PCR algorithm for the parallel computation of the solution of a class of singular linear systems

Gu, Chao; Wang, Guorong; Xu, Zhaoliang

doi:10.1016/j.amc.2005.09.060

Cited by 4 publications

(5 citation statements)

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“…Using the determinantal representation of the Drazin inverse solution ( 16), the group inverse (11) and the identity (12) we evidently obtain the following theorem.…”

Section: Examplesmentioning

confidence: 99%

Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations

Kyrchei

2013

Applied Mathematics and Computation

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The Drazin inverse solutions of the matrix equations AX = B, XA = B and AXB = D are considered in this paper. We use both the determinantal representations of the Drazin inverse obtained earlier by the author and in the paper. We get analogs of the Cramer rule for the Drazin inverse solutions of these matrix equations and using their for determinantal representations of solutions of some differential matrix equations, X ′ + AX = B and X ′ + XA = B, where the matrix A is singular.

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“…Using the determinantal representation of the Drazin inverse solution ( 16), the group inverse (11) and the identity (12) we evidently obtain the following theorem.…”

Section: Examplesmentioning

confidence: 99%

Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations

Kyrchei

2013

Applied Mathematics and Computation

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“…(iii) In [6], the unique solution of the singular linear equations W AW x = b is the solution of the nonsingular equations the nonsingular equations…”

Section: )mentioning

confidence: 99%

“…Our aim is to transform computing A (2) T ,S b into computing the solution of the nonsingular linear equations (GA + V U * )x = Gb using the PCR algorithm which generalizes the results in [6,11,14].…”

Section: Introductionmentioning

confidence: 99%

“…Since the PCR algorithm provides approximately twice the speedup over PGE with only twice the number of processors, it offers exciting possibilities for VLSI implementation as well as MIMD parallel processing structures. Y [11] presented a PCR algorithm for computing the minimum-norm least squares solution of inconsistent linear equations, Wei and Wang [14] put forward a PCR algorithm for parallel computing minimum-norm (N ) least-squares (M) solution of inconsistent linear equations, and Gu et al [6] studied a PCR algorithm for the parallel computation of the solution of a class of singular linear systems. These algorithms are the extension of the PCR algorithm for computing the solution of nonsingular linear equations given by Sridhar. In this paper, we present a PCR algorithm for parallel computing the solution of the general restricted linear equations.…”

Section: Introductionmentioning

confidence: 99%

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PCR algorithm for parallel computing the solution of the general restricted linear equations

2008

J. Appl. Math. Comput.

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“…However, in [13] gave counter example. Gauss elimination, Jacobi method and Gauss-Jordan elimination are efficient iterative and numerical methods that have succeeded Cramer's rule [14] including parallel Cramer's rule (PCR) for solving singular linear systems [15].…”

Section: Introductionmentioning

confidence: 99%

Modification of Cramer’s rule

Babarinsa

Kamarulhaili

2018

J. Fundam and Appl Sci.

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While Cramer's rule allows complete substitution of constant terms to the coefficient matrix in the system of linear equations, the modified methods of Cramer's rule consider the constant terms as well as the coefficients of the matrix at the same time. The methods are derived from one of the properties of determinants. Furthermore, we prove the two methods to be equivalent and provide MATLAB codes for the modified methods. However, the methods are not practically suitable for higher system of linear equations and instability of Cramer's rule.

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PCR algorithm for the parallel computation of the solution of a class of singular linear systems

Cited by 4 publications

References 10 publications

Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations

Explicit formulas for determinantal representations of the Drazin inverse solutions of some matrix and differential matrix equations

PCR algorithm for parallel computing the solution of the general restricted linear equations

Modification of Cramer’s rule

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