| Title:
|
Convergence of extrapolation coefficients (English) |
| Author:
|
Zítko, Jan |
| Language:
|
English |
| Journal:
|
Aplikace matematiky |
| ISSN:
|
0373-6725 |
| Volume:
|
29 |
| Issue:
|
2 |
| Year:
|
1984 |
| Pages:
|
114-133 |
| Summary lang:
|
English |
| Summary lang:
|
Czech |
| Summary lang:
|
Russian |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $x_{k+1}=Tx_k+b$ be an iterative process for solving the operator equation $x=Tx+b$ in Hilbert space $X$. Let the sequence $\{x_k\}^\infty _{k=o}$ formed by the above described iterative process be convergent for some initial approximation $x_o$ with a limit $x^*=Tx^*+b$. For given $l>1,m_0,m_1,\dots ,m_l$ let us define a new sequence $\{y_k\}^\infty _{k=m_1}$ by the formula $y_k=\alpha^{(k)}_0x_k+\alpha^{(k)}_1x_{k-m_1}+\ldots +\alpha^{(k)}_lx_{k-m_l}$, where $\alpha^{(k)}_i$ are obtained by solving a minimization problem for a given functional.
In this paper convergence properties of $\alpha^{(k)}_i$ are investigated and on the basis of the results thus obtainded it is proved that $\lim_{k\rightarrow \infty} \left\|x^*-y_k\right\|/\left\|x^*-x_k\right\|^p=0$ for some $p\geq 1$. (English) |
| Keyword:
|
iterative methods |
| Keyword:
|
convergence acceleration |
| Keyword:
|
Hilbert space |
| MSC:
|
47A50 |
| MSC:
|
65J10 |
| idZBL:
|
Zbl 0577.65044 |
| idMR:
|
MR0738497 |
| DOI:
|
10.21136/AM.1984.104075 |
| . |
| Date available:
|
2008-05-20T18:24:22Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104075 |
| . |
| Reference:
|
[1] J. Zítko: Improving the convergence of iterative methods.Apl. Mat. 28 (1983), 215-229. MR 0701740 |
| Reference:
|
[2] J. Zítko: Kellogg's iterations for general complex matrix.Apl. Mat. 19 (1974), 342-365. Zbl 0315.65025, MR 0368406 |
| Reference:
|
[3] G. Maess: Iterative Lösung linear Gleichungssysteme.Deutsche Akademie der Naturforscher Leopoldina Halle (Saale), 1979. MR 0558164 |
| Reference:
|
[4] G. Maess: Extrapolation bei Iterationsverfahren.ZAMM 56, 121-122 (1976). MR 0426417, 10.1002/zamm.19760560210 |
| Reference:
|
[5] I. Marek J. Zítko: Ljusternik Acceleration and the Extrapolated S.O.R. Method.Apl. Mat. 22 (1977), 116-133. MR 0431667 |
| Reference:
|
[6] I. Marek: On a method of accelerating the convergence of iterative processes.Journal Соmр. Math. and Math. Phys. 2 (1962), N2, 963-971 (Russian). MR 0152112 |
| Reference:
|
[7] I. Marek: On Ljusternik's method of improving the convergence of nonlinear iterative sequences.Comment. Math. Univ. Carol, 6 (1965), N3, 371-380. MR 0196901 |
| Reference:
|
[8] A. E. Taylor: Introduction to Functional Analysis.J. Wiley Publ. New York 1958. Zbl 0081.10202, MR 0098966 |
| . |
