International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 979245 | https://doi.org/10.1155/2012/979245

Diyashvir Kreetee Rajiv Babajee, Rajinder Thukral, "On a 4-Point Sixteenth-Order King Family of Iterative Methods for Solving Nonlinear Equations", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 979245, 13 pages, 2012. https://doi.org/10.1155/2012/979245

On a 4-Point Sixteenth-Order King Family of Iterative Methods for Solving Nonlinear Equations

Academic Editor: Songxiao Li
Received23 Mar 2012
Accepted14 May 2012
Published16 Aug 2012

Abstract

A one-parameter 4-point sixteenth-order King-type family of iterative methods which satisfy the famous Kung-Traub conjecture is proposed. The convergence of the family is proved, and numerical experiments are carried out to find the best member of the family. In most experiments, the best member was found to be a sixteenth-order Ostrowski-type method.

1. Introduction

Solutions of nonlinear equations by iterative methods have been of great interest to numerical analysts. One of the popular methods is the classic Newton method (Newton Raphson method). It has quadratic convergence close to the root, that is the number of good digits is roughly doubled at each iteration. Higher order methods which require the second or higher order derivatives can be costly and thus time consuming. Also, the Newton method can suffer from numerical instabilities. It is consequently important to study higher order variants of Newton's method, which require only function and first derivative calculation and are more robust as compared to Newton's method. Such methods are known as multipoint Newton-Like methods in the Traub sense [1]. Multipoint methods without memory are methods that use new information at a number of points. It is an efficient way of generating higher order methods free from second and higher order derivatives. For a survey of these methods, please refer to [2–4]. In this work, we develop a one-parameter 4-point sixteenth-order King-type family of iterative methods, which satisfy the famous Kung-Traub conjecture. We prove the local convergence of the methods and its asymptotic error constant. We test our methods by varying the parameter of the family in a suitable interval and obtain the best value of the parameter for the methods with the highest computational order of convergence. We also compare our methods with other optimal sixteenth order methods. Furthermore, we test the family with two more nonlinear functions and find the best member based on the highest number of successful converging points and lowest mean iteration number.

2. Preliminaries

Let 𝑥𝑛+1=𝜓(𝑥𝑛) define an iterative function (IF).

Definition 2.1 (see [6]). If the sequence {𝑥𝑛} tends to a limit 𝑥∗ in such a way that limğ‘›â†’âˆžğ‘¥ğ‘›+1−𝑥∗𝑥𝑛−𝑥∗𝑝=𝐶(2.1) for 𝑝≥1, then the order of convergence of the sequence is said to be 𝑝, and 𝐶 is known as the asymptotic error constant. If 𝑝=1, 𝑝=2, or 𝑝=3, the convergence is said to be linear, quadratic, or cubic, respectively.
Letting 𝑒𝑛=𝑥𝑛−𝑥∗, then the relation 𝑒𝑛+1=𝐶𝑒𝑝𝑛𝑒+𝑂𝑛𝑝+1𝑒=𝑂𝑝𝑛(2.2) is called the error equation. The value of 𝑝 is called the order of convergence of the method.

Definition 2.2 (see [6]). The efficiency index is given by 𝐼𝐸=𝑝1/𝑑,(2.3) where 𝑑 is the total number of new function evaluations (the values of 𝑓 and its derivatives) per iteration.

Kung-Traub Conjecture (see [7])
Let 𝜓 be an IF without memory with 𝑑 evaluations. Then 𝑝(𝜓)≤𝑝opt=2𝑑−1,(2.4) where 𝑝opt is the maximum order.
We use the approximate computational-order of convergence, COC [8] given by ||𝑥COC≈log𝑛+1−𝑥∗||/||𝑥𝑛−𝑥∗||||𝑥log𝑛−𝑥∗||/||𝑥𝑛−1−𝑥∗||.(2.5)

3. Developments of the Methods

The second-order Newton-Raphson method is given by 𝜓2ndNR(𝑥)=𝑥−𝑓(𝑥)ğ‘“î…ž.(𝑥)(3.1)

It is an optimal 1-point IF with efficiency index of 1.414.

A one-parameter King family of fourth-order IF [9] is given by 𝜓4thFK(𝑥)=𝜓2ndNR𝑓𝜓(𝑥)−2ndNR(𝑥)ğ‘“î…ž(𝑥)1+𝛽𝑡11+(𝛽−2)𝑡1.(3.2)

The members of the family are 2-point I.F.s with efficiency index of 1.587. The case 𝛽=0 corresponds to the famous Ostrowski method [6].

Several optimal eight-order methods are developed in [10–12]. Recently, Thukral and Petković [13] developed a family of optimal eighth-order King-type IF given by 𝜓8thFTPK(𝑥)=𝜓4thFK𝜙𝑡(𝑥)−1+4𝑡2+𝑡3𝑓𝜓4thFK(𝑥)ğ‘“î…ž,(𝑥)(3.3) where 𝑡1=𝑓𝜓2ndNR(𝑥)𝑓(𝑥),𝑡2=𝑓𝜓4thFK(𝑥)𝑓(𝑥),𝑡3=𝑓𝜓4thFK(𝑥)𝑓𝜓2ndNR(𝑥),(3.4) and 𝜙 is a weight function satisfying 𝜙(0)=1,ğœ™î…ž(0)=2,ğœ™î…žî…ž(0)=10−4𝛽,ğœ™î…žî…žî…ž(0)=12𝛽2−72𝛽+72.(3.5)

If we choose 𝜙𝑡1=1+𝛽𝑡1+(3/2)𝛽𝑡211+(𝛽−2)𝑡1+((3/2)𝛽−1)𝑡21(3.6) satisfying (3.5), we get a family of optimal eighth-order IF given by 𝜓8thFK(𝑥)=𝜓4thFK(𝑥)−3𝑖=0𝜃𝑖𝑓𝜓4thFK(𝑥)ğ‘“î…ž(,𝑥)(3.7) where 𝜃0𝜃=1,1=1+𝛽𝑡1+(3/2)𝛽𝑡211+(𝛽−2)𝑡1+((3/2)𝛽−1)𝑡21𝜃−1,2=𝑡3,𝜃3=4𝑡2.(3.8)

The members of the family are 3-point eighth-order I.F.s with efficiency index of 1.682.

Geum and Kim [14] developed a biparametric family of optimally convergent sixteenth-order 4-point I.F. with their fourth-step weighting function as a sum of a rational and a generic two-variable function: 𝑦=𝑥−𝑓(𝑥)ğ‘“î…ž,(𝑥)𝑧=𝑦−𝐾1𝑢1𝑓(𝑦)ğ‘“î…ž,(𝑥)𝑠=𝑧−𝐾2𝑢1,𝑢2,𝑢3𝑓(𝑠)ğ‘“î…ž,𝜓(𝑥)16thFGK1(𝑥)=𝑠−𝐾3𝑢1,𝑢2,𝑢3,𝑢4𝑓(𝑠)ğ‘“î…ž,(𝑥)(3.9) where 𝐾1𝑢1=1+𝛽𝑢1+(−9+5𝛽/2)𝑢211+(𝛽−2)𝑢1,𝐾+(−4+𝛽/2)2𝑢1,𝑢2,𝑢3=1+2𝑢1+(2+ğœŽ)𝑢31−𝑢2+ğœŽğ‘¢3,𝐾3𝑢1,𝑢2,𝑢3,𝑢4=1+2𝑢1+(2+ğœŽ)𝑢2𝑢31−𝑢2−2𝑢3−𝑢4+2(1+ğœŽ)𝑢2𝑢3+𝐾4𝑢1,𝑢3(3.10) are weighting functions, 𝐾4∶ℂ2→ℂ is an analytic function in a region containing the region (0,0), 𝛽,ğœŽ are to be chosen freely, and 𝑢1=𝑓(𝑦)𝑓(𝑥),𝑢2=𝑓(𝑧)𝑓(𝑦),𝑢3=𝑓(𝑧)𝑓(𝑥),𝑢4=𝑓(𝑠).𝑓(𝑧)(3.11)

We consider the case (𝛽,ğœŽ)=(2,−2) and 𝐾4𝑢1,𝑢3=1−6𝑢1−𝑢21−24𝑢31−12−2𝑢1𝑢+232(3.12) for numerical experiments and term the I.F. as 16th GK1.

Geum and Kim [15] proposed another family of optimal sixteenth-order 4-point I.F.s with a linear fraction plus a trivariate polynomial as the fourth-step weighting function. Their family is given by 𝜓16thFGK2(𝑥)=𝑠−1+2𝑢11−𝑢2−2𝑢3−𝑢4𝑓(𝑠)ğ‘“î…ž(𝑥)+𝐾5𝑢1,𝑢2,𝑢3,(3.13) where 𝐾5∶ℂ3→ℂ is an analytic function in a region containing the region (0,0,0) and 𝛽,ğœŽ are to be chosen freely.

We consider the case (𝛽,ğœŽ)=(24/11,−2) and 𝐾5𝑢1,𝑢2,𝑢3=−6𝑢31𝑢2−244𝑢1141𝑢3+6𝑢23+𝑢12𝑢22+4𝑢32+𝑢3−2𝑢22(3.14) for numerical experiments and term the I.F. as 16th GK2.

We observe that the 16th FGK1 and 16th FGK2 family of IFs require two parameters and an analytic function. Therefore, we develop a simplified one-parameter optimal 4-point sixteenth-order King-type family of IFs based on 4th FK and 8th FK families. We propose the following family: 𝜓16thFK(𝑥)=𝜓8thFK(𝑥)−7𝑖=0𝜃𝑖𝑓𝜓8thFK(𝑥)ğ‘“î…ž,(𝑥)(3.15) where 𝜃4=𝑡5+𝑡1𝑡2,𝜃5=2𝑡1𝑡5+4(1−𝛽)𝑡31𝑡3+2𝑡2𝑡3,𝜃6=2𝑡6+7𝛽2−472𝑡𝛽+143𝑡41+(2𝛽−3)𝑡22+(5−2𝛽)𝑡5𝑡21−𝑡33,𝜃7=8𝑡4+−12𝛽+12+2𝛽2𝑡5𝑡31−4𝑡33+𝑡1−2𝛽2𝑡−22+12𝛽23𝑡31+46+1272𝛽2−105𝛽−10𝛽3𝑡2𝑡41,𝑡(3.16)4=𝑓𝜓8th𝐹𝐾(𝑥)𝑓(𝑥),𝑡5=𝑓𝜓8th𝐹𝐾(𝑥)𝑓𝜓4th𝐹𝐾(𝑥),𝑡6=𝑓𝜓8th𝐹𝐾(𝑥)𝑓𝜓2nd𝑁𝑅(𝑥).(3.17)

4. Convergence Analysis of the 16th FK Family of IFs

In this section, we prove the local and sixteenth-order of the 16th FK family of I.F.s using classical Taylor expansion.

Theorem 4.1. Let a sufficiently smooth function 𝑓∶𝐷⊂ℝ→ℝ have a simple root 𝑥∗ in the open interval 𝐷. Then the class of methods without memory (3.15) is of local sixteenth-order convergence.

Proof. Let 𝑐𝑗=𝑓(𝑗)(𝑥∗)𝑗!𝑓′(𝑥∗),𝑗=2,3,4,….(4.1)
Using the Taylor series and the symbolic software such as Maple we have 𝑓(𝑥)=ğ‘“î…žî€·ğ‘¥âˆ—ğ‘’î€¸î€ºğ‘›+𝑐2𝑒2𝑛+𝑐3𝑒3𝑛+𝑐4𝑒4𝑛,𝑓+⋯(4.2)(𝑥)=ğ‘“î…žî€·ğ‘¥âˆ—î€¸î€º1+2𝑐2𝑒𝑛+3𝑐3𝑒2𝑛+4𝑐4𝑒3𝑛,+⋯(4.3) so that 𝑢(𝑥)=𝑒𝑛−𝑐2𝑒2𝑛𝑐+222−𝑐3𝑒3𝑛+7𝑐2𝑐3−4𝑐32−3𝑐4𝑒4𝑛𝜓+⋯,(4.4)2ndNR(𝑥)−𝑥∗=𝑐2𝑒2𝑛𝑐−222−𝑐3𝑒3𝑛−7𝑐2𝑐3−4𝑐32−3𝑐4𝑒4𝑛+⋯.(4.5)
Now, the Taylor expansion of 𝑓(𝑦) about 𝑥∗ gives 𝑓(𝑦)=ğ‘“î…žî€·ğ‘¥âˆ—î€¸î‚ƒî€·ğ‘¦âˆ’ğ‘¥âˆ—î€¸+𝑐2𝑦−𝑥∗2+𝑐3𝑦−𝑥∗3+𝑐4𝑦−𝑥∗4+⋯.(4.6)
Using (4.2), (4.6), and (4.5), we have 𝑡1=𝑐2𝑒𝑛+2𝑐3−3𝑐22𝑒2𝑛+3𝑐4−10𝑐2𝑐3+8𝑐32𝑒3𝑛+−14𝑐2𝑐4+37𝑐3𝑐22−20𝑐42−8𝑐23+4𝑐5𝑒4𝑛+⋯,(4.7) so that 𝜓4thFK(𝑥)−𝑥∗=(1+2𝛽)𝑐32−𝑐2𝑐3𝑒4𝑛+⋯.(4.8)
Similarly, we have 𝑡2=(1+2𝛽)𝑐32−𝑐2𝑐3𝑒3𝑛+−2𝑐23+(9+12𝛽)𝑐22𝑐3+−5−14𝛽−2𝛽2𝑐42−2𝑐2𝑐4𝑒4𝑛𝑡+⋯,(4.9)3=(1+2𝛽)𝑐32−𝑐2𝑐3𝑒2𝑛+(4+8𝛽)𝑐2𝑐3+−2−8𝛽−2𝛽2𝑐32−2𝑐4𝑒3𝑛+⋯(4.10) so that 𝜃0𝜃=1,1=(4𝛽+2)𝑐22−2𝑐3𝑒2𝑛+(16𝛽+8)𝑐2𝑐3+−16𝛽−4𝛽2𝑐−432−4𝑐4𝑒3𝑛𝜃+⋯,2=(1+2𝛽)𝑐32−𝑐2𝑐3𝑒2𝑛+(4+8𝛽)𝑐2𝑐3+−2−8𝛽−2𝛽2𝑐32−2𝑐4𝑒3𝑛𝜃+⋯,3=(4+8𝛽)𝑐32−4𝑐2𝑐3𝑒3𝑛+⋯,(4.11) and finally we get 𝜓8thFK(𝑥)−𝑥∗=−𝑐3𝑐22+(1+2𝛽)𝑐42𝑐4−𝑐2𝑐33+(16+4𝛽)𝑐23𝑐32+−27−1112𝛽−𝛽2𝑐52𝑐3+12+952𝛽+44𝛽2−6𝛽3𝑐72𝑒8𝑛+….(4.12)
By a similar argument, we have 𝑡4=−𝑐3𝑐22+(1+2𝛽)𝑐42𝑐4−𝑐2𝑐33+(16+4𝛽)𝑐32𝑐23+−27−1112𝛽−𝛽2𝑐52𝑐3+12+952𝛽+44𝛽2−6𝛽3𝑐72𝑒7𝑛𝑡+⋯,5=𝑐23+(−15−2𝛽)𝑐22𝑐3+12−3𝛽2+472𝛽𝑐42+𝑐2𝑐4𝑒4𝑛𝑡+⋯,6=−𝑐2𝑐3+(1+2𝛽)𝑐32𝑐4−𝑐33+(16+4𝛽)𝑐22𝑐23+−27−1112𝛽−𝛽2𝑐24𝑐3+12+952𝛽+44𝛽2−6𝛽3𝑐62𝑒6𝑛+⋯,(4.13) so that 𝜃4=𝑐32+(−16−2𝛽)𝑐22𝑐3+13−3𝛽2+512𝛽𝑐42+𝑐2𝑐4𝑒4𝑛𝜃+⋯,5=4𝑐2𝑐23+(−38−8𝛽)𝑐32𝑐3+30−6𝛽2𝑐+59𝛽52+2𝑐22𝑐4𝑒5𝑛𝜃+⋯,6=−2𝑐2𝑐3+(7+2𝛽)𝑐32𝑐4−𝑐33+(31+2𝛽)𝑐22𝑐23+−134−𝛽2−952𝛽𝑐42𝑐3+−30𝛽2+94+8𝛽3𝑐+177𝛽62𝑒6𝑛𝜃+⋯,7=−8𝑐3𝑐22+20+4𝛽+2𝛽2𝑐42𝑐4−4𝑐2𝑐33+(8𝛽+106)𝑐32𝑐23+−1472𝛽2+14𝛽3𝑐−386−71𝛽52𝑐3+−3572𝛽2−34𝛽4+260+405𝛽+160𝛽3𝑐72𝑒7𝑛+⋯,(4.14) and finally we get 𝜓16thFK(𝑥)−𝑥∗=−𝑐32𝑐43+−𝑐23𝑐42+(4𝛽+2)𝑐62𝑐3+−1−4𝛽−4𝛽2𝑐82𝑐4+(6𝛽+17)𝑐52𝑐33+−12−82𝛽3−139𝛽2+12𝛽4−1432𝛽𝑐211+−9𝛽2−43−1832𝛽𝑐72𝑐23+39+156𝛽2+157𝛽−4𝛽3𝑐92𝑐3𝑐5+−𝑐42𝑐3+(1+2𝛽)𝑐62𝑐34+(16𝛽+66)𝑐52𝑐23+−4𝛽4+81+219𝛽+124𝛽2+18𝛽3𝑐92+−189𝛽+2𝛽3−26𝛽2𝑐−14372𝑐3−4𝑐32𝑐33𝑐42+(22𝛽+140)𝑐42𝑐43+9𝛽4−210𝛽3+68054𝛽2𝑐+3933+4784𝛽82𝑐23+2𝛽3−699𝛽−1349−34𝛽2𝑐62𝑐33−4𝑐22𝑐53+−195394𝛽2+891𝛽4−4255−10278𝛽−2682𝛽3−72𝛽5𝑐210𝑐3+1535+92𝛽6+118232𝛽2+5963𝛽+28292𝛽3+1863𝛽4−882𝛽5𝑐212𝑐4+−20627𝛽6+4028𝛽7−240𝛽8+880332𝛽+2603214𝛽2+1343638𝛽3+401714𝛽4+588392𝛽5𝑐+8484215−𝑐73𝑐2+(6𝛽+60)𝑐32𝑐63+𝛽2−7772𝑐𝛽−113452𝑐53+−3778𝛽6−32541−2349972𝛽+609292𝛽5−60462𝛽4+62𝛽7−95165𝛽2−754994𝛽3𝑐213𝑐3+125672𝛽−242𝛽3+45854𝛽2+17𝛽4𝑐+876972𝑐43+177𝛽6−1332𝛽5−529234𝛽4+46415+3549478𝛽3+103054𝛽+1675054𝛽2𝑐211𝑐23+−35257𝛽−30052−108𝛽5−748754𝛽2+7252𝛽3+920𝛽4𝑐92𝑐33𝑒𝑛16+….(4.15)
In the next section, we carry out numerical experiments to find the best member of the family and compare it to the Geum and Kim sixteenth-order IFs

5. Numerical Experiments

The test functions and their exact root 𝑥∗ are displayed in Table 1. The approximation 𝑥𝑛 is calculated by using the same total number of function evaluations (TNFE) for all I.F.s considered. In the calculations, 15 TNFE are used by each I.F. For the 16th FK family, we choose a suitable range of values of 𝛽, which are based on the initial approximation of the root. ğ‘Ž indicates the values of 𝛽 excluded in the range because of invalid estimate. The best value of 𝛽 is chosen based on the smallest value of |𝑥3−𝑥∗| and the highest computational-order of convergence (COC). The range and best value of 𝛽 are given in Table 2 for each function with its starting point. For most functions, the best value of 𝛽=0, which corresponds to optimal 4-point sixteenth-order Ostrowski-type I.F. Furthermore, the approximation 𝑥3−𝑥∗ and the computational order of convergence (COC) for the best member of the 16th FK family and the 16thGK1 and 16thGK2 IFs are displayed in Table 3. The results show that the best member of the 16th FK family gives the smallest value of |𝑥3−𝑥∗| for 𝑓4,𝑓6,𝑓7 when compared to Geum and Kim sixteenth order IF.


𝑓 1 ( 𝑥 ) = e x p ( 𝑥 ) s i n ( 𝑥 ) + l n ( 1 + 𝑥 2 ) 𝑥 ∗ = 0
𝑓 2 ( 𝑥 ) = 𝑥 1 5 + 𝑥 4 + 4 𝑥 2 − 1 5 𝑥 ∗ = 1 . 1 4 8 5 3 8 …
𝑓 3 ( 𝑥 ) = ( 𝑥 − 2 ) ( 𝑥 1 0 + 𝑥 + 1 ) e x p ( − 𝑥 − 1 ) 𝑥 ∗ = 2
𝑓 4 ( 𝑥 ) = ( 𝑥 + 1 ) e x p ( s i n ( 𝑥 ) ) − 𝑥 2 e x p ( c o s ( 𝑥 ) ) − 1 𝑥 ∗ = 0
𝑓 5 ( 𝑥 ) = s i n 2 ( 𝑥 ) − 𝑥 2 + 1 𝑥 ∗ = 1 . 4 0 4 4 9 1 6 5 …
𝑓 6 ( 𝑥 ) = e x p ( − 𝑥 ) − c o s ( 𝑥 ) 𝑥 ∗ = 0
𝑓 7 ( 𝑥 ) = l n ( 𝑥 2 + 𝑥 + 2 ) − 𝑥 + 1 𝑥 ∗ = 4 . 1 5 2 5 9 0 7 4 …


FunctionRangeBest value of 𝛽

𝑓 1 , 𝑥 0 = 1 [ − 5 , 5 ] a , ğ‘Ž = − 3 , − 2 0
𝑓 2 , 𝑥 0 = 1 . 3 [ − 5 , 5 ] b , 𝑏 = − 3 , − 2 , − 1 0
𝑓 3 , 𝑥 0 = 2 . 5 [ − 5 , 5 ] c , 𝑐 = − 1 0
𝑓 4 , 𝑥 0 = 0 . 2 5 [ − 5 , 5 ] 1
𝑓 5 , 𝑥 0 = 2 . 5 [ − 2 , 5 ] 0
𝑓 6 , 𝑥 0 = 1 / 6 [ − 5 , 4 ] 0
𝑓 7 , 𝑥 0 = 3 . 5 [ − 5 , 5 ] − 1


function16th FK16th G K 1 16th G K 2
| 𝑥 3 − 𝑥 ∗ | COC | 𝑥 3 − 𝑥 ∗ | COC | 𝑥 3 − 𝑥 ∗ | COC

𝑓 _ 1 , 𝑥 _ 0 = 1 0.137e-36215.9890.353e-58115.9810.103e-52215.998
𝑓 _ 2 , 𝑥 _ 0 = 1 . 3 0.898e-67016.0000.994e-78216.0000.723e-81115.987
𝑓 _ 3 , 𝑥 _ 0 = 2 . 5 0.479e-20015.928failfail0.819e-31615.978
𝑓 _ 4 , 𝑥 _ 0 = 0 . 2 5 0.492e-315516.0000.241e-245516.0000.516e-207116.000
𝑓 _ 5 , 𝑥 _ 0 = 2 . 5 0.142e-81016.0000.164e-114715.9930.452e-102016.000
𝑓 _ 6 , 𝑥 _ 0 = 1 / 6 0.224e-170216.0000.172e-100415.9920.782e-89216.000
𝑓 _ 7 , 𝑥 _ 0 = 3 . 5 0.927e-446416.0000.241e-414416.0000.136e-376316.000

We next test the 16th FK family by varying the starting points. Let us consider the functions 𝑓2 and 𝑓7. We focus on the behaviour of the IFs with the starting points, which are equally spaced with Δ𝑥=0.1 in the intervals (−3.9,6.1] for 𝑓2 and (−0.9,9.1] for 𝑓7 to check the robustness of the IFs. A starting point was considered as divergent if it does not satisfy the condition |𝑥𝑛+1−𝑥𝑛|<10−13 in at most 100 iterations. We denote the quantity 𝜔𝑐 as the mean number of iterations from a successful starting point until convergence with |𝑥𝑛+1−𝑥𝑛|<10−13. Let 𝑁𝑠 denote the number of successful points of 100 starting points. We test for 101 𝛽 of the family with Δ𝛽=0.1 in the interval [−5,5]. Figure 1 shows the variation of the converging points and mean iteration number with respect to 𝛽 for the function 𝑓2. We can observe the family is globally convergent for the values of 𝛽∈[−1.2,−0.7] and 𝛽=−0.3,0.1,0.5,0.9. It is the member 𝛽=−0.8 that has the smallest mean iteration number and is the most efficient member for 𝑓2. In Figure 2, we observe that the family is globally convergent for the function 𝑓7 for all given values of 𝛽. It is the member 𝛽=0.2 which has the lowest mean iteration number. Figure 3 shows the number of iterations needed to achieve convergence is 2 for any starting point in the interval [2,8.3] enclosing the root. This illustrates the high speed of convergence of the method. That is, higher order I.F. can converge in few iterations even if the starting point is not very close to the root. We consider two more test functions, one of which is of simple cubic type [16, 17] 𝑓8(𝑥)=𝑥3+ln𝑥,𝑥>0,𝑥∈ℝ(5.1) for which the logarithm restricts the function to be positive and its convex properties of the function are favorable for global convergence [16, 17]. We test for 100 starting points in the interval (0,10]. The root 𝑥∗=0.704709490254913 correct to 14 digits. A starting point was considered as divergent if it does not satisfy the convergence condition in at most 100 iterations together with 𝑥≤0 at any iterates. A similar analysis is performed for another test function, the Oscillatory Cubic [16, 17]: 𝑓9(𝑥)=𝑥3+ln𝑥+0.15cos(50𝑥)(5.2) in which the single root has been moved marginally to 𝑥∗=0.717519716444759 but many local extrema have been introduced on a small scale [17]. This means that when the iterates of the I.F.s fall in the region where 𝑓′(𝑥)=0, they become zero or negative, causing them to diverge.

Figure 4 shows the variation of the converging points and mean iteration number with respect to 𝛽 for the Cubic function. It can be observed that the 16th FK family is globally convergent for 𝛽=−5,−0.1,0,0.1, and 𝛽≥0.4. This is the member 𝛽=0, which is the most efficient I.F. since it is globally convergent with the smallest mean iteration number of 3.25. We note that the family has many diverging points for negative values of 𝛽.

Figure 5 shows the variation of the converging points and mean iteration number with respect to 𝛽 for the Oscillatory Cubic function. It can be observed that due to the perturbations the 16th FK family has difficulty with this function because its members have less than 40% of starting points successfully converging. The mean of the IFs has also risen. The most efficient member of the family with the highest number of converging points (38) is the member 𝛽=3 with mean 17.

6. Conclusion

We develop a 4-point sixteenth order King family of iterative methods. We prove the local convergence of the methods. We test the family via some numerical experiments to find the best member, which corresponds to a sixteenth-order Ostrowski method for most cases.

Acknowledgments

The authors are grateful to the unknown referees for their valuable comments to improve the paper. D. K. R. Babajee is an IEEE member.

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Copyright © 2012 Diyashvir Kreetee Rajiv Babajee and Rajinder Thukral. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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