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Applying approximate LU-factorizations as preconditioners in eight iterative methods for solving systems of linear algebraic equations

Zahari Zlatev, Krassimir Georgiev (2013)

Open Mathematics

Many problems arising in different fields of science and engineering can be reduced, by applying some appropriate discretization, either to a system of linear algebraic equations or to a sequence of such systems. The solution of a system of linear algebraic equations is very often the most time-consuming part of the computational process during the treatment of the original problem, because these systems can be very large (containing up to many millions of equations). It is, therefore, important...

Block Factorization of Hankel Matrices and Euclidean Algorithm

S. Belhaj (2010)

Mathematical Modelling of Natural Phenomena

It is shown that a real Hankel matrix admits an approximate block diagonalization in which the successive transformation matrices are upper triangular Toeplitz matrices. The structure of this factorization was first fully discussed in [1]. This approach is extended to obtain the quotients and the remainders appearing in the Euclidean algorithm applied to two polynomials u(x) and v(x) of degree n and m, respectively, whith m < ...

Computing and Visualizing Solution Sets of Interval Linear Systems

Krämer, Walter (2007)

Serdica Journal of Computing

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006The computation of the exact solution set of an interval linear system is a nontrivial task [2, 13]. Even in two and three dimensions a lot of work has to be done. We demonstrate two different realizations. The first approach (see [16]) is based on Java, Java3D, and the BigRational package [21]. An applet allows modifications of the matrix coefficients and/or the coefficients...

Convergence of Rump's method for computing the Moore-Penrose inverse

Yunkun Chen, Xinghua Shi, Yi Min Wei (2016)

Czechoslovak Mathematical Journal

We extend Rump's verified method (S. Oishi, K. Tanabe, T. Ogita, S. M. Rump (2007)) for computing the inverse of extremely ill-conditioned square matrices to computing the Moore-Penrose inverse of extremely ill-conditioned rectangular matrices with full column (row) rank. We establish the convergence of our numerical verified method for computing the Moore-Penrose inverse. We also discuss the rank-deficient case and test some ill-conditioned examples. We provide our Matlab codes for computing the...

Directed forests with application to algorithms related to Markov chains

Piotr Pokarowski (1999)

Applicationes Mathematicae

This paper is devoted to computational problems related to Markov chains (MC) on a finite state space. We present formulas and bounds for characteristics of MCs using directed forest expansions given by the Matrix Tree Theorem. These results are applied to analysis of direct methods for solving systems of linear equations, aggregation algorithms for nearly completely decomposable MCs and the Markov chain Monte Carlo procedures.

Dynamical systems method for solving linear finite-rank operator equations

N. S. Hoang, A. G. Ramm (2009)

Annales Polonici Mathematici

A version of the dynamical systems method (DSM) for solving ill-conditioned linear algebraic systems is studied. An a priori and an a posteriori stopping rules are justified. An iterative scheme is constructed for solving ill-conditioned linear algebraic systems.

Currently displaying 41 – 60 of 195