IJMMS International Journal of Mathematics and Mathematical Sciences 1687-0425 0161-1712 Hindawi Publishing Corporation 979245 10.1155/2012/979245 979245 Research Article On a 4-Point Sixteenth-Order King Family of Iterative Methods for Solving Nonlinear Equations Babajee Diyashvir Kreetee Rajiv 1 Thukral Rajinder 2 Li Songxiao 1 Department of Applied Mathematical Sciences School of Innovative Technologies and Engineering University of Technology Mauritius La Tour Koenig Pointe aux Sables Mauritius uts.edu.au 2 Padé Research Centre 39 Deanswood Hill Leeds West Yorkshire LS17 5JS UK 2012 16 08 2012 2012 23 03 2012 14 05 2012 2012 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.

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 . 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 . 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 [<xref ref-type="bibr" rid="B12">6</xref>]).

If the sequence {xn} tends to a limit x* in such a way that (2.1)limnxn+1-x*(xn-x*)p=C 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=xn-x*, then the relation (2.2)en+1=C    enp+O(enp+1)=O(enp) is called the error equation. The value of p is called the order of convergence of the method.

Definition 2.2 (see [<xref ref-type="bibr" rid="B12">6</xref>]).

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

Kung-Traub Conjecture (see [<xref ref-type="bibr" rid="B9">7</xref>])

Let ψ be an IF without memory with d evaluations. Then (2.4)p(ψ)popt=2d-1, where popt is the maximum order.

We use the approximate computational-order of convergence, COC  given by (2.5)COClog{|xn+1-x*|/|xn-x*|}  log{|xn-x*|/|xn-1-x*|}.

3. Developments of the Methods

The second-order Newton-Raphson method is given by (3.1)ψ2ndNR(x)=x-f(x)f(x).

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

A one-parameter King family of fourth-order IF  is given by (3.2)ψ4thFK(x)=ψ2ndNR(x)-f(ψ2ndNR(x))f(x)1+βt11+(β-2)t1.

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 .

Several optimal eight-order methods are developed in . Recently, Thukral and Petković  developed a family of optimal eighth-order King-type IF given by (3.3)ψ8thFTPK(x)=ψ4thFK(x)-(ϕ(t1)+4t2+t3)f(ψ4thFK(x))f(x), where (3.4)t1=f(ψ2ndNR(x))f(x),t2=f(ψ4thFK(x))f(x),t3=f(ψ4thFK(x))f(ψ2ndNR(x)), and ϕ is a weight function satisfying (3.5)ϕ(0)=1,ϕ(0)=2,ϕ(0)=10-4β,ϕ′′(0)=12β2-72β+72.

If we choose (3.6)ϕ(t1)=1+βt1+(3/2)βt121+(β-2)t1+((3/2)β-1)t12 satisfying (3.5), we get a family of optimal eighth-order IF given by (3.7)ψ8thFK(x)=ψ4thFK(x)-(i=03    θi)    f(ψ4thFK(x))f(x), where (3.8)θ0=1,θ1=1+βt1+(3/2)βt121+(β-2)t1+((3/2)β-1)t12-1,θ2=t3,θ3=4t2.

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

Geum and Kim  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: (3.9)y=x-f(x)f(x),z=y-K1(u1)f(y)f(x),s=z-K2(u1,u2,u3)f(s)f(x),ψ16thFGK1(x)=s-K3(u1,u2,u3,u4)f(s)f(x), where (3.10)K1(u1)=1+βu1+(-9+5β/2)u121+(β-2)u1+(-4+β/2),K2(u1,u2,u3)=1+2u1+(2+σ)u31-u2+σu3,K3(u1,u2,u3,u4)=1+2u1+(2+σ)u2u31-u2-2u3-u4+2(1+σ)u2u3+K4(u1,u3) are weighting functions, K4:2 is an analytic function in a region containing the region (0,0), β,σ are to be chosen freely, and (3.11)u1=f(y)f(x),u2=f(z)f(y),u3=f(z)f(x),u4=f(s)f(z).

We consider the case (β,σ)=(2,-2) and (3.12)K4(u1,u3)=1-6  u1-u12-24  u13-12  (-2  u1+2)u32 for numerical experiments and term the I.F. as 16th GK1.

Geum and Kim  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 (3.13)ψ16thFGK2(x)=s-1+2u11-u2-2u3-u4f(s)f(x)+K5(u1,u2,u3), where K5:3 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 (3.14)K5(u1,u2,u3)=-6u13u2-24411u14u3+6u32+u1(2u22+4u23+u3-2u22) 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: (3.15)ψ16thFK(x)=ψ8thFK(x)-(i=07    θi)f(ψ8thFK(x))f(x), where (3.16)θ4=t5+t1t2,θ5=2t1t5+4(1-β)t13t3+2t2t3,θ6=2  t6+(7  β2-472  β+14)t3t14+(2  β-3)t22+(5-2  β)t5t12-t33,θ7=8  t4+(-12  β+12+2  β2)t5t13-4t33  t1+(-2  β2-22+12  β)t32t13+(46+1272  β2-105  β-10  β3)t2t14,(3.17)t4=f(ψ8thFK(x))f(x),t5=f(ψ8thFK(x))f(ψ4thFK(x)),t6=f(ψ8thFK(x))f(ψ2ndNR(x)).

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 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 (4.1)cj=f(j)(x*)j!f(x*),j=2,3,4,.

Using the Taylor series and the symbolic software such as Maple we have (4.2)f(x)=f(x*)[en+c2en2+c3en3+c4en4+],(4.3)f(x)=f(x*)[1+2c2en+3c3en2+4c4en3+], so that (4.4)u(x)=en-c2en2+2(c22-c3)en3+(7c2c3-4c23-3c4)en4+,(4.5)ψ2ndNR(x)-x*=c2en2-2(c22-c3)en3-(7c2c3-4c23-3c4)en4+.

Now, the Taylor expansion of f(y) about x* gives (4.6)f(y)=f(x*)[(y-x*)+c2(y-x*)2+c3(y-x*)3+c4(y-x*)4+].

Using (4.2), (4.6), and (4.5), we have (4.7)t1=c2en+(2  c3-3  c22)en2+(3  c4-10  c2c3+8  c23)en3+(-14  c2c4+37  c3c22-20  c24-8  c32+4  c5)en4+, so that (4.8)ψ4thFK(x)-x*=((1+2  β)c23-c2c3)en4+.

Similarly, we have (4.9)t2=((1+2  β)c23-c2c3)en3+(-2  c32+(9+12  β)c22c3+(-5-14  β-2  β2)c24-2  c2c4)en4+,(4.10)t3=((1+2  β)c23-c2c3)en2+((4+8  β)c2c3+(-2-8  β-2  β2)c23-2  c4)en3+ so that (4.11)θ0=1,θ1=((4  β+2)c22-2  c3)en2+((16  β+8)c2c3+(-16  β-4  β2-4)c23-4  c4)en3+,θ2=((1+2  β)c23-c2c3)en2+((4+8  β)c2c3+(-2-8  β-2  β2)c23-2  c4)en3+,θ3=((4+8  β)c23-4  c2c3)en3+, and finally we get (4.12)ψ8thFK(x)-x*=({952}(-c3c22+(1+2  β)c24)c4-c2c33+(16+4  β)c23c32+(-27-1112  β-β2)c25c3+(12+952  β+44  β2-6  β3)c27)en8+.

By a similar argument, we have (4.13)t4=({1112}(-c3c22+(1+2  β)c24)c4-c2c33+(16+4  β)c23c32+(-27-1112  β-β2)c25c3+(12+952  β+44  β2-6  β3)c27)en7+,t5=(c32+(-15-2  β)c22c3+(12-3  β2+472  β)c24+c2c4)en4+,t6=({952}(-c2c3+(1+2  β)c23)c4-c33+(16+4  β)c22c32+(-27-1112  β-β2)c24c3+(12+952  β+44  β2-6  β3)c26)en6+, so that (4.14)θ4=(c32+(-16-2  β)c22c3+(13-3  β2+512  β)c24+c2c4)en4+,θ5=(4  c2c32+(-38-8  β)c23c3+(30-6  β2+59  β)c25+2  c22c4)en5+,θ6=({952}(-2  c2c3+(7+2  β)c23)c4-c33+(31+2  β)c22c32+(-134-β2-952  β)c24c3+(-30  β2+94+8  β3+177  β)c26)en6+,θ7=({3572}(-8  c3c22+(20+4  β+2  β2)c24)c4-4  c2c33+(8  β+106)c23c32+(-1472  β2+14  β3-386-71  β)c25c3+(-3572  β2-34  β4+260+405  β+160  β3)c27)en7+, and finally we get (4.15)ψ16thFK(x)-x*=({2603214}({1832}-c23c34+(-c32c24+(4  β+2)c26c3+(-1-4  β-4  β2)c28)c4+(6  β+17)c25c33+(-12-82  β3-139  β2+12  β4-1432  β)c211+(-9  β2-43-1832  β)c27c32+(39+156  β2+157  β-4  β3)c29c3)c5+(-c24c3+(1+2  β)c26)c43+((16  β+66)c25c32+(-4  β4+81+219  β+124  β2+18  β3)c29+(-189  β+2  β3-26  β2-143)c27c3-4  c23c33)c42+({28292}(22  β+140)c24c34+(9  β4-210  β3+68054  β2+3933+4784  β)c28c32+(2  β3-699  β-1349-34  β2)c26c33-4  c22c35+(-195394  β2+891  β4-4255-10278  β-2682  β3-72  β5)c210c3+(1535+92  β6+118232  β2+5963  β+28292  β3+1863  β4-882  β5)c212)c4+(-20627  β6+4028  β7-240  β8+880332  β+2603214  β2+1343638  β3+401714  β4+588392  β5+8484)c215-c37c2+(6  β+60)c23c36+(β2-7772  β-1134)c25c35+(-3778  β6-32541-2349972  β+609292  β5-60462  β4+62  β7-95165  β2-754994  β3)c213c3+(125672  β-242  β3+45854  β2+17  β4+8769)c27c34+(177  β6-1332  β5-529234  β4+46415+3549478  β3+103054  β+1675054  β2)c211c32+(-35257  β-30052-108  β5-748754  β2+7252  β3+920  β4)c29c33{2603214})en16+.

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 |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.

Test functions and their roots.

 f 1 ( x ) = exp   ( x ) sin   ( x ) + ln   ( 1 + x 2 ) x * = 0 f 2 ( x ) = x 15 + x 4 + 4 x 2 - 15 x * = 1.148538 … f 3 ( x ) = ( x - 2 ) ( x 10 + x + 1 ) exp   ( - x - 1 ) x * = 2 f 4 ( x ) = ( x + 1 ) exp   ( sin   ( x ) ) - x 2 exp   ( cos   ( x ) ) - 1 x * = 0 f 5 ( x ) = sin   2 ( x ) - x 2 + 1 x * = 1.40449165 … f 6 ( x ) = exp   ( - x ) - cos   ( x ) x * = 0 f 7 ( x ) = ln   ( x 2 + x + 2 ) - x + 1 x * = 4.15259074 …

Range and the best value of β for the 16th FK family.

Function Range Best value of β
f 1 , x 0 = 1 [ - 5,5 ] a , a = - 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 = 0.25 [ - 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

Comparison of optimal 4-point sixteenth-order I.F.s.

function 16th FK 16th GK1 16th GK2
| x 3 - x * | COC | x 3 - x * | COC | x 3 - x * | COC
f _ 1 , x _ 0 = 1 0.137e-362 15.989 0.353e-581 15.981 0.103e-522 15.998
f _ 2 , x _ 0 = 1.3 0.898e-670 16.000 0.994e-782 16.000 0.723e-811 15.987
f _ 3 , x _ 0 = 2.5 0.479e-200 15.928 fail fail 0.819e-316 15.978
f _ 4 , x _ 0 = 0.25 0.492e-3155 16.000 0.241e-2455 16.000 0.516e-2071 16.000
f _ 5 , x _ 0 = 2.5 0.142e-810 16.000 0.164e-1147 15.993 0.452e-1020 16.000
f _ 6 , x _ 0 = 1 / 6 0.224e-1702 16.000 0.172e-1004 15.992 0.782e-892 16.000
f _ 7 , x _ 0 = 3.5 0.927e-4464 16.000 0.241e-4144 16.000 0.136e-3763 16.000

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] (5.1)f8(x)=x3+lnx,x>0,xR 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]: (5.2)f9(x)=x3+lnx+0.15cos(50x) in which the single root has been moved marginally to x*=0.717519716444759 but many local extrema have been introduced on a small scale . 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.

Behaviour of 16th FK family for the function f2.

Behaviour of 16th FK family for the function f7.

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.

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.

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.

Traub J. F. Iterative Methods for the Solution of Equations 1964 New Jersey, NJ, USA Prentice-Hall 0169356 Babajee D. K. R. Analysis of Higher Order Variants of Newton's Method and Their Applications to Differential and Integral Equations and in Ocean Acidification [Ph.D. thesis] 2010 University of Mauritius McNamee J. M. Numerical Methods for Roots of Polynomial: Part 1 2007 Amsterdam, The Netherlands Elsevier Petković M. S. Petković L. D. Families of optimal multipoint methods for solving nonlinear equations: a survey Applicable Analysis and Discrete Mathematics 2010 4 1 1 22 10.2298/AADM100217015P 2654926 Wait R. The Numerical Solution of Algebraic Equations 1979 New York, NY, USA John Wiley & Sons 544992 Ostrowski A. M. Solutions of Equations and System of Equations 1960 New York, NY, USA Academic Press Kung H. T. Traub J. F. Optimal order of one-point and multipoint iteration Journal of the Association for Computing Machinery 1974 21 4 643 651 0353657 10.1145/321850.321860 ZBL0289.65023 Weerakoon S. Fernando T. G. I. A variant of newton's method with accelerated third-order convergence Applied Mathematics Letters 2000 13 8 87 93 10.1016/S0893-9659(00)00100-2 1791767 ZBL0973.65037 King R. F. A family of fourth order methods for nonlinear equations SIAM Journal on Numerical Analysis 1973 10 876 879 0343585 10.1137/0710072 ZBL0266.65040 Bi W. Ren H. Wu Q. Three-step iterative methods with eighth-order convergence for solving nonlinear equations Journal of Computational and Applied Mathematics 2009 225 1 105 112 10.1016/j.cam.2008.07.004 2490175 ZBL1161.65039 Bi W. Wu Q. Ren H. A new family of eighth-order iterative methods for solving nonlinear equations Applied Mathematics and Computation 2009 214 1 236 245 10.1016/j.amc.2009.03.077 2541063 ZBL1173.65030 Neta B. On a family of multipoint methods for nonlinear equations International Journal of Computer Mathematics 1981 9 4 353 361 10.1080/00207168108803257 642112 Thukral R. Petković M. S. A family of three-point methods of optimal order for solving nonlinear equations Journal of Computational and Applied Mathematics 2010 233 9 2278 2284 10.1016/j.cam.2009.10.012 2577765 ZBL1180.65058 Geum Y. H. Kim Y. I. A biparametric family of optimally convergent sixteenth-order multipoint methods with their fourth-step weighting function as a sum of a rational and a generic two-variable function Journal of Computational and Applied Mathematics 2011 235 10 3178 3188 10.1016/j.cam.2011.01.003 2773302 ZBL1215.65093 Geum Y. H. Kim Y. I. A family of optimal sixteenth-order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth-step weighting function Computers and Mathematics with Applications 2011 61 11 3278 3287 10.1016/j.camwa.2011.04.006 2801993 ZBL1222.65046 Babajee D. K. R. Dauhoo M. Z. An analysis of the properties of the variants of newton's method with third order convergence Applied Mathematics and Computation 2006 183 1 659 684 10.1016/j.amc.2006.05.116 2286230 ZBL1123.65036 Drexler M. Newton's Method as a Global Solver for Non-Linear Problems [Ph.D. thesis] 1997 Oxford, UK University of Oxford