# Convergence of extrapolation coefficients

Aplikace matematiky (1984)

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

top Access to full text Full (PDF)

## Abstract

top
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 $p\ge 1$.

## Citations in EuDML Documents

top

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.