Error estimations in the numerical solving of the systems of equations

9 years ago

Abstract

Let $\left( x_{i},\rho_{i}\right) ,\ i=1,2,$ be two complete metric space and $F_{1}:X_{1}\times X_{2}\rightarrow X_{1},\ F_{2}:X_{1}\times X_{2}\rightarrow X_{2}$ two nonlinear mappings. We study the solving of the system \begin{align}
x_{1} & =F_{1}\left( x_{1},x_{2}\right) \label{f.1}\\
x_{2} & =F_{2}\left( x_{1},x_{2}\right) ,\qquad \left( x_{1},x_{2}\right)
\in X.\nonumber
\end{align} by the Gauss-Seidel type method \begin{align}
x_{1}^{\left( n+1\right) } & =F_{1}\left( x_{1}^{\left( n\right)
},x_{2}^{\left( n\right) }\right) \label{f.2}\\
x_{2}^{\left( n+1\right) } & =F_{2}\left( x_{1}^{\left( n+1\right)
},x_{2}^{\left( n\right) }\right) ,\qquad n=0,1,\ldots;\left( x_{1}^{\left(
0\right) },x_{2}^{\left( 0\right) }\right) \in X\nonumber
\end{align} We give sufficient conditions for convergence and some error estimations. We also study the case when the mappings $F_{1}$ and $F_{2}$ are replaced by some approximations.

Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Title

Original title (in French)

Délimitation des erreur dans la résolution numérique des systèmes d’equations

English translation of the title

Error estimations in the numerical solving of the systems of equations

Keywords

nonlinear system in metric space; Gauss-Seidel type method; convergence; approximate value

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Cite this paper as:

I. Păvăloiu, Délimitation des erreur dans la résolution numérique des systèmes d’equations, Seminar on mathematical analysis, Preprint no. 7 (1988), pp. 167-178 (in French).

About this paper

Journal

Seminar on mathematical analysis,
Preprint

Publisher Name

“Babes-Bolyai” University,
Faculty of Mathematics,
Research Seminars

DOI

Not available yet.

References

[1] Pavaloiu, I., Introducere in teoria aproximarii solutiilor ecuatiilor, Editura Dacia, Cluj-Napoca, 1976.

[2] Pavaloiu, I., La resolution des systemes operationnelles a l’aide des methodes iteratives, Mathematica, 11(34), (1969), 137–141.

[3] Pavaloiu, I., Estimation des erreurs dans la resolution numerique des systemes d’equations dans des espaces metriques, Seminar on Functional Analysis and Numerical Methods, Preprint Nr. 1, (1987), 121–129.

[4] Pavaloiu, I., La convergence de certaines methodes iteratives pour resoudre certaines equations operatorielles, Seminar on Functional analysis and Numerical Methods, Preprint Nr. 1 (1986), 127–132.

[5] Traub, J. F., Iterative Methods for the Solution of Equations, Prentice Hall Series in Automatic Computation, Englewood Cliffs, N. J. (1964).

[6] Urabe, M., Convergence of numerical iteration in solution of equations, J. Sci. Hiroshima Univ. Ser. A, 19 (1956), 479–489.

[7] Urabe, M., Error estimation in numerical solution of equations by iteration process, J. Sci. Hiroshima Univ. Ser. A-I, 26, (1962), 77–91.

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"Babeş-Bolyai" University

Faculty of Mathematics and Physics

Research Seminars

Seminar on Mathematical Analysis

Preprint Nr.7, 1988, pp.167-178

Error estimations in the numerical solving of the systems of equations

Ion Pavaloiu

Let us designate by $(X_{i}, r_{i}), i = 1, 2$ two complete metric spaces and by $X = X_{1} \times X_{2}$ the Cartesian product of these spaces. We will denote by $F_{1} : X \to X_{1}$ And $F_{2} : X \to X_{2}$ two applications of space $X$ in $X_{1}$ , respectively in $X_{2}$ and we consider the following system of equations:

(1)		$x_{1}$	$= F_{1} (x_{1}, x_{2})$
	$x_{2}$	$= F_{2} (x_{1}, x_{2}), (x_{1}, x_{2}) \in X .$

To solve the system ( 1 ) we will adopt the following Gauss-Seidel type iterative process:

(2)		$x_{1}^{(n + 1)}$	$= F_{1} (x_{1}^{(n)}, x_{2}^{(n)})$
	$x_{2}^{(n + 1)}$	$= F_{2} (x_{1}^{(n + 1)}, x_{2}^{(n)}), n = 0, 1, \dots; (x_{1}^{(0)}, x_{2}^{(0)}) \in X$

In works [ 2 ] , [ 3 ] we studied the convergence of method ( 2 ) under the assumption that the applications $F_{1}$ And $F_{2}$ satisfy Lipschitz-type conditions throughout the space $X$ .

In work [ 4 ] we obtained bounds of errors in the numerical resolution of system ( 1 ) using a method of type ( 2 ), in which the applications $F_{1}$ And $F_{2}$ are replaced by two other applications $F_{1}^{*}$ And $F_{2}^{*}$ , which meet certain conditions for bringing together $F_{2}$ And $F_{1}$ throughout space $X$ .

In the applications of this theory to the resolution of a class of concrete equations, the conditions imposed on the applications $F_{1}, F_{2}, F_{1}^{*}$ And $F_{2}^{*}$ throughout space $X$ are awkward, because of the fact that the space $X$ may not be limited.

In the following we will study this problem on the assumption that $F_{1}$ And $F_{2}$ meet Lipschitz-type conditions and rapprochement conditions $F_{1}^{*}$ And $F_{2}^{*}$ in certain bounded subsets $D_{1} \subset X_{1}$ And $D_{2} \subset X_{2}$ .

Let us therefore designate by $D_{1}$ And $D_{2}$ two bounded sets of spaces $X_{1}$ respectively $X_{2}$ and by $D = D_{1} \times D_{2}$ their Cartesian product. Consider two sequences ${(f_{n})}_{n = 0}^{\infty}$ And ${(g_{n})}_{n = 0}^{\infty}$ whose elements meet the conditions:

(3)		$f_{n}$	$\leq a f_{n - 1} + b g_{n - 1}$
	$g_{n}$	$\leq a f_{n} + b g_{n - 1}, n = 1, 2, \dots,$

Or $a, b,$ $a$ And $b$ are non-negative real numbers and $f_{n} \geq 0,$ $g_{n} \geq 0$ for everything $n = 0, 1, \dots$ .

We will associate to the system ( 3 ) the following system of equations in the unknowns $h$ And $k$ :

(4)		$a + b h$	$= h k$
	$a k + b$	$= n k$

In works [ 2 ] , [ 3 ] and [ 4 ] we have shown that if the numbers $a, b, a$ And $b$ check the relationships

(5)		$a + b + a b$	$< 2$
	$(1 - a) (1 - b) - a b$	$> 0$
	$b$	$> 0, a > 0$

then the system ( 4 ) admits real solutions $(h_{i}, k_{i}), i = 1, 2$ for which $0 < h_{i} k_{i} < 1$ and that one of the solutions verifies the conditions $h_{1} > 0, k_{1} > 0$ . In addition, the elements of the suites ${(f_{n})}_{n = 0}^{\infty}$ ${(g_{n})}_{n = 0}^{\infty}$ check the relationships

(6)		$f_{n}$	$\leq C h_{1}^{n - 1} k_{1}^{n - 1}$
	$g_{n}$	$\leq C h_{1}^{n} k_{1}^{n - 1}, n = 1, 2, \dots,$

Or $C = \max {a f_{0} + b g_{0}, \frac{a f_{1} + b g_{0}}{h_{1}}} .$

If we write $p_{1} = h_{1}, k_{1}$ then we easily see using ( 4 ) that $p_{1}$ check the equation

(7)

p^{2} - (b + b a + a) p + b a = 0 .

Let us designate by $d_{1} > 0$ a number such that $S_{1} \subseteq D_{1}$ , $S_{2} \subseteq D_{2}$ , Or

(8)		$S_{1}$	$= {x \in X_{1} : r_{1} (x, x_{1}^{(0)}) \leq \frac{d_{1}}{1 - p_{1}}}$
	$S_{2}$	$= {x \in X_{2} : r_{2} (x, x_{2}^{(0)}) \leq \frac{d_{1} h_{1}}{1 - p_{1}}}$

Regarding the convergence of sequences ${(x_{1}^{(n)})}_{n = 0}^{\infty}, {(x_{2}^{(n)})}_{n = 0}^{\infty}$ we have the following theorem:

Theorem 1 .

If the applications $F_{1}, F_{2}$ check the conditions

i.

$r_{1} (F_{1} (x_{1}, {and}_{1}), F_{1} (x_{2}, {and}_{2}))$ $\leq a r_{1} (x_{1}, x_{2}) + b r_{2} (x_{2}, {and}_{2})$

$r_{2} (F_{2} (x_{1}, {and}_{1}), F_{2} (x_{2}, {and}_{2}))$ $\leq a r_{1} (x_{1}, x_{2}) + b r_{2} (x_{2}, {and}_{2})$

for all $(x_{1}, x_{2}), ({and}_{1}, {and}_{2}) \in D;$
ii.

The names $a, b,$ $a$ And $b$ check the relationships ( 5 );
iii.

The elements $x_{1}^{(1)}$ And $x_{2}^{(1)}$ suites ${(x_{1}^{(n)})}_{n = 0}^{\infty}, {(x_{2}^{(n)})}_{n = 0}^{\infty}$ check the conditions $r_{1} (x_{1}^{(0)}, x_{1}^{(1)}) < d_{1}$ And $r_{2} (x_{2}^{(0)}, x_{2}^{(1)}) \leq d_{1} h_{1}$ .

Then the following properties are true:

j.

These suites ${(x_{1}^{(n)})}_{n = 0}^{\infty}, {(x_{2}^{(n)})}_{n = 0}^{\infty}$ constructed using method ( 2 ) are convergent;
jj.

If we write ${\bar{x}}_{1} = lim_{n \to \infty} x_{1}^{(n)}$ And ${\bar{x}}_{2} = lim_{n \to \infty} x_{2}^{(n)}$ SO $({\bar{x}}_{1}, {\bar{x}}_{2}) \in S$ Or $S = S_{1} \times S_{2}$ And $({\bar{x}}_{1}, {\bar{x}}_{2})$ is the only solution of the system ( 1 ) in the set $S$ ;
jjj.

The following relationships take place

$r_{1} ({\bar{x}}_{1}, x_{1}^{(n)})$ $\leq \frac{d_{1} p_{1}^{n}}{1 - p_{1}}$

$r_{2} ({\bar{x}}_{2}, x_{2}^{(n)})$ $\leq \frac{d_{1} h_{1} p_{1}^{n}}{1 - p_{1}}$

Demonstration.

From i. and ( 2 ) we deduce the following inequalities

	$r_{1} (x_{1}^{(2)}, x_{1}^{(1)})$	$\leq a r_{1} (x_{1}^{(1)}, x_{1}^{(0)}) + b r_{2} (x_{1}^{(1)}, x_{2}^{(0)})$
		$\leq a d_{1} + b d_{1} h_{1} \leq d_{1} p_{1}$

And

	$r_{2} (x_{2}^{(2)}, x_{2}^{(1)})$	$\leq a r_{1} (x_{1}^{(2)}, x_{1}^{(1)}) + b r_{2} (x_{2}^{(1)}, x_{2}^{(0)})$
		$\leq a d_{1} p_{1} + b d_{1} h_{1} \leq d_{1} h_{1} p_{1} .$

We now show that $x_{1}^{(2)} \in S_{1}$ And $x_{2}^{(2)} \in S_{2}$ . We have indeed

	$r_{1} (x_{1}^{(2)}, x_{1}^{(0)})$	$\leq r_{1} (x_{1}^{(2)}, x_{1}^{(1)}) + r_{1} (x_{1}^{(1)}, x_{1}^{(0)})$
	$d_{1} + d_{1} p_{1}$	$< \frac{d_{1}}{1 - p_{1}},$

And

	$r_{2} (x_{2}^{(2)}, x_{2}^{(0)})$	$\leq r_{2} (x_{2}^{(2)}, x_{2}^{(1)}) + r_{2} (x_{2}^{(1)}, x_{2}^{(0)})$
		$\leq d_{1} h_{1} + d_{1} p_{1} h_{1} \leq \frac{h_{1} d_{1}}{1 - p_{1}}$

from which it follows that $x_{1}^{(2)} \in S_{1}, x_{2}^{(2)} \in S_{2}$ .

We will now assume that the following inequalities hold

(9)		$r_{1} (x_{1}^{(i)}, x_{1}^{(i - 1)})$	$\leq d_{1} p_{1}^{i - 1}$
	$r_{2} (x_{2}^{(i)}, x_{2}^{(i - 1)})$	$\leq d_{1} h_{1} p_{1}^{i - 1}$

For $i = 1, 2, \dots, k$ And

(10)

x_{1}^{(k)} \in S_{1}, x_{2}^{(k)} \in S_{2} .

Taking into account ii., we deduce from hypotheses ( 9 ) and ( 10 ) and from i.

	$r_{1} (x_{1}^{(k + 1)}, x_{1}^{(k)})$	$= r_{1} (F_{1} (x_{1}^{(k)}, x_{2}^{(k)}), F_{1} (x_{1}^{(k - 1)}, x_{2}^{(k - 1)}))$
		$\leq a r_{1} (x_{1}^{(k)}, x_{1}^{(k - 1)}) + b r_{2} (x_{2}^{(k)}, x_{2}^{(k - 1)})$
		$\leq d_{1} p_{1}^{k - 1} (a + b h_{1}) = d_{1} p_{1}^{k}$

and taking into account the above inequality we deduce in an analogous manner

r_{2} (x_{2}^{(k + 1)}, x_{2}^{(k)}) \leq d_{1} \cdot h_{1} \cdot p_{1}^{k}

from which it follows that relations ( 9 ) also hold for $i = k + 1$ . We will now show that $x_{1}^{(k + 1)} \in S_{1}$ And $x_{2}^{(k + 1)} \in S_{2}$ . It follows from the above inequalities

	$r_{1} (x_{1}^{(k + 1)}, x_{1}^{(0)})$	$\leq r_{1} (x_{1}^{(k + 1)}, x_{1}^{(k)}) + r_{1} (x_{1}^{(k)}, x_{1}^{(k - 1)}) + \dots r_{1} (x_{1}^{(1)}, x_{1}^{(0)})$
		$\leq d_{1} + d_{1} p_{1} + d_{1} p_{1}^{2} + \dots + d_{1} p_{1}^{k} < \frac{d_{1}}{1 - p_{1}}$

and similarly

r_{2} (x_{2}^{(k + 1)}, x_{2}^{(0)}) < \frac{d_{1} h_{1}}{1 - p_{1}} .

Since inequalities ( 9 ) hold for all $i = 1, 2, \dots,$ it results that the suites ${(x_{1}^{(n)})}_{n = 0}^{\infty}$ And ${(x_{2}^{(n)})}_{n = 0}^{\infty}$ are fundamental and that the following inequalities are true

(11)		$r_{1} (x_{1}^{(n + s)}, x_{1}^{(n)})$	$\leq \frac{d_{1} p_{1}^{n}}{1 - p_{1}}$
	$r_{2} (x_{2}^{(n + s)}, x_{2}^{(n)})$	$\leq \frac{d_{1} h_{1} p_{1}^{n}}{1 - p_{1}}$

for everything $n = 0, 1, \dots;$ $s = 1, 2, \dots$ .

From the hypothesis that the spaces $X_{1}$ And $X_{2}$ are complete spaces it follows that the sequences ${(x_{n}^{(n)})}_{n = 0}^{\infty}$ And ${(x_{2}^{(n)})}_{n = 0}^{\infty}$ are convergent.

Passing the limit for $s \to \infty$ we deduce from ( 11 )

(12)		$r_{1} ({\bar{x}}_{1}, x_{1}^{(n)})$	$\leq \frac{d_{1} p_{1}^{n}}{1 - p_{1}}$
	$r_{2} ({\bar{x}}_{2}, x_{2}^{(n)})$	$\leq \frac{d_{1} h_{1} p_{1}^{n}}{1 - p_{1}}$

It results, for $n = 0,$ what ${\bar{x}}_{1} \in S_{1}$ And ${\bar{x}}_{2} \in S_{2}$ . We will show that $({\bar{x}}_{1}, {\bar{x}}_{2})$ is the only solution of the system ( 1 ) to the property that ${\bar{x}}_{1} \in S_{1}$ And ${\bar{x}}_{2} \in S_{2}$ .

Let us suppose by contradiction that system ( 1 ) has another solution $({\bar{and}}_{1}, {\bar{and}}_{2})$ to the property ${\bar{and}}_{1} \in S_{1}$ , ${\bar{and}}_{2} \in S_{2}$ . Taking into account i., the inequalities result

	$r_{1} ({\bar{x}}_{1}, {\bar{and}}_{1})$	$\leq \frac{b a}{(1 - a) (1 - b)} \cdot r_{1} ({\bar{x}}_{1}, {\bar{and}}_{1})$
	$r_{2} ({\bar{x}}_{2}, {\bar{and}}_{2})$	$\leq \frac{b a}{(1 - a) (1 - b)} \cdot r_{2} (x_{2}, {and}_{2})$

and from ( 5 ) it follows that $\frac{b a}{(1 - a) (1 - b)} < 1$ . Therefore the above inequalities hold if and only if ${\bar{x}}_{1} = {\bar{and}}_{1}$ And ${\bar{x}}_{2} = {\bar{and}}_{2}$ . ∎

We now consider two applications $F_{1}^{*} : D \to X_{1}$ And $F_{2}^{*} : D \to X_{2}$ , Or $D = D_{1} \times D_{2}$ .

We will assume that $F_{1}^{*},$ $F_{2}^{*},$ $F_{1}$ And $F_{2}$ check the relationships

(13)		$r_{1} (F_{1}^{*} (in, in), F_{1} (in, in))$	$\leq d_{1}$
	$r_{2} (F_{2}^{*} (in, in), F_{2} (in, in))$	$\leq d_{2}$

for everything $(in, in) \in D,$ Or $d_{1} > 0,$ $d_{2} > 0$ are given numbers. In order to solve the system ( 1 ) we will now consider instead of the procedure ( 2 ) the following iterative procedure:

(14)		$x_{1}^{(n + 1)}$	$= F_{1} (x_{1}^{(n)}, x_{2}^{(n)})$
	$x_{2}^{(n + 1)}$	$= F_{2} (x_{1}^{(n + 1)}, x_{2}^{(n)}), n = 0, 1, \dots x_{1}^{(0)} = x_{1}^{(0)}, x_{2}^{(0)} = x_{2}^{(0)},$

In the following we will proceed to the delimitation of errors in the case where for the resolution of the system ( 1 ) we use instead of the method ( 2 ) the method ( 14 ). We write

(15)		$i_{1}$	$= \frac{b d_{2} + (1 - b) d_{1}}{(1 - a) (1 - b) - a b}$
	$i_{2}$	$= \frac{(1 - a) d_{2} + a d_{1}}{(1 - a) (1 - b) - a b}$

and designate by $S_{1}^{*}, S_{2}^{*}$ the following sets:

(16)		$S_{1}^{*}$	$= {x \in X_{1} : r_{1} (x, x_{1}^{(0)}) \leq d_{1} + \frac{d_{1}}{1 - p_{1}} + i_{1}}$
	$S_{2}^{*}$	$= {x \in X_{2} : r_{2} (x, x_{2}^{(0)}) \leq d_{1} h_{1} + \frac{d_{1} h_{1}}{1 - p_{1}} + i_{2}} .$

In these notations we will demonstrate the following theorem:

Theorem 2 .

If the assumptions of Theorem 1 are satisfied and if in addition the following conditions are met:

i $_{1} .$

The applications $F_{1}, F_{2}, F_{1}^{*}$ And $F_{2}^{*}$ meet the conditions ( 13 );
i $_{2} .$

$S_{1}^{*} \subseteq D_{1}, S_{2}^{*} \subseteq D_{2}$ ,

so whatever the real numbers are $e_{1},$ $e_{2}$ who verify the relationships $e_{1} > 2 i_{1}, e_{2} > 2 i_{2}$ , there is a $n^{'} \in ℕ$ such as $r_{1} (x_{1}^{(n + 1)}, x_{1}^{(n)}) \leq e_{1}$ , And $r_{2} (x_{2}^{(n + 1)}, x_{2}^{(n)}) \leq e_{2}$ for everything $n > n^{'}$ and, furthermore, the following properties are true:

j $_{1} .$

$x_{1}^{(n)} \in S_{1},$ $x_{2}^{(n)} \in S_{2}$ for everything $n = 1, 2, \dots;$
j $_{2} .$

The following inequalities are true:

$r_{1} ({\bar{x}}_{1}, x_{1}^{(n + 1)})$ $\leq \frac{b (a e_{1} + b e_{2}) + a e_{1} (1 - b)}{(1 - a) (1 - b) - a b} + \frac{e_{1}}{2}$

$r_{2} ({\bar{x}}_{2}, x_{2}^{(n + 1)})$ $\leq \frac{a (a e_{1} + b e_{2}) + b e_{2} (1 - a)}{(1 - a) (1 - b) - a b} + \frac{e_{2}}{2}$

for everything $n > n^{'}$ , Or $({\bar{x}}_{1}, {\bar{x}}_{2})$ is the solution of the system ( 1 ).

Demonstration.

We will first show that properties j ₁ are true. Since we have assumed that $x_{1}^{(0)} = x_{1}^{(0)}, x_{2}^{(0)} = x_{2}^{(0)},$ it follows from ( 14 ) and i.

	$r_{1} (x_{1}^{(1)}, x_{1}^{(1)})$	$= r_{1} (F_{1} (x_{1}^{(0)}, x_{2}^{(0)}), F_{1}^{*} (x_{1}^{(0)}, x_{2}^{(0)}))$
		$\leq d_{1} + a r_{1} (x_{1}^{(0)} x_{1}^{(0)}) + b r_{2} (x_{2}^{(0)} x_{2}^{(0)}) = d_{1}$

Taking into account iii., we deduce

	$r_{1} (x_{1}^{(1)}, x_{1}^{(0)})$	$\leq r_{1} (x_{1}^{(1)}, x_{1}^{(1)}) + r_{1} (x_{1}^{(1)}, x_{1}^{(0)})$
		$\leq d_{1} + d_{1} \leq d_{1} + \frac{d_{1}}{1 - p_{1}} + i_{1}$

since obviously $d_{1} \leq i_{1}$ , from which it results $x_{1}^{(1)} \in S_{1}^{*}$ .

We have in a similar way

	$r_{2} (x_{2}^{(1)}, x_{2}^{(1)})$	$= r_{2} (F_{2} (x_{1}^{(1)}, x_{2}^{(0)}), F_{2}^{*} (x_{1}^{(1)}, x_{2}^{(0)}))$
		$\leq d_{2} + a r_{1} (x_{1}^{(1)}, x_{1}^{(1)}) + b r_{2} (x_{2}^{(0)}, x_{2}^{(0)}) \leq d_{2} + a d_{1}$

Taking into account iii. we deduce

r_{2} (x_{2}^{(1)}, x_{2}^{(0)}) \leq r_{2} (x_{2}^{(1)}, x_{2}^{(1)}) + r_{2} (x_{2}^{(1)}, x_{2}^{(0)}) \leq d_{2} + a d_{1} + d_{1} h_{1} .

Taking into account that $d_{2} + a d_{1} \leq i_{2}$ it follows that

r_{2} (x_{2}^{(1)}, x_{2}^{(0)}) \leq h_{1} d_{1} + \frac{h_{1} d_{1}}{1 - p_{1}} + i_{2}

and therefore $x_{2}^{(1)} \in S_{2}^{*}$ .

We will now assume by induction that $x_{1}^{(n - 1)} \in S_{1}^{*}$ And $x_{2}^{(n - 1)} \in S_{2}^{*}$ .

We then have

(17)		$r_{1} (x_{1}^{(n)}, x_{1}^{(n)})$	$= r_{1} (F_{1} (x_{1}^{(n - 1)}, x_{2}^{(n - 1)}), F^{*} (x_{1}^{(n - 1)}, x_{2}^{(n - 1)}))$
		$\leq a r_{1} (x_{1}^{(n - 1)}, x_{1}^{(n - 1)}) + b r_{2} (x_{2}^{(n - 1)}, x_{2}^{(n - 1)}) + d_{1}$

And

(18)		$r_{2} (x_{2}^{(n)}, x_{2}^{(n)})$	$= r_{2} (F_{2} (x_{1}^{(n)}, x_{2}^{(n - 1)}), F_{2}^{*} (x_{1}^{(n)}, x_{2}^{(n - 1)}))$
		$\leq a r_{1} (x_{1}^{(n)}, x_{1}^{(n)}) + b r_{2} (x_{2}^{(n - 1)}, x_{2}^{(n - 1)}) + d_{2} .$

We show by induction, from the above relations, that

(19)		$r_{1} (x_{1}^{(n)}, x_{1}^{(n)})$	$\leq d_{1} h_{1}^{n - 1}, k_{1}^{n - 1} + i_{1}$
	$r_{2} (x_{2}^{(n)}, x_{2}^{(n)})$	$\leq d_{1} h_{1}^{n}, k_{1}^{n - 1} + i_{2},$

from which we deduce:

r_{1} (x_{1}^{(n)}, x_{1}^{(0)}) \leq r_{1} (x_{1}^{(n)}, x_{1}^{(n)}) + r_{1} (x_{1}^{(n)}, x_{1}^{(0)}) \leq d_{1} + \frac{d_{1}}{1 - p_{1}} + i_{1}

And

r_{2} (x_{2}^{(n)}, x_{2}^{(0)}) \leq r_{2} (x_{2}^{(n)}, x_{2}^{(n)}) + r_{2} (x_{2}^{(n)}, x_{2}^{(0)}) \leq d_{1} h_{1} + \frac{d_{1} h_{1}}{1 - p_{1}} + i_{2}

that's to say $x_{1}^{(n)} \in S_{1}^{*},$ $x_{2}^{(n)} \in S_{2}^{*}$ .

From the assumptions of Theorem 2 and the fact that $x_{1}^{(n)} \in S_{1}^{*}, x_{2}^{(n)} \in S_{2}^{*}$ for everything $n \in N$ , the following relationships result:

r_{1} (x_{1}^{(n + 1)}, x_{1}^{(n)}) \leq a r_{1} (x_{1}^{(n)}, x_{1}^{(n - 1)}) + b r_{2} (x_{2}^{(n)}, x_{2}^{(n - 1)}) + 2 d_{1}

And

r_{2} (x_{2}^{(n + 1)}, x_{2}^{(n)}) \leq a r_{1} (x_{1}^{(n + 1)}, x_{1}^{(n)}) + b r_{2} (x_{2}^{(n)}, x_{2}^{(n - 1)}) + 2 d_{2}

For $n = 0, 1, \dots$ .

From the above inequalities we deduce the following inequalities:

(20)		$r_{1} (x_{1}^{(n + 1)}, x_{1}^{(n)})$	$\leq d_{1} h_{1}^{n - 1} \cdot k_{1}^{n - 1} + 2 i_{1}$
	$r_{2} (x_{2}^{(n + 1)}, x_{2}^{(n)})$	$\leq d_{1} h_{1}^{n} \cdot k_{1}^{n - 1} + 2 i_{2}, n = 1, 2, \dots$

We immediately deduce from ( 20 ) that if $e_{1} > 2 i_{1},$ And $e_{2} > 2 i_{2},$ so it exists $n^{'} \in ℕ$ such as for $n > n^{'}$ $r_{1} (x_{1}^{(n + 1)}, x_{1}^{(n)}) \leq e_{1}$ And $r_{2} (x_{2}^{(n + 1)}, x_{2}^{(n)}) \leq e_{2}$ that is, the iterative process ( 14 ) can be stopped when the distance between two successive iterations is sufficiently small.

We will now evaluate the distances between the solutions ${\bar{x}}_{1}$ And ${\bar{x}}_{2}$ of the system ( 1 ) and the elements of the sequences ${(x_{1}^{(n)})}_{s = 0}^{\infty},$ respectively ${(x_{2}^{(n)})}_{s = 0}^{\infty}$ . We will assume that the iterative process ( 14 ) is stopped when $r_{1} (x_{1}^{(n + 1)}, x_{1}^{(n)}) \leq e_{1}$ And $r_{2} (x_{2}^{(n + 1)}, x_{2}^{(n)}) \leq e_{2}$ .

Taking into account what has been demonstrated above and the assumptions of Theorem 2 , we have:

	$r_{1} ({\bar{x}}_{1}, x_{1}^{(n + 1)})$	$\leq a r_{1} ({\bar{x}}_{1}, x_{1}^{(n)}) + b r_{2} ({\bar{x}}_{2}, x_{2}^{(n)}) + d_{1}$
	$r_{2} ({\bar{x}}_{2}, x_{2}^{(n + 1)})$	$\leq a r_{1} ({\bar{x}}_{1}, x_{1}^{(n + 1)}) + b r_{2} ({\bar{x}}_{2}, x_{2}^{(n)}) + d_{2}$

It follows from the above inequalities

	$(1 - a) r_{1} ({\bar{x}}_{1}, x_{1}^{(n + 1)})$	$\leq a e_{1} + b r_{2} ({\bar{x}}_{2}, x_{2}^{(n)}) + d_{1}$
	$(1 - b) r_{2} ({\bar{x}}_{2}, x_{2}^{(n + 1)})$	$\leq a r_{1} ({\bar{x}}_{1}, x_{1}^{(n + 1)}) + b e_{2} + d_{2}$

that's to say

(21)		$r_{1} ({\bar{x}}_{1}, x_{1}^{(n + 1)})$	$\leq \frac{a e_{1}}{1 - a} + \frac{b}{1 - a} r_{2} ({\bar{x}}_{2}, x_{2}^{(n)}) + \frac{d_{1}}{1 - a}$
	$r_{2} ({\bar{x}}_{2}, x_{2}^{(n + 1)})$	$\leq \frac{a}{1 - b} r_{1} ({\bar{x}}_{1}, x_{1}^{(n + 1)}) + \frac{b e_{2}}{1 - b} + \frac{d_{2}}{1 - b}$

from which it results

(22)

r_{2} ({\bar{x}}_{2}, x_{2}^{(n + 1)}) \leq \frac{a (a e_{1} + b e_{2}) + b (1 - a) e_{2}}{(1 - a) (1 - b) - a b} + \frac{e_{2}}{2} .

Because $n > n^{'} + 1$ , it follows that the second inequality of ( 21 ) is also true if we replace $n + 1$ about $n$ , that is to say that

r_{2} ({\bar{x}}_{2}, x_{2}^{(n)}) \leq \frac{a}{1 - b} r_{1} ({\bar{x}}_{1}, x_{1}^{(n)}) + \frac{b e_{2}}{1 - b} + \frac{d_{2}}{1 - b}

inequality which associated with the inequality of ( 21 ) gives us

(23)

r_{1} ({\bar{x}}_{1}, x_{1}^{(n + 1)}) \leq \frac{b (a e_{1} + b e_{2}) + a (1 - b) e_{1}}{(1 - a) (1 - b) - a b} + \frac{e_{1}}{2}

∎

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Résumé

In this work, bounds are provided for the errors committed in the numerical resolution using the Gauss-Seidel method - of a system of two equations with two unknowns in metric spaces.

And $(X_{i}, r_{i}), i = 1, 2,$ are two complete metric spaces and $F_{1} : X_{1} \times X_{2} \to X_{1}$ And $F_{2} : X_{1} \times X_{2} \to X_{2}$ two applications, then we apply for the resolution of the system $x_{1} = F_{1} (x_{1}, x_{2}), x_{2} = F_{2} (x_{1}, x_{2}),$ the Gauss-Seidel method and sufficient conditions are given for the convergence of the process used.

We designate by $D_{1} \subset X_{1}$ And $D_{2} \subset X_{2}$ two bounded sets of $X_{1}$ And $X_{2}$ and then we consider two operators $F_{1}^{*} : D_{1} \times D_{2} \to X_{1} F_{2}^{*} : D_{1} \times D_{2} \to X_{2}$ which check against $F_{1}$ And $F_{2}$ the conditions: $r_{1} (F_{1} (x_{1}, x_{2}), F_{1}^{*} (x_{1}, x_{2})) \leq e_{1}, r_{2} (F_{2} (x_{1}, x_{2}), F_{2}^{*} (x_{1}, x_{2})) \leq e_{2}$ for each $(x_{1}, x_{2}) \in D_{1} \times D_{2}$ . Under these conditions, the Gauss-Seidel method is applied to the system for the resolution of the initial system. $x_{1} = F_{1}^{*} (x_{1}, x_{2}), x_{2} = F_{2}^{*} (x_{1}, x_{2})$ . Under these conditions, we give delimitations of the distance between the solution of the system and the approximate solution.

Ion Pavaloiu

Institute of Mathematics

Post Office 1

C.P. 68

3400 Cluj-Napoca

Romania

This Note is in final form and no version of it is or will be submitted for publication elsewhere.

1988

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	$r_{1} (F_{1} (x_{1}, {and}_{1}), F_{1} (x_{2}, {and}_{2}))$	$\leq a r_{1} (x_{1}, x_{2}) + b r_{2} (x_{2}, {and}_{2})$
	$r_{2} (F_{2} (x_{1}, {and}_{1}), F_{2} (x_{2}, {and}_{2}))$	$\leq a r_{1} (x_{1}, x_{2}) + b r_{2} (x_{2}, {and}_{2})$

	$r_{1} ({\bar{x}}_{1}, x_{1}^{(n + 1)})$	$\leq \frac{b (a e_{1} + b e_{2}) + a e_{1} (1 - b)}{(1 - a) (1 - b) - a b} + \frac{e_{1}}{2}$
	$r_{2} ({\bar{x}}_{2}, x_{2}^{(n + 1)})$	$\leq \frac{a (a e_{1} + b e_{2}) + b e_{2} (1 - a)}{(1 - a) (1 - b) - a b} + \frac{e_{2}}{2}$

Error estimations in the numerical solving of the systems of equations

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Error estimations in the numerical solving of the systems of equations

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