Remarks on polynomial methods for solving systems of linear algebraic equations

Moszyński, Krzysztof. "Remarks on polynomial methods for solving systems of linear algebraic equations." Applications of Mathematics 37.6 (1992): 419-436. <http://eudml.org/doc/15725>.

@article{Moszyński1992,
abstract = {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.},
author = {Moszyński, Krzysztof},
journal = {Applications of Mathematics},
keywords = {Fourier expansion; orthogonal polynomials on $L^2(\Omega )$ space; approximate solution of linear algebraic equations; Richardson iteration; preconditioning; polynomial methods; numerical examples; polynomial methods; Fourier expansions; Richardson iteration; numerical examples},
language = {eng},
number = {6},
pages = {419-436},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Remarks on polynomial methods for solving systems of linear algebraic equations},
url = {http://eudml.org/doc/15725},
volume = {37},
year = {1992},
}

TY - JOUR
AU - Moszyński, Krzysztof
TI - Remarks on polynomial methods for solving systems of linear algebraic equations
JO - Applications of Mathematics
PY - 1992
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 37
IS - 6
SP - 419
EP - 436
AB - 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.
LA - eng
KW - Fourier expansion; orthogonal polynomials on $L^2(\Omega )$ space; approximate solution of linear algebraic equations; Richardson iteration; preconditioning; polynomial methods; numerical examples; polynomial methods; Fourier expansions; Richardson iteration; numerical examples
UR - http://eudml.org/doc/15725
ER -