Journal of Applied Mathematics, Islamic AzadUniversity of Lahijan, Vol.7, No.4(27), Winter 2011, pp 17-24

Numerical Solution of FuzzyPolynomials by

Newton-Raphson Method

T. Allahviranloo*, S. Asari

Department of Mathematics, Science and Research Branch, Islamic AzadUniversity,Tehran, Iran

Received:3 June 2009

Accepted: 19 November 2009

Abstract

The main purpose of this paper is to find fuzzy root of fuzzy polynomials (if exists) by using Newton-Raphson method.The proposed numerical method has capability to solve fuzzy polynomials as well as algebric ones. For this purpose, by using parametric form of fuzzy coefficients of fuzzy polynomial and Newton-Rphson method we can find its fuzzy roots.

Finally, we illustrate our approach by numerical examples.

Keywords:Fuzzy Numbers, Fuzzy Polynomials, Newton-Raphson Method.

1 Introduction

Polynomials play a major role in various areas such as pure and applied mathematics, engineering and social sciences. In this paper we propose to find fuzzy roots of a fuzzy polynomial like wherefor (if exists). The set of all the fuzzy numbers is denoted by. The applications of fuzzy polynomials are considered by [1].

The concept of fuzzy numbers and arithmetic operation with this numbers were first introduced by Zadeh, [2]. Many researchers have studied on solution methods of fuzzy polynomials. Buckley and Eslami, [3] considered neural net solutions to fuzzy problems. Otadi [4, 5] proposed architecture of fuzzy neural networks with crisp weights for fuzzy input vector and fuzzy target. Abasbandy and Asady, [6] considered Newton’s method for solving fuzzy nonlinear equations. Linear and nonlinear fuzzy equations are solved by [5, 6, 7, 8, 9, 10]. In this paper we want to solve fuzzy polynomials with fuzzy coefficients and fuzzy variable, numerically. The fuzzy quantities are presented in parametric form. We first convert the polynomial fuzzy coefficients into parametric form then apply Newton method on each limit. Finally, in order to finding root, which is also fuzzy number/point, we numerically calculate level sets (i.e.,-cuts) of fuzzy coefficients on each limits. (Note that fuzzy polynomials may have no root).

In this paper the algorithm is illustrated by solving several numerical examples in last section.

2 Preliminaries

A popular fuzzy number is the triangular fuzzy numberfor with membership function

Its parametric form is .

An arbitrary fuzzy number is represented by an ordered pair of function which satisfies the following requirements.

is bounded left continuous no decreasing function over [0,1].

is bounded left continuous no increasing function over [0,1].

Let be the set of all upper semi continuous normal convex fuzzy numbers with bounded -level intervals. Meaning if then the -level set

For,and. Since level sets of fuzzy numbers become closed intervals, we denote as ,

Where and are the lower limit and the upper limit of the -level set, respectively.

2.1Interval arithmetic

Let A and B be fuzzy numbers with and. Then interval operations are as follow

(+)=

(-)=

In general case, we obtain a very complicated expression for -level sets of the product and division.

(.)=

m

(/)=[]

Definition 2.1 (see[11] ). Consider If there exists z such that, then is called the H-difference of and and is denoted by .

Definition 2.2 We say that is fuzzy-valued function if .

Definition 2.3 The supremum metric on is defined by

And is a complete metric space. is a real interval.

Definition 2.4 (see [12]). Let and . We say that if strongly generalized differentiable at (Bede-Gal differentiability), if there exists an element,such that

i) for all sufficiently small, , and the limits (in the metric ):

Or

ii) for all sufficiently small, ,and the following limits hold (in the metric ):

or

iii) for all sufficiently small, , and the following limits hold (in the metric ):

or

iv) for all sufficiently small, ,and the following limits hold (in the metric ):

Theorem 2.1(see [11]) letbe a function and setfor eachThen

1) If is differentiable in the first form (i), then and are differentiable functions and

2) If is differentiable in the second form (ii), then and are differentiable functions and

3Fuzzy polynomials

As mentioned before we are interested in finding solution of

(1)

where for i=1,2,…,n

let . is called fuzzy polynomial of degree at most n.

Therefore, Eq. (1) can be written such as . We use of parametric form of the , So, Eq. (1) can be in the form of following formulas

(2)

Assume and has at least two continuous derivatives for all in some intervals about the roots and respectively.

By using (2), has such form as follows

From (3), we obtain

(4)

Now, by applying Newton’s formula we have

(5)

=-[] / []

=

=- []

By applying Newton's method for parametric Eqs (3) with initial points and, we can obtain roots of and respectively.

Two sequences {} and {} must convergent to and respectively.

Where is the root of the polynomial?

3.1 Convergence of method

We are computing a sequence of iteration, and we would like to estimate their accuracy to know when to stop the iteration.

Assume r be the root of polynomial and be the computed solution of it. To estimate, note that, since, we have

For some between and. Therefore

, which is continuous and (6)

By considering (6) and Taylor’s theorem we have

Since, then

. (7)

From (6), (7), we obtain following result

. (8)

This formula says that the error in is proportional to square of the error in and method has at least second degree convergence. And if , then it will be second degree convergence.

That means, when the initial error is sufficiently small, the error in the succeeding iterates will decrease very rapidly.

Proof of Newton’s method convergence is [13].

Since both upper limit and lower limit of fuzzy polynomial satisfies in Newton’s method theorem we can say by starting from arbitrary initial point, sequence of iteration will be convergent to the root of polynomial (if exists).

4Numerical examples

Example 4.1

Consider the following fuzzy polynomial:

The exact solution is . The initial point is and n=5. we have fuzzy point 1. with e

Example 4.2

Consider the following fuzzy polynomial:

With the exact solution x=-1.

The initial point is and n=5. We have fuzzy point 1 with e.

4 Conclusions

In this paper, we considered to fined root of fuzzy polynomial and saw we can get the solution in initial steps with almost high accuracy

Refrences

[1] S. Abbasbandy, M. Amirfakhrnian, Numerical approximation of fuzzy function by fuzzy polynomials, Appl. Math. Comput.174(2006) 669-675

[2] L.A. Zadeh, Fuzzy sets, Inform. Control 8 (1986) 338-353

[3] J.J. Bucklyey, E. Eslami, Neural net solution of fuzzy problems: the quadratic equation, Fuzzu Sets Syst. 86 (1997) 289-298.

[4] S. Abbasbandy, M. Otadi, Numerical solution of fuzzy polynomials by fuzzy neural network, Appl. Math. Comput. 181 (2006) 1084-1089

[5] S. Abbasbandy, M. Otadi, Numerical solution of a system of fuzzy polynomials by fuzzy neural network, Inform. Sci. 178 (2008) 1948-1960

[6] S. Abbasbandy, B. Asady, Newton’s method for solving fuzzy nonlinear equations, Appl. Math. Comput. 159 (2004) 379-356.

[7] S. Abbasbandy, M. Alavi, A method for solving fuzzy linear systems, Iran. J. Fuzzy Setst. 2 (2005) 37-43

[8] S. Abbasbandy, R. Ezzati, Newton’s method for solving fuzzy nonlinear equations Appl. Math. Comput. 175 (2006) 1189-1199.

[9] B. Asady, S. Abbasbandy, M. Alavi, Fuzzy general linear systems, Appl. Math. Comput. 169 (2005) 34-40.

[10] J.J. Bucklyey, Y. Qu, Solving linear and Neural net solution of fuzzy problems: the quadratic equation, Fuzzy Sets Syst. 86 (1997) 289-298.

[11] T. Allahviranloo, N.A. Kiani, M. Barkhordari, toward the existence and uniqueness of solutions of second-order fuzzy differential equations, Information Sciences 179 (2009) 1207-1215.

[12] B. Bedde, ImreJ. Rudas, L. Attila, First order linear fuzzy differential equations under generalized differentiability, Information Science 177 (2007) 3627-3635.

[13] M. Karbasi, numerical analysis, published in YazdUniversity (1381).

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