Some multistep higher order iterative methods for non-linear equations

We establish here new three and four-step iterative methods of convergence order seven and fourteen to find roots of non-linear equations. The new developed iterative methods are variants of Newton's method that use different approximations of first derivatives in terms of previously known function values, thus improving efficiency indices of the methods. The seventh and fourteenth order iterative methods use four and five function values including one derivative of a function. So, the efficiency indices of these methods are , respectively. Numerical examples are given to show the performance of described methods.


INTRODUCTION
Newton's method is a well known and commonly used quadratically convergent iterative method for finding roots of non-linear equation: which is given by   +1 =   + (  ) ′(  ) In recent years, researchers have made many modifications in this method to get higher order iterative methods.These methods are developed using various techniques by introducing some more steps to Newton's method.In this way, not only the convergence order but efficiency index of the method may also be increased (Chun, 2007;Mir and Rafiq, 2013;Osada, 1998;Potra and Ptak, 1984;Sharma, 2005).
Recently, the researchers introduced three-step iterative methods of convergence order seven to eight (Kou and Wang, 2007;Mir and Rafiq, 2014;Bi et al., 2009).Most recently, Neta and Petkovic (2010) introduced four-step iterative method of optimal convergence order sixteen for solving non-linear equations.In 2011, Sargolzaei and Soleymani (2011) introduced 4-step method of convergence order fourteen with five function evaluation and thus have efficiency index 14 5 = 1.6952 .In 1981, Neta introduced 4-step method but did not prove its convergence order.In 2010 *Corresponding author.E-mail: nazirahmad.mir@gmail.comAuthor(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Mathematics Subject Classification (2010): 65H05, 65B99.
(Geum and Kim, 2010) prove that the order of convergence of Neta's method is fourteen.To see more in this direction, we refer to Neta (1981), Sargolzaei and Soleymani (2011), Soleymani and Sharifi (2011).
Motivated in this direction, we also introduce here three-step and four-step iterative methods of convergence order seven and fourteen with efficiency indices, namely 7 4 = 1.6265  14 5 = 1.6952 respectively.

THE ITERATIVE METHODS
Consider two-step optimal convergent order four methods for solution of non-linear equation by Mir et al. (2013): Where and At third step, we add Newton's method with an approximation of derivative from Mir et al. (2014) which is given by: (3) where    ,   and [  ,   ,   ] are the divided differences which are defined by: We therefore propose the following three-step algorithms: (4) and Mir and Rafiq 753 (5) We can further improve the efficiency indices of Methods ( 4) and ( 5) by adding Newton's method as one more step with the approximation of derivative which is given by ( 6) from Sargolzaei and Soleymani (2011) and Soleymani and Sharifi (2011) which increase just one function value.Thus, we propose the following four-step algorithms: (7) and ( 8)

Convergence analysis
We now use Maple 10.0 to derive error equations of the iterative methods described by Equations ( 4), ( 5), ( 7) and ( 8).We prove that the iterative Methods ( 4) and ( 5) are of convergence order seven and Methods ( 7) and ( 8) are of convergence order fourteen.
Theorem 2: Let be a simple root of a sufficiently differentiable function in an open interval .If is sufficiently close to then the convergence order of three-step method described by algorithm ( 5) is seven and the error equation is given by: Proof: Similar to Theorem 1.

Comparison of 7th order convergent methods
Now, we compare our seventh order convergent methods, namely 7 1 MN  and 7 2 MN  described by ( 4) and ( 5) respectively, with second order convergent Newton's method NW (11) sixth order convergent method (G6) Grau and Diaz-Barrero (2006) and seventh order convergent method (G7) Kou et al. (2007).The result of the numerical comparison of various methods is shown in Table 3 on the same number of function evaluations (TNFE=12), that is, on the third iteration with 350 significant digits and convergence criterion as follows:   300 10 .
Examples are taken from (Bi et al., 2009) which are used for comparison as follows (Table 1):

Comparison of 14th order methods
We compare our methods namely 14 1 MN  and 14 2 MN  described by ( 7) and ( 8) with Neta's fourteenth order method (N14) (1981) at the same number of function evaluations TNFE=12.The absolute values of the given test functions at first three iterations are given in Table 4.The computation is carried out with 2500 significant digits and with the following stopping criteria: Mir and Rafiq 755

Example Roots
Here, we use the following test functions (Parviz and Soleymani, 2011) for comparison (Table 2).We observe that the numerical results are comparable or better in some cases.

Conclusion
In this article, we modified the existing methods of Mir et al. (2014) with the introduction of one and two steps more in such a way that the modified methods have improved convergence order that is, from fourth order to seventh order and then to fourteenth order as well as with their efficiencies improved.Modified methods are comparable and have better results as compared to the existing methods shown in Tables 3 and 4. The three-step methods are of seventh order convergent with four function evaluations per iteration and thus having computational efficiency .The four-step methods are of fourteenth order convergent methods with five function evaluations per iteration and thus having computational efficiency .

Theorem 1 :
Let ∈  be a simple root of a sufficiently differentiable function :  ⊆ ℝ → ℝ in an open interval  .If  0 is sufficiently close to  then the convergence order of three-step method described by (4) is seven and the error equation is given by: Proof: Let be a simple root of and .By Taylor's expansion, we have:

Theorem 3 :
Let  ∈  be a simple root of a sufficiently differentiable function :  ⊆ ℝ → ℝ in an open interval .If is sufficiently close to then the convergence order of three-step method described by algorithm (7) has order of convergence fourteen and satisfy the error equation:where Proof: Let us consider the error Equation (13) as follows: Let be a simple root of a sufficiently differentiable functionin an open interval .If is sufficiently close to then the convergence order of three-step method described by algorithm (8) has order of convergence fourteen:Proof: Similar to Theorem 3.

Table 1 .
Examples for 7 th order methods.

Table 2 .
Examples for 14 th order methods.

Table 3 .
TNFE=Total number of function evaluation.

Table 4 .
TNFE=Total number of function evaluation.