# View of The convergence of the Euler's method

## Full text

(1)

Rev. Anal. Num´er. Th´eor. Approx., vol. 39 (2010) no. 1, pp. 87–92 ictp.acad.ro/jnaat

THE CONVERGENCE OF THE EULER’S METHOD

RALUCA ANAMARIA (POMIAN) SALAJAN

Abstract. 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.

MSC 2000. 37C25, 12D10.

Keywords. Euler’s method, fixed point, one-point iteration method.

1. INTRODUCTION

We consider the problem of finding a zero of the equation

(1) f(x) = 0,

where f : [a, b]⊂R→ Ris an analytic function with simple roots. This zero can be determined as a fixed point of some iteration functionsg: [a, b]→[a, b], by means of the one-point iteration method

(2) xn+1=g(xn), x0 ∈[a, b], n= 0,1, ..., n∈N, wherex0 is the starting value and g is a function of form

g(x) =x+ϕ(x).

In this article we analyze the Euler’s method for approximating the solution x ∈[a, b] of the equation (1). This method is defined by the relation

(3) xn+1=xn2f(xn)

f0(xn)+

[f0(xn)]2−2f(xn)f00(xn), x0∈[a, b], n≥0, n∈N. The Euler’s method has been rediscovered by several authors, see for exam- ple [1], [2], [5], [6], [7], [8], and references therein.

Secondary School Vasile Alecsandri, Strada P˘a¸sunii, Nr. 2A, Baia Mare, Romania, e- mail: [email protected]

(2)

2. THEOREMS OF CONVERGENCE

Next we will study sufficient conditions in order that the sequence{xn}n≥0 generated through (3) would be convergent, and if x = lim

n→∞xn, then f(x) = 0.

In order to prove the convergence of the method of form (3), we would use the next result.

Theorem1. ([3], [4])If we consider the functionf, the real number δ >0 and x0 ∈∆, where ∆ ={x ∈R:|x−x0| ≤ δ} ⊆[a, b], we could assure that the following relations hold

a) the function f is of class Cs(∆), s ≥ 2, s ∈ N and sup

x∈∆

f(s)(x) = M <∞;

b) we have the relation

s−1

P

i=0 1

i!f(i)(x)ϕi(x)

≤γ|f(x)|s for every x∈∆,where γ ∈R,γ ≥0;

c) the functionϕverifies the relation |ϕ(x)| ≤η|f(x)|,for everyx∈∆, where η∈R, η >0;

d) the numbers λ, η, M and δ verify the relations:

µ0 =λ|f(x0)|<1, where λ=

γ+M ηs!s s−11

and λ(1−µηµ0

0) ≤δ;

then the sequence {xn}n≥0 generated by (2)has the following properties:

i) it is convergent, and if x = lim

n→∞xn thenf(x) = 0 and x ∈∆;

ii) |xn+1−xn| ≤ ηµλsn0 , for anyn= 0,1, ..., n∈N; iii) |x−xn| ≤ ηµsn0

λ(1−µsn0 ), n= 0,1,2, ..., n∈N.

Proof. See [3], [4].

Based on Theorem 1, in our next result we would analyze the convergence of sequence {xn}n≥0 given by (3).

Theorem 2. If the function f, the real number δ >0 and x0 ∈∆,where

∆ ={x∈R:|x−x0| ≤δ} ⊆[a, b], verify the relations a) the functionf is of class C3(∆) and sup

x∈∆

|f000(x)|=M <∞;

b)

1 f0(x)

≤β for everyx∈∆, β ∈R, β >0;

c) f[f(x)f0(x)]00(x)2

not= Lf(x)≤ 12 for everyx∈∆;

d) λ= q8M

3! β3>0;

e) µ0 =λ|f(x0)|<1;

f) λ(1−µ2βµ0

0) ≤δ;

then the sequence{xn}n≥0 generated by(3)is convergent, and ifx = lim

n→∞xn, the next relations hold

(3)

i) f(x) = 0 and x∈∆;

ii) xn∈∆, n= 0,1,2..., n∈N; iii) |f(xn)| ≤ µ3

n 0

λ , n= 0,1,2, ..., n∈N;

iv) |x−xn| ≤ 2βµ3

n 0

λ(1−µ30n), n= 0,1,2, ..., n∈N. Proof. We consider the functionϕof form

ϕ(x) =− 2f(x)

f0(x)+

[f0(x)]2−2f(x)f00(x), x∈∆.

We’ll show that the elements of the sequence{xn}n≥0 generated by (3) are in

∆.

By conditions b), c) and f) we have

|x1−x0|=

f(x0) f0(x0)

2 1+

1−2Lf(x0)

≤2

f(x0) f0(x0)

≤2β|f(x0)|= 2λβ|f(xλ 0)|< λ(1−µ2βµ0

0) ≤δ⇒x1 ∈∆.

Applying the Taylor expansion of function f around x0 and taking into account that

ϕ(x) = −f

0(x)+

[f0(x)]2−2f(x)f00(x)

f00(x) =

=−f0(x)−

[f0(x)]2−2f(x)f00(x) f00(x) ,

∀x∈∆, andϕ(x) is verifying the parable

ax2+bx+c= 0, wherec=f(x), b=f0(x), a= f002(x), we get

|f(x1)| ≤

f(x1)− f(x0) +f0(x0)(x1−x0) +12f00(x0)(x1−x0)2 + +

f(x0) +f0(x0)(x1−x0) +12f00(x0)(x1−x0)2

f000(ξ)

3! (x1−x0)3 +

f(x0) +f0(x0)ϕ(x0) +12f00(x02(x0) ≤

M3! |x1−x0|38M β3! 3 |f(x0)|3= µλ30, ξ∈∆.

Because

1 f0(x1)

≤β, we have that

|x2−x1|=

f(x1) f0(x1)

2 1+

r

1−2f(x1)f00(x1) [f0(x1)]2

≤2

f(x1) f0(x1)

≤2β|f(x1)| ≤ 2βµλ30. From all that we have proved above, by using the induction, it results that the property iii) holds for every n∈N,

(4) |f(xn)| ≤ µ3

n 0

λ .

(4)

Analogously, from b), c) and (4) we can prove the following relation (5) |xn+1−xn|=

f(xn) f0(xn)

2 1+

r

1−2f(xn)f00(xn) [f0(xn)]2

2βµ3

n 0

λ , n= 0,1, ..., n∈N. From (5), e) and f) we get the relation ii)

|xn+1−x0| ≤

n

X

i=0

|xi+1−xi| ≤ (6)

n

X

i=0 2βµ30i

λ2βµλ0(1 +µ3−10302−1+...+µ30n−1)

< λ(1−µ2βµ0

0) ≤δ⇒xn+1 ∈∆, n= 0,1,2, ..., n∈N.

For the convergence of the sequence given by (3) we shall use the Cauchy’s theorem. By relation (5) and e) we deduce that

|xn+p−xn| ≤

n+p−1

X

i=n

|xi+1−xi| ≤

n+p−1

X

i=n 2βµ30i

λ <

(7)

< 2βµ3

n 0

λ (1 +µ30n+1−3n+...+µ30n+p−1−3n)

< 2βµ3

n 0

λ(1−µ30n), p∈N, n= 0,1,2, ..., n∈N.

Because µ0 < 1, it results that the sequence {xn}n≥0 is fundamental, so according to the Cauchy’s theorem, it is convergent.

If x = lim

n→∞xn, for p → ∞, from the inequality (7) we obtain the relation iv)

(8) |x−xn| ≤ 2βµ3

n 0

λ(1−µ30n), 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 functionf and from (4) for n→ ∞,it results 0≤ |f(x)| ≤ lim

n→∞

µ30n λ = 0, that is, f(x) = 0.

From f) and the inequality (8) forn= 0, we obtain

|x−x0| ≤ 2βµ300

λ(1−µ300 ) ≤δ,

so, x ∈∆.

It is evidently that all the assumptions of Theorem 1 are verified for s= 3, γ = 0 and η= 2β.

(5)

3. 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 zerosx found.

f1(x) = ln(x2−3), x =−2, x∈[−2.35,−1.9], f2(x) =x3−3x2−13x+ 15, x= 5, x∈[4.5,5.5], f3(x) =x5−1, x = 1, x∈[0.88,1.38].

For the derivatives of order 1, 2 and 3 offi, i= 1,2,3, we have the relations f10(x) = −3+x2x 2, f100(x) = (−3+x−4x22)2 +−3+x2 2,

f1000(x) = (−3+x16x32)3(−3+x12x2)2, from which we get β = 0.536702 andM = 422.22;

f20(x) = 3x2−6x−13, f200(x) = 6x−6, f2000(x) = 6, from which we get β = 0.0481928 andM = 6;

f30(x) = 5x4, f300(x) = 20x3, f3000(x) = 60x2, from which we get β = 0.333503 andM = 114.264.

In the Table 1 are listed the values for x0,M,β,λ,µ0, δ and λ(1−µ2βµ0

0), for each test functions.

i x0 M β λ µ0 δ λ(1−µ2βµ0

0) < δ 1 -1.989 422.22 0.53670 9.32908 0.41860 0.089 0.08284 2 4.875 6 0.04819 0.02992 0.11414 0.625 0.41503 3 1.035 114.264 0.33350 2.37724 0.33873 0.353 0.14372

Table 1.

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) converges atx.

i x0 x1 x2 x3=x

1 -1.989 -2.0000063482540900 -1.9999999999999990 -2 2 4.875 5.0000611233335144 4.999999999999993 5 3 1.027 0.999958832170524 1.000000000000139 1

Table 2.

(6)

REFERENCES

[1] Amat, S.,Busquier, S.andPlaza, S.,Review of some iterative root-finding methods from a dynamical point of view, Scientia, Series A: Mathematical Sciences,10, pp. 3–35, 2004.

[2] Osada, N.,A one parameter family of locally quartically convergent zero-finding methods, J. Comput. Appl. Math.,205, pp. 116–128, 2007.

[3] av˘aloiu, I.,Sur les proced´ees it´erative `a un order ´elev´e de convergence, Math´ematica, 12(35), no. 2, pp. 309–324, 1970.

[4] av˘aloiu, I. and Pop, N., Interpolare ¸si aplicat¸ii , Editura Risoprint, Cluj-Napoca, 2005.

[5] Petkovi´c, L. D., Petkovi´c, M. S.andZivikovi´˘ c, D.,Hansen-Patrick’s family is of Laguerre’s type, Novi Sad J. Math.,33, no. 1, pp. 109–115, 2003.

[6] Petrovi´c, M.,Triˇckovi´c, S.andHerceg, D.,Higher order Euler-like methods, Novi Sad J. Math.,28, no. 3, pp. 129–136, 1998.

[7] Varona, J.,Graphic and numerical comparison between iterative methods, The Mathe- matical Intelligencer,24, no. 1, pp. 37–46, 2002.

[8] Ye, X. and Li, C., Convergence of the family of the deformed Euler-Halley iterations under the H¨older condition of the second derivative, Journal of Computational and Applied Mathematics,194, pp. 294–308, 2006.

Received by the editors: October 23, 2010.

Updating...

## References

Related subjects :