AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2016.76043AM-64928ArticlesPhysics&Mathematics A New Approach for Solving Nonlinear Equations by Using of Integer Nonlinear Programming rminGhane-Kanafi1*SohrabKordrostami1*Department of Mathematics, Lahijan Branch, Islamic Azad University, Lahijan, Iran* E-mail:arminghane@liau.ac.ir(RG);krostami@liau.ac.ir(SK);24032016070647348117 January 2016accepted 21 March 24 March 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

One of the most important issues in numerical calculations is finding simple roots of nonlinear equations. This topic is one of the oldest challenges in science and engineering. Many important problems in engineering, to achieve the result need to solve a nonlinear equation. Thus, the formulation of a recursive relationship with high order of convergence and low time complexity is very important. This paper provides a modification to the Weerakoon-Fernando and Parhi-Gupta methods. It is shown that, in each iterate, the improved method requires three evaluations of the function and two evaluations of the first derivatives of function. The proposed with the Kou et al., Neta, Parhi-Gupta, Thukral and Mir et al. methods have been applied to a collection of 12 test problem. The results show that proposed approach significantly reduces the number of function calls when compared to the above methods. The numerical examples show that the proposed method is more efficiency than other methods in this class, such as sixth-order method of Parhi-Gupta or eighth-order method of Mir et al. and Thukral. We show that the order of convergence the proposed method is 9 and also, the modified method has the efficiency of .

Newton Method Nonlinear Equations Convergence Theorem Efficiency Index
1. Introduction

In the real world, many of the complex problems after simplification lead to solving nonlinear problems. Find an approximation of the simple roots of the equations is one of the important problems on this issue. The rapid development of technology has led to different of algorithms. Over time, many algorithms have been developed. In this state, one of the ways for comparison of different algorithms is finding of complexity of time and index efficiency of algorithms. MAPLE software is one of the powerful algebraic systems from Maplesoft company, such that in this article it has been used for the calculation. The boundary value problems appearing in kinetic theory of gases, elasticity and other applied areas are reduced to solve these equations. Many optimization problems also lead such equations. Hence, one of the most important problems in numerical analysis is to find a simple root a of a nonlinear equation, where for an open interval is a scalar function. In this study, in order to find a, we should start with an initial approximation which is near to the root and generates successive iterates converging to simple root a of nonlinear equation. In all iteration, the improved method requires three evaluations of the function and two evaluations of the first order derivatives of function. Therefore, the modified method has the efficiency index. The numerical examples show that, the proposed method has more efficient with respect to the Newton method and other methods in this class. The effectiveness of the modified ninth-order method will be examined by approximation the simple root of a given non-linear equation. The suggested method is comparable to the sixth-order methods   ; also the eighth-order methods  and  .

In the reminder, we proceed as follows: In Section 2, we recall the basic concepts. The proposed method is described in Section 3. In Section 4, the convergence analysis is carried out to establish the ninth-order of convergence of our method. In Section 5, as is shown in the numerical examples, this method is more efficient than Newton method and other methods of lower or same order. We conclude with some remarks on the presented approaches in Section 6.

2. Several Basic Definitions

Our goal is to find the value of x that satisfies the following equation.

where is a nonlinear equation. The value of x that satisfies (1) is called a root of and denoted by a. Therefore, the procedure used of to find x is called root-finding. Let a is a simple root of Equation (1) and is a real sequence.

Definition 1. See  : The sequence is said to converge to a if

Furthermore, if there exists positive constant c and p such that:

we say that converges to a of order p. Larger values of p correspond to faster convergence. Let be error in the nth iterate of the method which produces the sequence. The relation

is called the error equation. The value of p is called the order of convergence of method, see  .

Definition 2. Let a be a root of the function f and suppose that, and are three consecutive iterations closer to the root a. The the computational order of convergence can be approximated using the formula:

3. New Proposed Scheme

The new method is based on  method. With a simple manipulation, and a new approach to get the following equations.

and

This is four-step method. It is not necessary to compute the first-order derivative at the point since a good approximation can be obtained. In order to approximate use the linear interpolation on two points and, so we have:

Therefore,

Now using Equations (2), we have:

Substituting the relation of (6) into the relation (3), in this case, we obtain the following formula:

where,

Obviously this method requires evaluations of three function f and two derivatives.

4. Convergence Analysis

To determine order of convergence of proposed method, we must be solving integer nonlinear programming as follow:

where C is a special coefficient of. This is equivalent to the bellow theorem, i.e. we show that the convergence of the proposed method is of the order of 9.

Theorem 1. Let has continuous derivative function and is a simple root of f. If the initial point is sufficiently close1 to a, then the method defined by (2) converges to a in the ninth-order. Furthermore, the error in the method given by (2) satisfies the equation:

where and for.

Proof. Let be the error term in the iterate. Using Taylor expansion, we have:

and

Quotient relations (9) and (10), gives the following results:

Thus we have

Taylor expansion of the function around the point a to get the following result (i.e (11)):

Substituting (9), (10) and (11) into the zn section of the Equation (2), we have:

Furthermore, the Taylor expansion of about a is

Since from (10), (12) and (13) we get:

Again, using the Taylor expansion of function about the point a, in this case we have:

Taylor expansion of the function around the point a to get the following result (i.e (16)):

In this case, using the above result (i.e (15), (16) and (14)) and corresponding to the relation (3), we get:

Therefore, we have:

Thus, the ninth order of convergence of the method is established.

Numerical Examples

In order to demonstrate the performance, accuracy and effectiveness of the proposed ninth-order method, we take 12 special nonlinear equation test problems from   and  . We compare the proposed method with Wang-Liu’s third-order method  , Weerakoon-Fernando and Parhi-Gupta’s sixth-order methods   and Kou et al. and Neta’s eight-order methods as  and  , respectively. The computing results displayed in Tables 1-5. In every problem we try to seek an approximation of the root a of Equation (1) after n times

Comparison of result of proposed method (PM) with Kou and Li (KL) method
FunctionsnRun timeNFE
PMKLPMKLPMKL
1290.2030.0311036
2280.1560.0321032
13160.3900.4681564
3390.4210.3431536
0.511DIV0.187-55-
1.53320.1250.04715128
2.5390.2650.0471536
3.53130.2490.0161552
3.253680.4060.31215272
3.54DIV0.484-20-
1.53210.1560.0471584
24DIV0.234-20-
3240.2490.1251016
3.551630.2030.12525652
4.574330.1560.343351732
13DIV0.296-15-
33100.4680.3121540
1.2260.5150.1561024
−0.8510DIV0.827-50-
03DIV0.577-15-
0.84230.1710.0312092
05510.2810.01625204
Comparison of result of proposed method (PM) with Parhi and Gupta (PG) method
FunctionsnRun timeNFE
PMPGPMPGPMPG
1220.2030.031108
2220.1560.093108
1330.3900.2181512
3330.4210.3121512
0.511110.1870.1245544
1.5330.1250.0311512
2.5330.2650.0941512
3.5330.2490.0781512
3.25330.4060.2501512
3.5440.4840.2652016
1.5330.1560.0471512
2440.2340.1722016
3220.2490.141108
3.5530.2030.1252512
4.5770.1560.1403528
1330.2960.0631512
3330.4680.3121512
1.2220.5150.249108
−0.8510150.8270.6565060
0330.5770.2501512
0.8440.1710.0782016
0550.2810.0622520
Comparison of result of proposed method (PM) with Neta (NM) method
FunctionsnRun timeNFE
PMNMPMNMPMNM
1220.2030.109108
2220.1560.156108
1330.3900.3121512
3330.4210.3741512
0.511DIV0.187-55-
1.5330.1250.0941512
2.5320.2650.124158
3.5330.2490.0931512
3.25330.4060.3901512
3.5450.4840.4372020
1.5330.1560.1411512
24DIV0.234-20-
3220.2490.141108
3.5550.2030.0932520
4.5770.1560.1253528
1330.2960.1251512
3330.4680.3901512
1.2220.5150.359108
−0.851030.8270.4215012
0320.5770.437158
0.8430.1710.1412012
0540.2810.1562516
Comparison of result of proposed method (PM) with Thaukral (TM) method
FunctionsnRun timeNFE
PMTMPMTMPMTM
1230.2030.1251012
2230.1560.1411012
13DIV0.390-15-
3330.4210.2811512
0.511DIV0.187-55-
1.5340.1250.1401516
2.5330.2650.1091512
3.53DIV0.249-15-
3.25340.4060.3741512
3.5460.4840.4842024
1.531220.1560.65515488
24DIV0.234-20-
3220.2490.219108
3.5570.2030.1562528
4.57120.1560.2183548
13DIV0.296-15-
3330.4680.5601512
1.2230.5150.4211012
−0.8510DIV0.827-50-
0330.5770.3751512
0.84DIV0.171-20-
0570.2810.1252528
Comparison of result of proposed method (PM) with Mir (MM) method
FunctionsnRun timeNFE
PMMMPMMMPMMM
1220.2030.1251010
2220.1560.1401010
1320.3900.2971510
3320.4210.2181510
0.511DIV0.187-55-
1.5330.1250.1091515
2.5320.2650.0941510
3.53DIV0.249-15-
3.25330.4060.4371515
3.5440.4840.4372020
1.5320.1560.1881510
24DIV0.234-20-
32DIV0.249-10-
3.5550.2030.1252525
4.5780.1560.2033540
1320.2960.1091510
3330.4680.2961515
1.2220.5150.4841010
−0.8510DIV0.827-50-
03DIV0.577-15-
0.8430.1710.2182015
0550.2810.1722520

iteration. In this paper, the stoping criterion is. the Run time and the Number of function evaluations (NFE) are also given in Tables 1-5. “DIV” in the tables implies that the corresponding method is diverges. Furthermore, a comparison of the rate of convergence of the proposed method and Kou-Li method  for function at point is shown in Figure 1. The comparison is clearly marked on Figure 1. It should be noted that, Numerical computations reported here have been carried out in the MAPLE 18 environment. The results show that the speed of convergence in all methods discussed in this article, are depends on proper selection of the initial point. For example, in Table 1 for, choose an initial point is leading to the divergence of Kou et al. method, whereas the choose the initial point in the same method is leading to the coverage to the simple root a, see Table 1. In all examples, it is evident that the proposed approach, for any initial point is coverage to simple root of a.

The test functions are listed as follows: