a square matrix was introduced by Roberts in 1971. We show that it is useful to regard S = sign(A) as being part of a matrix sign decomposition A = SN , where N = (A 2 ) 1=2 . This decomposition leads to the new representation sign(A) = A(A 2 ) \Gamma1=2 . Most results f ..."
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Cited by 35 (12 self)
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The sign function of a square matrix was introduced by Roberts in 1971. We show that it is useful to regard S = sign(A) as being part of a matrix sign decomposition A = SN , where N = (A 2 ) 1=2 . This decomposition leads to the new representation sign(A) = A(A 2 ) \Gamma1=2 . Most results for the matrix sign decomposition have a counterpart for the polar decomposition A = UH, and vice versa. To illustrate this, we derive best approximation properties of the factors U , H and S, determine bounds for kA \Gamma Sk and kA \Gamma Uk, and describe integral formulas for S and U . We also derive explicit expressions for the condition numbers of the factors S and N . An important equation expresses the sign of a block 2 \Theta 2 matrix involving A in terms of the polar factor U of A. We apply this equation to a family of iterations for computing S by Pandey, Kenney and Laub, to obtain a new family of iterations for computing U . The iterations have some attractive properties, includin...
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 ..."
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Cited by 32 (11 self)
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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 ..."
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Cited by 27 (6 self)
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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 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...
Numerical methods for algebraic Riccati equations
 In Proc. Workshop on the Riccati Equation in Control, Systems, and Signals
, 1989
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Parallel performance of a symmetric eigensolver based on the invariant subspace decomposition approach
 In Proceedings of Scalable High Performance Computing Conference
, 1994
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Numerical stability and instability in matrix sign function based algorithms
 Computational and Combinatorial Methods in Systems Theory
, 1986
"... This paper uses a forward and backward error analysis to try to identify some classes of matrices for . P which the matrix sign function is a numerically stable algorithm for extracting invariant subspaces roper scaling is essential to numerical stability as well as to rapid convergence. g a Roberts ..."
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Cited by 14 (5 self)
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This paper uses a forward and backward error analysis to try to identify some classes of matrices for . P which the matrix sign function is a numerically stable algorithm for extracting invariant subspaces roper scaling is essential to numerical stability as well as to rapid convergence. g a Roberts [21] and Beavers and Denman [7] introduced the matrix sign function as a means of solvin lgebraic Riccati equations and Lyapunov equations. The matrix sign function has since attracted the ] a attention of control engineers and some applied mathematicians ([1] to [21]). Balzer [3], Barraud [5 nd Byers [9] have suggested strategies for accelerating convergence. Denman and Beavers [11]   0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000002 extended matrix sign function algorithms to a list of invariant subspace related calculations. Howland e a [16] used the matrix sign function to count eigenvalues in boxes in the complex plane. Some of th lgorithms have been refined and extended by Attarzadeh [2], Bierman [8], Byers [9]. Gardiner and d d Laub [14] have extended the use of the matrix sign function to generalized Riccati equations an iscrete Riccati equations. Higham [15] has used matrix sign function techniques to calculate polar decompositions
The PRISM Project: Infrastructure and Algorithms for Parallel Eigensolvers
, 1994
"... The goal of the PRISM project is the development of infrastructure and algorithms for the parallel solution of eigenvalue problems. We are currently investigating a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA). After briefly revie ..."
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Cited by 13 (6 self)
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The goal of the PRISM project is the development of infrastructure and algorithms for the parallel solution of eigenvalue problems. We are currently investigating a complete eigensolver based on the Invariant Subspace Decomposition Algorithm for dense symmetric matrices (SYISDA). After briefly reviewing SYISDA, we discuss the algorithmic highlights of a distributedmemory implementation of this approach. These include a fast matrixmatrix multiplication algorithm, a new approach to parallel band reduction and tridiagonalization, and a harness for coordinating the divideandconquer parallelism in the problem. We also present performance results of these kernels as well as the overall SYISDA implementation on the Intel Touchstone Delta prototype. 1. Introduction Computation of eigenvalues and eigenvectors is an essential kernel in many applications, and several promising parallel algorithms have been investigated [29, 24, 3, 27, 21]. The work presented in this paper is part of the PRI...