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Applied Numerical Linear Algebra
 Society for Industrial and Applied Mathematics
, 1997
"... We survey general techniques and open problems in numerical linear algebra on parallel architectures. We rst discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing e cient algorithms. We illustrate ..."
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Cited by 532 (26 self)
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We survey general techniques and open problems in numerical linear algebra on parallel architectures. We rst discuss basic principles of parallel processing, describing the costs of basic operations on parallel machines, including general principles for constructing e cient 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.
Stability of the diagonal pivoting method with partial pivoting
 SIAM J. Matrix Anal. Appl
, 1995
"... Abstract. LAPACK and LINPACK both solve symmetric indefinite linear systems using the diagonal pivoting method with the partial pivoting strategy of Bunch and Kaufman [Math. Comp., 31 (1977), pp. 163–179]. No proof of the stability of this method has appeared in the literature. It is tempting to arg ..."
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Cited by 22 (9 self)
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Abstract. LAPACK and LINPACK both solve symmetric indefinite linear systems using the diagonal pivoting method with the partial pivoting strategy of Bunch and Kaufman [Math. Comp., 31 (1977), pp. 163–179]. No proof of the stability of this method has appeared in the literature. It is tempting to argue that the diagonal pivoting method is stable for a given pivoting strategy if the growth factor is small. We show that this argument is false in general and give a sufficient condition for stability. This condition is not satisfied by the partial pivoting strategy because the multipliers are unbounded. Nevertheless, using a more specific approach we are able to prove the stability of partial pivoting, thereby filling a gap in the body of theory supporting LAPACK and LINPACK.
Techniques in Computational Stochastic Dynamic Programming
 in Control and Dynamic Systems
, 1996
"... INTRODUCTION When Bellman introduced dynamic programming in his original monograph [8], computers were not as powerful as current personal computers. Hence, his description of the extreme computational demands as the Curse of Dimensionality [9] would not have had the super and massively parallel p ..."
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Cited by 12 (8 self)
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INTRODUCTION When Bellman introduced dynamic programming in his original monograph [8], computers were not as powerful as current personal computers. Hence, his description of the extreme computational demands as the Curse of Dimensionality [9] would not have had the super and massively parallel processors of today in mind. However, massive and super computers can not overcome the Curse of Dimensionality alone, but parallel and vector computation can permit the solution of higher dimension than was previously possible and thus permit more realistic dynamic programming applications. Today such large problems are called Grand and National Challenge problems [45, 46] in high performance computing. Today's availability of high performance vector supercomputers and massively parallel processors have made it possible to compute optimal policies and values of control systems for much larger dimensions than was possible earlier. Advance
Numerical Analysis
, 1989
"... Introduction. Numerical anC"Cfix is the area of mathematics a n computerscienB that creates, a n lyzes, a n implemen ts algorithms for solvin n umerically the problems of con tin uous mathematics. Such problems origi n te genfi ally from realworld application s of algebra, geometryan d calculus,an ..."
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Cited by 2 (0 self)
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Introduction. Numerical anC"Cfix is the area of mathematics a n computerscienB that creates, a n lyzes, a n implemen ts algorithms for solvin n umerically the problems of con tin uous mathematics. Such problems origi n te genfi ally from realworld application s of algebra, geometryan d calculus,an d they in volve variables which vary con tin uously; these problems occur throughout the n tural scienfiB( social scien(Ifi e n infiC in , medicinI a n businB(I Durin the past halfcen tury, the growthin power a n availability of digital computers has led to an inE easin use of realistic mathematical models in scien cean d en gin eerin g,an dn umericalan alysis of infifiBBE(I sophistication has been nnAC to solve these more detailed mathematical models of the world. The formal academic area ofn umerical an alysis varies from quite theoretical mathematical studies (e.g. see [5]) to computer scienr issues (e.g. see [1], [11]). With the growth in importa n( of usin computers to carry outn umeri