# Convergence of extrapolation coefficients

Aplikace matematiky (1984)

• Volume: 29, Issue: 2, page 114-133
• ISSN: 0862-7940

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## Abstract

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Let ${x}_{k+1}=T{x}_{k}+b$ be an iterative process for solving the operator equation $x=Tx+b$ in Hilbert space $X$. Let the sequence ${\left\{{x}_{k}\right\}}_{k=o}^{\infty }$ formed by the above described iterative process be convergent for some initial approximation ${x}_{o}$ with a limit ${x}^{*}=T{x}^{*}+b$. For given $l>1,{m}_{0},{m}_{1},\cdots ,{m}_{l}$ let us define a new sequence ${\left\{{y}_{k}\right\}}_{k={m}_{1}}^{\infty }$ by the formula ${y}_{k}={\alpha }_{0}^{\left(k\right)}{x}_{k}+{\alpha }_{1}^{\left(k\right)}{x}_{k-{m}_{1}}+...+{\alpha }_{l}^{\left(k\right)}{x}_{k-{m}_{l}}$, where ${\alpha }_{i}^{\left(k\right)}$ are obtained by solving a minimization problem for a given functional. In this paper convergence properties of ${\alpha }_{i}^{\left(k\right)}$ are investigated and on the basis of the results thus obtainded it is proved that ${lim}_{k\to \infty }∥{x}^{*}-{y}_{k}∥/{∥{x}^{*}-{x}_{k}∥}^{p}=0$ for some $p\ge 1$.

## How to cite

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Zítko, Jan. "Convergence of extrapolation coefficients." Aplikace matematiky 29.2 (1984): 114-133. <http://eudml.org/doc/15339>.

@article{Zítko1984,
abstract = {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 $\lbrace x_k\rbrace ^\infty _\{k=o\}$ 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 $\lbrace y_k\rbrace ^\infty _\{k=m_1\}$ 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 $\lim _\{k\rightarrow \infty \} \left\Vert x^*-y_k\right\Vert /\left\Vert x^*-x_k\right\Vert ^p=0$ for some $p\ge 1$.},
author = {Zítko, Jan},
journal = {Aplikace matematiky},
keywords = {iterative methods; convergence acceleration; Hilbert space; iterative methods; convergence acceleration; Hilbert space},
language = {eng},
number = {2},
pages = {114-133},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Convergence of extrapolation coefficients},
url = {http://eudml.org/doc/15339},
volume = {29},
year = {1984},
}

TY - JOUR
AU - Zítko, Jan
TI - Convergence of extrapolation coefficients
JO - Aplikace matematiky
PY - 1984
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 29
IS - 2
SP - 114
EP - 133
AB - 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 $\lbrace x_k\rbrace ^\infty _{k=o}$ 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 $\lbrace y_k\rbrace ^\infty _{k=m_1}$ 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 $\lim _{k\rightarrow \infty } \left\Vert x^*-y_k\right\Vert /\left\Vert x^*-x_k\right\Vert ^p=0$ for some $p\ge 1$.
LA - eng
KW - iterative methods; convergence acceleration; Hilbert space; iterative methods; convergence acceleration; Hilbert space
UR - http://eudml.org/doc/15339
ER -

## References

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1. J. Zítko, Improving the convergence of iterative methods, Apl. Mat. 28 (1983), 215-229. (1983) Zbl0528.65029MR0701740
2. J. Zítko, Kellogg's iterations for general complex matrix, Apl. Mat. 19 (1974), 342-365. (1974) Zbl0315.65025MR0368406
3. G. Maess, Iterative Lösung linear Gleichungssysteme, Deutsche Akademie der Naturforscher Leopoldina Halle (Saale), 1979. (1979) Zbl0416.65029MR0558164
4. G. Maess, 10.1002/zamm.19760560210, ZAMM 56, 121-122 (1976). (1976) MR0426417DOI10.1002/zamm.19760560210
5. I. Marek J. Zítko, Ljusternik Acceleration and the Extrapolated S.O.R. Method, Apl. Mat. 22 (1977), 116-133. (1977) Zbl0367.65016MR0431667
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). (1962) MR0152112
7. I. Marek, On Ljusternik's method of improving the convergence of nonlinear iterative sequences, Comment. Math. Univ. Carol, 6 (1965), N3, 371-380. (1965) MR0196901
8. A. E. Taylor, Introduction to Functional Analysis, J. Wiley Publ. New York 1958. (1958) Zbl0081.10202MR0098966

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