_{1}

^{*}

In a recent paper, Noor and Khan [M. Aslam Noor, & W. A. Khan, (2012) New Iterative Methods for Solving Nonlinear Equation by Using Homotopy Perturbation Method, Applied Mathematics and Computation, 219, 3565-3574], suggested a fourth-order method for solving nonlinear equations. Per iteration in this method requires two evaluations of the function and two of its first derivatives; therefore, the efficiency index is 1.41421 as Newton’s method. In this paper, we modified this method and obtained a family of iterative methods for appropriate and suitable choice of the parameter. It should be noted that per iteration for the new methods requires two evaluations of the function and one evaluation of its first derivatives, so its efficiency index equals to 1.5874. Analysis of convergence shows that the methods are fourth-order. Several numerical examples are given to illustrate the performance of the presented methods.

In this paper, we consider iterative methods to find a simple root of a nonlinear equation f(x) = 0, where

This is an important and basic method [

where

It is clear that to implement (2), one has to evaluate the second derivative of the function. This can create some problems. In order to overcome this drawback, several techniques have been developed [

In a recent paper, Noor and Khan [

In this paper, we rederive the method in (4) to obtain a family of fourth-order method free from second derivative. Moreover, per iteration in these new methods requires two evaluations of the function and just one of its first derivatives.

The rest of this paper is organized as follows. The proposed methods are described in Section 2. In Section 3, the convergence analysis is carried out to establish the order of convergence. Finally, in Section 4, the methods are tested on some numerical examples and comparisons of the results of our methods.

The following approximations of

where

Using (6) in (4), we get a new family of iterative method

Essentially, we consider the convergence criteria of the new methods in (7) and (8), and this is the main motivation of our next result.

Theorem 1. Let

and

Proof.

Using Taylor expansion of

Furthermore, we have

and

Substituting (11) in

Expanding

Using Equations (9)-(13) in method (7) we have the following error equation:

this means that the method defined by (7) is fourth order. Also, using Equations (9)-(13) in (8) we get the following error equation:

which means that the family defined by (8) is of order four

This completes the proof of the theorem.

If we consider the definition of efficiency index as

number of function evaluations per iteration required by the method, then the fourth-order method (4) has the efficiency index equal to

All computations were done using the Mathematica package using 64 digit floating point arithmetic’s. We accept an approximate solution rather than the exact root, depending on the precision (ϵ) of the computer. We use the following stopping criteria for computer programs:

It is well-known that the convergence of iteration formula is guaranteed only when the initial approximation is sufficiently near to root. In general, it may be divergent when initial approximation is far from the root.

We employ the present methods to solve some nonlinear equations, which not only illustrate the methods practically but also serve to check the validity of theoretical results we have derived, the following

Displayed in

Compared with the Newton method (NM), the method in (4) (NOR), the new methods in (7) (MNR1), and as an example of (8) we take β = 0 (MNR2), and β = 1 (MNR3). The test results in

N | TNFE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

NM | NOR | MNR1 | MNR2 | MNR3 | NM | NOR | MNR1 | MNR2 | MNR3 | |

5 | 3 | 3 | 3 | 3 | 10 | 12 | 9 | 9 | 9 | |

53 | 38 | 22 | 56 | 17 | 106 | 152 | 66 | 168 | 51 | |

5 | 3 | 3 | 4 | 3 | 10 | 12 | 9 | 12 | 9 | |

6 | 3 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 12 | |

4 | 2 | 2 | 2 | 2 | 8 | 8 | 6 | 6 | 6 | |

5 | 3 | 3 | 3 | 3 | 10 | 12 | 9 | 9 | 9 | |

4 | 2 | 2 | 2 | 2 | 8 | 8 | 6 | 6 | 6 | |

4 | 3 | 3 | 3 | 3 | 8 | 12 | 9 | 9 | 9 | |

NC | 4 | 3 | 4 | 3 | - | 16 | 9 | 12 | 9 | |

12 | 5 | 6 | 5 | 6 | 24 | 20 | 18 | 15 | 18 | |

5 | 3 | 4 | 3 | 4 | 10 | 12 | 12 | 9 | 12 | |

8 | 5 | 5 | 5 | 5 | 16 | 20 | 15 | 15 | 15 | |

8 | 5 | 5 | 5 | 5 | 16 | 20 | 15 | 15 | 15 | |

4 | 2 | 2 | 2 | 2 | 8 | 8 | 6 | 6 | 6 | |

12 | 7 | 7 | 7 | 7 | 24 | 28 | 21 | 21 | 21 | |

19 | 11 | 12 | 11 | 12 | 38 | 44 | 36 | 33 | 36 |

As a conclusion, we can infer that the present method has better performance in accordance with the theoretical analysis of the order. However, it should be noted that per iteration the methods (MNR1), (MNR2) and (MNR32) do require two evaluations of the function and one of its first derivative, whereas the method (4) does require two evaluations of the function and two of its first derivative, costing more expensive computation. Thus, the present methods can be of practical interest.

We have proposed new fourth order methods of iterative methods for solving nonlinear equations. Numerical results show that the number of iterations of the new method is always less than that of the classical Newton’s method and the method in (4).

Analysis of convergence of methods is supplied in Theorem 2. Analysis of efficiency shows that these methods are preferable to Newton’s method and the fourth order method in (4). The number of function evaluations of the new methods is comparable.

This research is funded by the Deanship of Research in Zarqa University/Jordan.

Osama Y.Ababneh, (2016) New Fourth Order Iterative Methods Second Derivative Free. Journal of Applied Mathematics and Physics,04,519-523. doi: 10.4236/jamp.2016.43058