CUBO A Mathematical Journal
Vol.20, No¯ 3, (65–79). October 2018

http: // dx. doi. org/ 10. 4067/ S0719-06462018000300065

Ball comparison between Jarratt’s and other fourth order
method for solving equations

Ioannis K. Argyros
1
and Santhosh George

2

1Department of Mathematical Sciences,

Cameron University,

Lawton, OK 73505, USA.
2Department of Mathematical and Computational Sciences,

National Institute of Technology Karnataka,

India-575 025.

iargyros@cameron.edu, sgeorge@nitk.edu.in

ABSTRACT

The convergence order of iterative methods is determined using high order derivatives

and Taylor series, and without providing computable error bounds, uniqueness of the

solution results or information on how to choose the initial point. We address all these

problems by using hypotheses only on the first derivative. Moreover, to achieve all

these we present our technique using a comparison between the convergence radii of

Jarratt’s fourth order method and another method of the same convergence order.

RESUMEN

El orden de convergencia de métodos iterativos es determinado usando derivadas de

orden alto y series de Taylor, y sin poder entregar cotas de error calculables, resultados

de unicidad de soluciones o información de cómo elegir el punto inicial. Tratamos estos

problemas usando hipótesis sólo en la primera derivada. Más aún, para responder

todos los anteriores, presentamos una técnica que usa una comparación entre el radio

de convergencia del método de cuarto orden de Jarratt y otro método con el mismo

orden de convergencia.

http://dx.doi.org/10.4067/S0719-06462018000300065


66 Ioannis K. Argyros and Santhosh George CUBO
20, 3 (2018)

Keywords and Phrases: Jarratt method; Banach space; Ball convergence.

2010 AMS Mathematics Subject Classification: 65D10, 65D99.



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Ball comparison between Jarratt’s and other fourth order method . . . 67

1 Introduction

Let B1 and B2 stand for Banach spaces, with Ω ⊆ B1 being nonempty, open and convex. Consider

an equation

F(x) = 0, (1.1)

where F : Ω −→ B2 is a differentiable in the of Fréchet-sense. The task of finding a solution p

of equation (1.1) is very difficult in general. It is even harder to find a solution p in closed form,

since this can be achieved in some special cases. That explains why most authors develop iterative

methods, to generate a sequence approximating p under some initial conditions.

Notice that, solution methods for equation (1.1) is an important area of research, since a

plethora of problems from diverse disciplines such that Mathematics, Optimization, Mathematical

Programming, Chemistry, Biology, Physics, Economics, Statistics, Engineering and other disci-

plines can be modeled into an equation of the form (1.1) using mathematical modeling [1, 2, 3, 4,

5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The most popular method is without a doubt Newton’s

method (NM)

xn+1 = xn − F
′(xn)

−1F(xn), x0 ∈ Ω, and all n = 0, 1, 2, . . . . (1.2)

NM converges quadratically to p for x0 sufficiently close to p [10]. To increase the convergence

order numerous methods have been proposed [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The

order of these methods is almost exclusively been obtained using Taylor series, and hypotheses on

high order derivatives. No computable error bounds or uniqueness results are given, and the choice

of the initial point is a shot in the dark.

Iterative methods are usually studied based on: semi-local and local convergence. The semi-

local convergence matter is, based on the information around an initial point, to give conditions

ensuring the convergence of the iterative procedure; while the local one is, based on the information

around a solution, to find estimates of the radii of convergence balls [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11,

12, 13, 14, 15, 16].

A radius of convergence about p determines a ball such that if an initial point is selected from

that ball convergence of the method to p is guaranteed. To deal with all these problems we have

selected two popular fourth order methods. In particular, we compare the radii of convergence of

fourth order Jarratt’s iterative method defined [9, 12] for n = 0, 1, 2, . . . , as

yn = xn −
2

3
F′(xn)

−1F(xn)

xn+1 = xn −
1

2
[(3F′(yn) − F

′(xn))
−1(3F′(yn) + F

′(xn))]

×F′(xn)
−1F(xn), (1.3)



68 Ioannis K. Argyros and Santhosh George CUBO
20, 3 (2018)

to the fourth order Sharma’s method [13] defined for n = 0, 1, 2, . . . , as

yn = xn −
2

3
F′(xn)

−1F(xn)

xn+1 = xn −
1

2
[−I +

9

4
F′(yn)

−1F′(xn) +
3

4
F′(xn)

−1F′(yn)]

×F′(xn)
−1F(xn). (1.4)

Earlier convergence analysis of these methods, in the special case when B1 = B2 = R
k used,

assumptions of the Fréchet derivatives of F of order up to five [9, 12, 13]. But these assumptions

limit the applicability of methods (1.3) and (1.4).

Let as an example, B1 = B2 = R, Ω = [−
1
2
, 3
2
]. Define F on Ω as

F(x) = x3 log x2 + x5 − x4

Then, we have p = 1, and

F′(x) = 3x2 log x2 + 5x4 − 4x3 + 2x2,

F′′(x) = 6x log x2 + 20x3 − 12x2 + 10x,

F′′′(x) = 6 log x2 + 60x2 = 24x + 22.

Clearly, F′′′(x) is not bounded on Ω. So, methods (1.3) and (1.4) cannot be applied to solve the

above example, if we use the analysis in the earlier studies. In this study, our analysis uses only

the assumptions on the first Fréchet derivative of F.

Moreover, we provide computable upper estimates on ‖xn −p‖, a radius of convergence as well

as uniqueness results based on generalized Lipschitz conditions. Hence, we extend the applicability

of these methods. Our technique can be used to extend the applicability of other high order methods

along the same lines.

The rest of the study is organized as follows. In Section 2 , the local convergence analysis is

given and numerical examples are given in the last Section 4.

2 Local convergence

It is convenient for the local convergence analysis of method (1.3) and method (1.4) to introduce

some fucntions and parameters. First for method (1.3): Let ω0 : S −→ S be a continuous and

increasing function with w0(0) = 0, where S = [0, ∞). Suppose that equation

ω0(t) = 1 (2.1)

has at least one positive solution. Denote by ρ0 the smallest such solution. Set S0 = [0, ρ0). Let

also ω : S0 −→ S and ω1 : S0 −→ S be continuous and increasing functions with ω(0) = 0. Define



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Ball comparison between Jarratt’s and other fourth order method . . . 69

functions ϕ1 and ϕ̄1 on the interval S0 by

ϕ1(t) =

∫1
0
ω((1 − θ)t)dθ + 1

3

∫1
0
ω1(θt)dθ

1 − ω0(t)

and

ϕ̄1(t) = ϕ1(t) − 1.

Suppose that

ω0(0) < 3. (2.2)

Then, we get by (2.2) that ϕ̄1(0) < 0 and ϕ̄1(t) −→ ∞ as t −→ ρ
−

0 . The intermediate value

theorem guarantees that equation ϕ̄1(t) = 0 has at least one solution in (0, ρ0). Denote by R1 the

smallest such solution. Suppose that equation

ω0(ϕ1(t)t) = 1 (2.3)

has at least one positive solution. Denote by ρ1 the smallest such solution. Set ρ = min{ρ0, ρ1}

and S1 = [0, ρ). Define functions ϕ2 and ϕ̄2 on S1 by

ϕ2(t) =

∫1
0
ω((1 − θ)t)dθ

1 − ω0(t)
+

3

8
[

(ω0(ϕ1(t)t) + ω0(t))
2

(1 − ω0(t))(1 − ω0(ϕ1(t)t))

+2
w0(ϕ1(t)t) + ω0(t)

1 − ω0(ϕ1(t)t)
]

∫1
0
ω1(θt)dθ

1 − ω0(t)

and

ϕ̄2(t) = ϕ2(t) − 1.

We get that ϕ̄2(0) = −1 and ϕ̄2(t) −→ ∞ as t −→ ρ
−. Denote by R2 the smallest such solution

of equation ϕ̄2(t) = 0. Moreover, define a radius of convergence R by

R = min{R1, R2}. (2.4)

It follows that for each t ∈ [0, R)

0 ≤ ω0(t) < 1 (2.5)

0 ≤ ω0(ϕ1(t)t) < 1 (2.6)

0 ≤ ϕ1(t) < 1 (2.7)

and

0 ≤ ϕ2(t) < 1. (2.8)

Let us introduce conditions (A):

(a1) F : Ω −→ B2 is continuously differentiable in the sense of Fréchet and there exists p ∈ Ω

such that F(p) = 0 and F′(p)−1 ∈ L(B2, B1).



70 Ioannis K. Argyros and Santhosh George CUBO
20, 3 (2018)

(a2) There exists function ω0 : S −→ S continuous and increasing with ω0(0) = 0 and for each

x ∈ Ω

‖F′(p)−1(F′(x) − F′(p))‖ ≤ ω0(‖x − p‖)

and (2.2) holds. Set Ω0 = Ω ∩ U(p, ρ0), where ρ0 is given in (2.1).

(a3) There exist functions ω : S0 −→ S, ω1 : S0 −→ S continuous and increasing with ω(0) = 0

such that for each x, y ∈ Ω0

‖F′(p)−1(F′(y) − F′(x))‖ ≤ ω(‖y − x‖)

and

‖F′(p)−1F′(x)‖ ≤ ω1(‖x − p‖).

(a4) Ū(p, R) ⊂ Ω, ρ0, ρ1 exist and are given by (2.1) and (2.3), respectively.

(a5) There exists R∗ ≥ R such that
∫1

0

ω0(θR
∗
)dθ < 1.

Set Ω1 = Ω ∩ Ū(p, R
∗).

Next, the local convergence analysis is given for method (1.3) based on the conditions (A) and the

preceding notation.

Theorem 2.1. Suppose that the conditions (A) hold. Then, sequence {xn} generated by (1.3),

starting at x0 ∈ U(p, R) − {p} is well defined, remains in U(p, R) for each n = 0, 1, 2, 3, . . . and

converges to p. Moreover, the following error bounds hold

‖yn − p‖ ≤ ϕ1(‖x − p‖)‖x − p‖ ≤ ‖x − p‖ < R (2.9)

and

‖xn+1 − p‖ ≤ ϕ2(‖x − p‖)‖x − p‖ ≤ ‖x − p‖, (2.10)

where functions ϕ1 and ϕ2 are given previously and R is defined in (2.4). Furthermore, the limit

point p is the only solution of equation F(x) = 0 in the set Ω1, which is defined in (a5).

Proof. Mathematical induction is utilized to show (2.9) and (2.10). Let x ∈ U(p, R) − {p}.

Then, by (a1), (a2), (2.1), (2.4) and (2.5), we obtain in turn that

‖F′(p)−1(F′(x) − F′(p))‖ ≤ ω0(‖x − p‖) ≤ ω0(R) < 1. (2.11)

In view of (2.11) and the Banach lemma on invertible operators [7, 8, 10], F′(x)−1 ∈ L(B2, B1) and

‖F′(x)−1F′(p)‖ ≤
1

1 − ω(‖x − p‖)
. (2.12)



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Ball comparison between Jarratt’s and other fourth order method . . . 71

The point y0 is well defined by the first substep of method (1.3) and (2.12) for x = x0. We can

write by (a1)

F(x) = F(x) − F(p) =

∫1

0

F′(p + θ(x − p))(x − p)dθ. (2.13)

Then, by the second hypothesis in (a3), we get by (2.13) that

‖F′(p)−1F′(p)‖ ≤

∫1

0

ω1(θ‖x − p‖)dθ‖x − p‖. (2.14)

Using the first substep of method (1.3) for n = 0, (a3), (2.4), (2.7), (2.12) (for x = x0) and (2.14),

we have in turn from

y0 − p = x0 − p − F
′(x0)

−1F(x0) +
1

3
F′(x0)

−1F(x0)

that

‖y0 − p‖ ≤ ‖F
′(x0)

−1F′(p)‖‖

∫1

0

F′(p)−1(F′(p + θ(x0 − p)) − F
′(x0))dθ(x − p)‖

1

3
‖F′(x0)

−1F′(p)‖‖F′(p)−1F(x0)‖

≤
[
∫1
0
ω((1 − θ)‖x0 − p‖)dθ +

1
3

∫1
0
ω1(θ‖x0 − p‖)dθ]

1 − ω0(‖x0 − p‖)

×‖x0 − p‖

= ϕ1(‖x0 − p‖)‖x0 − p‖ ≤ ‖x0 − p‖ < R, (2.15)

which implies that (2.9) holds for n = 0 and y0 ∈ U(p, R). Moreover, F
′(y0)

−1 ∈ L(B2, B1), so x1

is well defined by the second substep of method (1.3) for n = 0 and (2.6). Furthermore, by (2.4),

(2.8), (2.12) (for x = y0), (2.14) (for x = y0) and the estimate

x1 − p = x0 − p − F
′
(x0)

−1F(x0)

−
1

2
[−3I +

9

4
F′(y0)

−1F′(x0)

+
3

4
F′(x0)

−1F′(y0)]F
′(x0)

−1F(x0)

= x0 − p − F
′
(x0)

−1F(x0)

−
3

2
[−I +

3

4
F′(y0)

−1F′(x0)

+
1

4
F′(x0)

−1F′(y0)]F
′(x0)

−1F(x0)

= x0 − p − F
′(x0)

−1F(x0)

−
3

8
[F′(x0)

−1(F′(y0) − F
′(x0))F

′(y0)
−1(F′(y0) − F

′(x0))

−2F′(y0)
−1

(F′(y0) − F
′
(x0)]F

′
(x0)

−1F(x0), (2.16)



72 Ioannis K. Argyros and Santhosh George CUBO
20, 3 (2018)

we have in turn that

‖x1 − p‖ ≤ ‖x0 − p − F
′
(x0)

−1F(x0)‖

+
3

8
[‖F′(x0)

−1F′(p)‖(‖F′(p)−1(F′(y0) − F
′(x0))‖

+‖F′(p)−1(F′(x0) − F
′
(p))‖)2

‖F′(y0)
−1F′(p)‖

+2‖F′(y0)
−1F′(p)‖(‖F′(p)−1(F′(y0) − F

′(x0)‖

+‖F′(p)−1(F′(x0) − F
′(p))‖)]

‖F′(x0)
−1F′(p)‖‖F′(p)−1F(x0)‖

≤

{∫1
0
ω((1 − θ)‖x0 − p‖)dθ

1 − ω0(‖x0 − p‖)

3

8

[

(ω0(‖y0 − p‖) + ω0(‖x0 − p‖))
2

(1 − ω0(‖x0 − p‖))(1 − ω0(‖y0 − p‖))

+2
ω0(‖x0 − p‖) + ω0(‖y0 − p‖)

1 − ω0(‖y0 − p‖)

]

∫1
0
ω1(θ‖x0 − p‖)dθ

1 − ω0(‖x0 − p‖)

}

‖x0 − p‖

≤ ϕ2(‖x0 − p‖)‖x0 − p‖ ≤ ‖x0 − p‖, (2.17)

so (2.10) holds for n = 0 and x1 ∈ U(p, R), where we also used the following estimates in the

derivativation of (2.16):

−I +
3

4
F′(y0)

−1F′(x0) +
1

4
F′(x0)

−1F′(y0) (2.18)

= −
3

4
I +

3

4
F′(yn)

−1F′(xn) −
1

4
I +

1

4
F′(x0)

−1F′(y0)

=
3

4
[F′(y0)

−1F′(x0) − I] +
1

4
[F′(x0)

−1F′(y0) − I]

=
3

4
F′(y0)

−1(F′(x0) − F
′(y0)) +

1

4
F′(x0)

−1(F′(y0) − F
′(x0))

=
1

4
F′(x0)

−1
(F′(y0) − F

′
(x0)) −

1

4
F′(y0)

−1
(F′(y0) − F

′
(x0))

−
2

4
F′(y0)

−1(F′(y0) − F
′(x0))

=
1

4
(F′(x0)

−1 − F′(y0)
−1)(F′(y0) − F

′(x0))

−
1

2
F′(y0)

−1
(F′(y0) − F

′
(x0))

=
1

4
F′(x0)

−1(F′(y0) − F
′(x0))F

′(y0)
−1(F′(y0) − F

′(x0))

−
1

2
F′(y0)

−1(F′(y0) − F
′(x0)). (2.19)



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Ball comparison between Jarratt’s and other fourth order method . . . 73

The induction for (2.9) and (2.10) is completed, if xm, ym, xm+1 replace x0, y0, x1 in the preceding

estimations. Then, from the estimate

‖xm+1 − p‖ ≤ r‖xm − p‖ < R, r = ϕ2(‖x0 − p‖) ∈ [0, 1) (2.20)

we conclude that limm−→∞ xm = p and xm+1 ∈ U(p, R). Finally, let G =
∫1
0
F′(p1 + θ(p − p1))dθ

for p1 ∈ Ω1 with F(p1) = 0. Then, by (a2), we get that

‖F′(p)−1(G − F′(p))‖ ≤

∫1

0

ω0(θ‖p − p1‖)dθ

≤

∫1

0

ω0(θR
∗)dθ < 1 (2.21)

leading to G−1 ∈ L(B2, B1). Then, from the identity

0 = F(p) − F(p1) = G(p − p1),

we deduce that p1 = p.

Next, we study the local convergence analysis of method (1.4) in an analogous way. Let

ω0, ω, ω1, ρ0, ϕ1 and ϕ̄1 are previously. Suppose that equation

q(t) = 1 (2.22)

where q(t) = 1
2
(3ω0(ϕ1(t)t)+ω0(t)) has at least one positive solution. Denote by ρ1 the smallest

such solution. Set D1 = [0, ρ) where ρ = min{ρ0, ρ1}. Define functions ϕ3 and ϕ̄3 on D1 by

ϕ3(t) =

∫1
0
ω((1 − θ)t)dθ

1 − ω0(t)

+
3

2

(ω0(t) + ω0(ϕ1(t)t))
∫1
0
ω1(θt)dθ

(1 − q(t))(1 − ω0(t))

and

ϕ̄3 = ϕ3 − 1.

We get ϕ̄3(t) = −1 and ϕ̄3(t) −→ ∞ as t −→ ρ
−. Denote by R3 the smallest solution of equation

ϕ̄3 = 0 in (0, ρ). Define a radius of convergence R by

R = min{R1, R3}. (2.23)

Consider the conditions (A) again but with R given in (2.23) and ρ1 given in (2.22). Call these

conditions (A)’. Then, for each t ∈ [0, R), we have

0 ≤ ω0(t) < 1 (2.24)

0 ≤ q(t) < 1 (2.25)

0 ≤ ϕ1(t) < 1 (2.26)

and

0 ≤ ϕ3(t) < 1. (2.27)



74 Ioannis K. Argyros and Santhosh George CUBO
20, 3 (2018)

Theorem 2.2. Suppose that the conditions (A) hold. Then, sequence {xn} defined by (1.4),

starting at x0 ∈ U(p, R) − {p} is well defined, remains in U(p, R) for each n = 0, 1, 2, 3, . . . and

converges to p. Moreover, the following error bounds hold

‖yn − p‖ ≤ ϕ1(‖x − p‖)‖x − p‖ ≤ ‖x − p‖ < R (2.28)

and

‖xn+1 − p‖ ≤ ϕ3(‖x − p‖)‖x − p‖ ≤ ‖x − p‖, (2.29)

where functions ϕ1 and ϕ3 are given previously and R is defined in (2.23). Furthermore, the limit

point p is the only solution of equation F(x) = 0 in the set Ω1, which is defined previously.

Proof. It follows as in Theorem 2.1 but notice

‖(2F′(p))−1(3F′(y0) − F
′(x0) − 3F

′(p) + F′(p))‖

≤
1

2
(3‖F′(p)−1(F′(y0) − F

′
(p))‖

+‖F′(p)−1(F′(x0) − F
′(p))‖)

≤
1

2
(3ω0(‖y0 − p‖) + ω0(‖x0 − p‖))

≤
1

2
(3ω0(ϕ1(‖x0 − p‖)‖x0 − p‖) + ω0(‖x0 − p‖)

= q(‖x0 − p‖) < 1 (2.30)

and

x1 − p = x0 − p −
1

2
[(3F′(y0) − F

′(x0))
−1(3F′(y0) − F

′(x0))

+2F′(x0)]F
′(x0)

−1F(x0)

= x0 − p − F
′(x0)

−1F(x0)

−
1

2
[−I + 2(3F′(y0) − F

′
(x0))

−1
]F′(x0)

−1F(x0)

= x0 − p − F
′(x0)

−1F(x0) −
3

2
(3F′(y0) − F

′(x0))
−1

×(F′(x0) − F
′(y0))F

′(x0)
−1F(x0), (2.31)

where for the derivation of (2.31), we also used the estimate

−I + 2(3F′(y0) − F
′(x0))

−1F′(x0)

= (3F′(y0) − F
′
(x0))

−1
[−(3F′(y0) − F

′
(x0)) + 2F

′
(x0)]

= 3(3F′(y0) − F
′
(x0))

−1
[F′(x0) − F

′
(y0)],



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Ball comparison between Jarratt’s and other fourth order method . . . 75

so we get by (2.31)

‖x1 − p‖ ≤ ‖x0 − p − F
′(x0)

−1F(x0)‖

+
3

2
‖(3F′(y0) − F

′(x0))
−1F′(p)‖

×[‖F′(p)−1(F′(y0) − F
′(p))‖ + ‖F′(p)−1(F′(x0) − F

′(p))‖]

×‖F′(x0)
−1F′(p)‖‖F′(p)−1F(p)‖

≤

[∫1
0
ω0((1 − θ)‖x0 − p‖)dθ

1 − ω0(‖x0 − p‖)

+
3

2

(ω0(‖x0 − p‖) + ω0(‖y0 − p‖))
∫1
0
ω1(θ‖x0 − p‖)dθ

(1 − q(‖x0 − p‖))(1 − ω0(‖x0 − p‖))

]

‖x0 − p‖

≤ ϕ3(‖x0 − p‖)‖x0 − p‖ ≤ ‖x0 − p‖ < R, (2.32)

which shows (2.29) for n = 0 and x1 ∈ U(p, R). The rest of the proof as identical to the one in

Theorem 2.1 is omitted.

✷

Remark 2.3. (a) Let ω0(t) = L0t, ω(t) = Lt. Then, the radius rA =
2

2L0+L
was obtained by

Argyros in [4] as the convergence radius for Newton’s method under condition (2.12)-(2.14).

Notice that the convergence radius for Newton’s method given independently by Rheinboldt

[14] and Traub [16] is given by

ρ =
2

3L
< rA,

where ω1(t) = L1t replaces ω(t), and L1 is the Lipschitz constant on Ω. Notice that Ω0 ⊆ Ω,

so L0 ≤ L1 and L ≤ L1. As an example, let us consider the function f(x) = e
x − 1. Then

p = 0. Set D = U(0, 1). Then, we have that L0 = e − 1 < L = e
1

e−1 < L1 = e, so

ρ = 0.24252961 < rA = 0.3827.

Moreover, the new error bounds [4, 5, 6, 7, 8] are:

‖xn+1 − p‖ ≤
L

1 − L0‖xn − p‖
‖xn − p‖

2,

whereas the old ones [10, 14, 16]

‖xn+1 − p‖ ≤
L

1 − L‖xn − p‖
‖xn − p‖

2.

Clearly, the new error bounds are more precise, if L0 < L. Then, the radius of convergence

of method (1.3) or method (1.4) cannot be larger than rA.

(b) The local results can be used for projection methods such as Arnoldi’s method, the generalized

minimum residual method(GMREM), the generalized conjugate method(GCM) for combined

Newton/finite projection methods and in connection to the mesh independence principle in

order to develop the cheapest and most efficient mesh refinement strategy [4, 5, 6, 7, 8, 10].



76 Ioannis K. Argyros and Santhosh George CUBO
20, 3 (2018)

(c) The results can be also be used to solve equations where the operator F′ satisfies the au-

tonomous differential equation [4, 5, 6, 7, 8, 10]:

F′(x) = p(F(x)),

where p is a known continuous operator. Since F′(p) = p(F(p)) = p(0), we can apply the

results without actually knowing the solution p. Let as an example F(x) = ex − 1. Then, we

can choose p(x) = x + 1 and p = 0.

(d) It is worth noticing that method (1.3) or method (1.4) are not changing, if we use the new

instead of the old conditions [9, 12, 13]. Moreover, for the error bounds in practice we can

use the computational order of convergence (COC)

ξ =
ln

‖xn+2−xn+1‖
‖xn+1−xn‖

ln
‖xn+1−xn‖
‖xn−xn−1‖

, for all n = 1, 2, . . .

or the approximate computational order of convergence (ACOC)

ξ∗ =
ln

‖xn+2−p‖
‖xn+1−p‖

ln
‖xn+1−p‖
‖xn−p‖

, for all n = 0, 1, 2, . . .

instead of the error bounds obtained in Theorem 2.1. Notice that these formulae do not require

high order derivatives and in the case of ACOC not even knowledge of p. The convergence

radii are optimum under conditions (A).

(e) In view of (a2) and the estimate

‖F′(p)−1F′(x)‖ = ‖F′(p)−1(F′(x) − F′(p)) + I‖

≤ 1 + ‖F′(p)−1(F′(x) − F′(p))‖ ≤ 1 + L0‖x − p‖

second condition in (a3) can be dropped and M can be replaced by

M(t) = 1 + L0t

or

M(t) = M = 2,

since t ∈ [0, 1
L0

).

3 Numerical examples

Example 3.1. Let B1 = B2 = R
3, Ω = Ū(0, 1), x∗ = (0, 0, 0)T . Define function F on Ω for

u = (x, y, z)T by

F(u) = (ex − 1,
e − 1

2
y2 + y, z)T .



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Ball comparison between Jarratt’s and other fourth order method . . . 77

Then, the Fréchet-derivative is given by

F′(v) =









ex 0 0

0 (e − 1)y + 1 0

0 0 1









.

Notice that using the (2.8)-(2.12), conditions, we get ω0(t) = (e−1)t, ω(t) = e
1

e−1 t, ω1(t) = e
1

e−1 .

Then using the definition of r, we have that

R1 = 0.15440695135715407082521721804369 = R

and

R2 = 0.17352535186531112265662102345232.

Example 3.2. Let B1 = B2 = C[0, 1], the space of continuous functions defined on [0, 1] and be

equipped with the max norm. Let Ω = U(0, 1). Define function F on Ω by

F(ϕ)(x) = ϕ(x) − 5

∫1

0

xθϕ(θ)3dθ. (3.1)

We have that

F′(ϕ(ξ))(x) = ξ(x) − 15

∫1

0

xθϕ(θ)2ξ(θ)dθ, for each ξ ∈ Ω.

Then, we get that x∗ = 0, ω0(t) = 7.5t, ω(t) = 15t, ω1(t) = 2. This way, we have that

R1 = 0.022222222222222222222222222222222 = R

and

R2 = 0.18929637111931424398036938328005.

Example 3.3. Let B1 = B2 = R, Ω = [−
1
2
, 3
2
]. Define F on Ω by

F(x) = x3 log x2 + x5 − x4

Then

F′(x) = 3x2 log x2 + 5x4 − 4x3 + 2x2,

Then, we get that ω0(t) = ω(t) = 147t, ω1(t) = 2. So, we obtain

R1 = 0.0015117157974300831443688586545729 = R

and

R2 = 0.01297295712377562193484692443235.

Example 3.4. Let B1 = B2 = C[0, 1], Ω = Ū(x
∗, 1) and consider the nonlinear integral equation

of the mixed Hammerstein-type [1, 2, 3, 5, 11] defined by

x(s) =

∫1

0

G(s, t)(x(t)3/2 +
x(t)2

2
)dt,



78 Ioannis K. Argyros and Santhosh George CUBO
20, 3 (2018)

where the kernel G is the Green’s function defined on the interval [0, 1] × [0, 1] by

G(s, t) =

{
(1 − s)t, t ≤ s

s(1 − t), s ≤ t.

The solution x∗(s) = 0 is the same as the solution of equation (1.1), where F : C[0, 1] −→ C[0, 1])

is defined by

F(x)(s) = x(s) −

∫1

0

G(s, t)(x(t)3/2 +
x(t)2

2
)dt.

Notice that

‖

∫1

0

G(s, t)dt‖ ≤
1

8
.

Then, we have that

F′(x)y(s) = y(s) −

∫1

0

G(s, t)(
3

2
x(t)1/2 + x(t))dt,

so since F′(x∗(s)) = I,

‖F′(x∗)−1(F′(x) − F′(y))‖ ≤
1

8
(
3

2
‖x − y‖1/2 + ‖x − y‖).

Then, we get that ω0(t) = ω(t) =
1
8
(3
2
t1/2 + t), ω1(t) = 1 + ω0(t). So, we obtain

R1 = 1.2

and

R2 = 0.82757632634917221992054692236707 = R.



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Ball comparison between Jarratt’s and other fourth order method . . . 79

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[15] Sharma, J.R., Guha , R. K., Sharma, R., An efficient fourth order weighted Newton method

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	Introduction
	Local convergence
	Numerical examples