Convergence of extrapolation coefficients

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 -