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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 979245, 13 pages
http://dx.doi.org/10.1155/2012/979245
Research Article

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

1Department of Applied Mathematical Sciences, School of Innovative Technologies and Engineering, University of Technology, Mauritius, La Tour Koenig, Pointe aux Sables, Mauritius
2Padé Research Centre, 39 Deanswood Hill, Leeds, West Yorkshire LS17 5JS, UK

Received 23 March 2012; Accepted 14 May 2012

Academic Editor: Songxiao Li

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.

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 [24]. 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 lim𝑛𝑥𝑛+1𝑥𝑥𝑛𝑥𝑝=𝐶(2.1) for 𝑝1, 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=𝑥𝑛𝑥, then the relation 𝑒𝑛+1=𝐶𝑒𝑝𝑛𝑒+𝑂𝑛𝑝+1𝑒=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=2𝑑1,(2.4) where 𝑝opt is the maximum order.
We use the approximate computational-order of convergence, COC [8] given by ||𝑥COClog𝑛+1𝑥||/||𝑥𝑛𝑥||||𝑥log𝑛𝑥||/||𝑥𝑛1𝑥||.(2.5)

3. Developments of the Methods

The second-order Newton-Raphson method is given by 𝜓2ndNR(𝑥)=𝑥𝑓(𝑥)𝑓.(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)=𝜓2ndNR𝑓𝜓(𝑥)2ndNR(𝑥)𝑓(𝑥)1+𝛽𝑡11+(𝛽2)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 [1012]. Recently, Thukral and Petković [13] developed a family of optimal eighth-order King-type IF given by 𝜓8thFTPK(𝑥)=𝜓4thFK𝜙𝑡(𝑥)1+4𝑡2+𝑡3𝑓𝜓4thFK(𝑥)𝑓,(𝑥)(3.3) where 𝑡1=𝑓𝜓2ndNR(𝑥)𝑓(𝑥),𝑡2=𝑓𝜓4thFK(𝑥)𝑓(𝑥),𝑡3=𝑓𝜓4thFK(𝑥)𝑓𝜓2ndNR(𝑥),(3.4) and 𝜙 is a weight function satisfying 𝜙(0)=1,𝜙(0)=2,𝜙(0)=104𝛽,𝜙(0)=12𝛽272𝛽+72.(3.5)

If we choose 𝜙𝑡1=1+𝛽𝑡1+(3/2)𝛽𝑡211+(𝛽2)𝑡1+((3/2)𝛽1)𝑡21(3.6) satisfying (3.5), we get a family of optimal eighth-order IF given by 𝜓8thFK(𝑥)=𝜓4thFK(𝑥)3𝑖=0𝜃𝑖𝑓𝜓4thFK(𝑥)𝑓(,𝑥)(3.7) where 𝜃0𝜃=1,1=1+𝛽𝑡1+(3/2)𝛽𝑡211+(𝛽2)𝑡1+((3/2)𝛽1)𝑡21𝜃1,2=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=𝑥𝑓(𝑥)𝑓,(𝑥)𝑧=𝑦𝐾1𝑢1𝑓(𝑦)𝑓,(𝑥)𝑠=𝑧𝐾2𝑢1,𝑢2,𝑢3𝑓(𝑠)𝑓,𝜓(𝑥)16thFGK1(𝑥)=𝑠𝐾3𝑢1,𝑢2,𝑢3,𝑢4𝑓(𝑠)𝑓,(𝑥)(3.9) where 𝐾1𝑢1=1+𝛽𝑢1+(9+5𝛽/2)𝑢211+(𝛽2)𝑢1,𝐾+(4+𝛽/2)2𝑢1,𝑢2,𝑢3=1+2𝑢1+(2+𝜎)𝑢31𝑢2+𝜎𝑢3,𝐾3𝑢1,𝑢2,𝑢3,𝑢4=1+2𝑢1+(2+𝜎)𝑢2𝑢31𝑢22𝑢3𝑢4+2(1+𝜎)𝑢2𝑢3+𝐾4𝑢1,𝑢3(3.10) are weighting functions, 𝐾42 is an analytic function in a region containing the region (0,0), 𝛽,𝜎 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,𝜎)=(2,2) and 𝐾4𝑢1,𝑢3=16𝑢1𝑢2124𝑢31122𝑢1𝑢+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)=𝑠1+2𝑢11𝑢22𝑢3𝑢4𝑓(𝑠)𝑓(𝑥)+𝐾5𝑢1,𝑢2,𝑢3,(3.13) where 𝐾53 is an analytic function in a region containing the region (0,0,0) and 𝛽,𝜎 are to be chosen freely.

We consider the case (𝛽,𝜎)=(24/11,2) and 𝐾5𝑢1,𝑢2,𝑢3=6𝑢31𝑢2244𝑢1141𝑢3+6𝑢23+𝑢12𝑢22+4𝑢32+𝑢32𝑢22(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)7𝑖=0𝜃𝑖𝑓𝜓8thFK(𝑥)𝑓,(𝑥)(3.15) where 𝜃4=𝑡5+𝑡1𝑡2,𝜃5=2𝑡1𝑡5+4(1𝛽)𝑡31𝑡3+2𝑡2𝑡3,𝜃6=2𝑡6+7𝛽2472𝑡𝛽+143𝑡41+(2𝛽3)𝑡22+(52𝛽)𝑡5𝑡21𝑡33,𝜃7=8𝑡4+12𝛽+12+2𝛽2𝑡5𝑡314𝑡33+𝑡12𝛽2𝑡22+12𝛽23𝑡31+46+1272𝛽2105𝛽10𝛽3𝑡2𝑡41,𝑡(3.16)4=𝑓𝜓8th𝐹𝐾(𝑥)𝑓(𝑥),𝑡5=𝑓𝜓8th𝐹𝐾(𝑥)𝑓𝜓4th𝐹𝐾(𝑥),𝑡6=𝑓𝜓8th𝐹𝐾(𝑥)𝑓𝜓2nd𝑁𝑅(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 𝑓𝐷 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)=𝑓𝑥𝑒𝑛+𝑐2𝑒2𝑛+𝑐3𝑒3𝑛+𝑐4𝑒4𝑛,𝑓+(4.2)(𝑥)=𝑓𝑥1+2𝑐2𝑒𝑛+3𝑐3𝑒2𝑛+4𝑐4𝑒3𝑛,+(4.3) so that 𝑢(𝑥)=𝑒𝑛𝑐2𝑒2𝑛𝑐+222𝑐3𝑒3𝑛+7𝑐2𝑐34𝑐323𝑐4𝑒4𝑛𝜓+,(4.4)2ndNR(𝑥)𝑥=𝑐2𝑒2𝑛𝑐222𝑐3𝑒3𝑛7𝑐2𝑐34𝑐323𝑐4𝑒4𝑛+.(4.5)
Now, the Taylor expansion of 𝑓(𝑦) about 𝑥 gives 𝑓(𝑦)=𝑓𝑥𝑦𝑥+𝑐2𝑦𝑥2+𝑐3𝑦𝑥3+𝑐4𝑦𝑥4+.(4.6)
Using (4.2), (4.6), and (4.5), we have 𝑡1=𝑐2𝑒𝑛+2𝑐33𝑐22𝑒2𝑛+3𝑐410𝑐2𝑐3+8𝑐32𝑒3𝑛+14𝑐2𝑐4+37𝑐3𝑐2220𝑐428𝑐23+4𝑐5𝑒4𝑛+,(4.7) so that 𝜓4thFK(𝑥)𝑥=(1+2𝛽)𝑐32𝑐2𝑐3𝑒4𝑛+.(4.8)
Similarly, we have 𝑡2=(1+2𝛽)𝑐32𝑐2𝑐3𝑒3𝑛+2𝑐23+(9+12𝛽)𝑐22𝑐3+514𝛽2𝛽2𝑐422𝑐2𝑐4𝑒4𝑛𝑡+,(4.9)3=(1+2𝛽)𝑐32𝑐2𝑐3𝑒2𝑛+(4+8𝛽)𝑐2𝑐3+28𝛽2𝛽2𝑐322𝑐4𝑒3𝑛+(4.10) so that 𝜃0𝜃=1,1=(4𝛽+2)𝑐222𝑐3𝑒2𝑛+(16𝛽+8)𝑐2𝑐3+16𝛽4𝛽2𝑐4324𝑐4𝑒3𝑛𝜃+,2=(1+2𝛽)𝑐32𝑐2𝑐3𝑒2𝑛+(4+8𝛽)𝑐2𝑐3+28𝛽2𝛽2𝑐322𝑐4𝑒3𝑛𝜃+,3=(4+8𝛽)𝑐324𝑐2𝑐3𝑒3𝑛+,(4.11) and finally we get 𝜓8thFK(𝑥)𝑥=𝑐3𝑐22+(1+2𝛽)𝑐42𝑐4𝑐2𝑐33+(16+4𝛽)𝑐23𝑐32+271112𝛽𝛽2𝑐52𝑐3+12+952𝛽+44𝛽26𝛽3𝑐72𝑒8𝑛+.(4.12)
By a similar argument, we have 𝑡4=𝑐3𝑐22+(1+2𝛽)𝑐42𝑐4𝑐2𝑐33+(16+4𝛽)𝑐32𝑐23+271112𝛽𝛽2𝑐52𝑐3+12+952𝛽+44𝛽26𝛽3𝑐72𝑒7𝑛𝑡+,5=𝑐23+(152𝛽)𝑐22𝑐3+123𝛽2+472𝛽𝑐42+𝑐2𝑐4𝑒4𝑛𝑡+,6=𝑐2𝑐3+(1+2𝛽)𝑐32𝑐4𝑐33+(16+4𝛽)𝑐22𝑐23+271112𝛽𝛽2𝑐24𝑐3+12+952𝛽+44𝛽26𝛽3𝑐62𝑒6𝑛+,(4.13) so that 𝜃4=𝑐32+(162𝛽)𝑐22𝑐3+133𝛽2+512𝛽𝑐42+𝑐2𝑐4𝑒4𝑛𝜃+,5=4𝑐2𝑐23+(388𝛽)𝑐32𝑐3+306𝛽2𝑐+59𝛽52+2𝑐22𝑐4𝑒5𝑛𝜃+,6=2𝑐2𝑐3+(7+2𝛽)𝑐32𝑐4𝑐33+(31+2𝛽)𝑐22𝑐23+134𝛽2952𝛽𝑐42𝑐3+30𝛽2+94+8𝛽3𝑐+177𝛽62𝑒6𝑛𝜃+,7=8𝑐3𝑐22+20+4𝛽+2𝛽2𝑐42𝑐44𝑐2𝑐33+(8𝛽+106)𝑐32𝑐23+1472𝛽2+14𝛽3𝑐38671𝛽52𝑐3+3572𝛽234𝛽4+260+405𝛽+160𝛽3𝑐72𝑒7𝑛+,(4.14) and finally we get 𝜓16thFK(𝑥)𝑥=𝑐32𝑐43+𝑐23𝑐42+(4𝛽+2)𝑐62𝑐3+14𝛽4𝛽2𝑐82𝑐4+(6𝛽+17)𝑐52𝑐33+1282𝛽3139𝛽2+12𝛽41432𝛽𝑐211+9𝛽2431832𝛽𝑐72𝑐23+39+156𝛽2+157𝛽4𝛽3𝑐92𝑐3𝑐5+𝑐42𝑐3+(1+2𝛽)𝑐62𝑐34+(16𝛽+66)𝑐52𝑐23+4𝛽4+81+219𝛽+124𝛽2+18𝛽3𝑐92+189𝛽+2𝛽326𝛽2𝑐14372𝑐34𝑐32𝑐33𝑐42+(22𝛽+140)𝑐42𝑐43+9𝛽4210𝛽3+68054𝛽2𝑐+3933+4784𝛽82𝑐23+2𝛽3699𝛽134934𝛽2𝑐62𝑐334𝑐22𝑐53+195394𝛽2+891𝛽4425510278𝛽2682𝛽372𝛽5𝑐210𝑐3+1535+92𝛽6+118232𝛽2+5963𝛽+28292𝛽3+1863𝛽4882𝛽5𝑐212𝑐4+20627𝛽6+4028𝛽7240𝛽8+880332𝛽+2603214𝛽2+1343638𝛽3+401714𝛽4+588392𝛽5𝑐+8484215𝑐73𝑐2+(6𝛽+60)𝑐32𝑐63+𝛽27772𝑐𝛽113452𝑐53+3778𝛽6325412349972𝛽+609292𝛽560462𝛽4+62𝛽795165𝛽2754994𝛽3𝑐213𝑐3+125672𝛽242𝛽3+45854𝛽2+17𝛽4𝑐+876972𝑐43+177𝛽61332𝛽5529234𝛽4+46415+3549478𝛽3+103054𝛽+1675054𝛽2𝑐211𝑐23+35257𝛽30052108𝛽5748754𝛽2+7252𝛽3+920𝛽4𝑐92𝑐33𝑒𝑛16+.(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. 𝑎 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 |𝑥3𝑥| 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 𝑥3𝑥 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 |𝑥3𝑥| for 𝑓4,𝑓6,𝑓7 when compared to Geum and Kim sixteenth order IF.

tab1
Table 1: Test functions and their roots.
tab2
Table 2: Range and the best value of 𝛽 for the 16th FK family.
tab3
Table 3: Comparison of optimal 4-point sixteenth-order I.F.s.

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+1𝑥𝑛|<1013 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+1𝑥𝑛|<1013. 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(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 𝑥0 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.

fig1
Figure 1: Behaviour of 16th FK family for the function 𝑓2.
fig2
Figure 2: Behaviour of 16th FK family for the function 𝑓7.
979245.fig.003
Figure 3: Behaviour of the number of iterations for convergence with the starting point 𝑥0 of the member 𝛽=0.2 of the 16th FK family for the function 𝑓7.

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 𝛽0.4. 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.

fig4
Figure 4: Behaviour of 16th FK family for the Cubic function.

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.

fig5
Figure 5: Behaviour of 16th FK family for the Oscillatory Cubic function.

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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