THE CONVERGENCE OF THE EULER’S METHOD

. In this article we study the Euler’s iterative method. For this method we give a global theorem of convergence. In the last section of the paper we give a numerical example which illustrates the result exposed in this work.


INTRODUCTION
We consider the problem of finding a zero of the equation by means of the one-point iteration method (2) x n+1 = g(x n ), x 0 ∈ [a, b], n = 0, 1, ..., n ∈ N, where x 0 is the starting value and g is a function of form g(x) = x + ϕ(x).

THEOREMS OF CONVERGENCE
Next we will study sufficient conditions in order that the sequence {x n } n≥0 generated through (3) would be convergent, and if In order to prove the convergence of the method of form (3), we would use the next result.
Based on Theorem 1, in our next result we would analyze the convergence of sequence {x n } n≥0 given by (3).
Theorem 2. If the function f , the real number δ > 0 and x 0 ∈ ∆, where We'll show that the elements of the sequence {x n } n≥0 generated by (3) are in ∆.
By conditions b), c) and f) we have Applying the Taylor expansion of function f around x 0 and taking into account that , ∀x ∈ ∆, and ϕ(x) is verifying the parable From all that we have proved above, by using the induction, it results that the property iii) holds for every n ∈ N, Analogously, from b), c) and ( 4) we can prove the following relation (5) From ( 5), e) and f) we get the relation ii) For the convergence of the sequence given by (3) we shall use the Cauchy's theorem.By relation ( 5) and e) we deduce that Because µ 0 < 1, it results that the sequence {x n } n≥0 is fundamental, so according to the Cauchy's theorem, it is convergent.
If x * = lim n→∞ x n , for p → ∞, from the inequality (7) we obtain the relation iv) , n = 0, 1, 2, ..., n ∈ N. We show now that the relations i) hold, that is, x * is a root of equation ( 1) and x * ∈ ∆.
From the continuity of function f and from (4) for n → ∞, it results From f) and the inequality (8) for n = 0, we obtain It is evidently that all the assumptions of Theorem 1 are verified for s = 3, γ = 0 and η = 2β.

NUMERICAL EXAMPLE
We shall present a numerical example, which illustrates the result exposed in Theorem 2.
Example 3. We used the following test functions and display the zeros x * found.
For the derivatives of order 1, 2 and 3 of f i , i = 1, 2, 3, we have the relations from which we get β = 0.536702 and M = 422.22; from which we get β = 0.0481928 and M = 6; from which we get β = 0.333503 and M = 114.264.
In the Table 1 are listed the values for x 0 , M , β, λ, µ 0 , δ and 2βµ 0 λ(1−µ 0 ) , for each test functions.The implementations were done in Mathematica 7.0 with double precision.From the Table 1 we can conclude that all the assumptions a)-f) of Theorem 2 are verified.
In the next Table 2 we can observe that, the convergence is faster and the method (3)

( 1 )
f (x) = 0, where f : [a, b] ⊂ R → R is an analytic function with simple roots.This zero can be determined as a fixed point of some iteration functions g : [a, b] → [a, b],