Notes on the Solution of Algebraic Linear Simultaneous Equations

Fox, L.; Huskey, Harry D.; Wilkinson, J. H.

doi:10.1093/qjmam/1.1.149

Cited by 82 publications

(39 citation statements)

References 6 publications

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“…Some references on this kind of algorithm are [21,31,46,47]. If A is SPD, only the process for Z need to be carried out (since in this case W = Z), and the algorithm is just a conjugate Gram-Schmidt process, i.e., orthogonalization of the unit vectors with respect to the "energy" inner product x, y := x T Ay.…”

Section: Factorized Sparse Approximate Inversesmentioning

confidence: 99%

A comparative study of sparse approximate inverse preconditioners

Benzi

Tůma

1999

Applied Numerical Mathematics

224

200

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A number of recently proposed preconditioning techniques based on sparse approximate inverses are considered. A description of the preconditioners is given, and the results of an experimental comparison performed on one processor of a Cray C98 vector computer using sparse matrices from a variety of applications are presented. A comparison with more standard preconditioning techniques, such as incomplete factorizations, is also included. Robustness, convergence rates, and implementation issues are discussed.

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Section: Factorized Sparse Approximate Inversesmentioning

confidence: 99%

A comparative study of sparse approximate inverse preconditioners

Benzi

Tůma

1999

Applied Numerical Mathematics

224

200

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“…The contemporary form of the former process which was described in that paper is the LU decomposition, and the latter process is called biconjugation, cf. [4,22], see also the nice survey [14]. A convenient equivalent form of biconjugation is the inverse decomposition, in which we get the inverse factors Z and W such that A = W −T DZ −1 , directly from A.…”

Section: Introduction Consider the System Of Linear Algebraic Equatimentioning

confidence: 99%

“…The biconjugation scheme described in the paper by Hestenes and Stiefel is leftlooking and stabilized as well as the algorithm given by Fox, Huskey and Wilkinson [22]. The escalator method by Morris [31] is the left-looking biconjugation process with one-sided computation of diagonal entries, Purcell [33] came with the one-sided and right-looking algorithm.…”

Section: Introduction Consider the System Of Linear Algebraic Equatimentioning

confidence: 99%

Improved Balanced Incomplete Factorization

Bru¹,

Marı́n²,

Mas³

et al. 2010

SIAM J. Matrix Anal. & Appl.

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Abstract. In this paper we improve the BIF algorithm which computes simultaneously the LU factors (direct factors) of a given matrix, and their inverses (inverse factors). This algorithm was introduced in [R. Bru, J. Marín, J. Mas and M. Tůma, SIAM J. Sci. Comput., 30 (2008Comput., 30 ( ), pp. 2302Comput., 30 ( -2318. The improvements are based on a deeper understanding of the Inverse Sherman-Morrison (ISM) decomposition and they provide a new insight into the BIF decomposition. In particular, it is shown that a slight algorithmic reformulation of the basic algorithm implies that the direct and inverse factors influence numerically each other even without any dropping for incompleteness. Algorithmically, the nonsymmetric version of the improved BIF algorithm is formulated. Numerical experiments show very high robustness of the incomplete implementation of the algorithm used for preconditioning nonsymmetric linear systems.

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“…Algorithms (8.6) and (10.2) are also related to Morris' escalator method [24], [25] and its equivalent, the method of orthogonal vectors [13], especially when A is symmetric. In the escalator method, the matrix ßfn) is also computed.…”

Section: Relationship To Other Methodsmentioning

confidence: 99%

Modification Methods for Inverting Matrices and Solving Systems of Linear Algebraic Equations

Goldfarb¹

1972

Mathematics of Computation

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Abstract.Modification methods for inverting matrices and solving systems of linear algebraic equations are developed from Broyden's rank-one modification formula. Several algorithms are presented that take as few, or nearly as few, arithmetic operations as Gaussian elimination and are well suited for the handling of data. The effect of rounding errors is discussed briefly.Some of these algorithms are essentially equivalent to, or "compact" forms of, such known methods as Sherman and Morrison's modification method, Hestenes' biorthogonalization method, Gauss-Jordan elimination, Aitken's below-the-line elimination method, Purcell's vector method, and its equivalent, Pietrzykowski's projection method, and the bordering method. These methods are thus shown to be directly related to each other.Iterative methods and methods for inverting symmetric matrices are also given, as are the results of some computational experiments. Introduction.This paper is concerned with the related problems of inverting matrices and solving linear nonhomogeneous algebraic systems of equations by a class of direct methods which we shall call modification methods. The adjective modification is used to describe these methods, for they are all based upon the modification of a matrix and its inverse by a matrix of rank one. Most of these methods are direct-that is, a solution to the problem is obtained by using a finite number of elementary arithmetic operations. The general formula that these methods are based upon, however, is iterative in nature, and iterative methods are also given.Modification

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Notes on the Solution of Algebraic Linear Simultaneous Equations

Cited by 82 publications

References 6 publications

A comparative study of sparse approximate inverse preconditioners

A comparative study of sparse approximate inverse preconditioners

Improved Balanced Incomplete Factorization

Modification Methods for Inverting Matrices and Solving Systems of Linear Algebraic Equations

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