# A Family of Iterative Methods with Accelerated Eighth-Order Convergence

Authors: Alicia Cordero; Mojtaba Fardi; Mehdi Ghasemi; Juan R. Torregrosa
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We propose a family of eighth-order iterative methods without memory for solving nonlinear equations. The new iterative methods are developed by using weight function method and using an approximation for the last derivative, which reduces the required number of functional evaluations per step. Their efficiency indices are all found to be 1.682. Several examples allow us to compare our algorithms with known ones and confirm the theoretical results.


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## Body

## 1. Introduction

Finding the solution of nonlinear equations is one of the most important problems in numerical analysis. There are some texts that have become classic, as the one of Traub (see [1]) and Neta (see [2]) which include a vast collection of methods and their efficiency, or the paper by Neta and Johnson, [3]), Jarratt (see [4]), or Homeier (see [5]), among others.As the order of an iterative method increases, so does the number of functional evaluations per step. The efficiency index (see [6]) gives a measure of the balance between those quantities, according to the formula I=p1/d, where p is the order of convergence of the method and d is the number of functional evaluations per step. Kung and Traub conjectured in [7] that the order of convergence of any multipoint method without memory cannot exceed the bound 2d-1 (called the optimal order). Thus, the optimal order for a method with 3 functional evaluations per step would be 4. King's method [8], Chun's schemes (see [9, 10]), Chun et al. [11], Maheshwari's procedure (see [12]), and Jarratt's method [4] are some of optimal fourth-order methods, because they only perform three functional evaluations per step.More recently, optimal eighth-order iterative methods have been investigated by many researchers (see, e.g., [13] where the authors show optimal methods of order eight and sixteen). Bi et al. in [14] developed a new family of eighth-order iterative methods for solving nonlinear equations, which is denoted by BRW8 and whose iterative expression is
(1.1)ym=xm-f(xm)f′(xm),zm=ym-f(xm)-(1/2)f(ym)f(xm)-(5/2)f(ym)f(ym)f′(xm),xm+1=zm-f(xm)+(θ+2)f(zm)f(xm)+θf(zm)f(zm)f[zm,ym]+f[zm,xm,xm](zm-ym),
where θ∈ℝ.Also Cordero et al. in [15] developed a parametric family of eighth-order methods based on Ostrowski's scheme, that we will denote by M8. It can be expressed as
(1.2)ym=xm-f(xm)f′(xm),zm=xm-f(xm)f′(xm)f(xm)-f(ym)f(xm)-2f(ym),um=zm-f(zm)f′(xm)(f(xm)-f(ym)f(xm)-2f(ym)+12f(zm)f(ym)-2f(zm))2,xm+1=um-f(zm)f′(xm)3(β2+β3)(um-zm)β1(um-zm)+β2(ym-xm)+β3(zm-xm),
where β2+β3≠0. Other family of variants of Ostrowski's method with eighth-order convergence was developed by Liu and Wang in [16]. We will denote it by LW8 and its iterative expression is
(1.3)ym=xm-f(xm)f′(xm),zm=xm-f(xm)(f(xm)-f(ym))f′(xm)(f(xm)-2f(ym)),xm+1=zm-f(zm)f′(xm)[(f(xm)-f(ym)f(xm)-2f(ym))2+f(zm)f(xm)-αf(ym)+G(f(zm)f(xm))],
where α is constant and G denotes a real-valued function.Let us note that all these methods are optimal in the sense of Kung-Traub's conjecture (see [7]) for methods without memory; that is, they reach eighth-order convergence with only four functional evaluations.It is usual to design high-order methods from Ostrowski-type schemes, as they are optimal and use few operations per step, trying to obtain procedures as efficient as possible by using different techniques (see, for instance [13, 15, 16]). If we compose Newton and Ostrowski's methods, and estimate the last derivative by divided differences, it is necessary to use divided differences of second order to reach eighth order of convergence (see [14]). Nevertheless, as we will see in the next section, it is possible to obtain an optimal eighth-order scheme by composing King and Newton's methods by using only divided differences of first order.In this paper, we design a family of eighth-order iterative methods to find a simple rootx* of the nonlinear equation f(x)=0, where f:D→ℝ is a smooth function, and D is an open interval. We present in Section 2 a family of eighth-order iterative methods, based on Chun's method. In Section 3, different numerical examples confirm the theoretical results and allow us to compare the new methods with other known methods mentioned in the introduction. Finally, some conclusions are presented in Section 4.


## 2. Development of Eighth-Order Algorithm

Let us consider the family of optimal methods proposed by Chun in [9], that is a variant of King's family,
(2.1)ym=xm-f(xm)f′(xm),zm=ym-f2(xm)f2(xm)-2f(xm)f(ym)+2βf2(ym)f(ym)f′(xm),
where β∈ℝ. We consider now a three-step iterative scheme composed by Chun's scheme (2.1) and a third step designed by using Newton's method and weight functions.
(2.2)xm+1=zm-H(μm)f(zm)f′(zm),
where μm=f(zm)/f(xm) and H(t) represents a real-valued function.However, this procedure is not optimal, as it uses two new functional evaluations in the last step. So, we expressf′(zm) as a linear combination of f[ym,xm], f[zm,ym], and f[zm,xm].
(2.3)f′(zm)=θ1f[ym,xm]+θ2f[zm,ym]+(1-θ1-θ2)f[zm,xm],
where θ1 and θ2 are real numbers.By using (2.3), we propose the following iterative scheme:
(2.4)ym=xm-f(xm)f′(xm),zm=ym-f2(xm)f2(xm)-2f(xm)f(ym)+2βf2(ym)f(ym)f′(xm),xm+1=zm-H(μm)f(zm)θ1f[ym,xm]+θ2f[zm,ym]+(1-θ1-θ2)f[zm,xm],
whose convergence analysis will be made in the following result.Theorem 2.1.
Letx* be a simple zero of a sufficiently differentiable function f:D⊆ℝ→ℝ. If the initial point x0 is sufficiently close to x*, then the sequence {xm}m≥0 generated by any method of the family (2.4) converges to x*. If H(t) is any function with H(0)=1, H'(0)=1, H′′(0)<∞, and β=1/2, then the convergence order of any method of the family (2.4) is eight if and only if θ1=-1, θ2=1.Proof.
Letem=xm-x* be the error at the mth iteration and cm=(1/m!)(f(m)(x*)/f′(x*)),m=2,3,…. By using Taylor expansions, we have
(2.5)f(xm)=f′(x*)[em+c2em2+c3em3+c4em4+c5em5+c6em6+c7em7+c8em8+O(em9)],f′(xm)=f′(x*)[1+2c2em+3c3em2+4c4em3+5c5em4+6c6em5+7c7em6+8c8em7+O(em8)].
Now, from (2.5), we have
(2.6)ym=x*+c2em2+(2c3-2c22)em3+(3c4-3c2c3-2(2c3-2c22)c2)em4+(4c5-10c2c4-6c32+20c3c22-8c24)em5+(-17c4c3+28c4c22-13c2c5+33c2c32+5c6-52c3c23+16c25)em6+O(em7),
and then, we get
(2.7)f(ym)=f′(x*)[c2e2+(2c3-2c22)em3+(3c4-7c2c3+5c23)em4f′(x*)+(-6c32+24c3c22-10c2c4+4c5-12c24)e5f′(x*)+(-17c4c3+34c4c22-13c2c5+5c6+37c2c32-73c3c23+28c25)em6+O(em7)].
Combining (2.5), (2.6) and (2.7), we obtain
(2.8)zm=x*+(-c2c3+c23+2βc23)em4-2((2+6β)c24-2(2+3β)c22c3+c32+c2c4)em5+((10+44β-4β2)c25-6(5+14β)c23c3+6(2+3β)c22c4x*x-7c3c4+3c2((6+8β)c32-c5))em6+O(em7).
So, from (2.8), we get
(2.9)f(zm)=f′(x*)[((1+2β)c23-c2c3)em4-2(((2+6β)c24-2(2+3β)c22c3+c32+c2c4))em5f′(x*)+((10+44β-4β2)c25-6(5+14β)c23c3+6(2+3β)c22c4f′(x*)x-7c3c4+3c2((6+8β)c32-c5))em6+O(em7)].
Using the Taylor expansion of H around 0 and considering H′′(0)<∞, we get
(2.10)H(μm)=H(0)+H′(0)f(zm)f(xm)+O((μm)2)=H(0)+H′(0)c2(-c3+c22+2βc22)em3+H′(0)(-2c2c4+9c3c22-5c24+12βc3c22-14βc44-2c32)em4+O(em5).
In these terms, the error equation of the method can be expressed as
(2.11)em+1=-(-1+H(0))c2((1+2β)c22-c3)em4+(-(4+12β+H(0)(-5+2β(-7+θ2)+θ2))c24(0)+(8+12β+H(0)(-9-12β+θ2))c22c3(0)+2(-1+H(0))c32+2(-1+H(0))c2c4)em5+O(em6),
which shows that the convergence order of any method of the family (2.4) is at least five if H(0)=1. Then,
(2.12)em+1=-(-1+θ2)c22((1+2β)c22-c3)em5+c2((-5+θ1+7θ2-θ22+2β(-7+θ1+9θ2-θ22))c24+c2c2+(10-θ1-14β(-1+θ2)-12θ2+θ22)c22c3+c2c2+3(-1+θ2)c32+2(-1+θ2)c2c4)em6+O(em7).
and it is necessary that θ1=-1 and θ2=1 in order to reach order of convergence seven. Then, the error equation is
(2.13)em+1=-c2((H′(0)+2βH′(0))c22-(-1+H′(0))c3)((1+2β)c23-c2c3)em7+O(em8),
and finally, if β=-1/2 and H'(0)=1, it can be concluded that the order is eight and
(2.14)em+1=c3c22(2c2c3-c4+2c23)em8+O(em9).Remark 2.2.
Any method of the family (2.4) has the efficiency index equals to 81/4≈1.682, which is better than the Newton's method with efficiency index equals to 21/2≈1.414and equal to BRW8, M8, and LW8.In what follows, we give some concrete optimal iterative methods of family (2.4) for different functions H.  (F1)
H(t)=(1+βt)γ,β·γ=1, where β,γ∈ℝ. Hence we get a new eighth-order method whose last step is
(2.15)xm+1=zm-(1+βμm)γf(zm)-f[ym,xm]+f[zm,ym]+f[zm,xm].(F2)
H(t)=1+(t/(1+λt)), where λ∈ℝ. So,
(2.16)xm+1=zm-(1+μm1+λμm)f(zm)-f[ym,xm]+f[zm,ym]+f[zm,xm].(F3)
H(t)=1/(1-t+ωt2), where ω∈ℝ. Then we obtain a new scheme
(2.17)xm+1=zm-(11-μm+ωμm2)f(zm)-f[ym,xm]+f[zm,ym]+f[zm,xm].


## 3. Numerical Results

We present some examples to illustrate the efficiency of the iterative algorithm. All computations were done using MAPLE. We have used as stopping criteria that|xm+1-xm|≤10-200 or |f(xm)|≤10-200. The test functions are listed below:  (a)
f1(x)=x3+4x2-15;x*≈1.6319808055661;(b)
f2(x)=xex2-sin2(x)+3cos(x)+5;x*≈-1.2076478271309;(c)
f3(x)=sin(x)-(x/2);x*≈1.8954942670339;(d)
f4(x)=10xe-x2-1;x*≈1.6796306104285;(e)
f5(x)=cos(x)-x;x*≈0.73908513321516;(f)
f6(x)=sin2(x)-x2+1;x*≈1.4044916482153.(g)
f7(x)=e-x+cos(x);x*≈1.7461395304080.We compare the classical Newtons (CN), BRW8 method withθ=1; M8 scheme with β1=0, β3=0, and β2=1; LW8 method with α=1 and G(t)=4t, and our methods with ω=1, λ=1, β=1, and γ=1.In Table1, the following elements appear for each test function and each iterative method: the value of the elements involved in the stopping criterium, |xm+1-xm| and |f(xm)|, the number of iterations, iter, needed to converge to the solution, and the approximated computational order of convergence ρ, that can be calculated by using the formula (see [17])
(3.1)ρ=ln(|xm+1-xm|/|xm-xm-1|)ln(|xm-xm-1|/|xm-1-xm-2|),
where xm+1,xm,xm-1, and xm-2 are iterations close to a zero of the nonlinear equation.Table 1
Comparison of various iterative methods.
f
1, x0=2
CN
RWB8
M8
LW8
F1
F2
F3
|
x
m
+
1
-
x
m
|
6.4
e
-
110
7.9
e
-
59
7.1
e
-
54
7.5
e
-
49
2.9
e
-
62
2.9
e
-
62
2.9
e
-
62
|
f
(
x
m
)
|
3.7
e
-
218
0
0
0
0
0
0
iter
8
3
3
3
3
3
3
ρ
2.0
7.9
8.0
8.0
8.0
8.0
8.0
f
2, x0=-1
CN
RWB8
M8
LW8
F1
F2
F3
|
x
m
+
1
-
x
m
|
1.9
e
-
128
4.1
e
-
28
1.1
e
-
50
3.9
e
-
43
8.5
e
-
27
4.8
e
-
27
4.8
e
-
27
|
f
(
x
m
)
|
1.1
e
-
254
9.7
e
-
217
0
0
0
2.9
e
-
208
2.1
e
-
206
iter
9
3
3
3
3
3
3
ρ
2.0
8.0
8.0
8.0
8.1
8.1
8.1
f
3, x0=1.9
CN
RWB8
M8
LW8
F1
F2
F3
|
x
m
+
1
-
x
m
|
6.1
e
-
166
3.5
e
-
168
4.8
e
-
161
7.1
e
-
155
7.5
e
-
170
7.5
e
-
170
7.5
e
-
170
|
f
(
x
m
)
|
0
0
0
0
0
0
0
iter
7
3
3
3
3
3
3
ρ
2.0
7.8
7.6
7.8
8.0
8.0
8.0
f
4, x0=1.5
CN
RWB8
M8
LW8
F1
F2
F3
|
x
m
+
1
-
x
m
|
2.0
e
-
108
6.6
e
-
55
5.3
e
-
52
3.5
e
-
45
5.0
e
-
56
5.0
e
-
56
5.0
e
-
56
|
f
(
x
m
)
|
1.1
e
-
215
0
0
0
0
0
0
iter
8
3
3
3
3
3
3
ρ
2.0
7.9
8.0
8.0
8.0
8.0
8.0
f
5, x0=1
CN
RWB8
M8
LW8
F1
F2
F3
|
x
m
+
1
-
x
m
|
7.1
e
-
167
3.3
e
-
83
5.3
e
-
82
1.7
e
-
66
1.8
e
-
86
2.0
e
-
86
1.9
e
-
81
|
f
(
x
m
)
|
0
0
0
0
0
0
0
iter
8
9
3
3
3
3
3
ρ
2.0
8.0
8.0
8.0
8.0
8.0
8.0
f
6, x0=1.5
CN
RWB8
M8
LW8
F1
F2
F3
|
x
m
+
1
-
x
m
|
2.6
e
-
148
6.2
e
-
86
3.8
e
-
72
2.3
e
-
66
4.5
e
-
87
4.5
e
-
87
4.5
e
-
87
|
f
(
x
m
)
|
1.3
e
-
295
0
0
0
0
0
0
iter
8
3
3
3
3
3
3
ρ
2.0
7.8
8.0
8.0
8.0
8.0
8.0
f
7, x0=2
CN
RWB8
M8
LW8
F1
F2
F3
|
x
m
+
1
-
x
m
|
9.6
e
-
170
2.7
e
-
80
5.3
e
-
78
2.8
e
-
61
1.3
e
-
76
1.5
e
-
76
1.3
e
-
76
|
f
(
x
m
)
|
0
0
0
0
0
0
0
iter
8
3
3
3
3
3
3
ρ
2.0
7.9
7.9
8.0
8.0
8.0
8.0


## 4. Conclusions

In this work, we have constructed a new general eighth-order iterative family of methods without memory for solving nonlinear equations. Convergence analysis shows that the order of convergence of the methods is eight. Per iteration the present methods require three evaluations of the function and one evaluation of its first derivative and therefore have the efficiency index equal to81/4=1.682. Some of the obtained methods were also compared in their performance and efficiency to various other iteration methods of the same order, and it was observed that they demonstrate at least equal behavior.



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