Fourier expansion; orthogonal polynomials on $L^2(\Omega)$ space; approximate solution of linear algebraic equations; Richardson iteration; preconditioning; polynomial methods; numerical examples
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.
 Doan Van Ban, Moszyński K., Pokrzywa A.: Semiiterative methods for linear equations
. Matematyka Stosowana-Applied Mathematics, Vol. 35, Warszawa, 1992. MR 1221221
 Gaier D.: Lectures on Complex Approximation
. Birkhäuser, 1989. MR 0894920