# A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders

Authors: Young Ik Kim; Young Hee Geum
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We construct a biparametric family of fourth-order iterative methods to compute multiple roots of nonlinear equations. This method is verified to be optimally convergent. Various
nonlinear equations confirm our proposed method with order of convergence of four and show that the computed asymptotic error constant agrees with the theoretical one.


---

## Body

## 1. Introduction

It is not surprising that modified Newton’s method [1] in the simple form
(1)xn+1=xn-mf(xn)f′(xn)
is most widely used to find the approximate multiple root α of known multiplicity m for a given nonlinear equation f(x)=0. Recall that numerical scheme (1) is a one-point optimal method with quadratic convergence. In order to find numerical solution for multiple roots of nonlinear equations more accurately, many researchers have made enormous efforts in developing higher-order methods with improved convergence.In this paper, we extend modified Newton’s method and propose two-point optimal fourth-order multiple-root finders by evaluating two derivatives and one function per iteration. The optimality will be pursued based on Kung-Traub’s conjecture [2] in which the convergence order of any multipoint method [3] without memory can reach at most 2k-1 for k evaluations of functions or derivatives.The contents of this paper consist of what follows. Described in Section2 are previous studies on multiple-root finders. Section 3 proposes a new biparametric family of two-point optimal fourth-order multiple-root finders. It fully treats method development and convergence analysis. Derivation of the error equations for the proposed schemes is an important task for ensuring convergence behavior. In Section 4, a variety of numerical examples are presented for a wide selection of test functions. It is important to compare the convergence behavior of the proposed schemes with that of existing methods. We confirm that the proposed methods well show the convergence behavior predicted by the developed theory.


## 2. Preliminary Review of Previous Studies

A number of interesting fourth-order multiple-root finders can be found in papers [4–16]. Among these, we especially introduce five studies as follows. Shengguo et al. [14] introduced the following fourth-order method which needs evaluations of one function and two derivatives per iteration for x0 chosen in a neighborhood of the sought zero α of f(x) with known multiplicity m≥1:
(2)xn+1=xn-βf′(xn)+ϕf′(yn)f′(xn)+δf′(yn)f(xn)f′(xn),n=0,1,2,…,
where yn=xn-(2m/(m+2))(f(xn)/f′(xn)), β=-m2/2, ϕ=(1/2)(m(m-2)/(m/(m+2))m), and δ=-(m/(m+2))-m.J. R. Sharma and R. Sharma [12] constructed the following fourth-order scheme with A=(1/8)m(m3-4m+8), B=-(1/4)m(m-1)(m+2)2(m/(m+2))2, C=(1/8)m(m+2)3(m/(m+2))2m, and yn=xn-(2m/(m+2))(f(xn)/f′(xn)):(3)xn+1=xn-Af(xn)f′(xn)-Bf(xn)f′(yn)-C(f(xn)f′(yn))2(f(xn)f′(xn))-1,n=0,1,2,….
Li et al. [15] presented the fourth-order method with yn=xn-(2m/(m+2))(f(xn)/f′(xn)):
(4)xn+1=xn-(m-m22)f(xn)f′(xn)-mf(xn)-f′(xn)+(m/(m+2))-mf′(yn).
Zhou et al. [16] proposed the following fourth-order iterative scheme with yn=xn-(2m/(m+2))(f(xn)/f′(xn)):
(5)xn+1=xn-m8[m3(m+2m)2m(f′(yn)f′(xn))2hhhhhhh-2m2(m+3)(2+mm)mf′(yn)f′(xn)hhhhhhh+(m3+6m2+8m+8)(m+2m)2m(f′(yn)f′(xn))2f′(yn)f′(xn)]f(xn)f′(xn).
Kanwar et al. [8] developed the fourth-order optimal multipoint iterative method for multiple zeros:
(6)xn+1=xn-(mf(xn)(m2(3m-2){f′(yn)}2hhhhhhhhhhhhhhhh-C1f′(xn)f′(yn)+C2{f′(xn)}2))×((mf′(yn)-(mm+2)m(m+8)f′(xn))2f′(xn)((m-1)(mm+2)mf′(xn)-mf′(yn))hhh×(mf′(yn)-(mm+2)m(m+8)f′(xn)))-1,
where yn=xn-(2m/(m+2))(f(xn)/f′(xn)), C1=m(m/(m+2))m(6m2+17m-14), and C2=(m/(m+2))2m(3m3+19m2+16m+16).


## 3. Method Development and Convergence Analysis

We first suppose that a functionf:C→C has a multiple root α with integer multiplicity m≥1 and is analytic in a small neighborhood of α. Then a new iteration method free of second derivatives is proposed below to find an approximate root α of multiplicity m, given an initial guess x0 sufficiently close to α:
(7)xn+1=yn-af(xn)f′(xn)-cf(xn)f′(yn)-dF(yn)f′(yn),n=0,1,2,…,
where
(8)yn=xn-γf(xn)f′(xn),F(yn)=f(xn)-(yn-xn)λf′(xn)f′(yn)f′(xn)+ρf′(yn),
with a, c, d, γ, λ, and ρ are parameters to be chosen for maximal order of convergence [17, 18]. We establish a main theorem describing the convergence analysis regarding proposed scheme (7) and find out how to select parameters a, c, and d for optimal fourth-order convergence.Definition 1 (error equation, asymptotic error constant, and order of convergence).
Letx0,x1,x2,…,xn,… be a sequence converging to α and let en=xn-α be the nth iterate error. If there exist real numbers p∈R and b∈R-{0} such that the following error equation holds
(9)en+1=benp+O(enp+1),
then b or |b| is called the asymptotic error constant and p is called the order of convergence [17, 18].Theorem 2.
Letf:C→C have a zero α with integer multiplicity m≥1 and be analytic in a small neighborhood of α. Let κ=(m/(m+2))m and θj=f(m+j)(α)/f(m)(α) for j∈N. Let x0 be an initial guess chosen in a sufficiently small neighborhood of α. Let λ, ρ∈R be two free constant parameters. Let a=m3(-4κρ+m(m+2)(1+κρ))/8(m+2)κρ, c=(m+2){-m3+m(m+2)κ(-3m+2(m+2)κλ)ρ+m(m+2)2κ2(-3+2κρ)ρ2-(m+2)3κ3ρ3}/16κλρ, d=(2+m)(m+(2+m)κρ)3/16κλρ, and γ=2m/(2+m). Then iterative methods (7) are optimal and of order four and possess the following error equation:
(10)en+1=ψ4en4+O(en5),
where ψ4=((m4+4m3+6m2+2m+8-(24κρ/(m+(m+2)κρ)))/3m2(m+1)3(m+2))θ13-(1/m(m+1)2(m+2))θ1θ2+(m/(m+2)3(m+1)(m+3))θ3.Proof.
The optimality on convergence order of proposed scheme (7) is clear in the sense of Kung-Traub due to three functional evaluations. Hence, it suffices to determine the constant parameters for fourth-order convergence.
Applying the Taylor’s series expansion aboutα, we get the following relations:
(11)f(xn)=f(m)(α)m!enm[1+A1en+A2en2+A3en3+A4en4+O(en5)],(12)f′(xn)=f(m)(α)(m-1)!enm-1[1+B1en+B2en2+B3en3+B4en4+O(en5)],
where Ak=(m!/(m+k)!)θk, Bk=((m-1)!/(m+k-1)!)θk, and θk=f(m+k)(α)/f(m)(α) for k∈N.
Dividing (11) by (12), we have
(13)f(xn)f′(xn)=1m[en-K1en2-K2en3-K3en4+O(en5)],
where K1=-A1+B1, K2=-A2+A1B1-B12+B2, K3=-A3+A2B1-A1B12+B13+A1B2-2B1B2+B3.
For algebraic convenience, we introduce a parametert defined by γ=m(1-t); that is, t=1-γ/m to obtain
(14)yn=xn-γf(xn)f′(xn)=en+α-γf(xn)f′(xn)=α+te+K1(1-t)en2+K2(1-t)en3+K3(1-t)en4+O(en5).
Evaluating f(yn) from (11) with en being replaced by yn-α in (14), we find(15)f(yn)=f(m)(α)emm!hhhhhh×{tm+tm-1(A1t2+K1m(1-t))enhhhhhhhh+12(tm-2(K12(m-1)m(t-1)2hhhhhhhhhhhhhhhh-2A1K1(m+1)(t-1)t2hhhhhhhhhhhhhhhh+2t(A2t3+K2m(1-t))))en2hhhhhhhh+16tm-3(+A3t3)))en3-K13m(2-3m+m2)(t-1)3hhhhhhhhdhhhhhhh+3A1K12m(m+1)(t-1)2t2hhhhhhhhhdhhhhhh-6K1(t-1)thhhhhhhhhhdhhhhh×(+A3t3)))A2(m+2)t3hhhhhhhhhhhdhhhhhhh+K2m(m-1)(1-t)+6t2hhhhhhhhhhhhdhhhhhh×(+A3t3))K3m(1-t)hhhhhhhhhhhhhdhhhhhhhh+t(+A3t3))A1K2(m+1)(t-1)hhhhhhhhhhhhhhdhhhhhhhhhh+A3t3)))en3)+O(en4)tm+tm-1(A1t2+K1m(1-t))}.
Substituting (11)–(15) into (7), we obtain the error equation:(16)en+1=yn-α-Kf=ψ1en+ψ2en2+ψ3en3+ψ4en4+O(en5),
where Kf=a(f(xn)/f′(xn))+c(f(xn)/f′(yn))+d(F(yn)/f′(yn)), ψ1=-(a/m)+t(1-((c+d)t-m/m)-(d(t-1)λ/(t+tmρ))), and coefficients ψi, (i=2,3,4) depend on the parameters t, a, c, d, λ, and ρ and the function f(x).
Solvingψ1=0, ψ2=0 for a and c, respectively, we get
(17)a=mt(1-(c+d)t-mm-d(t-1)λt+tmρ),c=-((t2+2tm+1ρ+t2mρ(m(t-1)λ+ρ))mtm(t+tmρ)2+d(t-1)(1+m(t-1)+t)hhh×(t2+2tm+1ρ+t2mρ(m(t-1)λ+ρ)))×((t-1)(1+m(t-1)+t)(t+tmρ)2)-1.
We substitute a, c into ψ3 and put ψ3=ψ31θ12+ψ32θ2. Solving ψ3=0 independently of θ1 and θ2, that is, solving ψ31=ψ32=0 for d and t, we obtain
(18)d=-(t-1-m(2t2(2+t)+m2(t-1)(1+2(t-1)t)hhhhhhhh+m(1-3t+4t3))(t+tmρ)3)×(2(t-1)3(1+m(t-1)+t)3λρ)-1,hhhhhhhhhhhhhhhhhhhhht=mm+2.
Substituting t=m/(m+2) into (17) and (18) with κ=(m/(2+m))m, we get the following relations:
(19)a=m3(-4κρ+m(m+2)(1+κρ))8(m+2)κρ,c=((m+2){-m3+m(m+2)κ(-3m+2(m+2)κλ)ρhhhhhhhh+m(m+2)2κ2(-3+2κρ)ρ2hhhhhhhh-m3+m(m+2)κ(-3m+2(m+2)κλ)-(m+2)3κ3ρ3}-m3+m(m+2)κ(-3m+2(m+2)κλ))×(16κλρ)-1,d=(2+m)(m+(2+m)κρ)316κλρ.
By the aid of symbolic computation ofMathematica [19], we arrive at the relation below:
(20)en+1=ψ4en4+O(en5),
where ψ4=((m4+4m3+6m2+2m+8-(24κρ/(m+(m+2)κρ)))/3m2(m+1)3(m+2))θ13-(1/m(m+1)2(m+2))θ1θ2+(m/(m+2)3(m+1)(m+3))θ3. As a result, the proof is completed.Remark 3.
We observe that error equation (20) contains only one free parameter ρ, being independent of λ. Table 1 shows typically chosen parameters λ and ρ and defines various methods Yk, (k=1,2,3,4).Table 1
Typical methods with interesting parameters(λ,ρ) and constants (a,c,d).
Method
(
λ
,
ρ
)
(
a
,
c
,
d
)
Y1
(
-
10,10
)
a
=
m
3
{
-
40
κ
+
m
(
m
+
2
)
(
10
κ
+
1
)
}
80
(
m
+
2
)
κ,  d=-(m+2)(m+10(m+2)κ)31600κ c=(m+2){m3+30m2(m+2)κ+500m(m+2)2κ2+1000(m+2)2(3m+2)κ3}1600κ
Y2
(
-
1
,
-
m
(
m
+
2
)
(
-
4
+
2
m
+
m
2
)
κ
)
(
0
,
-
m
{
m
(
m
+
4
)
3
-
(
m
+
2
)
3
(
m
2
+
2
m
-
4
)
κ
}
2
(
m
2
+
2
m
-
4
)
2
,
-
m
2
(
m
+
4
)
3
2
(
m
2
+
2
m
-
4
)
2
)
Y3
(
-
m
(
m
+
4
)
3
(
m
+
2
)
3
(
m
2
+
2
m
-
4
)
κ
,
-
m
(
m
+
2
)
(
m
2
+
2
m
-
4
)
κ
)
(
0,0
,
-
m
(
m
+
2
)
3
κ
2
(
m
2
+
2
m
-
4
)
)
Y4
(
(
m
+
(
m
+
2
)
κ
ρ
)
3
2
m
(
m
+
2
)
2
κ
2
ρ
(
1
+
κ
ρ
)
,
-
1
2
)
(
m
3
(
m
(
m
+
2
)
(
κ
-
2
)
-
4
κ
)
8
(
m
+
2
)
κ
,
0
,
-
1
16
m
(
m
+
2
)
3
κ
(
κ
+
2
)
)


## 4. Numerical Examples and Conclusion

We have performed a variety of numerical experiments with Mathematica Version 5 [19] to confirm the theory developed in Section 3. In these experiments, we assign 300, via Mathematica command$MinPrecision =300, as the minimum number of precision digits to achieve the specified sufficient accuracy. It is crucial to compute en=xn-α with high accuracy for desired numerical results. When zero α is not exactly known, it is replaced by a highly accurate value α~ which has larger number of significant digits than the assigned minimum number of precision digits. To deal with numerical results more effectively, we first define
(21)α-={α,ifαisexactlyknown,α~,ifαisnotexactlyknown.
To properly display numerical results, we need to define the nth computational error e-n=xn-α- for n=0,1,2…. We need further terminologies as defined below.Definition 4 (computational asymptotic error constant and computational convergence order).
Assume that theoretical asymptotic error constantη=lim⁡n→∞|en|/|en-1|p and convergence order p≥1 are known (usually via main theorem). Define qn=|e-n|/|e-n-1|p as the computational asymptotic error constant and pn=log⁡|e-n/η|/log⁡|e-n-1| as the computational convergence order. Then we find that lim⁡n→∞qn is equal or close to η, while lim⁡n→∞pn is equal or close to p.
Ifα~ has the same accuracy of$MinPrecision as that of xn, then e-n=xn-α~ would be nearly zero and hence computing |e-n+1/e-n4| would unfavorably break down. Computed values of xn are accurate up to 300 significant digits. For current experiments α~ is found to be accurate enough about up to 400 significant digits. To supply such α~, a set of following Mathematica commands are used:
(22)sol=FindRoot[f[x],{x,x0},PrecisionGoal⟶100hhhhhhhhh+$MinPrecision,hhhhhhhhhhWorkingPrecision⟶2hhhhhhhhhh*$MinPrecisionf[x],{x,x0},PrecisionGoal⟶100];hhhhhhhhhhhhhα~=sol[[1,2]].
Although the number of significant digits of xn and α~ is 300 and 400, respectively, the limited paper space allows us to list both of them only up to 15 significant digits. We set the error bound ε to 10-250 for |xn-α-|<ε.As a first example, we select a functionf(x)=(x2+7)2cscxlog⁡(x2+8) having a multiple zero α=-7i with i=-1. We choose x0=-2.71i as an initial guess. We take another function f(x)=(x2+4x-1)4e-x with a root α=-2-5. We select x0=-4.35 as an initial value. The order of convergence and the asymptotic error constant are clearly shown in Tables 2 and 3 revealing a good agreement with the theory in Section 3. Taking another function f(x)=(ex2+3x+7-1)5(x2+3x+7)3 with a root α=(-3-19i)/2 with multiplicity m=8, we select x0=-1.46-2.09i as an initial value. In this example, we also find that the order of convergence is four and the computational asymptotic error constant |e-n+1/e-n4| well approaches the theoretical value η. The computational convergence order and the computational asymptotic error constant are certainly shown in Tables 2–4 reaching a good agreement with the theory. It is certain that these methods need one evaluation of the function f and two evaluations of the first derivative f′.Table 2
Convergence behavior withf(x)=(x2+7)2cscxlog⁡⁡(x2+8). (m,λ,ρ)=(3,1/2,-1),α=-7i=-2.64i.
n
x
n
|
f
(
x
n
)
|
|
x
n
-
α
-
|
|
e
-
n
+
1
/
e
-
n
4
|
η
log
⁡
⁡
(
e
-
n
/
η
)
/
log
⁡
⁡
(
e
-
n
-
1
)
0
-
2.71
i
0.00667483
0.0642487
0.2482221894
1
-
2.64575363508446
i
2.65244
×
1
0
-
16
2.32402
×
1
0
-
6
0.1363900716
4.21814
2
-
2.64575131106459
i
8.02272
×
1
0
-
69
7.24101
×
1
0
-
24
0.2482218636
4.00000
3
-
2.64575131106459
i
6.71488
×
1
0
-
279
6.82399
×
1
0
-
94
0.2482221894
4.00000
4
-
2.64575131106459
i
0.×10899
0
.
×
1
0
-
299Table 3
Convergence behavior withf(x)=(x2+4x-1)4e-x. (m,λ,ρ)=(4,-10,10),α=-2-5=-4.23607.
n
x
n
|
f
(
x
n
)
|
|
x
n
-
α
-
|
|
e
-
n
+
1
/
e
-
n
4
|
η
log
⁡
⁡
(
e
-
n
/
η
)
/
log
⁡
⁡
(
e
-
n
-
1
)
0
-
4.35000000000000
5.77466
0.113932
0.2202805193
1
-
4.23609825091956
2.32291
×
1
0
-
14
0.0000302734
0.1796711117
4.09332
2
-
4.23606797749979
3.24005
×
1
0
-
71
1.85011
×
1
0
-
19
0.2202681897
4.00001
3
-
4.23606797749979
1.22696
×
1
0
-
298
2.58087
×
1
0
-
76
0.2202805193
4.00000
4
-
4.23606797749979
2.52311
×
1
0
-
1208
0
.
×
1
0
-
299Table 4
Convergence behavior withf(x)=(ex2+3x+7-1)5(x2+3x+7)3. (m,λ,ρ)=(8,1/7,-3),α=(-3-19i)/2.
n
x
n
|
f
(
x
n
)
|
|
x
n
-
α
-
|
|
e
-
n
+
1
/
e
-
n
4
|
η
log
⁡
⁡
(
e
-
n
/
η
)
/
log
⁡
⁡
(
e
-
n
-
1
)
0
-
1.46000000000000
-
2.09000000000000
i
0.00250952
0.0979858
2.458883957
1
-
1.49981340826556
-
2.17945296114250
i
1.91755×10-25
0.000186624
2.024494357
4.08382
2
-
1.50000000000000
-
2.17944947177033
i
8.16451
×
1
0
-
112
2.98273
×
1
0
-
15
2.458901785
4.00000
3
-
1.50000000000000
-
2.17944947177034
i
2.68276
×
1
0
-
457
1.94624
×
1
0
-
58
2.458883957
4.00000
4
-
1.50000000000000
-
2.17944947177034
i
3.12745
×
1
0
-
1839
3.52795
×
1
0
-
231
2.458883957
4.00000
5
-
1.50000000000000
-
2.17944947177034
i
0
.
×
1
0
-
3888
0
.
×
1
0
-
299Additional test functions below are used to display the convergence behavior of proposed scheme (7):(23)f1(x)=(-1+3x+x2)log⁡(3+x2),α~=0.302775637731995,m=1,x0=0.29,f2(x)=(x2-5)log⁡(x2-4),α=-5,m=2,x0=-2.19,f3(x)=(x2-25)2ex2+1cscxlog⁡(x2+3x-9),α=-5,m=3,x0=-5,f4(x)=(x-7x)4,α=7,m=4,x0=2.77,f5(x)=log⁡3(x2-1)(x2-2)2,α=-2,m=5,x0=-1.38,f6(x)=(x2+3)4(log⁡(x4+x2-5))2,α=3i=1.73i,m=6,x0=1.68i,f7(x)=log⁡(x2-8)(ex2+7x-30-1)3(x-3)3,α=3,m=7,x0=2.88.
In Table 5, we compare numerical errors |xn-α| of proposed methods Y1–Y4 with those of existing optimal fourth-order multiple-root finders. Abbreviations S, J, L, Z, and K denote optimal fourth-order multiple-root finders obtained by Shengguo et al., Sharma et al., Li et al., Zhou et al., and Kanwar et al., respectively.Table 5
Comparison of|xn-α-| for various multiple-root finders.
f
(
x
)
x
0
|
x
n
-
α
|
S
J
L
Z
K
Y1
Y2
Y3
Y4
f
1
0.29
|
x
1
-
α
-
|
1.35
e
-
9
2.90
e
-
10
2.90
e
-
10
1.85
e
-
9
2.93
e
-
9
9.70
e
-
11
7.76
e
-
11
7.76
e
-
11
4.02
e
-
9
|
x
2
-
α
-
|
1.63
e
-
37
6.07
e
-
41
6.07
e
-
41
8.60
e
-
37
8.46
e
-
36
9.92
e
-
44
1.38
e
-
44
1.38
e
-
44
4.07
e
-
35
|
x
3
-
α
-
|
3.53
e
-
149
1.16
e
-
163
1.16
e
-
163
3.99
e
-
146
5.83
e
-
142
1.09
e
-
175
1.39e − 179
1.39
e
-
179
4.24
e
-
139
|
x
4
-
α
-
|
0
.
e
-300
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
f
2
-
2.19
|
x
1
-
α
-
|
1.83
e
-
6
1.67
e
-
6
3.03
e
-
6
3.09
e
-
7
7.02
e
-
7
1.45
e
-
6
1.45
-
6
2.12
-
6
2.46
e
-
8
|
x
2
-
α
-
|
2.60
e
-
23
1.97
e
-
24
3.06
e
-
23
1.62
e
-
27
2.66
e
-
26
9.92
e
-
25
9.92
e
-
24
6.24
e
-
24
3.80
e
-
33
|
x
3
-
α
-
|
1.05
e
-
91
3.76
e
-
96
7.24
e
-
91
4.92
e
-
109
5.51
e
-
104
2.12
e
-
97
2.12
e
-
94
4.69
e
-
94
2.15e − 132
|
x
4
-
α
-
|
0
.
e
-
300
0
.
e
-
299
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
0
.
e
-
300
f
3
-
5
|
x
1
-
α
-
|
0.011
0.013
0.011
0.013
0.013
0.012
0.011
0.012
0.014
|
x
2
-
α
-
|
2.73
e
-
8
1.26
e
-
7
2.73
e
-
8
2.26
e
-
7
1.50
e
-
7
6.32
e
-
8
2.22
e
-
8
6.67
e
-
8
2.51
e
-
7
|
x
3
-
α
-
|
1.05
e
-
30
6.94
e
-
29
1.05
e
-
30
4.47
e
-
27
4.01
e
-
28
9.72
e
-
30
4.61
e
-
31
1.70
e
-
29
8.06
e
-
27
|
x
4
-
α
-
|
2.35
e
-
120
6.28
e
-
114
2.35
e
-
120
6.83
e
-
106
2.02
e
-
110
5.42
e
-
117
8.51e − 122
7.20
e
-
116
8.55
e
-
105
|
x
5
-
α
-
|
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
f
4
2.77
|
x
1
-
α
-
|
9.18
e
-
7
7.67
e
-
7
9.18
e
-
7
6.90
e
-
7
7.67
e
-
7
7.48
e
-
7
8.62
e
-
7
8.62
e
-
7
6.78
e
-
7
|
x
2
-
α
-
|
2.93
e
-
27
1.17
e
-
27
2.93
e
-
27
6.86
e
-
28
1.16
e
-
27
1.03
e
-
27
2.12
e
-
27
2.12
e
-
27
6.21
e
-
28
|
x
3
-
α
-
|
3.07
e
-
109
6.39
e
-
111
3.07
e
-
109
6.65
e
-
112
6.32
e
-
111
3.70
e
-
111
7.85
e
-
110
7.85
e
-
110
4.36e − 112
|
x
4
-
α
-
|
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
f
5
-
1.38
|
x
1
-
α
-
|
1.04
e
-
7
8.69
e
-
8
1.04
e
-
7
8.00
e
-
8
8.91
e
-
8
8.49
e
-
8
9.88
e
-
8
9.88
e
-
8
7.89
e
-
8
|
x
2
-
α
-
|
6.35
e
-
30
2.50
e
-
30
6.35
e
-
30
1.63
e
-
30
2.85
e
-
30
2.23
e
-
30
4.83
e
-
30
4.83
e
-
30
1.53
e
-
30
|
x
3
-
α
-
|
8.75
e
-
119
1.72
e
-
120
8.75
e
-
119
2.86
e
-
121
2.99
e
-
120
1.06
e
-
120
2.77
e
-
119
2.77
e
-
119
2.16e − 121
|
x
4
-
α
-
|
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
f
6
1.68
i
|
x
1
-
α
-
|
1.79
e
-
3
1.32
e
-
3
1.79
e
-
3
1.20
e
-
3
1.43
e
-
3
1.29
e
-
3
1.16
e
-
3
1.65
e
-
3
1.18
e
-
3
|
x
2
-
α
-
|
1.36
e
-
10
4.38
e
-
11
1.36
e
-
10
2.99
e
-
11
5.88
e
-
11
3.91
e
-
11
2.66
e
-
11
1.008
e
-
10
2.84
e
-
11
|
x
3
-
α
-
|
4.76
e
-
39
5.29
e
-
41
4.76
e
-
39
1.18
e
-
41
1.69
e
-
40
3.39
e
-
41
7.38
e
-
42
1.42
e
-
39
9.63
e
-
42
|
x
4
-
α
-
|
6.94
e
-
153
1.13
e
-
160
6.94
e
-
153
2.84
e
-
163
1.18
e
-
158
1.92
e
-
161
4.37e − 164
5.67
e
-
155
1.26
e
-
163
|
x
5
-
α
-
|
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
f
7
2.88
|
x
1
-
α
-
|
1.380
e
-
2
1.372
e
-
2
1.380
e
-
2
1.370
e
-
2
1.374
e
-
2
1.372
e
-
2
1.378
e
-
2
1.378
e
-
2
1.370
e
-
2
|
x
2
-
α
-
|
2.21
e
-
7
1.92
e
-
7
2.21
e
-
7
1.84
e
-
7
2.00
e
-
7
1.91
e
-
7
2.14
e
-
7
2.14
e
-
7
1.83
e
-
7
|
x
3
-
α
-
|
4.23
e
-
27
1.48
e
-
27
4.23
e
-
27
1.02
e
-
27
2.02
e
-
27
1.33
e
-
27
3.40
e
-
27
3.40
e
-
27
9.73
e
-
28
|
x
4
-
α
-
|
5.69
e
-
106
5.27
e
-
108
5.69
e
-
106
9.61
e
-
109
2.11
e
-
107
3.18
e
-
108
2.16
e
-
106
2.16
e
-
106
7.64e − 109
|
x
5
-
α
-
|
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
0
.
e
-
299
Here,1.35e-9 denotes 1.35×10-9.The least errors within the prescribed error bound are highlighted in boldface. Method Y2 shows best convergence forf1, f3, f6, while method Y4 does show for f2, f4, f5, f7. It must be kept in mind that such favorable performance is shown only in the current numerical experiments with the particular choice of test functions. For a different choice of test functions and initial values, it is hardly expected that each of listed methods would always give rise to better convergence behavior. One should be aware that no numerical method exhibits better performance for all the test functions as compared with other numerical methods. With the same order of convergence, one should notice that the speed of local convergence is dependent on the function f(x), an initial value x0, and a multiple zero α itself.For efficiency check of multipoint iterative methods, we need to calculate the efficiency index [17] defined by EI=p1/d, where p is the order of convergence and d is the number of distinct functional or derivative evaluations per iteration. The proposed iterative methods (7) have EI=41/3≈1.587 describing the optimality, which is the same as that of any other listed method. This paper confirms optimal fourth-order convergence and derives the correct error equation for proposed iterative methods, using the weighted harmonic mean of two derivatives to find approximate multiple zeros of nonlinear equations. Proposed optimal fourth-order schemes efficiently solve given problems without any difficulty, with a wide selection of free parameters λ and ρ. In future work, we will pursue higher-order optimal methods by extending the methods developed here.



---

*Source: 737305-2014-09-14.xml*