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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...
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 <
...
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...
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...
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.
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.
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