An upper <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" id="d1e1610" altimg="si539.gif"><mml:mi>J</mml:mi></mml:math>-Hessenberg reduction of a matrix through symplectic Householder transformations

Salam, A.; Kahla, Haithem Ben

doi:10.1016/j.camwa.2019.02.025

Cited by 4 publications

(7 citation statements)

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“…The matrix R11 is 2j-by-2j upper triangular matrix, and from relation (12), we have det((P T A T JAP ) [2j,2j] ) = [r 1,1 r n+1,n+1 . .…”

Section: Discussion : Existence Of Sr Decomposition Link With Srdeco ...mentioning

confidence: 99%

“…We obtain The breakdown in JHMSH occurs exactly in the same conditions as in JHESS, and is located in the call of the function osh2. A slight different version of JHMSH is JHMSH2 (see [12]).…”

Section: Breakdowns or Near-breakdowns In Jhess Jhmsh Algorithmsmentioning

confidence: 99%

“…This leads to the algorithm JHESS. In a similar way, the algorithm SRSH or its variant SRMSH may be adapted for handling the reduction of a matrix to J-Hessenberg form, see [12].…”

Section: Breakdowns or Near-breakdowns In Jhess Jhmsh Algorithmsmentioning

confidence: 99%

“…Hence, the J-Hessenberg form may be obtained. Numerical experiments are given, demonstrating the efficiency of our strategies to cure and treat breakdowns or near breakdowns.In [12], the algorithm JHSH, based on an adaptation of SRSH, is introduced, for reducing a general matrix, to an upper J-Hessenberg form. Variants of JHSH, named JHOSH, JHMSH and JHMSH2 are then derived, motivated by the numerical stability.…”

mentioning

confidence: 99%

“…In [12], the algorithm JHSH, based on an adaptation of SRSH, is introduced, for reducing a general matrix, to an upper J-Hessenberg form. Variants of JHSH, named JHOSH, JHMSH and JHMSH2 are then derived, motivated by the numerical stability.…”

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confidence: 99%

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A treatment of breakdowns and near breakdowns in a reduction of a matrix to upper J-Hessenberg form and related topics

Salam

Kahla

2019

Journal of Computational and Applied Mathematics

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The reduction of a matrix to an upper J-Hessenberg form is a crucial step in the SR-algorithm (which is a QR-like algorithm), structure-preserving, for computing eigenvalues and vectors, for a class of structured matrices. This reduction may be handled via the algorithm JHESS or via the recent algorithm JHMSH and its variants.The main drawback of JHESS (or JHMSH) is that it may suffer from a fatal breakdown, causing a brutal stop of the computations and hence, the SR-algorithm does not run. JHESS may also encounter near-breakdowns, source of serious numerical instability.In this paper, we focus on these aspects. We first bring light on the necessary and sufficient condition for the existence of the SR-decomposition, which is intimately linked to J-Hessenberg reduction. Then we will derive a strategy for curing fatal breakdowns and also for treating near breakdowns. Hence, the J-Hessenberg form may be obtained. Numerical experiments are given, demonstrating the efficiency of our strategies to cure and treat breakdowns or near breakdowns.In [12], the algorithm JHSH, based on an adaptation of SRSH, is introduced, for reducing a general matrix, to an upper J-Hessenberg form. Variants of JHSH, named JHOSH, JHMSH and JHMSH2 are then derived, motivated by the numerical stability. The algorithms JHESS as well as JHSH and its variants have O(n 3 ) as complexity.The algorithm JHESS (and also SRDECO), as described in [2] may be subject of a fatal breakdown, causing a brutal stop of the computations. As consequence, the J-Hessenberg reduction can not be computed and the SR-algorithm does not run. Moreover, we demonstrate that the algorithm JHESS may breaks down while a condensed J Hessenberg form exists. These algorithms also may suffer from severe form of near-breakdowns, source of serious numerical instability.In this paper, we restrict ourselves to the study of such aspects, bringing significant insights on SRDECO and JHESS algorithms.We will show derive a strategy for curing fatal breakdowns and treating near breakdowns. To this aim, we first bring light on the SR-decomposition and SRDECO algorithm, in connection with the theory developed by Elsner in [5]. Then, a strategy for remedying to such breakdowns is proposed. The same strategy is used for treating the near-breakdowns. Numerical experiments are given, demonstrating the efficiency of our strategies to cure breakdowns or to treat near breakdowns.The remainder of this paper is organized as follows. Section 2, is devoted to the necessary preliminaries. In the section 3, the algorithms SRDECO, SRSH or SRMSH are presented. Then, we establish a connection between some coefficients of the current matrix produced by the SRDECO algorithm, when applied to a matrix A and the necessary and sufficient condition for the existence of the SR decomposition of A, as given in [5]. In section 4, we recall the algorithms JHESS and JHMSH. We present then an example, for which a fatal breakdown is meet, in JHESS algorithm (also for JHMSH), for reducing the matrix to an upper J-Hessenber...

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“…The matrix R11 is 2j-by-2j upper triangular matrix, and from relation (12), we have det((P T A T JAP ) [2j,2j] ) = [r 1,1 r n+1,n+1 . .…”

Section: Discussion : Existence Of Sr Decomposition Link With Srdeco ...mentioning

confidence: 99%

Section: Breakdowns or Near-breakdowns In Jhess Jhmsh Algorithmsmentioning

confidence: 99%

“…This leads to the algorithm JHESS. In a similar way, the algorithm SRSH or its variant SRMSH may be adapted for handling the reduction of a matrix to J-Hessenberg form, see [12].…”

Section: Breakdowns or Near-breakdowns In Jhess Jhmsh Algorithmsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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A treatment of breakdowns and near breakdowns in a reduction of a matrix to upper J-Hessenberg form and related topics

Salam

Kahla

2019

Journal of Computational and Applied Mathematics

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Solving of Eigenvalue and Singular Value Problems via Modified Householder Transformations on Shared Memory Parallel Computing Systems

Andreev

Егунов

2019

Communications in Computer and Information Science

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A Theoretical Scientific Programming Framework for Application of Linear Matrix Transformation in Plane Computer Graphics

Chen

2022

Scientific Programming

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Plane computer graphics are basic information carriers in many industrial scenarios, such as engineering simulation, automatic control, and software design. Plane computer graphics are generally a kind of digital signals guided by mathematical symbols, and each vertex of a plane computer graph forms a graph matrix. Therefore, linear matrix transformation serves as the most common algorithmic unit to realize various information processing operations. To improve ease of graph matrix computing in practical engineering scenarios, this paper proposes a theoretical scientific programming framework for application of linear matrix transformation in plane computer graphics. Firstly, theoretical basis of linear matrix transformation in homogeneous plane coordinates is displayed and analyzed. Then, the universal theorem about linear transformation of graph matrices is deduced, and corresponding proofs are also given. Finally, a case study is set up to demonstrate the main workflow of the proposed theoretical scientific programming framework. The simulative results reveal feasibility of the proposal.

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An upper J-Hessenberg reduction of a matrix through symplectic Householder transformations

Cited by 4 publications

References 10 publications

A treatment of breakdowns and near breakdowns in a reduction of a matrix to upper J-Hessenberg form and related topics

A treatment of breakdowns and near breakdowns in a reduction of a matrix to upper J-Hessenberg form and related topics

Solving of Eigenvalue and Singular Value Problems via Modified Householder Transformations on Shared Memory Parallel Computing Systems

A Theoretical Scientific Programming Framework for Application of Linear Matrix Transformation in Plane Computer Graphics

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