"Babeş-Bolyai" University

Faculty of Mathematics and Physics

Research Seminars

Seminar on Functional Analysis and Numerical Methods

Preprint nr.1, 1986, pp.133-136

Sur l’estimation des erreurs en convergence numérique de certaines méthodes d’itération
(English translation)
On the estimation of errors in numerical convergence of certain iteration methods

Ion Păvăloiu

Let $E$ be a Banach space and

(1)

x=\lambda D\left(x\right)+y,\qquad D\left(\theta\right)=\theta,

an operatorial equation, where $\lambda\in\mathbb{R},\ D:E\rightarrow E,\ x,y\in E,$ and $\theta$ is the null element of the space $E .$

In order to solve equation (1) we consider the following iterative process:

(2)

x_{n+1}=\lambda D\left(x_{n}\right)+y,\qquad n=0,1,\ldots,x_{0}=y.

We denote by $\ S=\{x\in E:\left\|x\right\|\leq\rho\}$ the ball of radius $\rho$ and center $\theta$ .

Regarding the convergence of iterations (2) we shall consider the following theorem

Theorem 1.

If the application $D$ and the element $y$ from equation (1) verify the following conditions:

i.

$\left\|D\left(x_{1}\right)-D\left(x_{2}\right)\right\|\leq C\left(\rho\right)% \left\|x_{1}-x_{2}\right\|$ for all $x_{1},x_{2}\in S\left(\theta\right),$ where $C:\left(0,+\infty\right)\rightarrow\left(0,+\infty\right)$ is a functional;
ii.

$\gamma=\left|\lambda\right|C\left(\rho\right)<1;$
iii.

$\left\|y\right\|\leq\left(1-\gamma\right)\rho,$

then equation (1) admits a unique solution $\bar{x}\in S\left(\theta,\rho\right).$

This solution is obtained as the limit of the sequence $\left(x_{n}\right)_{n=0}^{\infty}$ generated by method (2), and the following estimation holds:

(3)

\left\|\bar{x}-x_{n}\right\|\leq\tfrac{\gamma^{n+1}}{1-\gamma}\left\|y\right\|% ,\qquad n=0,1,\ldots\ .

Let $D_{\varepsilon}:E\rightarrow E$ be an application which verifies the conditions:

i ${}_{1}.$

$\left\|D\left(x\right)-D_{\varepsilon}\left(x\right)\right\|\leq\eta_{1}\left(% \varepsilon,\rho\right),\forall x\in B\left(\theta,\rho\right),$ where $\eta_{1}:[0,+\infty)\times[0,+\infty)\$ and $\ \lim\limits_{\varepsilon_{1}\rightarrow 0}\ \eta_{1}\left(\varepsilon,\rho% \right)=0,\ \forall\rho>0$
ii ${}_{1}.$

$\left\|D_{\varepsilon}\left(x_{1}\right)-D_{\varepsilon}\left(x_{2}\right)% \right\|\leq C_{\varepsilon}\left(\rho\right)\left\|x_{1}-x_{2}\right\|$ for every $x_{1},x_{2}\in S\left(\theta,\rho\right)$ where $C_{\varepsilon}:(0,+\infty)\rightarrow\left(0,+\infty\right);$
iii ${}_{1}.$

$\left|C\left(\rho\right)-C_{\varepsilon}\left(\rho\right)\right|<\eta_{2}\left% (\varepsilon\right)\$ where $\ \eta_{2}:[0,+\infty)\rightarrow[0,+\infty)\$ and $\lim\limits_{\varepsilon\rightarrow 0}\eta_{2}\left(\varepsilon\right)=0;$
iv ${}_{1}.$

We consider an element for which $\left\|y-y_{\varepsilon}\right\|\leq\eta_{3}\left(\varepsilon\right)\$ where $\ \eta_{3}:[0,+\infty)\rightarrow[0,+\infty)\$ and $\lim\eta_{3}\left(\varepsilon\right)=0.$

In order to solve equation (1) we consider, instead of iterative method (2) the following iterative procedure:

(4)

\xi_{n+1}=\lambda D_{\varepsilon}\left(\xi_{n}\right)+y_{\varepsilon},\qquad n% =0,1,\ldots,\xi_{0}=y_{\varepsilon}.

Regarding the convergence of method (4) we obtain the following theorem:

Theorem 2.

If the conditions of Theorem 1 are fulfilled, the operator $D_{\varepsilon}$ and the element $y_{\varepsilon}$ verify conditions $i_{1}$ – $iv_{1},$ and if

(5)

\delta=\left(1-\gamma\right)\rho-y\left\|y\right\|>0,

then there exists an $\bar{\varepsilon}>0,$ so that for all $\varepsilon<\bar{\varepsilon}$ we have

(6)

\gamma_{\varepsilon}=\left|\lambda\right|C_{\varepsilon}\left(\rho\right)\leq% \gamma+\left|\lambda\right|\eta_{2}\left(\varepsilon\right)<1;

and

(7)

\left\|y_{\varepsilon}\right\|\leq\left(1-\gamma_{\varepsilon}\right)\rho.

Proof.

Indeed, from the fact that $\gamma=\left|\lambda\right|C\left(\rho\right)<1$ and $\lim\limits_{\varepsilon\rightarrow 0}\eta_{2}\left(\varepsilon\right)=0$ it follows that there exists a number $\bar{\varepsilon}_{1}>0$ so that for $\varepsilon<\bar{\varepsilon}_{1}$ we have

\gamma_{\varepsilon}=\left|\lambda\right|C_{\varepsilon}\left(\rho\right)\leq% \left|\lambda\right|C\left(\rho\right)+\left|\lambda\right|\eta_{2}\left(% \varepsilon\right)<1

and

	$\displaystyle\left\\|y_{\varepsilon}\right\\|$	$\displaystyle\leq\left\\|y\right\\|+\eta_{3}\left(\varepsilon\right)\leq\left(1-% \gamma\right)\rho-\delta+\eta_{3}\left(\varepsilon\right)$
		$\displaystyle\leq\left(1-\gamma_{\varepsilon}\right)\rho+\left\|\lambda\right\|% \eta_{2}\left(\varepsilon\right)\rho+\eta_{3}\left(\varepsilon\right)-\delta% \leq\left(1-\gamma_{\varepsilon}\right)\rho$

for $\varepsilon<\bar{\varepsilon}_{2}\$ because $\eta_{3}\left(\varepsilon\right)\rightarrow 0\$ for $\ \varepsilon\rightarrow 0$ and $\eta_{2}\left(\varepsilon\right)\rightarrow 0$ as $\varepsilon\rightarrow 0$ .

If we take now

\bar{\varepsilon}=\min\{\bar{\varepsilon}_{1},\bar{\varepsilon}_{2}\}

then the theorem is proved. ∎

Relations (6) and (7) assure the convergence of sequence $\left(\xi_{n}\right)_{n=0}^{\infty}$ determined by method (4).

We show now that in the conditions of Theorem 2 we have the following estimation:

(8)

\left\|\bar{x}-\xi_{n}\right\|\leq\frac{\left|\lambda\right|C\left(\rho\right)% \left\|\xi_{n}-\xi_{n-1}\right\|+\left|\lambda\right|\eta_{1}\left(\varepsilon% ,\rho\right)+\eta_{3}\left(\varepsilon\right)}{1-\gamma}

Indeed, we have

	$\displaystyle\left\\|\bar{x}-\xi_{n}\right\\|$	$\displaystyle\leq\left\|\lambda\right\|\left\\|D\left(\bar{x}\right)-D_{% \varepsilon}\left(\xi_{n-1}\right)\right\\|+\left\\|y-y_{\varepsilon}\right\\|$
		$\displaystyle\leq\left\|\lambda\right\|C\left(\rho\right)\left\\|\bar{x}-\xi_{n}% \right\\|+\left\|\lambda\right\|C\left(\rho\right)\left\\|\xi_{n}-\xi_{n-1}\right% \\|+\left\|\lambda\right\|\eta_{1}\left(\varepsilon,\rho\right)+\eta_{3}\left(% \varepsilon\right)$

so inequality (8) follows. From (8) it results

\left\|\bar{x}-\overline{\xi}\right\|\leq\frac{\left|\lambda\right|\eta_{1}% \left(\varepsilon,\rho\right)+\eta_{3}\left(\varepsilon\right)}{1-\gamma},% \qquad\text{as\ }n\rightarrow\infty,

where $\overline{\xi}=\lim\limits_{n\rightarrow\infty}\xi_{n}.$

References

[1]
[2] Babici, D.M., Ivanov, V.N., Oţenca polnoi progresnosti prireshenia nelineinyh operatornyh uravnenii metodov prostei iteraţii. Jurnal vycislitelnoi matematiki i matematiceskoi fisiki 7, 5 (1967), 988–1000.
[3] Păvăloiu, I., ^†^†margin: $\leftarrow$ clickable Introduction in the Theory of Approximating the Solutions of Equations, Ed. Dacia 1976 (in Romanian).
[4] Urabe, M., Error estimation in numerical solution of equations by iteration process, J. Sci. Hiroshima Univ. Ser. A-I, 26 (1962), 77–91.

On the error estimation in the numerical convergence of certain iterative methods

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Paper (preprint) in HTML form

Sur l’estimation des erreurs en convergence numérique de certaines méthodes d’itération
(English translation)
On the estimation of errors in numerical convergence of certain iteration methods

Theorem 1.

Theorem 2.

Proof.

References

Related Posts

On the error estimation in the numerical convergence of certain iterative methods

Abstract

Authors

Title

Original title (in French)

English translation of the title

Keywords

PDF

Cite this paper as:

About this paper

Journal

Publisher Name

DOI

References

Paper (preprint) in HTML form

Sur l’estimation des erreurs en convergence numérique de certaines méthodes d’itération (English translation) On the estimation of errors in numerical convergence of certain iteration methods

Theorem 1.

Theorem 2.

Proof.

References

Related Posts

Sur l’estimation des erreurs en convergence numérique de certaines méthodes d’itération
(English translation)
On the estimation of errors in numerical convergence of certain iteration methods