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12
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 773 (23 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.
Fernando's Solution to Wilkinson's Problem: an Application of Double Factorization
 Linear Algebra and Appl
, 1996
"... Suppose that one knows a very accurate approximation oe to an eigenvalue of a symmetric tridiagonal matrix T . A good way to approximate the eigenvector x is to discard an appropriate equation, say the rth, from the system (T \Gamma oeI)x = 0 and then to solve the resulting underdetermined system i ..."
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Cited by 50 (19 self)
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Suppose that one knows a very accurate approximation oe to an eigenvalue of a symmetric tridiagonal matrix T . A good way to approximate the eigenvector x is to discard an appropriate equation, say the rth, from the system (T \Gamma oeI)x = 0 and then to solve the resulting underdetermined system in any of several stable ways. However the output x can be completely inaccurate if r is chosen poorly and in the absence of a quick and reliable way to choose r this method has lain neglected for over 35 years. Experts in boundary value problems have known about the special structure of the inverse of a tridiagonal matrix since the 1960s and their double triangular factorization technique (down and up) gives directly the redundancy of each equation and so reveals the set of good choices for r. The relation of double factorization to the eigenvector algorithm of Godunov and his collaborates is described in Section 4. The results extend to band matrices (Section 7) and to zero entries in eigen...
A New O(n²) Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem
 In progress
, 1997
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On Computing an Eigenvector of a Tridiagonal Matrix
, 1995
"... We consider the solution of the homogeneous equation (J \Gamma I)x = 0 where J is a tridiagonal matrix, is a known eigenvalue, and x is the unknown eigenvector corresponding to . Since the system is underdetermined, x could be obtained by setting x k = 1 and solving for the rest of the elements o ..."
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Cited by 20 (2 self)
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We consider the solution of the homogeneous equation (J \Gamma I)x = 0 where J is a tridiagonal matrix, is a known eigenvalue, and x is the unknown eigenvector corresponding to . Since the system is underdetermined, x could be obtained by setting x k = 1 and solving for the rest of the elements of x. This method is not entirely new and it can be traced back to the times of Cauchy (1829). In 1958, Wilkinson demonstrated that, in finiteprecision arithmetic, the computed x is highly sensitive to the choice of k; the traditional practice of setting k = 1 or k = n can lead to disastrous results. We develop algorithms to find optimal k which require a LDU and a UDL factorisation of J \Gamma I and are based on the theory developed by Fernando for general matrices. We have also discovered new formulae (valid also for more general Hessenberg matrices) for the determinant of J \Gamma øI, which give better numerical results when the shifted matrix is nearly singular. These formulae could be ...
Developments and Trends in the Parallel Solution of Linear Systems
, 1999
"... In this review paper, we consider some important developments and trends in algorithm design for the solution of linear systems concentrating on aspects that involve the exploitation of parallelism. We briefly discuss the solution of dense linear systems, before studying the solution of sparse equat ..."
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Cited by 7 (0 self)
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In this review paper, we consider some important developments and trends in algorithm design for the solution of linear systems concentrating on aspects that involve the exploitation of parallelism. We briefly discuss the solution of dense linear systems, before studying the solution of sparse equations by direct and iterative methods. We consider preconditioning techniques for iterative solvers and discuss some of the present research issues in this field.
A TwoWay Parallel Partition Method for Solving Tridiagonal Systems
 Univ. of Leeds, School of Computer Studies Research Report Series
, 1993
"... The Parallel Partition Method for tridiagonal systems is described. It is noted that, in the local reduction phase, the inherent parallelism is not exploited to the full and so a TwoWay Parallel Partition Method is introduced. This new algorithm results in a reduced system of order P=2 0 1 compared ..."
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Cited by 4 (1 self)
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The Parallel Partition Method for tridiagonal systems is described. It is noted that, in the local reduction phase, the inherent parallelism is not exploited to the full and so a TwoWay Parallel Partition Method is introduced. This new algorithm results in a reduced system of order P=2 0 1 compared to P 0 1 previously and in particular for 4 processors, a much lower arithmetic count. Both versions are tested and the results compared. 1 Introduction 1.1 The Parallel Solution of Tridiagonal Systems Many nearest neighbour problems (for example those involving spatial finite difference approximations) have at their heart a tridiagonal matrix. Their sparsity pattern suggests that, for large values of n, such systems are ideal candidates for parallelisation. However, the common methods for solving tridiagonal systems, such as Gaussian elimination or matrix decomposition, tend to be inherently sequential in nature and as a result, this topic was one of the earliest subjects investigated in...
Accurate BABE Factorisation of Tridiagonal Matrices for Eigenproblems
"... Recently, Fernando successfully resurrected a classical method for computing eigenvectors which goes back to the times of Cauchy. This algorithm has been in the doldrums for nearly fofty years because of a fundamental difficulty highlighted by Wilkinson. The algorithm is based on the solution of a n ..."
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Cited by 4 (1 self)
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Recently, Fernando successfully resurrected a classical method for computing eigenvectors which goes back to the times of Cauchy. This algorithm has been in the doldrums for nearly fofty years because of a fundamental difficulty highlighted by Wilkinson. The algorithm is based on the solution of a nearly homogeneous system of equations (J; I)z = k ()ek � zk = 1 for the approximate eigenvector z where is an eigenvalue shift, k ( ) is a scalar and ek is a unit vector. The best (minimal residual) approximation for z is obtained by choosing the k, 1 k n, for which j k ()j is minimal and tiny. If the LDU factorisation is computed from the top of the matrix J; I and the UDL factorisation from the bottom then the residual k ()appear as the pivot where these two factorisations meet. We study the properties of this BABE (burn at both ends) factorisation which are closely related to the properties of LDU and UDL factorisations. We show that LDU, UDL and BABE factorisations possess mixed stability with tiny relative perturbations. However, it is demonstrated
Preconditioning and Parallel Preconditioning
, 1998
"... We review current methods for preconditioning systems of equations for their solution using iterative methods. We consider the solution of unsymmetric as well as symmetric systems and discuss techniques and implementations that exploit parallelism. We particularly study preconditioning techniques ba ..."
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Cited by 2 (1 self)
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We review current methods for preconditioning systems of equations for their solution using iterative methods. We consider the solution of unsymmetric as well as symmetric systems and discuss techniques and implementations that exploit parallelism. We particularly study preconditioning techniques based on incomplete LU factorization, sparse approximate inverses, polynomial preconditioning, and block and element by element preconditioning. In the parallel implementation, we consider the effect of reordering.
A PARALLEL COMPACT MULTIDIMENSIONAL NUMERICAL ALGORITHM WITH AEROACOUSTICS APPLICATIONS
, 1999
"... In this study we propose a novel method to parallelize highorder compact numerical algorithms for the solution of threedimensional PDEs in a spacetime domain. For this numerical integration most of the computer time is spent in computation of spatial derivatives at each stage of the RungeKutta ..."
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Cited by 1 (1 self)
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In this study we propose a novel method to parallelize highorder compact numerical algorithms for the solution of threedimensional PDEs in a spacetime domain. For this numerical integration most of the computer time is spent in computation of spatial derivatives at each stage of the RungeKutta temporal update. The most efficient direct method to compute spatial derivatives on a serial computer is aversion of Gaussian elimination for narrow linear banded systems known as the Thomas algorithm. In a straightforward pipelined implementation of the Thomas algorithm processors are idle due to the forward and backward recurrences of the Thomas algorithm. To utilize processors during this time, we propose to use them for either nonlocal data independent computations, solving lines in the next spatial direction, or local datadependent computations by the RungeKutta method. To achieve this goal, control of processor communication and computations by a static schedule is adopted. Thus, our parallel code is driven by a communication and computation schedule instead of the usual "creative programming" approach. The obtained parallelization speedup of the novel algorithm is about twice as much as that for the standard pipelined algorithm and close to that for the explicit DRP algorithm.
On Computing an Eigenvector of a Tridiagonal Matrix
"... We consider the solution of the homogeneous equation (J; I)x = 0 where J is a tridiagonal matrix, is a known eigenvalue, and x is the unknown eigenvector corresponding to. Since the system is underdetermined, x could be obtained by setting xk = 1 and solving for the rest of the elements of x. This ..."
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We consider the solution of the homogeneous equation (J; I)x = 0 where J is a tridiagonal matrix, is a known eigenvalue, and x is the unknown eigenvector corresponding to. Since the system is underdetermined, x could be obtained by setting xk = 1 and solving for the rest of the elements of x. This method is not entirely new and it can be traced back to the times of Cauchy (1829). In 1958, Wilkinson demonstrated that, in niteprecision arithmetic, the computed x is highly sensitive to the choice of k � the traditional practice of setting k =1ork = n can lead to disastrous results. We develop algorithms to nd optimal k which require a LDU and a UDL factorisation of J; I and are based on the theory developed by Fernando for general matrices. We have also discovered new formulae (valid also for more general Hessenberg matrices) for the determinant ofJ; I,whichgivebetter numerical results when the shifted matrix is nearly singular. These formulae could be used to compute eigenvalues (or to improve the accuracy of known estimates) based on standard zero nders such as Newton and Laguerre methods. The accuracy of the computed eigenvalues is crucial for obtaining small residuals for the computed eigenvectors. The algorithms for solving eigenproblems are embarrassingly parallel and hence suitable for modern architectures. 1