# Article

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Keywords:
Fourier expansion; orthogonal polynomials on $L^2(\Omega)$ space; approximate solution of linear algebraic equations; Richardson iteration; preconditioning; polynomial methods; numerical examples
Summary:
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.
References:
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[2] Gaier D.: Lectures on Complex Approximation. Birkhäuser, 1989. MR 0894920
[3] Reichel L.: Polynomials by conformal mapping for the Richardson iteration method for complex linear systems. SIAM NA 25 no. 6 (1988), 1359-1368. DOI 10.1137/0725077 | MR 0972459 | Zbl 0692.65011

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