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Remarks on polynomial methods for solving systems of linear algebraic equations
Applications of Mathematics, Tome 37 (1992) no. 6, pp. 419-436 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.21136/AM.1992.104521
Classification : 65F10
Keywords: Fourier expansion; orthogonal polynomials on $L^2(\Omega)$ space; approximate solution of linear algebraic equations; Richardson iteration; preconditioning; polynomial methods; numerical examples 
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Moszyński, Krzysztof. Remarks on polynomial methods for solving systems of linear algebraic equations. Applications of Mathematics, Tome 37 (1992) no. 6, pp. 419-436. doi: 10.21136/AM.1992.104521

[1] Doan Van Ban, Moszyński K., Pokrzywa A.: Semiiterative methods for linear equations. Matematyka Stosowana-Applied Mathematics, Vol. 35, Warszawa, 1992. | MR

[2] Gaier D.: Lectures on Complex Approximation. Birkhäuser, 1989. | MR

[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 | MR | Zbl

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