Journal of Applied Mathematics

Volume 2013 (2013), Article ID 850365, 5 pages

http://dx.doi.org/10.1155/2013/850365

Research Article

Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations

Malik Zaka Ullah,¹ A. S. Al-Fhaid,¹ and Fayyaz Ahmad²

¹Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

²Department de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, 08036 Barcelona, Spain

Received 10 April 2013; Accepted 27 July 2013

Academic Editor: Saeid Abbasbandy

Copyright © 2013 Malik Zaka Ullah et al. 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.

We present an iterative method for solving nonlinear equations. The proposed iterative method has optimal order of convergence sixteen in the sense of Kung-Traub conjecture (Kung and Traub, 1974); it means that the iterative scheme uses five functional evaluations to achieve 16(=) order of convergence. The proposed iterative method utilizes one derivative and four function evaluations. Numerical experiments are made to demonstrate the convergence and validation of the iterative method.

1. Introduction

According to Kung and Traub conjecture, a multipoint iterative method without memory could achieve optimal convergence order by performing evaluations of function or its derivatives [1]. In order to construct an optimal sixteenth-order convergent iterative method for solving nonlinear equations, we require four and eight optimal-order iterative schemes. Many authors have been developed the optimal eighth-order iterative methods, namely, Bi et al. [2], Bi et al. [3], Geum and Kim [4], Liu and Wang [5], Wang and Liu [6], and Soleymani et al. [7–9]. Some recent applications of nonlinear equation solvers in matrix inversion for regular or rectangular matrices have been introduced in [10–12].

For the proposed iterative method, we developed new optimal fourth- and eighth-orders iterative methods to construct optimal sixteenth-order iterative scheme. On the other hand, it is known that rational weight functions give a better convergence radius. By keeping this fact in mind, we introduced rational terms in weight functions to achieve optimal sixteenth order.

For the sake of completeness, we list some existing optimal sixteenth-order convergent methods. Babajee and Thukral [13] suggested 4-point sixteenth-order king family of iterative methods for solving nonlinear equations (BT):

where

In 2011, Geum and Kim [14] proposed a family of optimal sixteenth-order multipoint methods (GK2):

where

one of the choices for along with and :

In the same year, Geum and Kim [15] presented a biparametric family of optimally convergent sixteenth-order multipoint methods (GK1):

where

one of the choices for along with and :

2. A New Method and Convergence Analysis

The proposed sixteenth-order iterative method is described as follows (MA):

where

Theorem 1. Let be a sufficiently differentiable function, and is a simple root of , for an open interval . If is chosen sufficiently close to , then the iterative scheme (9) converges to and shows an order of convergence at least equal to sixteen.

Proof. Let error at step be denoted by and and , . We provided Maple based computer assisted proof in Algorithm 1 and got the following error equation:

fxn:= c1*(e+c2*e2+c3*e3+c4*e4+c5*e5+c6*e6+c7*e7+c8*e8+c9*e9+c10*e10+

          c11*e11+c13*e12+c13*e13+c14*e14+c15*e15+c16*e16+c17*e17);

dfxn:= diff (fxn, e);

ye:= simplify (taylor (e−fxn/dfxn, e = 0, 17));

fyn:= c1*(ye+c2*ye2+c3*ye3+c4*ye4+c5*ye5+c6*ye6+c7*ye7+c8*ye8+c9*ye9+

          c10*ye10+c11*ye11+c12*ye12+c13*ye13+c14*ye14+c15*ye15+c16*ye16+c17*ye17);

fyn:= simplify (taylor (fyn, e = 0, 17));

t1:= simplify (taylor (fyn/fxn, e = 0, 17));

fydfx:= simplify (taylor (fyn/dfxn, e = 0, 17));

ze:= factor (simplify (taylor (ye−(1+2*t1−t12)*fydfx/(1−6*t12), e = 0, 17)));

factor (simplify (taylor (ye−(1+2*t1−t12)*fydfx/(1−6*t12), e = 0, 7)));

  −c2*c3*e4+(−2*c32+2*c3*c22+2*c24−2*c2*c4)*e5+(−3*c2*c5−7*c3*c4+6*c2*c32+

      12*c3*c23+3*c4*c22−14*c25)*e6+O(e7)

fzn:= c1*(ze+c2*ze2+c3*ze3+c4*ze4+c5*ze5+c6*ze6+c7*ze7+c8*ze8+c9*ze9+

          c10*ze10+c11*ze11+c12*ze12+c13*ze13+c14*ze14+c15*ze15+c16*ze16+c17*ze17);

fzn:= simplify (taylor (fzn, e = 0, 17));

t2:= simplify (taylor (fzn/fyn, e = 0, 17));

t3:= simplify (taylor (fzn/fxn, e = 0, 17));

fzdfx:= simplify (taylor (fzn/dfxn, e = 0, 17));

we:= simplify (taylor (ze−(1−t1+t3)*fzdfx/(1−3*t1+2*t3−t2), e = 0, 17));

simplify (taylor (ze−(1−t1+t3)*fzdfx/(1−3*t1+2*t3−t2), e = 0, 10));

    −c4*c3*c22*e8+(−2*c5*c3*c22−4*c2*c4*c32+4*c3*c4*c23−2*c42*c22+2*c4*c25

    −3*c33*c22+4*c32*c24+4*c3*c26)*e9+O(e10)

fwn:= c1*(we+c2*we2+c3*we3+c4*we4+c5*we5+c6*we6+c7*we7+c8*we8+c9*we9+c10*we10+

           c12*we12+c13*we13+c14*we14+c15*we15+c16*we16+c17*we17);

fwn:= simplify (taylor (fwn, e = 0, 17));

fwdfx:= simplify (taylor (fwn/dfxn, e = 0, 17));

t4:= simplify (taylor (fwn/fxn, e = 0, 17));

t5:= simplify (taylor (fwn/fyn, e = 0, 17));

t6:= simplify (taylor (fwn/fzn, e = 0, 17));

q1:= simplify (taylor (1/(1−2*(t1+t12+t13+t14+t15+t16+t17)), e = 0, 17));

q2:= simplify (taylor (4*t3/(1−(31/4)*t3), e = 0, 17));

q3:= simplify (taylor (t2/(1−t2−20*t23), e = 0, 17));

q4:= simplify (taylor (8*t4/(1−t4)+2*t5/(1−t5)+t6/(1−t6), e = 0, 17));

q5:= simplify (taylor (15*t1*t3/(1−(131/15)*t3), e = 0, 17));

q6:= simplify (taylor (54*t12*t3/(1−t12*t3), e = 0, 17));

q7:= simplify (taylor (7*t2*t3+2*t1*t6+6*t6*t12+188*t3*t13+18*t6*t13+

                                   9*t22*t3+648*t14*t3, e = 0, 17));

x[n+1]:= simplify (taylor (we−fwdfx*(q1+q2+q3+q4+q5+q6+q7), e = 0, 17));

for i to 16 do p:= factor (simplify (coeff (x[n+1], e, i))) end do;

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

                                                     0

    −c4*c3*c22*(c5*c3*c22+2*c2*c4*c32−20*c34−51*c33*c22+522*c32*c24−2199*c3*c26

    +2*c28−30*c3*c4*c23+54*c4*c25)

Algorithm 1: The Maple code for finding the error equation.

3. Numerical Results

If the convergence order is defined as

then the following expression approximates the computational order of convergence (COC) [16] as follows:

where is the root of nonlinear equation. A set of five nonlinear equations are used for numerical computations in Table 1. Three iterations are performed to calculate the absolute error () and computational order of convergence. Table 2 shows absolute error and computational order of convergence, respectively.

Table 1: Set of six test nonlinear functions.

Functions
Roots

Table 2: Numerical comparison of absolute error , number of iterations = 3.

Iter/COC
MA
BT
GK1
GK2

, 1.0
1
0.00268
0.00183
0.0111
0.00230

2

3

COC
16
16
16
16

, 2.5
1
0.04086
0.0639
0.0296
0.00866

2

650.0

3

Divergent

COC
16.5
—
15.9
16.0

, 2.5
1
0.0000326
0.0000303
0.000497
0.0000677

2

3

COC
16
16
16
16

, 1/6
1

0.0000864

0.000167

2

3

COC
17
16
16
16

, 3.0
1
0.0486
0.135
0.0949
0.0133

2

3

COC
16.0
16.0
15.9
16.0

4. Conclusion

An optimal sixteenth-order iterative scheme has been developed for solving nonlinear equations. A Maple program is provided to calculate error equation, which actually shows the optimal order of convergence in the sense of Kung-Traub conjecture. The computational order of convergence also verifies our claimed order of convergence. The proposed scheme uses four functions and one derivative evaluation per full cycle, which gives 1.741 as the efficiency index. We also have shown the validity of our proposed iterative scheme by comparing it with other existing optimal sixteenth-order iterative methods. The numerical results show that the performance of iterative scheme is competitive as compared to other methods.

Acknowledgments

The first and the second authors were funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. They, therefore, acknowledge with thanks DSR technical and financial support while the third author is supported for this research under the Spanish MEC Grants AYA2010-15685.

References

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