Optimal fourth- and eighth-order of convergence derivative-free modifications of King’s method

Obadah Said Solaiman; Samsul Ariffin Abdul Karim; Ishak Hashim

doi:10.1016/j.jksus.2018.12.001

Optimal fourth- and eighth-order of convergence derivative-free modifications of King’s method

Obadah Said Solaiman^{^a,⁎}, Samsul Ariffin Abdul Karim^{^b}, Ishak Hashim^{^c}

a Preparatory Year Deanship, King Faisal University, 31982 Hofuf, Ahsaa, Saudi Arabia

b Fundamental and Applied Sciences Department, Center for Smart Grid Energy Research (CSMER), Institute of Autonomous System, Universiti Teknologi PETRONAS, Bandar Seri Iskandar, 32610 Seri Iskandar, Perak DR, Malaysia

c School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

⁎Corresponding author. obadahmass@kfu.edu.sa (Obadah Said Solaiman),

Received: 2018-10-28, Accepted: 2018-12-4, Published: 2018-10-01

Disclaimer:
This article was originally published by Elsevier and was migrated to Scientific Scholar after the change of Publisher.

Peer review under responsibility of King Saud University.

Abstract

Starting by King’s method, we propose a modified families of fourth- and eighth-order of convergence iterative methods for nonlinear equations. The fourth-order method requires at each iteration three function evaluations, while the eighth-order methods both need four function evaluations. The proposed methods are derivative-free. Based on the conjecture of Kung and Traub, the new methods attain the optimality with efficiency index 1.587 for the fourth-order method and 1.68 for the eighth-order methods. The convergence analyses of the methods are given, and comparisons with some well-known schemes having identical order of convergence demonstrate the efficiency of the present techniques.

Keywords

41-xx

65-xx

Root finding method

Iterative method

Order of convergence

King’s method

Nonlinear equations

1 Introduction

Searching out a solution of the equation $f (x) = 0$ , where $f (x)$ is nonlinear is highly significant in mathematics. The second-order well-known Newton’s iterative technique for solving nonlinear equations defined as Traub (1964)

(1)

x_{n + 1} = x_{n} - \frac{f (x_{n})}{f^{'} (x_{n})} .

Many researchers have improved Newton’s method in order to get more accurate results and higher order of convergence, see for example Behl et al. (2017), Chun (2008), Cordero et al. (2016), Pandey and Jaiswal (2017), Said Solaiman and Hashim (2019), Sharma and Goyal (2007), Waseem et al. (2018) and the references therein.

Recently, Kogan et al. (2017) proved that methods of order $p = 3$ are the most efficient methods among all one-point iterative methods without memory of order p. Besides, the efficiency index is a common method to compare the performance of different iterative methods. This index is defined as $p^{1 / m}$ , where p represents the convergence order and m is the number of functional evaluations needed at each iteration. Based on the conjecture of Kung and Traub (1974), the iterative scheme with m functional evaluations is optimal if its order of convergence equals $2^{m - 1}$ . Many authors have constructed the optimal iterative methods of different convergence orders. The standard way for constructing optimal method is the composition technique together with the usage of some interpolations and approximations to minimize the needed functional evaluations at each iteration. Different optimal fourth-order iterative methods were constructed, see for example Argyros and Magreñán (2015), Behl et al. (2015a), Chun et al. (2012), Cordero et al. (2010), Sharma and Bahl (2015), Soleymani et al. (2012). Optimal eighth-order of convergence was also reached by many authors also, for instance Behl et al. (2016), Behl et al. (2018), Cordero et al. (2015), Geum et al. (2018), Sharma and Arora (2016), Singh and Jaiswal (2016). Neta and Chun (2014) and Chun and Neta (2016) presented a comparison of several families of optimal iterative methods which are of fourth- and eighth-order of convergence based on their basins of attraction. Chun and Neta (2017) presented a quantitative comparison of many optimal iterative techniques of order eight beside the visual comparison of the methods by graphing their dynamics.

Since, to find the derivative value is not always an easy task, as well as requires more computation time, many authors have proposed and implemented various derivative-free optimal methods, see Cordero et al. (2013), Lee and Kim (2012), Sharma and Goyal (2007), Yasmin et al. (2016), Zafar et al. (2015). One of the most famous optimal fourth-order iterative techniques is the method proposed by King (1973). But the main weakness is that finding the first derivative is needed at each iteration. Many authors have modified King’s method. For example, Chun (2007) implemented some King’s like methods of order four, but computing the first derivative within the iteration is needed also. Behl et al. (2015b) proposed a fourth-order derivative-free modification of King’s method. Sharifi et al. (2014) implemented an optimal derivative-free fourth- and eighth-order modifications of King’s method.

In this work, by modifying King’s method, we propose a family of optimal fourth-order derivative-free iterative method for nonlinear equations. With the use of some approximations and the composition technique, we extend the new method to two new optimal schemes of order eight. The convergence analysis of all three methods are derived. The proposed optimal methods were tested on six different examples to show the efficiency of the methods with numerical comparison to other established methods of the same order.

The work of this paper is divided as follows. The new schemes are described below in Section 2. To show the order of convergence of the new schemes, thee convergence analysis is implemented in Section 3. The numerical examples with the comparisons with other techniques of identical orders are summarized in Section 4. Eventually, in Section 5 the conclusion is given.

2 The new methods

2.1

2.1 Derivative-free optimal fourth-order iterative method

We propose a modified family of optimal fourth-order derivative-free schemes. We start by Newton’s two steps method of order four:

(2)

\{\begin{matrix} y_{n} = x_{n} - \frac{f (x_{n})}{f^{'} (x_{n})}, \\ x_{n + 1} = y_{n} - \frac{f (y_{n})}{f^{'} (y_{n})} . \end{matrix}

This two steps method is not optimal as it needs the evaluation of two functions and two first derivatives. In order to reduce the number of functional evaluations, King (1973) replaced

f^{'} (y_{n})

by the following approximation:

(3)

f^{'} (y_{n}) = f^{'} (x_{n}) \frac{f (x_{n}) + (β - 2) f (y_{n})}{f (x_{n}) + β f (y_{n})} .

Substituting (3) into (2) produces the famous optimal King’s method of order four:

(4)

\{\begin{matrix} y_{n} = x_{n} - \frac{f (x_{n})}{f^{'} (x_{n})}, \\ x_{n + 1} = y_{n} - \frac{f (y_{n})}{f^{'} (x_{n})} \frac{f (x_{n}) + β f (y_{n})}{f (x_{n}) + (β - 2) f (y_{n})} . \end{matrix}

King’s method needs the computation of two functions and one first derivative. Based on the conjecture of Kung and Traub (1974), King’s method reached the optimality with efficiency index equals

{(4)}^{\frac{1}{3}} \approx 1.587

However, the computation of the first derivative is not always easy, in addition to that it costs extra time. To implement a derivative-free technique from King’s method (4), the next approximations for $f^{'} (x_{n})$ in both steps will be considered

(5)

f^{'} (x_{n}) \approx f [w_{n}, x_{n}],

(6)

f^{'} (x_{n}) \approx g (x_{n}) = f [w_{n}, x_{n}] + 2 (w_{n} - x_{n}) f [w_{n}, x_{n}, y_{n}] - f [y_{n}, w_{n}] + f [x_{n}, y_{n}] .

Here

w_{n} = x_{n} + f (x_{n}), f [x_{n}, y_{n}] = \frac{f (x_{n}) - f (y_{n})}{x_{n} - y_{n}}

, and

f [w_{n}, x_{n}, y_{n}] = \frac{f [w_{n}, x_{n}] - f [x_{n}, y_{n}]}{w_{n} - y_{n}}

. The first approximation (5) was firstly used by Steffensen (1933), and the second one (6) is obtained with the help of divided differences approximation. After substituting (5) and (6) into (4) one can obtain the first essential finding of this work:

Algorithm 1

Given $x_{0}$ , the approximate solution $x_{n + 1}$ of $f (x) = 0$ can be found by the following iterative scheme

(7)

\{\begin{matrix} y_{n} = x_{n} - \frac{f (x_{n})}{f [w_{n}, x_{n}]}, \\ x_{n + 1} = y_{n} - \frac{f (y_{n})}{g (x_{n})} \frac{f (x_{n}) + β f (y_{n})}{f (x_{n}) + (β - 2) f (y_{n})} . \end{matrix}

We call this family, modified King’s method MK₄. The order of convergence of this family is four. In each iteration, MK₄ requires only three evaluations of functions and no derivative evaluation is needed. Based on the conjecture of Kung and Traub (1974), MK₄ reached the optimality with efficiency index equals

{(4)}^{\frac{1}{3}} \approx 1.587

2.2

2.2 Derivative-free optimal eighth-order iterative method

In order to extend MK₄ method given by Algorithm 1 to the eighth-order of convergence, we will use the composition technique. The additional step of Algorithm 1 is produced using the idea of Zafar et al. (2015) based on rational interpolation. Consider the following Algorithm, which is the second main finding of this work:

Algorithm 2

Given $x_{0}$ , the approximate solution $x_{n + 1}$ of $f (x) = 0$ can be found by the following iterative scheme

(8)

\{\begin{matrix} y_{n} = x_{n} - \frac{f (x_{n})}{f [w_{n}, x_{n}]}, \\ z_{n} = y_{n} - \frac{f (y_{n})}{g (x_{n})} \frac{f (x_{n}) + β f (y_{n})}{f (x_{n}) + (β - 2) f (y_{n})}, \\ x_{n + 1} = x_{n} - \frac{f (x_{n}) (m_{1} + m_{2} + m_{3})}{m_{1} f [w_{n}, x_{n}] + m_{2} f [y_{n}, x_{n}] + m_{3} f [z_{n}, x_{n}]}, \end{matrix}

where

\begin{matrix} m_{1} = & f (y_{n}) f (z_{n}) (z_{n} - y_{n}), \\ m_{2} = & f (w_{n}) f (z_{n}) (w_{n} - z_{n}), \\ m_{3} = & f (w_{n}) f (y_{n}) (y_{n} - w_{n}) . \end{matrix}

We call this family, modified King’s method MK_8a. The convergence order of this family is eight. In each iteration MK_8a requires only four evaluations of functions and no derivative evaluation is needed. Based on the conjecture of Kung and Traub (1974), MK_8a reached the optimality with efficiency index equals

{(8)}^{\frac{1}{4}} \approx 1.68

2.3

2.3 Another derivative-free optimal eighth-order iterative method

Another extension of MK₄ method can be achieved by adding Newton’s technique as a third step of MK₄ method, and then using the derivative of the second degree Padé approximation for $f^{'} (z_{n})$ . This approximation was firstly proposed by Cordero et al. (2013). The following algorithm is the third essential finding of this work:

Algorithm 3

Given $x_{0}$ , the approximate solution $x_{n + 1}$ of $f (x) = 0$ can be found by the following iterative scheme

(9)

\{\begin{matrix} y_{n} = x_{n} - \frac{f (x_{n})}{f [w_{n}, x_{n}]}, \\ z_{n} = y_{n} - \frac{f (y_{n})}{g (x_{n})} \frac{f (x_{n}) + β f (y_{n})}{f (x_{n}) + (β - 2) f (y_{n})}, \\ x_{n + 1} = z_{n} - \frac{f (z_{n})}{c_{2} - c_{1} c_{4}}, \end{matrix}

where

\begin{matrix} c_{1} = & f (z_{n}), \\ c_{2} = & f [y_{n}, z_{n}] - c_{3} (y_{n} - z_{n}) + c_{4} f (y_{n}), \\ c_{3} = & f [y_{n}, z_{n}, w_{n}] + c_{4} f [y_{n}, w_{n}], \\ c_{4} = & \frac{f [y_{n}, z_{n}, x_{n}] - f [y_{n}, z_{n}, w_{n}]}{f [y_{n}, w_{n}] - f [y_{n}, x_{n}]} . \end{matrix}

We call this family, modified King’s method MK_8b. The order of convergence of this family is eight. Each iteration of MK_8b requires only four evaluations of functions and there is no derivative evaluation needed. Based on the conjecture of Kung and Traub (1974), MK_8b reached the optimality with efficiency index equals

{(8)}^{\frac{1}{4}} \approx 1.68

3 Convergence analysis

The convergence analysis of the proposed methods will be discussed in the following theorems.

Theorem 4

Consider that $α$ is a root of a sufficiently differentiable function $f : I \subseteq R \to R$ in an open interval I, and let $x_{0}$ be sufficiently close to $α$ . Also let $e_{n} = x_{n} - α$ be the error at the $n^{th}$ iteration. The method defined in Algorithm 1 is of fourth-order of convergence.

Proof

By the Taylor series expansion of $f (x)$ about $x = α$ one obtains

(10)

f (x_{n}) = [c_{1} e_{n} + c_{2} e_{n}^{2} + c_{3} e_{n}^{3} + c_{4} e_{n}^{4} + \dots],

where

c_{k} = \frac{f^{(k)} (α)}{k!}

k = 1, 2, 3, \dots

. Furthermore,

(11)

\begin{matrix} w_{n} = & x_{n} + f (x_{n}), \\ = & α + e_{n} + c_{1} e_{n} + c_{2} e_{n}^{2} + c_{3} e_{n}^{3} + c_{4} e_{n}^{4} + \dots . \end{matrix}

Expanding

f (w_{n})

about

α

, one obtains

(12)

f (w_{n}) = (c_{1} + c_{1}^{2}) e_{n} + (c_{2} + 3 c_{1} c_{2} + c_{1}^{2} c_{2}) e_{n}^{2} + \dots .

Hence, from Eqs. (10)–(12) we obtain

(13)

\begin{matrix} f [w_{n}, x_{n}] = & \frac{f (w_{n}) - f (x_{n})}{w_{n} - x_{n}} = \frac{f (w_{n}) - f (x_{n})}{f (x_{n})}, \\ = & c_{1} + (2 c_{2} + c_{1} c_{2}) e_{n} + (c_{2}^{2} + 3 c_{3} + 3 c_{1} c_{3} + c_{1}^{2} c_{3}) e_{n}^{2} + \dots . \end{matrix}

Now, substituting Eqs. (10) and (13) into the first step of scheme (7), we have

(14)

\begin{matrix} y_{n} = & x_{n} - \frac{f (x_{n})}{f [w_{n}, x_{n}]}, \\ = & α + \frac{(c_{1} c_{2} + c_{2}) e_{n}^{2}}{c_{2}} - \frac{(2 c_{2}^{2} + 2 c_{1} c_{2}^{2} + c_{1}^{2} c_{2}^{2} - 2 c_{1} c_{3} - 3 c_{1}^{2} c_{3} - c_{1}^{3} c_{3}) e_{n}^{3}}{c_{1}^{2}} + \dots . \end{matrix}

Expanding

f (y_{n})

about

α

, with the use of Eq. (14) one obtains

(15)

f (y_{n}) = (c_{2} + c_{1} c_{2}) e_{n}^{2} + \frac{(- 2 c_{2}^{2} - 2 c_{1} c_{2}^{2} - c_{1}^{2} c_{2}^{2} + 2 c_{1} c_{3} + 3 c_{1}^{2} c_{3}) e_{n}^{3}}{c_{1}^{3}} + \dots .

Also, we have

(16)

\begin{matrix} g (x_{n}) = & f [w_{n}, x_{n}] + 2 (w_{n} - x_{n}) f [w_{n}, x_{n}, y_{n}] - f [y_{n}, w_{n}] + f [x_{n}, y_{n}], \\ = & c_{1} + 2 (c_{2} + c_{1} c_{2}) e_{n} + (2 c_{2}^{2} + 3 c_{3} + 5 c_{1} c_{3} + 2 c_{1}^{2} c_{3}) e_{n}^{2} + \dots . \end{matrix}

After substituting Eqs. (10), (15) and (16) into the second step of scheme (7), we obtain

(17)

x_{n + 1} = α + \frac{{(1 + c_{1})}^{2} c_{2} ((1 + 2 β + 2 (- 1 + β) c_{1}) c_{2}^{2} - c_{1} c_{3}) e_{n}^{4}}{c_{1}^{3}} + \dots .

This implies that

e_{n + 1} = \frac{{(1 + c_{1})}^{2} c_{2} ((1 + 2 β + 2 (- 1 + β) c_{1}) c_{2}^{2} - c_{1} c_{3}) e_{n}^{4}}{c_{1}^{3}} + O (e_{n}^{5}) .

Hence, MK₄ method proposed in Algorithm 1 has fourth-order of convergence. □

Theorem 5

Consider the same assumptions in Theorem 4, then the method defined by Algorithm 2 is of eighth-order of convergence.

Proof

From Theorem 4, we have

(18)

z_{n} = α + \frac{{(1 + c_{1})}^{2} c_{2} ((1 + 2 β + 2 (- 1 + β) c_{1}) c_{2}^{2} - c_{1} c_{3}) e_{n}^{4}}{c_{1}^{3}} .

Expanding

f (z_{n})

about

α

, with the use of Eq. (18), we have

(19)

f (z_{n}) = [c_{2}^{3} + 2 β c_{2}^{3} + 6 β c_{1} c_{2}^{3} + (6 β - 3) c_{1}^{2} c_{2}^{3} + 2 (β - 1) c_{1}^{3} c_{2}^{3} - c_{1} c_{2} c_{3} - 2 c_{1}^{2} c_{2} c_{3} - c_{1}^{3} c_{2} c_{3}] e_{n}^{4} c_{1}^{- 2} + \dots .

From Eqs. (11), (12), (14), (15), (18), and (19) we have

(20)

m_{1} = - \frac{{(1 + c_{1})}^{4} c_{2}^{3} (c_{2}^{2} + 2 β c_{2}^{2} - 2 c_{1} c_{2}^{2} + 2 β c_{1} c_{2}^{2} - c_{1} c_{3}) e_{n}^{8}}{c_{1}^{3}} + \dots,

(21)

m_{2} = \frac{{(1 + c_{1})}^{4} (c_{2}^{3} + 2 β c_{2}^{3} - 2 c_{1} c_{2}^{3} + 2 β c_{1} c_{2}^{3} - c_{1} c_{2} c_{3}) e_{n}^{6}}{c_{1}} + \dots,

(22)

m_{3} = - (1 + c_{1}) (c_{1} + c_{1}^{2}) (c_{2} + c_{1} c_{2}) e_{n}^{4} + \dots .

Using Eqs. (10), (15), and (19), one obtains

(23)

f [y_{n}, x_{n}] = c_{1} + c_{2} e_{n} + \frac{(c_{2}^{2} + c_{1} c_{2}^{2} + c_{1} c_{3}) e_{n}^{2}}{c_{1}} + \dots,

(24)

f [z_{n}, x_{n}] = c_{1} + c_{2} e_{n} + c_{3} e_{n}^{2} + \dots .

Substituting Eqs. (10), (13), and (20)–(24) in the third step of scheme (8), we get

x_{n + 1} = α + \frac{{(1 + c_{1})}^{4} c_{2}^{2} ((1 + 2 β + 2 (- 1 + β) c_{1}) c_{2}^{2} - c_{1} c_{3}) (c_{2}^{3} - 2 c_{1} c_{2} c_{3} + c_{1}^{2} c_{4}) e_{n}^{8}}{c_{1}^{7}} + \dots,

which implies that

e_{n + 1} = \frac{{(1 + c_{1})}^{4} c_{2}^{2} ((1 + 2 β + 2 (- 1 + β) c_{1}) c_{2}^{2} - c_{1} c_{3}) (c_{2}^{3} - 2 c_{1} c_{2} c_{3} + c_{1}^{2} c_{4}) e_{n}^{8}}{c_{1}^{7}} + O (e_{n}^{9}) .

This shows that MK_8b method proposed in Algorithm 2 has eighth-order of convergence. □

Theorem 6

Consider the same assumptions of Theorem 4, then the method defined by Algorithm 3 is of eighth-order of convergence.

Proof

Based on the definitions of $c_{1}, c_{2}, c_{3}$ , and $c_{4}$ given in the scheme (9), and as the series expansion is too large and can’t be expressed in a few lines, we used Mathematica 9 to do the required computations. After some simplification we obtain $x_{n + 1} = α + \frac{{(1 + c_{1})}^{4} c_{2} ((1 + 2 β + 2 (- 1 + β) c_{1}) c_{2}^{2} - c_{1} c_{3}) ((1 + 2 β + 2 (- 1 + β) c_{1}) c_{2}^{4} - c_{1} c_{2}^{2} c_{3} - c_{1}^{2} c_{3}^{2} + c_{1}^{2} c_{2} c_{4}) e_{n}^{8}}{c_{1}^{7}} + \dots .$ This leads to $e_{n + 1} = \frac{{(1 + c_{1})}^{4} c_{2} ((1 + 2 β + 2 (- 1 + β) c_{1}) c_{2}^{2} - c_{1} c_{3}) ((1 + 2 β + 2 (- 1 + β) c_{1}) c_{2}^{4} - c_{1} c_{2}^{2} c_{3} - c_{1}^{2} c_{3}^{2} + c_{1}^{2} c_{2} c_{4}) e_{n}^{8}}{c_{1}^{7}} + O (e_{n}^{9}),$ which proves that MK_8b method proposed in Algorithm 3 is of eighth-order of convergence □

4 Numerical examples

To show the efficiency of the new optimal schemes MK₄, MK_8a and MK_8b, several examples will be tested. We compare the new schemes with the optimal fourth-order technique K₄ presented by King (1973), and with the derivative-free fourth- and eighth-order methods presented by Yasmin et al. (2016), Cordero et al. (2013); and Zafar et al. (2015) denoted respectively as: YZA₄, YZA₈, CHMT₄, CHMT₈, and ZYAJ₄, ZYAJ₈. In all examples, we consider that $α = 1$ whenever $w_{n} = x_{n} + α f (x_{n})$ , and that $β = 2$ in King’s method and our proposed methods.

Six test examples are considered below: $\begin{matrix} f_{1} (x) = & \cos (x) - x, f_{2} (x) = \sin^{2} (x) - x^{2} + 1, \\ f_{3} (x) = & \ln (x^{2} - x + 1) - 4 \sin (x - 1), f_{4} (x) = e^{- x^{2}} + \cos (x) - x^{2}, \\ f_{5} (x) = & \arctan (x) - x^{2} + 1, \\ f_{6} (x) = & \{\begin{matrix} x (x + 1), if x < 0 \\ - 2 x (x - 1), if x ⩾ 0 \end{matrix} . \end{matrix}$

We take $|x_{n} - x_{n - 1}| < 10^{- 15}$ as a stopping criterion of the computer programs. The computations here have been carried out using Mathematica version 9 with 10,000 significant digits.

Table 1 shows the number of iterations n needed so that the stopping criterion is satisfied, the approximate zero $x_{n}$ , the distance between two successive approximations with $|x_{n} - x_{n - 1}| < 10^{- 15}$ , the value of $f (x)$ at the approximate zero, and the computational order of convergence (COC) defined by Weerakoon and Fernando (2000), which can be estimated as follows $COC \approx \frac{\ln |(x_{n + 1} - x_{n}) / (x_{n} - x_{n - 1})|}{\ln |(x_{n} - x_{n - 1}) / (x_{n - 1} - x_{n - 2})|} .$

Table 1 Comparisons between different methods.

Method	n	$x_{n}$	$\|x_{n} - x_{n - 1}\|$	$f (x_{n})$	COC
$f_{1} (x), x_{0} = 0$
MK₄	4	0.73908513321516064	1.63E−52	$- 1.75$ E−209	4
K₄	4	0.73908513321516064	5.30E−18	$- 9.03$ E−71	4
YZA₄	4	0.73908513321516064	1.71E−32	1.60E−128	4
CHMT₄	4	0.73908513321516064	4.82E−52	$- 1.49$ E−207	4
ZYAJ₄	4	0.73908513321516064	2.69E−56	$- 1.02$ E−224	4

MK_8a	3	0.73908513321516064	3.12E−55	$- 4.94$ E−441	8
MK_8b	3	0.73908513321516064	2.75E−58	5.03E−466	8
YZA₈	3	0.73908513321516064	3.15E−28	9.51E−223	8
CHMT₈	3	0.73908513321516064	4.13E−58	5.73E−465	8
ZYAJ₈	3	0.73908513321516064	5.79E−57	$- 5.41$ E−455	8

$f_{2} (x), x_{0} = 1$
MK₄	4	1.4044916482153412	1.76E−44	2.69E−176	4
K₄	5	1.4044916482153412	7.84E−18	$- 2.19$ E−68	4
YZA₄	4	1.4044916482153412	2.18E−23	1.90E−90	4
CHMT₄	4	1.4044916482153412	1.76E−27	$- 4.63$ E−107	4
ZYAJ₄	4	1.4044916482153412	2.45E−35	$- 8.16$ E−139	4

MK_8a	3	1.4044916482153412	3.29E−42	1.44E−333	8
MK_8b	3	1.4044916482153412	2.01E−45	$- 2.42$ E−359	8
YZA₈	3	1.4044916482153412	8.99E−31	4.00E−240	8
CHMT₈	3	1.4044916482153412	4.21E−30	$- 6.26$ E−235	8
ZYAJ₈	3	1.4044916482153412	5.62E−39	$- 8.38$ E−307	8

$f_{3} (x), x_{0} = 1.5$
MK₄	3	1	9.64E−16	$- 4.80$ E−62	4
K₄	4	1	9.53E−41	5.73E−162	4
YZA₄	4	3.5302670187568383	1.53E−37	$- 2.30$ E−147	4
CHMT₄	4	1	1.44E−49	4.75E−197	4
ZYAJ₄	4	1	1.55E−53	3.22E−213	4

MK_8a	3	1	4.29E−54	$- 3.75$ E−430	8
MK_8b	3	1	7.57E−57	$- 3.14$ E−452	8
YZA₈	3	1	8.02E−53	$- 1.75$ E−419	8
CHMT₈	3	1	6.44E−49	2.04E−388	8
ZYAJ₈	3	1	1.28E−49	2.39E−394	8

$f_{4} (x), x_{0} = 1$
MK₄	3	0.97416230520054071	2.71E−32	8.46E−128	4
K₄	3	0.97416230520054071	7.45E−31	$- 1.34$ E−121	4
YZA₄	3	0.97416230520054071	3.46E−26	4.16E−102	4
CHMT₄	3	0.97416230520054071	8.90E−28	$- 9.65$ E−109	4
ZYAJ₄	3	0.97416230520054071	1.78E−28	$- 1.11$ E−111	4

MK_8a	3	0.97416230520054071	3.81E−118	1.93E−941	8
MK_8b	2	0.97416230520054071	3.81E−16	2.58E−126	8
YZA₈	3	0.97416230520054071	5.03E−107	2.18E−851	8
CHMT₈	3	0.97416230520054071	3.78E−112	$- 1.06$ E−892	8
ZYAJ₈	3	0.97416230520054071	2.32E−111	$- 2.54$ E−886	8

$f_{5} (x), x_{0} =$ 1.5
MK₄	3	1.3961536566409308	6.61E−23	$- 2.18$ E−90	4
K₄	3	1.3961536566409308	2.01E−18	$- 2.16$ E−71	4
YZA₄	4	1.3961536566409308	1.31E−58	8.75E−232	4
CHMT₄	3	1.3961536566409308	1.18E−17	$- 2.25$ E−68	4
ZYAJ₄	3	1.3961536566409308	4.49E−19	$- 2.48$ E−74	4

MK_8a	3	1.3961536566409308	3.50E−82	$- 3.52$ E−654	8
MK_8b	3	1.3961536566409308	9.22E−89	$- 1.65$ E−707	8
YZA₈	3	1.3961536566409308	2.76E−64	3.19E−509	8
CHMT₈	3	1.3961536566409308	1.28E−70	$- 1.89$ E−560	8
ZYAJ₈	3	1.3961536566409308	3.01E−75	$- 5.71$ E−598	8

$f_{6} (x), x_{0} = 0.6$
MK₄	4	1	3.53E−36	$- 3.09$ E−142	4
K₄	9	1	3.10E−27	$- 9.19$ E−106	4
YZA₄	4	1	5.01E−30	3.79E−117	4
CHMT₄	4	1	2.36E−34	$- 1.25$ E−134	4
ZYAJ₄	4	1	1.69E−42	$- 1.64$ E−167	4

MK_8a	3	1	2.13E−39	$- 8.52$ E−310	8
MK_8b	3	1	2.90E−36	$- 1.01$ E−284	8
YZA₈	3	1	1.62E−24	6.67E−190	8
CHMT₈	3	1	2.41E−35	$- 9.18$ E−277	8
ZYAJ₈	3	1	1.56E−42	$- 7.19$ E−335	8

The second column in Table 1 shows the number of iterations n needed to reach the stopping criterion. It is clear that the new methods need less iterations than the other methods to reach the stopping criterion, or the same number of iterations in some cases. Therefore, the approximate solutions obtained by the proposed techniques are as good as of those obtained by other existing methods of the same order.

Note that, even though the new proposed methods need the same number of iterations to satisfy the stopping criterion as with the other methods, but still they are superior to the other methods as $|x_{n} - x_{n - 1}|$ and $f (x_{n})$ are less for the new schemes than the other schemes of the same order.

Table 2 illustrates the number of iterations needed to achieve approximate solution using the stopping criterion $|x_{n} - x_{n - 1}| < 10^{- 200}$ . Setting the same convergence criterion for all methods, the required number of iterations for the new methods is either less than or equal the needed iterations by the other techniques with identical order.

Table 2 Comparisons of number of iterations needed for different methods with

|x_{n} - x_{n - 1}| < 10^{- 200}

	$f_{1} (x)$	$f_{2} (x)$	$f_{3} (x)$	$f_{4} (x)$	$f_{5} (x)$	$f_{6} (x)$
	$x_{0} = 0$	$x_{0} = 0$	$x_{0} = 1.5$	$x_{0} = 1$	$x_{0} = 1.5$	$x_{0} = 0.6$
MK₄	5	6	5	5	5	6
K₄	6	7	6	5	5	11
YZA₄	6	6	6	5	5	6
CHMT₄	5	6	6	5	5	6
ZYAJ₄	5	6	5	5	5	6

MK_8a	4	4	4	4	4	4
MK_8b	4	4	4	4	4	4
YZA₈	4	4	4	4	4	5
CHMT₈	4	4	4	4	4	4
ZYAJ₈	4	4	4	4	4	4

5 Conclusion

In this work we proposed new optimal three derivative-free root finding schemes for nonlinear equations. These methods are implemented via efficient algorithms. The first method has order four, and derived using King’s method with finite difference approximations. The second and the third optimal methods were of order eight. We implement the methods by using the composition technique combined with rational interpolation, and the idea of Padé approximation. The convergence analysis of the proposed optimal methods has been proved, with the convergence order has been established to be of the optimal fourth- and eighth-order, respectively. Six examples were tested, showing the capability of the new techniques. Overall, the new methods are comparable to other well-known schemes with same order of convergence.

References

Argyros I.K., Magreñán Á.A., . On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput.. 2015;252:336-346.
[Google Scholar]
Behl R., Argyros I.K., Motsa S.S., . A new highly efficient and optimal family of eighth-order methods for solving nonlinear equations. Appl. Math. Comput.. 2016;282:175-186.
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Behl R., Cordero A., Motsa S.S., Torregrosa J.R., . Construction of fourth-order optimal families of iterative methods and their dynamics. Appl. Math. Comput.. 2015;271:89-101.
[Google Scholar]
Behl R., González D., Maroju P., Motsa S.S., . An optimal and efficient general eighth-order derivative free scheme for simple roots. J. Comput. Appl. Math.. 2018;330:666-675.
[Google Scholar]
Behl R., Maroju P., Motsa S.S., . A family of second derivative free fourth order continuation method for solving nonlinear equations. J. Comput. Appl. Math.. 2017;318:38-46.
[Google Scholar]
Behl R., Motsa S.S., Kansal M., Kanwar V., . Fourth-order derivative-free optimal families of King’s and Ostrowski’s methods. In: Agrawal P., Mohapatra R., Singh U., Srivastava H., eds. Mathematical Analysis and Its Applications. New Delhi: Springer Proceedings in Mathematics & Statistics. Vol 143. Springer; 2015. p. :359-371.
[Google Scholar]
Chun C., . Some variants of King’s fourth-order family of methods for nonlinear equations. Appl. Math. Comput.. 2007;190:57-62.
[Google Scholar]
Chun C., . Some fourth-order iterative methods for solving nonlinear equations. Appl. Math. Comput.. 2008;195:454-459.
[Google Scholar]
Chun C., Lee M.Y., Neta B., Džunić J., . On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput.. 2012;218:6427-6438.
[Google Scholar]
Chun C., Neta B., . Comparison of several families of optimal eighth order methods. Appl. Math. Comput.. 2016;274:762-773.
[Google Scholar]
Chun C., Neta B., . Comparative study of eighth-order methods for finding simple roots of nonlinear equations. Numer. Algorithms. 2017;74(4):1169-1201.
[Google Scholar]
Cordero A., Hueso J.L., Martínez E., Torregrosa J.R., . New modifications of Potra-Pták’s method with optimal fourth and eighth orders of convergence. J. Comput. Appl. Math.. 2010;234:2969-2976.
[Google Scholar]
Cordero A., Hueso J.L., Martínez E., Torregrosa J.R., . A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations. J. Comput. Appl. Math.. 2013;252:95-102.
[Google Scholar]
Cordero A., Lotfi T., Mahdiani K., Torregrosa J.R., . A stable family with high order of convergence for solving nonlinear equations. Appl. Math. Comput.. 2015;254:240-251.
[Google Scholar]
Cordero A., Maimó J.G., Torregrosa J.R., Vassileva M.P., . Stability of a fourth order bi-parametric family of iterative methods. J. Comput. Appl. Math.. 2016;312:94-102.
[Google Scholar]
Geum Y.H., Kim Y.I., Neta B., . Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points. J. Comput. Appl. Math.. 2018;333:131-156.
[Google Scholar]
King R.F., . A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal.. 1973;10:876-879.
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Kogan T., Sapir L., Sapir A., Sapir A., . To the question of efficiency of iterative methods. Appl. Math. Lett.. 2017;66:40-46.
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Kung H.T., Traub J.F., . Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach.. 1974;21:643-651.
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Lee M.Y., Kim Y.I., . A family of fast derivative-free fourth-order multipoint optimal methods for nonlinear equations. Int. J. Comput. Math.. 2012;89(15):2081-2093.
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Neta B., Chun C., . Basins of attraction for several optimal fourth order methods for multiple roots. Math. Comput. Simulat.. 2014;103:39-59.
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Pandey B., Jaiswal J.P., . New seventh and eighth order derivative free methods for solving nonlinear equations. Tbilisi Math. J.. 2017;10(4):103-115.
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Said Solaiman O., Hashim I., . Two new efficient sixth order iterative methods for solving nonlinear equations. J. King Saud Univ.–Sci.. 2019;31:701-705.
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Sharma J.R., Arora H., . Some novel optimal eighth order derivative-free root solvers and their basins of attraction. Appl. Math. Comput.. 2016;284:149-161.
[Google Scholar]
Sharma J.R., Bahl A., . An optimal fourth order iterative method for solving nonlinear equations and its dynamics. J. Comp. Anal.. 2015;2015 Article ID 259167
[Google Scholar]
Sharma J.R., Goyal R.K., . Fourth-order derivative-free methods for solving non-linear equations. Int. J. Comput. Math.. 2007;83(1):101-106.
[Google Scholar]
Singh A., Jaiswal J.P., . A class of optimal eighth-order Steffensen-type iterative methods for solving nonlinear equations and their basins of attraction. Appl. Math. Inf. Sci.. 2016;10:251-257.
[Google Scholar]
Soleymani F., Khattri S.K., Vanani S.K., . Two new classes of optimal Jarratt-type fourth order methods. Appl. Math. Lett.. 2012;25(5):847-853.
[Google Scholar]
Steffensen J.F., . Remarks on iteration. Scand. Actuar. J.. 1933;1:64-72.
[Google Scholar]
Traub J.F., . Iterative Methods for Solution of Equations. Englewood, Cliffs, NJ: Prentice-Hall; 1964.
Waseem M., Noor M.A., Farooq A.S., Noor K.I., . An efficient technique to solve nonlinear equations using multiplicative calculus. Turk. J. Math.. 2018;42:679-691.
[Google Scholar]
Weerakoon S., Fernando T.G.I., . A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett.. 2000;13:87-93.
[Google Scholar]
Yasmin N., Zafar F., Akram S., . Optimal derivative-free root nding methods based on the Hermite interpolation. J. Nonlinear. Sci. Appl.. 2016;9:4427-4435.
[Google Scholar]
Zafar F., Yasmin N., Akram S., Junjua M., . A general class of derivative free optimal root finding methods based on rational interpolation. The Sci. World J.. 2015;2015 Article ID 934260
[Google Scholar]

Introduction

The new methods

Derivative-free optimal fourth-order iterative method
Derivative-free optimal eighth-order iterative method
Another derivative-free optimal eighth-order iterative method

Convergence analysis

Numerical examples

Conclusion

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[1] Argyros I.K., Magreñán Á.A., . On the convergence of an optimal fourth-order family of methods and its dynamics. Appl. Math. Comput.. 2015;252:336-346.
[Google Scholar]

[2] Behl R., Argyros I.K., Motsa S.S., . A new highly efficient and optimal family of eighth-order methods for solving nonlinear equations. Appl. Math. Comput.. 2016;282:175-186.
[Google Scholar]

[3] Behl R., Cordero A., Motsa S.S., Torregrosa J.R., . Construction of fourth-order optimal families of iterative methods and their dynamics. Appl. Math. Comput.. 2015;271:89-101.
[Google Scholar]

[4] Behl R., González D., Maroju P., Motsa S.S., . An optimal and efficient general eighth-order derivative free scheme for simple roots. J. Comput. Appl. Math.. 2018;330:666-675.
[Google Scholar]

[5] Behl R., Maroju P., Motsa S.S., . A family of second derivative free fourth order continuation method for solving nonlinear equations. J. Comput. Appl. Math.. 2017;318:38-46.
[Google Scholar]

[6] Behl R., Motsa S.S., Kansal M., Kanwar V., . Fourth-order derivative-free optimal families of King’s and Ostrowski’s methods. In: Agrawal P., Mohapatra R., Singh U., Srivastava H., eds. Mathematical Analysis and Its Applications. New Delhi: Springer Proceedings in Mathematics & Statistics. Vol 143. Springer; 2015. p. :359-371.
[Google Scholar]

[7] Chun C., . Some variants of King’s fourth-order family of methods for nonlinear equations. Appl. Math. Comput.. 2007;190:57-62.
[Google Scholar]

[8] Chun C., . Some fourth-order iterative methods for solving nonlinear equations. Appl. Math. Comput.. 2008;195:454-459.
[Google Scholar]

[9] Chun C., Lee M.Y., Neta B., Džunić J., . On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput.. 2012;218:6427-6438.
[Google Scholar]

[10] Chun C., Neta B., . Comparison of several families of optimal eighth order methods. Appl. Math. Comput.. 2016;274:762-773.
[Google Scholar]

[11] Chun C., Neta B., . Comparative study of eighth-order methods for finding simple roots of nonlinear equations. Numer. Algorithms. 2017;74(4):1169-1201.
[Google Scholar]

[12] Cordero A., Hueso J.L., Martínez E., Torregrosa J.R., . New modifications of Potra-Pták’s method with optimal fourth and eighth orders of convergence. J. Comput. Appl. Math.. 2010;234:2969-2976.
[Google Scholar]

[13] Cordero A., Hueso J.L., Martínez E., Torregrosa J.R., . A new technique to obtain derivative-free optimal iterative methods for solving nonlinear equations. J. Comput. Appl. Math.. 2013;252:95-102.
[Google Scholar]

[14] Cordero A., Lotfi T., Mahdiani K., Torregrosa J.R., . A stable family with high order of convergence for solving nonlinear equations. Appl. Math. Comput.. 2015;254:240-251.
[Google Scholar]

[15] Cordero A., Maimó J.G., Torregrosa J.R., Vassileva M.P., . Stability of a fourth order bi-parametric family of iterative methods. J. Comput. Appl. Math.. 2016;312:94-102.
[Google Scholar]

[16] Geum Y.H., Kim Y.I., Neta B., . Constructing a family of optimal eighth-order modified Newton-type multiple-zero finders along with the dynamics behind their purely imaginary extraneous fixed points. J. Comput. Appl. Math.. 2018;333:131-156.
[Google Scholar]

[17] King R.F., . A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal.. 1973;10:876-879.
[Google Scholar]

[18] Kogan T., Sapir L., Sapir A., Sapir A., . To the question of efficiency of iterative methods. Appl. Math. Lett.. 2017;66:40-46.
[Google Scholar]

[19] Kung H.T., Traub J.F., . Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach.. 1974;21:643-651.
[Google Scholar]

[20] Lee M.Y., Kim Y.I., . A family of fast derivative-free fourth-order multipoint optimal methods for nonlinear equations. Int. J. Comput. Math.. 2012;89(15):2081-2093.
[Google Scholar]

[21] Neta B., Chun C., . Basins of attraction for several optimal fourth order methods for multiple roots. Math. Comput. Simulat.. 2014;103:39-59.
[Google Scholar]

[22] Pandey B., Jaiswal J.P., . New seventh and eighth order derivative free methods for solving nonlinear equations. Tbilisi Math. J.. 2017;10(4):103-115.
[Google Scholar]

[23] Said Solaiman O., Hashim I., . Two new efficient sixth order iterative methods for solving nonlinear equations. J. King Saud Univ.–Sci.. 2019;31:701-705.
[CrossRef] [Google Scholar]

[24] Sharifi, S., Siegmund, S., Salimi, M., 2014. Solving nonlinear equations by a derivative-free form of the King’s family with memory. arXiv 1410.5867v1.

[25] Sharma J.R., Arora H., . Some novel optimal eighth order derivative-free root solvers and their basins of attraction. Appl. Math. Comput.. 2016;284:149-161.
[Google Scholar]

[26] Sharma J.R., Bahl A., . An optimal fourth order iterative method for solving nonlinear equations and its dynamics. J. Comp. Anal.. 2015;2015 Article ID 259167
[Google Scholar]

[27] Sharma J.R., Goyal R.K., . Fourth-order derivative-free methods for solving non-linear equations. Int. J. Comput. Math.. 2007;83(1):101-106.
[Google Scholar]

[28] Singh A., Jaiswal J.P., . A class of optimal eighth-order Steffensen-type iterative methods for solving nonlinear equations and their basins of attraction. Appl. Math. Inf. Sci.. 2016;10:251-257.
[Google Scholar]

[29] Soleymani F., Khattri S.K., Vanani S.K., . Two new classes of optimal Jarratt-type fourth order methods. Appl. Math. Lett.. 2012;25(5):847-853.
[Google Scholar]

[30] Steffensen J.F., . Remarks on iteration. Scand. Actuar. J.. 1933;1:64-72.
[Google Scholar]

[31] Traub J.F., . Iterative Methods for Solution of Equations. Englewood, Cliffs, NJ: Prentice-Hall; 1964.

[32] Waseem M., Noor M.A., Farooq A.S., Noor K.I., . An efficient technique to solve nonlinear equations using multiplicative calculus. Turk. J. Math.. 2018;42:679-691.
[Google Scholar]

[33] Weerakoon S., Fernando T.G.I., . A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett.. 2000;13:87-93.
[Google Scholar]

[34] Yasmin N., Zafar F., Akram S., . Optimal derivative-free root nding methods based on the Hermite interpolation. J. Nonlinear. Sci. Appl.. 2016;9:4427-4435.
[Google Scholar]

[35] Zafar F., Yasmin N., Akram S., Junjua M., . A general class of derivative free optimal root finding methods based on rational interpolation. The Sci. World J.. 2015;2015 Article ID 934260
[Google Scholar]

Optimal fourth- and eighth-order of convergence derivative-free modifications of King’s method

Abstract

Keywords

41-xx

65-xx

Root finding method

Iterative method

Order of convergence

King’s method

Nonlinear equations

1 Introduction

2 The new methods

2.1 Derivative-free optimal fourth-order iterative method

2.2 Derivative-free optimal eighth-order iterative method

2.3 Another derivative-free optimal eighth-order iterative method

3 Convergence analysis

4 Numerical examples

5 Conclusion

References

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