For a large system of linear algebraic equations $A_x=b$, the approximate solution $x_k$ is computed as the $k$-th order Fourier development of the function $1/z$, related to orthogonal polynomials in $L^2(\Omega)$ space. The domain $\Omega$ in the complex plane is assumed to be known. This domain contains the spectrum $\sigma(A)$ of the matrix $A$. Two algorithms for $x_k$ are discussed. Two possibilities of preconditioning by an application of the so called Richardson iteration process with a constant relaxation coefficient are proposed. The case when Jordan blocs of higher dimension are present is discussed, with the following conslusion: in such a case application of the Sobolev space $H^s(\Omega)$ may be resonable, with $s$ equal to the dimension of the maximal Jordan bloc. The paper contains several numerical examples.
@article{104521,
author = {Krzysztof Moszy\'nski},
title = {Remarks on polynomial methods for solving systems of linear algebraic equations},
journal = {Applications of Mathematics},
volume = {37},
year = {1992},
pages = {419-436},
zbl = {0802.65032},
mrnumber = {1185798},
language = {en},
url = {http://dml.mathdoc.fr/item/104521}
}
Moszyński, Krzysztof. Remarks on polynomial methods for solving systems of linear algebraic equations. Applications of Mathematics, Tome 37 (1992) pp. 419-436. http://gdmltest.u-ga.fr/item/104521/
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