Updating LU factorizations for computing stationary distributions
SIAM Journal on Algebraic and Discrete Methods
SIAM Journal on Algebraic and Discrete Methods
GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
An iterative aggregation-disaggregation algorithm for solving linear equations
Applied Mathematics and Computation
Aggregation/disaggregation methods for computing the stationary distribution of a Markov chain
SIAM Journal on Numerical Analysis
The WZ matrix factorisation method
Parallel Computing
BSP linear solvers for dense matrices
Parallel Computing
A Simultaneous Iteration Algorithm for Real Matrices
ACM Transactions on Mathematical Software (TOMS)
Parallel solution of large symmetric tridiagonal linear systems
Parallel Computing
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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.