a third-order newton-type method to solve systems of nonlinear equations - m.t. darvishi, a. barati

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A third-order Newton-type method to

solve systems of nonlinear equations

ARTICLE in APPLIED MATHEMATICS AND COMPUTATION · APRIL 2007

Impact Factor: 1.55 · DOI: 10.1016/j.amc.2006.08.080 · Source: DBLP

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2 AUTHORS:

M. T. Darvishi

Razi University

85 PUBLICATIONS 917 CITATIONS

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Ali Barati

Razi University

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A third-order Newton-type method to solve systemsof nonlinear equations

M.T. Darvishi *, A. Barati

Department of Mathematics, Razi University, Kermanshah 67149, Iran

Abstract

In this paper, we present a third-order Newton-type method to solve systems of nonlinear equations. In the first we

present theoretical preliminaries of the method. Secondly, we solve some systems of nonlinear equations. All test problems

show the third-order convergence of our method.

2006 Elsevier Inc. All rights reserved.

Keywords: Systems of nonlinear equations; Newton-type method; Third-order convergence

1. Introduction

In recent papers [4,5] a new family of third-order convergence methods have been obtained, by using

an integral interpolation of Newton’s method to solve equation f (x) = 0. There has been another approach

based on the Adomian decomposition method on developing iterative method to solve the equation f (x) =

0 (see [3]).

Recently, there has been some progress on Newton-type methods with cubic convergence that do not

require the computation of second derivatives [6]. Chun [2] by improving Newton–Raphson method presented

a new iterative method to solve nonlinear equations. His work is based on modification of the Abbasbandy’s

proposal [1] on improving the order of accuracy of Newton–Raphson method. Frontini and Sormani [4] pre-

sented a third-order iterative method for solving systems of nonlinear equations.

In this paper, we construct a high order iterative method based on Adomian decomposition method tosolve the systems of nonlinear equations. This paper is organized as follows: In the next section we introduce

an iterative method based on Adomian decomposition method to solve f (x) = 0, this method introduced by

Chun [2]. In Section 3 we extend the Chun’s method to solve systems of nonlinear equations. In that section

we state and prove a theorem that shows the cubic convergence of the method. Numerical results are in Sec-

tion 4. Finally, the paper is concluded is Section 5.

0096-3003/$ - see front matter 2006 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2006.08.080

* Corresponding author.

E-mail address: [email protected] (M.T. Darvishi).

Applied Mathematics and Computation 187 (2007) 630–635

www.elsevier.com/locate/amc

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2. Description of an iterative method

Consider the nonlinear equation

f ð xÞ ¼ 0; ð1Þ

we assume that f (x) has a simple root a and c is an initial guess sufficiently close to a. Let us convert the non-

linear equation (1) into the following coupled system:

f ðcÞ þ f 0ðcÞð x cÞ þ g ð xÞ ¼ 0; ð2Þ

g ð xÞ ¼ f ð xÞ f ðcÞ f 0ðcÞð x cÞ: ð3Þ

Eq. (2) can be rewritten in the following form:

x ¼ c þ N ð xÞ; ð4Þ

where c ¼ c f ðcÞ f 0ðcÞ

and N ð xÞ ¼ g ð xÞ f 0ðcÞ

is a nonlinear function.

The Adomian decomposition method looks for a solution having the series form

x ¼ X1

n¼0

xn ð5Þ

and the nonlinear function is decomposed as

N ð xÞ ¼X1n¼0

An; ð6Þ

where the Ans are functions called the Adomian’s polynomials depending on x0, x1, . . . , xn given by

An ¼ 1

n!

dn

dkn N

X1i¼0

ki xi

!" #k¼0

; n ¼ 0; 1; . . .

the first few polynomials are given by

A0 ¼ N ð x0Þ;

A1 ¼ x1 N 0ð x0Þ;

A2 ¼ x2 N 0ð x0Þ þ

1

2 x21 N

00ð x0Þ:

Upon substituting (5) and (6) into (4) yieldsX1n¼0

xn ¼ c þX1n¼0

An: ð7Þ

It follows from (7) that

x0 ¼ c;

x1 ¼ A0;

xnþ1 ¼ An; n ¼ 0; 1; . . .

An elementary calculation shows that

A0 ¼ N ð x0Þ ¼ g ð x0Þ

f 0ðcÞ ¼

f ð x0Þ

f 0ðcÞ ;

N 0ð x0Þ ¼ 1 f ð x0Þ

f 0ðcÞ ;

A1 ¼ x1 N 0ð x0Þ ¼ A0 N

0ð x0Þ ¼ f ð x0Þ

f 0

ðcÞ

þ f ð x0Þ f

0ð x0Þ

ð f 0ðcÞÞ2

:

M.T. Darvishi, A. Barati / Applied Mathematics and Computation 187 (2007) 630–635 631

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Note that x is approximated by

X m ¼ x0 þ x1 þ þ xm ¼ x0 þ A0 þ A1 þ þ Am1;

where limm!1 X m = x.

For m = 0

x X 0 ¼ x0 ¼ c ¼ c f

ðc

Þ f 0ðcÞ;

which yields the Newton method

xnþ1 ¼ xn f ð xnÞ

f 0ð xnÞ:

For m = 1

x X 1 ¼ x0 þ x1 ¼ c þ A0 ¼ c f ðcÞ

f 0ðcÞ

f ð x0Þ

f 0ðcÞ ;

which produces the following iteration scheme


f 0ð xnÞ f ð x

nþ1Þ

f 0ð xnÞ ; ð8Þ

where


f 0ð xnÞ:

3. The n-dimensional case

We know Newton iterative method for the nonlinear system F(x) = 0 where F : X Rn ! Rn is defined by

xnþ1 ¼ xn F0ð xnÞ

1Fð xnÞ; ð9Þ

where F 0(xn) is the Jacobian matrix in point xn.

We rewrite Eq. (8) to solve the nonlinear system F(x) = 0, this produces the following iteration scheme:


1ðFð xnÞ þ Fð xnþ1ÞÞ; ð10Þ

where


1Fð xnÞ:

We call iteration scheme (10) as the modified Newton method (mNm) to solve systems of nonlinear equations.

Theorem 1. Let F : X Rn ! Rn, is k-time Fré chet differentiable in a convex set X containing the root a of F(x) = 0. The modified Newton’s method (mNm) (10) has order of convergence three.

Proof. For any x, xn

2 X, we write the Taylor’s expansion for F:

Fð xÞ ¼ Fð xnÞ þ F0ð xnÞð x xnÞ þ

1

2!F00ð xnÞð x xnÞ

2 þ 1

3!Fð3Þð xnÞð x xnÞ

3 þ þ 1

k !Fðk Þð xnÞða xnÞ

k þ

If a be root of system F(x) = 0, we have:

FðaÞ ¼ Fð xnÞ þ F0ð xnÞða xnÞ þ

1

2!F00ð xnÞða xnÞ

2 þ 1

3!Fð3Þð xnÞða xnÞ

3 þ þ 1

k !Fðk Þð xnÞða xnÞ

k þ

Defining en

= xn a we have:

0 ¼ Fð xnÞ F0ð xnÞen þ

1

2!F00ð xnÞe

2n

1

3!Fð3Þð xnÞe

3n þ þ ð1Þ

k 1

k !Fðk Þð xnÞe

k n þ

632 M.T. Darvishi, A. Barati / Applied Mathematics and Computation 187 (2007) 630–635

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hence

Fð xnÞ ¼ F0ð xnÞen

1

2!F00ð xnÞe

2n þ

1

3!Fð3Þð xnÞe

3n þ ð1Þ

k 1

k !Fðk Þð xnÞe

k n þ

For k = 3

Fð xnÞ ¼ F0ð xnÞen 12!

F00ð xnÞe2n þ 13!Fð3Þð xnÞe3n þ Oðke4nkÞ ð11Þ

if we multiply Eq. (11) from left by F0

(xn)1 then we have:

F0ð xnÞ1

Fð xnÞ ¼ F0ð xnÞ

1F0ð xnÞen

1

2!F0ð xnÞ

1F00ð xnÞe

2n þ

1

3!F0ð xnÞ

1Fð3Þð xnÞe

3n þ Oðke

4nkÞ

or

F0ð xnÞ1

Fð xnÞ ¼ en 1

2!F0ð xnÞ

1F00ð xnÞe

2n þ

1

3!F0ð xnÞ

1Fð3Þð xnÞe

3n þ Oðke

4nkÞ ð12Þ

from iterative scheme (10) we have:

xnþ1 xn ¼ F0

ð xnÞ

1

ðFð xnÞ þ Fð xn F0

ð xnÞ

1

Fð xnÞÞÞ;

then

enþ1 ¼ en F0ð xnÞ

1Fð xnÞ F

0ð xnÞ1

Fð xn F0ð xnÞ

1Fð xnÞÞ: ð13Þ

By substituting (12) into (13) we obtain:

enþ1 ¼ en en 1

2!F0ð xnÞ

1F00ð xnÞe

2n þ

1

3!F0ð xnÞ

1Fð3Þð xnÞe

3n þ Oðke

4nkÞ

F0ð xnÞ

1Fð xn F

0ð xnÞ1

Fð xnÞÞ:

By Taylor’s expansion for F(xn F

0

(xn)1F(x

n)) around x

n we have:

enþ1 ¼ 1

2!

F0ð xnÞ1

F00ð xnÞe2n

1

3!

F0ð xnÞ1

Fð3Þð xnÞe3n

þ Oðke4nkÞ F0ð xnÞ

1Fð xnÞ F

0ð xnÞF0ð xnÞ

1Fð xnÞ

þ 1

2!F00ð xnÞðF

0ð xnÞ1

Fð xnÞÞ2

1

3!Fð3Þð xnÞðF

0ð xnÞ1

Fð xnÞÞ3 þ Oðke4nkÞ

enþ1 ¼ 1

2!F0ð xnÞ

1F00ð xnÞe

2n

1

3!F0ð xnÞ

1Fð3Þð xnÞe

3n

1

2!F0ð xnÞ

1F00ð xnÞ en

1

2!F0ð xnÞ

1F00ð xnÞe

2n þ Oðke

3nkÞ

2

þ 1

3!

F0ð xnÞ1

Fð3Þð xnÞ en 1

2!

F0ð xnÞ1

F00ð xnÞe2n þ Oðke

3nkÞ

3

þ Oðke4nkÞ

after some manipulations we obtain:

enþ1 ¼ 1

2½F0ð xnÞ

1F00ð xnÞ

2e3n þ Oðke

4nkÞ: ð14Þ

This shows the third-order convergence of the method. Hence, the proof is completed. h

4. Numerical examples

In this section we solve three systems of nonlinear equations. The following tables show the number of iter-

ations to receive the required solution, as these tables show the order of convergence is three. For all test prob-

lems the stop criteria is kF(xn)k < 1015.


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Example 1. Consider the following system of nonlinear equations

x21 þ x22 þ x

23 ¼ 1;

2 x21 þ x22 4 x3 ¼ 0;

3 x2

1

4 x22

þ x23

¼ 0:

We solve this system by mNm using initial approximation x0 = (0.5,0.5,0.5)T. Table 1 shows the values of x1,

x2 and x3. We can see the third-order convergence of the approximations.

Example 2. The second test problem is taken as

x21 þ 3 log x1 x22 ¼ 0;

2 x21 x1 x2 5 x1 þ 1 ¼ 0:

Its initial approximation is x0 = (3.4,2.2)T. Table 2 shows xi s approximations. We obtain the required solu-

tions only after four iterations.

Example 3. The last example has the following form:

x1 þ 2 x2 3 ¼ 0

2 x21 þ x22 5 ¼ 0:

The initial approximation of the solution is x0 = (1.5,1.0)T. Table 3 shows the approximation of the solution.

As we can see from this table, the convergency order is three.

Table 2

Approximations of x i s for Example 2

n x1 x2

1 3.48730136242187 2.26152939389406

2 3.48744278764273 2.26162863055323

3 3.48744278764295 2.26162863055359

4 3.48744278764295 2.26162863055359

Table 3


n x1 x2

1 1.48750000000000 0.75625000000000

2 1.48803387165546 0.75598306417227

3 1.48803387171258 0.75598306414371

4 1.48803387171258 0.75598306414371

Table 1


n x1 x2 x3

1 0.67625000000000 0.62300000000000 0.34325000000000

2 0.69827680505888 0.62852407959171 0.34256418976030

3 0.69828860997151 0.62852429796021 0.34256418968957

4 0.69828860997151 0.62852429796021 0.34256418968957

634 M.T. Darvishi, A. Barati / Applied Mathematics and Computation 187 (2007) 630–635

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7/7

5. Conclusion

In this paper, we introduced a cubic convergence iterative method to solve systems of nonlinear equations.

This method has simple implementation. As test examples show there is a fast convergence to receive the solu-

tion with a required accuracy.

References

[1] S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Appl.

Math. Comput. 145 (2003) 887–893.

[2] Ch. Chun, A new iterative method for solving nonlinear equations, Appl. Math. Comput. 178 (2) (2006) 415–422.

[3] Ch. Chun, Iterative methods improving Newton’s method by the decomposition method, Comput. Math. Appl. 50 (2005) 1559–1568.

[4] M. Frontini, E. Sormani, Third-order methods from quadrature formulae for solving systems of nonlinear equations, Appl. Math.

Comput. 149 (2004) 771–782.

[5] M. Frontini, E. Sormani, Some variants of Newton’s method with third-order convergence, Appl. Math. Comput. 140 (2003) 419–426.

[6] H.H.H. Homeier, On Newton-type methods with cubic convergence, J. Comput. Appl. Math. 176 (2005) 425–432.


a third-order newton-type method to solve systems of nonlinear equations - m.t. darvishi, a. barati

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