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The vector cross product from an algebraic point of view

100%
Trenkler G.
Discussiones Mathematicae - General Algebra and Applications
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2001
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tom 21
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nr 1
67-82
EN
The usual vector cross product of the three-dimensional Euclidian space is considered from an algebraic point of view. It is shown that many proofs, known from analytical geometry, can be distinctly simplified by using the matrix oriented approach. Moreover, by using the concept of generalized matrix inverse, we are able to facilitate the analysis of equations involving vector cross products.
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Acceleration properties of the hybrid procedure for solving linear systems

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Abkowicz A., Brezinski C.
Applicationes Mathematicae
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1995-1996
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tom 23
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nr 4
417-432
EN
The aim of this paper is to discuss the acceleration properties of the hybrid procedure for solving a system of linear equations. These properties are studied in a general case and in two particular cases which are illustrated by numerical examples.
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Influence of preconditioning and blocking on accuracy in solving Markovian models

88%
Bylina B., Bylina J.
International Journal of Applied Mathematics and Computer Science
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2009
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tom 19
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nr 2
207-217
EN
The article considers the effectiveness of various methods used to solve systems of linear equations (which emerge while modeling computer networks and systems with Markov chains) and the practical influence of the methods applied on accuracy. The paper considers some hybrids of both direct and iterative methods. Two varieties of the Gauss elimination will be considered as an example of direct methods: the LU factorization method and the WZ factorization method. The Gauss-Seidel iterative method will be discussed. The paper also shows preconditioning (with the use of incomplete Gauss elimination) and dividing the matrix into blocks where blocks are solved applying direct methods. The motivation for such hybrids is a very high condition number (which is bad) for coefficient matrices occuring in Markov chains and, thus, slow convergence of traditional iterative methods. Also, the blocking, preconditioning and merging of both are analysed. The paper presents the impact of linked methods on both the time and accuracy of finding vector probability. The results of an experiment are given for two groups of matrices: those derived from some very abstract Markovian models, and those from a general 2D Markov chain.
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