Additional Results on Convergence of Alternating Iterations Involving Rectangular Matrices

Giri, Chinmay Kumar; Mishra, Debasisha

doi:10.1080/01630563.2016.1232732

Cited by 11 publications

(5 citation statements)

References 15 publications

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“…Finally, a comparison result is proved which guarantees the fact that the three-step alternating iterations converges faster than the usual one, and is also shown through examples. The authors of [10], [24] and [26] studied the two-step alternating iterations for rectangular matrices using the Moore-Penrose inverses, very recently. However, their works lack computational implementation which is addressed in this paper.…”

Section: Discussionmentioning

confidence: 99%

“…The numerical results for the convergence analysis is provided in Table and comparison results discussed in Table . The next example shows the importance of the study of the alternating iteration scheme in the group inverse setting. Note that existing theory in the literature uses the nonnegativity of the Moore-Penrose inverse, see [10,24,26] which fails here. Clearly,…”

Section: Numerical Examplesmentioning

confidence: 96%

“…will converge to A # b, for any initial value x 0 . The numerical implementation of the iterative scheme (10) and its comparison are discussed in the next section.…”

Section: Applying Theorem 33 To the Pair Of The Splittingsmentioning

confidence: 99%

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Three-step alternating iterations for index 1 and non-singular matrices

2019

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Iterative methods based on matrix splittings are useful in solving large sparse linear systems. In this direction, proper splittings and its several extensions are used to deal with singular and rectangular linear systems. In this article, we introduce a new iteration scheme called three-step alternating iterations using proper splittings and group inverses to find an approximate solution of singular linear systems, iteratively. A preconditioned alternating iterative scheme is also proposed to relax some sufficient conditions and to obtain faster convergence as well. We then show that our scheme converges faster than the existing one. The theoretical findings are then validated numerically.

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Section: Discussionmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 96%

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Three-step alternating iterations for index 1 and non-singular matrices

2019

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“…motivated by the work of [4]. Convergence theory of (1.3) can be found in [30,28,15]. The idea of introducing alternating iteration scheme is inspired from the Alternating Direction Implicit (ADI) method proposed by Peaceman and Rachford [35] in 1955 to solve higher dimensional Partial Differential Equations(PDEs).…”

Section: Introductionmentioning

confidence: 99%

Alternating Stationary Iterative Methods Based on Double Splittings

Nandi¹,

Mishra²,

Mishra³

2020

Preprint

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Matrix double splitting iterations are simple in implementation while solving real nonsingular (rectangular) linear systems. In this paper, we present two Alternating Double Splitting (ADS) schemes formulated by two double splittings and then alternating the respective iterations. The convergence conditions are then discussed along with comparative analysis. The set of double splittings used in each ADS schemes induce a preconditioned system which helps in showing the convergence of the ADS schemes. We also show that the classes of matrices for which one ADS scheme is better than the other, are mutually exclusive. Numerical experiments confirm the proposed ADS schemes are superior to the existing methods in actual implementation. Though the problems are considered in the rectangular matrix settings, the same problems are even new in non-singular matrix settings.

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“…Later on, Climent and Perea [7], Climent et al [6] have introduced different classes of proper splittings and studied its convergence theory. Subsequently, it is carried forward by Mishra and Sivakumar [16], Jena et al [12], Mishra [13], Baliarsingh and Mishra [2], and Giri and Mishra [10], to name a few. Here we list three important classes of proper splittings.…”

Section: Introductionmentioning

confidence: 99%

Comparison results for proper multisplittings of rectangular matrices

Giri¹,

Mishra²

2016

Preprint

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The least square solution of minimum norm of a rectangular linear system of equations can be found out iteratively by using matrix splittings. However, the convergence of such an iteration scheme arising out of a matrix splitting is practically very slow in many cases. Thus, works on improving the speed of the iteration scheme have attracted great interest. In this direction, comparison of the rate of convergence of the iteration schemes produced by two matrix splittings is very useful. But, in the case of matrices having many matrix splittings, this process is time-consuming. The main goal of the current article is to provide a solution to the above issue by using proper multisplittings. To this end, we propose a few comparison theorems for proper weak regular splittings and proper nonnegative splittings first. We then derive convergence and comparison theorems for proper multisplittings with the help of the theory of proper weak regular splittings.

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Additional Results on Convergence of Alternating Iterations Involving Rectangular Matrices

Cited by 11 publications

References 15 publications

Three-step alternating iterations for index 1 and non-singular matrices

Three-step alternating iterations for index 1 and non-singular matrices

Alternating Stationary Iterative Methods Based on Double Splittings

Comparison results for proper multisplittings of rectangular matrices

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