Convergence of Matrix Iterations Subject to Diagonal Dominance

James, Kevin R.

doi:10.1137/0710042

Cited by 51 publications

(21 citation statements)

References 2 publications

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“…We refer the reader to [11,1] for results on the convergence of the Jacobi method for strictly diagonally dominant matrices or irreducible and weakly diagonally dominant matrices, as well as [7] for the corresponding notions in the case of block matrices. which amounts to the HS iteration (2.9).…”

Section: Appendix Bmentioning

confidence: 99%

A Proof of Convergence of the Horn--Schunck Optical Flow Algorithm in Arbitrary Dimension

Tarnec¹,

Destrempes²,

Cloutier³

et al. 2014

SIAM J. Imaging Sci.

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The Horn and Schunck (HS) method, which amounts to the Jacobi iterative scheme in the interior of the image, was one of the first optical flow algorithms. In this article, we prove the convergence of the HS method, whenever the problem is well-posed. Our result is shown in the framework of a generalization of the HS method in dimension n ≥ 1, with a broad definition of the discrete Laplacian. In this context, the condition for the convergence is that the intensity gradients are not all contained in a same hyperplane. Two other articles ([17] and [13]) claimed to solve this problem in the case n = 2, but it appears that both of these proofs are erroneous. Moreover, we explain why some standard results on the convergence of the Jacobi method do not apply for the HS problem, unless n = 1. It is also shown that the convergence of the HS scheme implies the convergence of the Gauss-Seidel and SOR schemes for the HS problem.

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Section: Appendix Bmentioning

confidence: 99%

A Proof of Convergence of the Horn--Schunck Optical Flow Algorithm in Arbitrary Dimension

Tarnec¹,

Destrempes²,

Cloutier³

et al. 2014

SIAM J. Imaging Sci.

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“…Previous convergence results [17], [22] have made the stronger assumption that either: (a) A is irreducible and E, la,• 5 1, for all i, with strict inequality for at least one i, or (b) j, lail < 1, for all i.…”

Section: X(t+ 1) =(I -Y)x(t)+ Y(ax(t)+ B)mentioning

confidence: 99%

Partially Asynchronous, Parallel Algorithms for Network Flow and Other Problems

Tseng

Bertsekas

Tsitsiklis

1990

SIAM J. Control Optim.

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Abstract. The problem of computing a fixed point of a nonexpansive function f is considered. Sufficient conditions are provided under which a parallel, partially asynchronous implementation of the iteration x:=f(x) converges. These results are then applied to (i) quadratic programming subject to box constraints, (ii) strictly convex cost network flow optimization, (iii) an agreement and a Markov chain problem, (iv) neural network optimization, and (v) finding the least element of a polyhedral set determined by a weakly diagonally dominant, Leontief system. Finally, simulation results illustrating the attainable speedup and the effects of asynchronism are presented.

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“…Many authors [2,5,13,14,16,19,21] discussed the use of local relaxation parameters in SOR-like methods for linear or nonlinear systems of equations. In particular, Russel [19], Brazier [2], Strikwerda [21] and Ehrlich [5] proposed special selections of the local relaxation parameters for two-dimensional problems of a discrete convection-diffusion equation (for example, [1] and [18]).…”

Section: Introductionmentioning

confidence: 99%

“…3 if there is a stable turning point in this block.2) Compute all Xik of the equation (12) whose coefficients are obtained from(13).…”

mentioning

confidence: 99%

Improved SOR method with orderings and direct methods

Ishiwata

Muroya

1999

Japan J. Indust. Appl. Math.

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A generalized SOR method with multiple relaxation parameters is considered for solving a linear system of equations. Optimal choices of the parameters are examined under the assumption that the coefficient matrix is tridiagonal and regular. It is shown that the spectral radius of the iterative matrix is reduced to zero for a pair of parameter values which is computed from the pivots of the Gaussian elimination applied to the system. A proper choice of orderings and starting vectors for the iteration is also proposed.When the system is well-conditioned, it is solved stably by Gaussian elimination; there is little advantage in using an iterative method. However, when the system is illconditioned, the direct method is not necessarily stable and it is often required to improve the numerical solution by a certain iterative algorithm. Some numerical examples are presented which show that the proposed method is more efficient than the standard iterative refinement method.It is also discussed how to apply our technique to a class of systems which includes Hessenberg systems.

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Convergence of Matrix Iterations Subject to Diagonal Dominance

Cited by 51 publications

References 2 publications

A Proof of Convergence of the Horn--Schunck Optical Flow Algorithm in Arbitrary Dimension

A Proof of Convergence of the Horn--Schunck Optical Flow Algorithm in Arbitrary Dimension

Partially Asynchronous, Parallel Algorithms for Network Flow and Other Problems

Improved SOR method with orderings and direct methods

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