| Title:
|
Remarks on polynomial methods for solving systems of linear algebraic equations (English) |
| Author:
|
Moszyński, Krzysztof |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
37 |
| Issue:
|
6 |
| Year:
|
1992 |
| Pages:
|
419-436 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
Fourier expansion |
| Keyword:
|
orthogonal polynomials on $L^2(\Omega)$ space |
| Keyword:
|
approximate solution of linear algebraic equations |
| Keyword:
|
Richardson iteration |
| Keyword:
|
preconditioning |
| Keyword:
|
polynomial methods |
| Keyword:
|
numerical examples |
| MSC:
|
65F10 |
| idZBL:
|
Zbl 0802.65032 |
| idMR:
|
MR1185798 |
| DOI:
|
10.21136/AM.1992.104521 |
| . |
| Date available:
|
2008-05-20T18:44:23Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104521 |
| . |
| Reference:
|
[1] Doan Van Ban, Moszyński K., Pokrzywa A.: Semiiterative methods for linear equations.Matematyka Stosowana-Applied Mathematics, Vol. 35, Warszawa, 1992. MR 1221221 |
| Reference:
|
[2] Gaier D.: Lectures on Complex Approximation.Birkhäuser, 1989. MR 0894920 |
| Reference:
|
[3] Reichel L.: Polynomials by conformal mapping for the Richardson iteration method for complex linear systems.SIAM NA 25 no. 6 (1988), 1359-1368. Zbl 0692.65011, MR 0972459, 10.1137/0725077 |
| . |
