Direct Methods for Solving Systems of Linear Equations

1996 ◽  
pp. 59-129
Author(s):  
Gisela Engeln-Müllges ◽  
Frank Uhlig
Keyword(s):  
Linear Equations ◽  
Direct Methods ◽  
Systems Of Linear Equations

Direct Methods for Solving Systems of Linear Equations

1996 ◽  
pp. 59-129
Author(s):  
Gisela Engeln-Müllges ◽  
Frank Uhlig
Keyword(s):  
Linear Equations ◽  
Direct Methods ◽  
Systems Of Linear Equations

scholarly journals Influence of Preconditioning and Blocking on Accuracy in Solving Markovian Models

2009 ◽  
Vol 19 (2) ◽  
pp. 207-217 ◽  
Author(s):  
Beata Bylina ◽  
Jarosław Bylina
Keyword(s):  
Markov Chains ◽  
Iterative Methods ◽  
Linear Equations ◽  
Direct Methods ◽  
Factorization Method ◽  
Markovian Models ◽  
The Matrix ◽  
Gauss Elimination ◽  
Systems Of Linear Equations ◽  
The Impact

Influence of Preconditioning and Blocking on Accuracy in Solving Markovian ModelsThe 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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