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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 .

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

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.

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

where

Definition 1. See [

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

we say that

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

The new method is based on [

and

This is four-step method. It is not necessary to compute the first-order derivative at the point

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

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

where C is a special coefficient of

Theorem 1. Let

where

Proof. Let

and

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

Thus we have

Taylor expansion of the function

Substituting (9), (10) and (11) into the z_{n} section of the Equation (2), we have:

Furthermore, the Taylor expansion of

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

Again, using the Taylor expansion of function

Taylor expansion of the function

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 ExamplesIn order to demonstrate the performance, accuracy and effectiveness of the proposed ninth-order method, we take 12 special nonlinear equation test problems from [

Functions | n | Run time | NFE | ||||||
---|---|---|---|---|---|---|---|---|---|

PM | KL | PM | KL | PM | KL | ||||

1 | 2 | 9 | 0.203 | 0.031 | 10 | 36 | |||

2 | 2 | 8 | 0.156 | 0.032 | 10 | 32 | |||

1 | 3 | 16 | 0.390 | 0.468 | 15 | 64 | |||

3 | 3 | 9 | 0.421 | 0.343 | 15 | 36 | |||

0.5 | 11 | DIV | 0.187 | - | 55 | - | |||

1.5 | 3 | 32 | 0.125 | 0.047 | 15 | 128 | |||

2.5 | 3 | 9 | 0.265 | 0.047 | 15 | 36 | |||

3.5 | 3 | 13 | 0.249 | 0.016 | 15 | 52 | |||

3.25 | 3 | 68 | 0.406 | 0.312 | 15 | 272 | |||

3.5 | 4 | DIV | 0.484 | - | 20 | - | |||

1.5 | 3 | 21 | 0.156 | 0.047 | 15 | 84 | |||

2 | 4 | DIV | 0.234 | - | 20 | - | |||

3 | 2 | 4 | 0.249 | 0.125 | 10 | 16 | |||

3.5 | 5 | 163 | 0.203 | 0.125 | 25 | 652 | |||

4.5 | 7 | 433 | 0.156 | 0.343 | 35 | 1732 | |||

1 | 3 | DIV | 0.296 | - | 15 | - | |||

3 | 3 | 10 | 0.468 | 0.312 | 15 | 40 | |||

1.2 | 2 | 6 | 0.515 | 0.156 | 10 | 24 | |||

−0.85 | 10 | DIV | 0.827 | - | 50 | - | |||

0 | 3 | DIV | 0.577 | - | 15 | - | |||

0.8 | 4 | 23 | 0.171 | 0.031 | 20 | 92 | |||

0 | 5 | 51 | 0.281 | 0.016 | 25 | 204 |

Functions | n | Run time | NFE | ||||||
---|---|---|---|---|---|---|---|---|---|

PM | PG | PM | PG | PM | PG | ||||

1 | 2 | 2 | 0.203 | 0.031 | 10 | 8 | |||

2 | 2 | 2 | 0.156 | 0.093 | 10 | 8 | |||

1 | 3 | 3 | 0.390 | 0.218 | 15 | 12 | |||

3 | 3 | 3 | 0.421 | 0.312 | 15 | 12 | |||

0.5 | 11 | 11 | 0.187 | 0.124 | 55 | 44 | |||

1.5 | 3 | 3 | 0.125 | 0.031 | 15 | 12 | |||

2.5 | 3 | 3 | 0.265 | 0.094 | 15 | 12 | |||

3.5 | 3 | 3 | 0.249 | 0.078 | 15 | 12 | |||

3.25 | 3 | 3 | 0.406 | 0.250 | 15 | 12 | |||

3.5 | 4 | 4 | 0.484 | 0.265 | 20 | 16 | |||

1.5 | 3 | 3 | 0.156 | 0.047 | 15 | 12 | |||

2 | 4 | 4 | 0.234 | 0.172 | 20 | 16 | |||

3 | 2 | 2 | 0.249 | 0.141 | 10 | 8 | |||

3.5 | 5 | 3 | 0.203 | 0.125 | 25 | 12 | |||

4.5 | 7 | 7 | 0.156 | 0.140 | 35 | 28 | |||

1 | 3 | 3 | 0.296 | 0.063 | 15 | 12 | |||

3 | 3 | 3 | 0.468 | 0.312 | 15 | 12 | |||

1.2 | 2 | 2 | 0.515 | 0.249 | 10 | 8 | |||

−0.85 | 10 | 15 | 0.827 | 0.656 | 50 | 60 | |||

0 | 3 | 3 | 0.577 | 0.250 | 15 | 12 | |||

0.8 | 4 | 4 | 0.171 | 0.078 | 20 | 16 | |||

0 | 5 | 5 | 0.281 | 0.062 | 25 | 20 |

Functions | n | Run time | NFE | ||||||
---|---|---|---|---|---|---|---|---|---|

PM | NM | PM | NM | PM | NM | ||||

1 | 2 | 2 | 0.203 | 0.109 | 10 | 8 | |||

2 | 2 | 2 | 0.156 | 0.156 | 10 | 8 | |||

1 | 3 | 3 | 0.390 | 0.312 | 15 | 12 | |||

3 | 3 | 3 | 0.421 | 0.374 | 15 | 12 | |||

0.5 | 11 | DIV | 0.187 | - | 55 | - | |||

1.5 | 3 | 3 | 0.125 | 0.094 | 15 | 12 | |||

2.5 | 3 | 2 | 0.265 | 0.124 | 15 | 8 | |||

3.5 | 3 | 3 | 0.249 | 0.093 | 15 | 12 | |||

3.25 | 3 | 3 | 0.406 | 0.390 | 15 | 12 | |||

3.5 | 4 | 5 | 0.484 | 0.437 | 20 | 20 | |||

1.5 | 3 | 3 | 0.156 | 0.141 | 15 | 12 | |||

2 | 4 | DIV | 0.234 | - | 20 | - | |||

3 | 2 | 2 | 0.249 | 0.141 | 10 | 8 | |||

3.5 | 5 | 5 | 0.203 | 0.093 | 25 | 20 | |||

4.5 | 7 | 7 | 0.156 | 0.125 | 35 | 28 | |||

1 | 3 | 3 | 0.296 | 0.125 | 15 | 12 | |||

3 | 3 | 3 | 0.468 | 0.390 | 15 | 12 | |||

1.2 | 2 | 2 | 0.515 | 0.359 | 10 | 8 | |||

−0.85 | 10 | 3 | 0.827 | 0.421 | 50 | 12 | |||

0 | 3 | 2 | 0.577 | 0.437 | 15 | 8 | |||

0.8 | 4 | 3 | 0.171 | 0.141 | 20 | 12 | |||

0 | 5 | 4 | 0.281 | 0.156 | 25 | 16 |

Functions | n | Run time | NFE | ||||||
---|---|---|---|---|---|---|---|---|---|

PM | TM | PM | TM | PM | TM | ||||

1 | 2 | 3 | 0.203 | 0.125 | 10 | 12 | |||

2 | 2 | 3 | 0.156 | 0.141 | 10 | 12 | |||

1 | 3 | DIV | 0.390 | - | 15 | - | |||

3 | 3 | 3 | 0.421 | 0.281 | 15 | 12 | |||

0.5 | 11 | DIV | 0.187 | - | 55 | - | |||

1.5 | 3 | 4 | 0.125 | 0.140 | 15 | 16 | |||

2.5 | 3 | 3 | 0.265 | 0.109 | 15 | 12 | |||

3.5 | 3 | DIV | 0.249 | - | 15 | - | |||

3.25 | 3 | 4 | 0.406 | 0.374 | 15 | 12 | |||

3.5 | 4 | 6 | 0.484 | 0.484 | 20 | 24 | |||

1.5 | 3 | 122 | 0.156 | 0.655 | 15 | 488 | |||

2 | 4 | DIV | 0.234 | - | 20 | - | |||

3 | 2 | 2 | 0.249 | 0.219 | 10 | 8 | |||

3.5 | 5 | 7 | 0.203 | 0.156 | 25 | 28 | |||

4.5 | 7 | 12 | 0.156 | 0.218 | 35 | 48 | |||

1 | 3 | DIV | 0.296 | - | 15 | - | |||

3 | 3 | 3 | 0.468 | 0.560 | 15 | 12 | |||

1.2 | 2 | 3 | 0.515 | 0.421 | 10 | 12 | |||

−0.85 | 10 | DIV | 0.827 | - | 50 | - | |||

0 | 3 | 3 | 0.577 | 0.375 | 15 | 12 | |||

0.8 | 4 | DIV | 0.171 | - | 20 | - | |||

0 | 5 | 7 | 0.281 | 0.125 | 25 | 28 |

Functions | n | Run time | NFE | ||||||
---|---|---|---|---|---|---|---|---|---|

PM | MM | PM | MM | PM | MM | ||||

1 | 2 | 2 | 0.203 | 0.125 | 10 | 10 | |||

2 | 2 | 2 | 0.156 | 0.140 | 10 | 10 | |||

1 | 3 | 2 | 0.390 | 0.297 | 15 | 10 | |||

3 | 3 | 2 | 0.421 | 0.218 | 15 | 10 | |||

0.5 | 11 | DIV | 0.187 | - | 55 | - | |||

1.5 | 3 | 3 | 0.125 | 0.109 | 15 | 15 | |||

2.5 | 3 | 2 | 0.265 | 0.094 | 15 | 10 | |||

3.5 | 3 | DIV | 0.249 | - | 15 | - | |||

3.25 | 3 | 3 | 0.406 | 0.437 | 15 | 15 | |||

3.5 | 4 | 4 | 0.484 | 0.437 | 20 | 20 | |||

1.5 | 3 | 2 | 0.156 | 0.188 | 15 | 10 | |||

2 | 4 | DIV | 0.234 | - | 20 | - | |||

3 | 2 | DIV | 0.249 | - | 10 | - | |||

3.5 | 5 | 5 | 0.203 | 0.125 | 25 | 25 | |||

4.5 | 7 | 8 | 0.156 | 0.203 | 35 | 40 | |||

1 | 3 | 2 | 0.296 | 0.109 | 15 | 10 | |||

3 | 3 | 3 | 0.468 | 0.296 | 15 | 15 | |||

1.2 | 2 | 2 | 0.515 | 0.484 | 10 | 10 | |||

−0.85 | 10 | DIV | 0.827 | - | 50 | - | |||

0 | 3 | DIV | 0.577 | - | 15 | - | |||

0.8 | 4 | 3 | 0.171 | 0.218 | 20 | 15 | |||

0 | 5 | 5 | 0.281 | 0.172 | 25 | 20 |

iteration. In this paper, the stoping criterion is

The test functions are listed as follows:

One can easily see from Tables 1-5 that our method behaves either similarly or better then the compared methods. The results show that the new method has advantages over the Kou et al. [

In numerical analysis, many methods produce sequences of real numbers, for example the iterative schemes for solving

ArminGhane-Kanafi,SohrabKordrostami, (2016) A New Approach for Solving Nonlinear Equations by Using of Integer Nonlinear Programming. Applied Mathematics,07,473-481. doi: 10.4236/am.2016.76043