Zheng Journal of Inequalities and Applications (2016) 2016:289
DOI 10.1186/s13660-016-1227-5
RESEARCH
Open Access
The convergence theorem for
fourth-order super-Halley method in weaker
conditions
Lin Zheng*
*
Correspondence:
[email protected]
School of Statistics and Applied
Mathematics, Anhui University of
Finance and Economics, Bengbu,
233030, China
Department of Mathematics,
Shanghai University, Shanghai,
200444, China
Abstract
In this paper, we establish the Newton-Kantorovich convergence theorem of a
fourth-order super-Halley method under weaker conditions in Banach space, which is
used to solve the nonlinear equations. Finally, some examples are provided to show
the application of our theorem.
MSC: 65J15; 65H10; 65G99; 47J25; 49M15
Keywords: nonlinear equations in Banach spaces; super-Halley method; semilocal
convergence; Newton-Kantorovich theorem; weaker conditions
1 Introduction
For a number of problems arising in scientific and engineering areas one often needs to
find the solution of nonlinear equations in Banach spaces
F(x) = ,
()
where F is a third-order Fréchet-differentiable operator defined on a convex subset of
a Banach space X with values in a Banach space Y .
There are kinds of methods to find a solution of equation (). Generally, iterative methods are often used to solve this problem []. The best-known iterative method is Newton’s
method
xn+ = xn – F (xn )– F(xn ),
()
which has quadratic convergence. Recently a lot of research has been carried out to provide improvements. Third-order iterative methods such as Halley’s method, Chebyshev’s
method, super-Halley’s method, Chebyshev-like’s method etc. [–] are used to solve
equation (). To improve the convergence order, fourth-order iterative methods are also
discussed in [–].
Kou et al. [] presented a variant of the super-Halley method which improves the order
of the super-Halley method from three to four by using the values of the second derivative
at (xn –  f (xn )/f (xn )) instead of xn . Wang et al. [] established the semilocal convergence
© Zheng 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
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indicate if changes were made.
Zheng Journal of Inequalities and Applications (2016) 2016:289
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of the fourth-order super-Halley method in Banach spaces by using recurrence relations.
This method in Banach spaces can be given by
–

xn+ = xn – I + KF (xn ) I – KF (xn )
n F(xn ),

()
where n = [F (xn )]– , KF (xn ) = n F (un )n F(xn ), and un = xn –  n F(xn ).
Let x ∈ and the nonlinear operator F : ⊂ X → Y be continuously third-order
Fréchet differentiable where is an open set and X and Y are Banach spaces. Assume
that
(C)  F(x ) ≤ η,
(C)  ≤ β,
(C) F (x) ≤ M, x ∈ ,
(C) F (x) ≤ N, x ∈ ,
(C) there exists a positive real number L such that
F (x) – F (y) ≤ Lx – y,
∀x, y ∈ .
Under the above assumptions, we apply majorizing functions to prove the semilocal
convergence of the method () to solve nonlinear equations in Banach spaces and establish
its convergence theorems in []. The main results is as follows.
Theorem  ([]) Let X and Y be two Banach spaces and F : ⊆ X → Y be a third-order
Fréchet differentiable on a non-empty open convex subset . Assume that all conditions
(C)-(C) hold and x ∈ , h = Kβη ≤ /, B(x , t ∗ ) ⊂ , then the sequence {xn } generated
by the method () is well defined, xn ∈ B(x , t ∗ ) and converges to the unique solution x∗ ∈
B(x , t ∗∗ ) of F(x), and xn – x∗ ≤ t ∗ – tn , where
√
 +  – h
 – h
∗∗
η,
t =
η,
t =
h
h
L
N
.
K ≥M +  +
M β M β 
∗
–
√
()
We know the conditions of Theorem  cannot be satisfied by some general nonlinear
operator equations. For example,
F(x) =

    
x + x – x + = .




Let the initial point x = , = [–, ]. Then we know

β = F (x )– = ,


M= ,
N = ,


η = F (x )– F(x ) = ,

L = .
From (), we can get K ≥ M, so
h = Kβη ≥ Mβη =
 
> .
 
()
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The conditions of Theorem  cannot be satisfied. Hence, we cannot know whether the
sequence {xn } generated by the method () converges to the solution x∗ .
In this paper, we consider weaker conditions and establish a new Newton-Kantorovich
convergence theorem. The paper is organized as follows: in Section  the convergence
analysis based on weaker conditions is given and in Section , a new Newton-Kantorovich
convergence theorem is established. In Section , some numerical examples are worked
out. We finish the work with some conclusions and references.
2 Analysis of convergence
Let x ∈ and nonlinear operator F : ⊂ X → Y be continuously third-order Fréchet
differentiable where is an open set and X and Y are Banach spaces. We assume that:
(C)  F(x ) ≤ η,
(C) F (x )– F (x ) ≤ γ ,
(C) F (x )– F (x) ≤ N, x ∈ ,
(C) there exists a positive real number L such that
F (x )– F (x) – F (y) ≤ Lx – y,
∀x, y ∈ .
()
Denote
g(t) =
   
Kt + γ t – t + η,


()
where K, γ , η are positive real numbers and
L
+ N ≤ K.
Nη + γ
Lemma  ([]) Let α =
()
√
γ+
γ  +K
, β = α –  Kα  –  γ α  =
√

√ γ +K) . If η ≤ β, then the
(γ +
(γ +
γ  +K )
polynomial equation g(t) has two positive real roots r , r (r ≤ r ) and a negative root –r
(r > ).
Lemma  Let r , r , –r be three roots of g(t) and r ≤ r , r > . Write u = r + t, a = r – t,
b = r – t, and
q(t) =
b – u (a – u)b + u(u – a)b + u (a – u)b – au
·
.
a – u (b – u)a + u(u – b)a + u (b – u)a – bu
()
Then as  ≤ t ≤ r , we have
q() ≤ q(t) ≤ q(r ) ≤ .
()
Proof Since g(t) = K abu and g (t) ≥  (t ≥ ), we have
u – a – b ≥ .
()
Differentiating q and noticing q (t) ≥  ( ≤ t ≤ r ), we obtain
q() ≤ q(t) ≤ q(r ).
()
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On the other hand, since
q(t) –  ≤ ,
the lemma is proved.
Now we consider the majorizing sequences {tn }, {sn } (n ≥ ), t = ,
⎧
g(tn )
⎪
⎪
⎨sn = tn – g (tn ) ,
hn = –g (tn )– g (rn )(sn – tn ),
⎪
⎪
⎩
hn
n)
tn+ = tn – [ +  –h
] g(t
,
n g (tn )
()
where rn = tn + /(sn – tn ).
Lemma  Let g(t) be defined by () and satisfy the condition η ≤ β, then we have
√
√
n
n
(  λ  θ )
(  λ  θ )
(r – r ) ≤ r – tn ≤ √
(r – r ),
√
√
√


λ – (  λ θ )n
λ – (  λ θ )n
where θ =
r
, λ
r
n = , , . . . ,
()
= q(r ), λ = q().
Proof Let an = r – tn , bn = r – tn , un = r + tn , then
K
an bn un ,

K
g (tn ) = – [an un + bn un – an bn ],

K
g (tn ) = [un – bn – an ].

g(tn ) =
()
()
()
Write ϕ(tn ) = an un + bn un – an bn , then we have
g(tn )
 hn
an+ = an +  +
  – hn g (tn )
an (bn – un )[(an – un )bn + un (un – an )bn + un (an – un )bn – an un ]
,
ϕ  (tn )(an un + bn un + an bn ) – an bn ϕ(tn )
 hn
g(tn )
bn+ = bn +  +
  – hn g (tn )
=
=
bn (an – un )[(bn – un )an + un (un – bn )bn + un (bn – un )an – bn un ]
.
ϕ  (tn )(an un + bn un + an bn ) – an bn ϕ(tn )
()
()
We can obtain

an+ bn – un (an – un )bn + un (un – an )bn + un (an – un )bn – an un
an
=
·
·
.




bn+ an – un (bn – un )an + un (un – bn )an + un (bn – un )an – bn un
bn
()
From Lemma , we have λ ≤ q(t) ≤ λ . Thus
n
n
 an
an– 
n– a

≤ λ
≤ · · · ≤ (λ )++···+
=√
λ θ .

bn
bn–
b
λ
()
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In a similar way,
n
an
 
≥√
λ θ .

bn
λ
That completes the proof of the lemma.
()
Lemma  Suppose tn , sn are generated by (). If η < β, then the sequences {tn }, {sn } increase
and converge to r , and
 ≤ tn ≤ sn ≤ tn+ < r .
()
Proof Let
U(t) = t – g (t)– g(t),

H(t) = g (t)– g (T)g(t),
 H(t) V (t) = t +  +
U(t) – t ,
  – H(t)
()
where T = (t + U(t))/.
When  ≤ t ≤ r , we can obtain g(t) ≥ , g (t) < , g (t) > . Hence
U(t) = t –
g(t)
≥ t ≥ .
g (t)
()
So ∀t ∈ [, r ], we always have U(t) ≥ t.
≥ t ≥ , we have
Since T = t+U(t)

H(t) =
g (T)g(t)
≥ .
g (t)
()
On the other hand g (T)g(t) – g (t) > , then
 ≤ H(t) =
g (T)g(t)
< .
g (t)
()
Thus
V (t) = U(t) +
 H(t) U(t) – t ≥ ,
  – H(t)
and ∀t ∈ [, r ], we always have V (t) ≥ U(t).
Since
V (t) =
g(t)[g (t) (g (T) – g (t))(g (T)g(t) – g (t) ) – Kg(t) g (t)g (t)
g (t) [g (T)g(t) – g (t) ]
+
=
–Kg(t) g (t)g (t) + g(t) g (t)g (T)]
g (t) [g (T)g(t) – g (t) ]
g(t)[–Kg(t) g (t)g (t) – Kg(t) g (t)g (t) + g(t) g (t)g (T)]
,
g (t) [g (T)g(t) – g (t) ]
()
Zheng Journal of Inequalities and Applications (2016) 2016:289
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we know V (t) >  for  ≤ t ≤ r . That is to say that V (t) is monotonically increasing. By
this we will inductively prove that
 ≤ tn ≤ sn ≤ tn+ < V (r ) = r .
()
In fact, () is obviously true for n = . Assume () holds until some n. Since tn+ < r ,
sn+ , tn+ are well defined and tn+ ≤ sn+ ≤ tn+ . On the other hand, by the monotonicity of
V (t), we also have
tn+ = V (tn+ ) < V (r ) = r .
Thus, () also holds for n + .
From Lemma , we can see that {tn } converges to r . That completes the proof of the
lemma.
Lemma  Assume F satisfies the conditions (C)-(C), then ∀x ∈ B(x , r ), F (x)– exists
and satisfies the inequality
(I) F (x )– F (x) ≤ g (x – x ),
(II) F (x)– F (x ) ≤ –g (x – x )– .
Proof (I) From the above assumptions, we have
F (x )– F (x) = F (x )– F (x ) + F (x )– F (x) – F (x ) ≤ γ + Nx – x ≤ γ + Kx – x = g x – x .
(II) When t ∈ [, r ), we know g (t) < . Hence when x ∈ B(x , r ),
F (x )– F (x) – I = F (x )– F (x) – F (x ) – F (x )(x – x ) + F (x )(x – x ) 
–
+ γ x – x F
x
F
(x
)
+
t(x
–
x
)
–
F
(x
)
dt(x
–
x
)
≤









≤ Nt dt(x – x ) + γ x – x ≤ Kx – x  + γ x – x 

=  + g x – x < .
By the Banach lemma, we know (F (x )– F (x))– = F (x)– F (x ) exists and
– F (x) F (x ) ≤

 – I
– F (x

)– F (x)
That completes the proof of the lemma.
–
≤ –g x – x .
Lemma  ([]) Assume that the nonlinear operator F : ⊂ X → Y is continuously thirdorder Fréchet differentiable where is an open set and X and Y are Banach spaces. The
Zheng Journal of Inequalities and Applications (2016) 2016:289
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sequences {xn }, {yn } are generated by (). Then we have


F(xn+ ) = F (un )(xn+ – yn ) + F (xn )(xn+ – yn )(xn+ – xn )


  
F xn + t(yn – xn ) – F (xn ) dt(yn – xn )(xn+ – xn )
–
 



F xn + t(xn+ – xn ) – F (xn ) ( – t) dt(xn+ – xn ) ,
+
 
where yn = xn – n F(xn ) and un = xn +  (yn – xn ).
3 Newton-Kantorovich convergence theorem
Now we give a theorem to establish the semilocal convergence of the method () in weaker
conditions, the existence and uniqueness of the solution and the domain in which it is
located, along with a priori error bounds, which lead to the R-order of convergence of at
least four of the iterations ().
Theorem  Let X and Y be two Banach spaces, and F : ⊆ X → Y be a third-order
Fréchet differentiable on a non-empty open convex subset . Assume that all conditions
(C)-(C) hold true and x ∈ . If η < β, B(x , r ) ⊂ , then the sequence {xn } generated by
() is well defined, {xn } ∈ B(x , r ) and converges to the unique solution x∗ ∈ B(x , α), and
xn – x∗ ≤ r – tn . Further, we have
√

n
θ )
xn – x∗ ≤ √ ( λ√
(r – r ),

λ – (  λ θ )n
where θ =
r
, λ
r
= q(r ), α =
√
γ+
γ  +K
n = , , . . . ,
()
.
Proof We will prove the following formula by induction:
(In ) xn ∈ B(x , tn ),
(IIn ) F (xn )– F (x ) ≤ –g (tn )– ,
(IIIn ) F (x )– F (xn ) ≤ g (xn – x ) ≤ g (tn ),
(IVn ) yn – xn ≤ sn – tn ,
(Vn ) yn ∈ B(x , sn ),
(VIn ) xn+ – yn ≤ tn+ – sn .
Estimate that (In )-(VIn ) are true for n =  by the initial conditions. Now, assume that
(In )-(VIn ) are true for all integers k ≤ n.
(In+ ) From the above assumptions, we have
xn+ – x ≤ xn+ – yn + yn – xn + xn – x ≤ (tn+ – sn ) + (sn – tn ) + (tn – t ) = tn+ .
()
(IIn+ ) From (II) of Lemma , we can obtain
F (xn+ )– F (x ) ≤ –g xn+ – x – ≤ –g (tn+ )– .
()
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(IIIn+ ) From (I) of Lemma , we can obtain
F (x )– F (xn+ ) ≤ g xn+ – x ≤ g (tn+ ).
()
(IVn+ ) From Lemma , we have
F (x )– F(xn+ ) ≤  g (rn )(tn+ – sn ) +  N(tn+ – sn )(tn+ – tn )


L
L
+
(sn – tn ) (tn+ – tn ) + (tn+ – tn )


 ≤ g (rn )(tn+ – sn )

L (sn – tn )

L (tn+ – tn )
(tn+ – sn )(tn+ – tn )
N+
·
·
+
+

 tn+ – sn  tn+ – sn


≤ g (rn )(tn+ – sn ) + K(tn+ – sn )(tn+ – tn )


≤ g(tn+ ).
()
Thus, we have
yn+ – xn+ ≤ –F (xn+ )F (x )F (x )– F(xn+ )
≤–
g(tn+ )
= sn+ – tn+ .
g (tn+ )
()
(Vn+ ) From the above assumptions and (), we obtain
yn+ – x ≤ yn+ – xn+ + xn+ – yn + yn – xn + xn – x ≤ (sn+ – tn+ ) + (tn+ – sn ) + (sn – tn ) + (tn – t ) = sn+ ,
()
so yn+ ∈ B(x , sn+ ).
(VIn+ ) Since
I – KF (xn+ ) ≥  – KF (xn+ ) ≥  – –g (tn+ ) – g (rn+ )(sn+ – tn+ )
=  + g (tn+ )– g (rn+ )(sn+ – tn+ ) =  – hn+ ,
we have
– 
xn+ – yn+ = KF (xn+ ) I – KF (xn+ ) F (xn+ )– F(xn+ )

g(tn+ )
 hn+
= tn+ – sn+ .
·
≤
  – hn+ –g (tn+ )
()
Further, we have
xn+ – xn+ ≤ xn+ – yn+ + yn+ – xn+ ≤ tn+ – tn+ ,
()
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and when m > n
xm – xn ≤ xm – xm– + · · · + xn+ – xn ≤ tm – tn .
()
It then follows that the sequence {xn } is convergent to a limit x∗ . Take n → ∞ in (),
we deduce F(x∗ ) = . From (), we also get
∗
x – xn ≤ r – tn .
()
Now, we prove the uniqueness. Suppose x∗∗ is also the solution of F(x) on B(x , α). By
Taylor expansion, we have
 = F x∗∗ – F x∗ =

F ( – t)x∗ + tx∗∗ dt x∗∗ – x∗ .
()

Since

∗
∗∗
F (x )–
F ( – t)x + tx – F (x ) dt 
 
∗∗ ∗ ∗
∗
∗∗
–
∗ x
+
t
x
ds
dt
x
≤
F
(x
)
F
+
t
x
–
x
–
t
–
x
+
t
x
–
x






 
≤
g sx∗ – x + t x∗∗ – x∗ ds dt x∗ – x + t x∗∗ – x∗ 
g x∗ – x + t x∗∗ – x∗ dt – g ()

g ( – t) x∗ – x + t x∗∗ – x + 
=



=

g (r ) + g (α)
+  ≤ ,
<

we can find that the inverse of
From Lemma , we get
()


F (( – t)x∗ + tx∗∗ ) dt exists, so x∗∗ = x∗ .
√

n
θ )
xn – x∗ ≤ √ ( λ√
(r – r ),

λ – (  λ θ )n
This completes the proof of the theorem.
n = , , . . . .
()
4 Numerical examples
In this section, we illustrate the previous study with an application to the following nonlinear equations.
Example  Let X = Y = R, and
F(x) =

    
x + x – x + = .




()
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We consider the initial point x = , = [–, ], we can get

η=γ = ,

N=

,

L = .
Hence, from (), we have K = N =
(γ +  γ  + K) 
= ,
β=
(γ + γ  + K) 


and
η < β.
This means that the hypotheses of Theorem  are satisfied, we can get the sequence
{xn }(n≥) generated by the method () is well defined and converges.
Example  Consider an interesting case as follows:

x(s) =  + x(s)



s
x(t) dt,
s+t
()
where we have the space X = C[, ] with norm
x = max x(s).
≤s≤
This equation arises in the theory of the radiative transfer, neutron transport and the kinetic theory of gases.
Let us define the operator F on X by
F(x) =

x(s)



s
x(t) dt – x(s) + .
s+t
()
Then for x =  we can obtain
F (x )– = .,
η = .,


s
ln 
γ = F (x )– F (x ) = . ×  · max dt = . ×
= .,
 ≤s≤  s + t

(γ +  γ  + K)
= . > η.
(γ + γ  + K)
N = ,
L = ,
K = ,
That means that the hypotheses of Theorem  are satisfied.
Example  Consider the problem of finding the minimizer of the chained Rosenbrock
function []:
g(x) =
m
  xi – xi+ +  – xi+ ,
x ∈ Rm .
()
i=
For finding the minimum of g one needs to solve the nonlinear system F(x) = , where
F(x) = ∇g(x). Here, we apply the method (), and compare it with Chebyshev method
(CM), the Halley method (HM), and the super-Halley method (SHM).
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Table 1 The iterative errors (xn – x∗ 2 ) of various methods
n
CM
HM
SHM
Method (3)
1
2
3
4
3.086657e–001
1.198959e–003
1.099899e–007
3.140185e–016
2.169261e–001
4.506048e–003
2.048709e–008
5.438960e–016
5.962251e–002
1.913033e–005
1.435454e–014
8.599751e–016
3.119924e–002
1.989913e–006
7.771561e–016
In a numerical tests, the stopping criterion of each method is xk – x∗  ≤ e – , where
x∗ = (, , . . . , )T is the exact solution. We choose m =  and x = .x∗ . Listed in Table 
are the iterative errors (xk – x∗  ) of various methods. From Table , we know that, as
tested here, the performance of the method () is better.
5 Conclusions
In this paper, a new Newton-Kantorovich convergence theorem of a fourth-order superHalley method is established. As compared with the method in [], the differentiability
conditions of the method in the paper are mild. Finally, some examples are provided to
show the application of the convergence theorem.
Competing interests
The author declares to have no competing interests.
Author’s contributions
Only the author contributed in writing this paper.
Acknowledgements
This work is supported by the National Natural Science Foundation of China (11371243, 11301001, 61300048), and the
Natural Science Foundation of Universities of Anhui Province (KJ2014A003).
Received: 26 August 2016 Accepted: 8 November 2016
References
1. Ortega, JM, Rheinbolt, WC: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York
(1970)
2. Ezquerro, JA, Gutiérrez, JM, Hernández, MA, Salanova, MA: Chebyshev-like methods and quadratic equations. Rev.
Anal. Numér. Théor. Approx. 28, 23-35 (1999)
3. Candela, V, Marquina, A: Recurrence relations for rational cubic methods I: the Halley method. Computing. 44,
169-184 (1990)
4. Candela, V, Marquina, A: Recurrence relations for rational cubic methods II: the Chebyshev method. Computing. 45,
355-367 (1990)
5. Amat, S, Busquier, S, Gutiérrez, JM: Third-order iterative methods with applications to Hammerstein equations:
a unified approach. J. Comput. Appl. Math. 235, 2936-2943 (2011)
6. Hernández, MA, Romero, N: General study of iterative processes of R-order at least three under weak convergence
conditions. J. Optim. Theory Appl. 133, 163-177 (2007)
7. Amat, S, Bermúdez, C, Busquier, S, Plaza, S: On a third-order Newton-type method free of bilinear operators. Numer.
Linear Algebra Appl. 17, 639-653 (2010)
8. Argyros, IK, Cho, YJ, Hilout, S: On the semilocal convergence of the Halley method using recurrent functions. J. Appl.
Math. Comput. 37, 221-246 (2011)
9. Argyros, IK: On a class of Newton-like method for methods for solving nonlinear equations. J. Comput. Appl. Math.
228, 115-122 (2009)
10. Parida, PK, Gupta, DK: Recurrence relations for semilocal convergence of a Newton-like method in Banach spaces.
J. Math. Anal. Appl. 345, 350-361 (2008)
11. Parida, PK, Gupta, DK: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206,
873-887 (2007)
12. Gutiérrez, JM, Hernández, MA: Recurrence relations for the super-Halley method. Comput. Math. Appl. 36, 1-8 (1998)
13. Ezquerro, JA, Hernández, MA: New iterations of R-order four with reduced computational cost. BIT Numer. Math. 49,
325-342 (2009)
14. Ezquerro, JA, Gutiérrez, JM, Hernández, MA, Salanova, MA: The application of an inverse-free Jarratt-type
approximation to nonlinear integral equations of Hammerstein-type. Comput. Math. Appl. 36, 9-20 (1998)
15. Wang, X, Gu, C, Kou, J: Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces.
Numer. Algorithms 56, 497-516 (2011)
Zheng Journal of Inequalities and Applications (2016) 2016:289
Page 12 of 12
16. Amat, S, Busquier, S, Gutiérrez, JM: An adaptative version of a fourth-order iterative method for quadratic equations.
J. Comput. Appl. Math. 191, 259-268 (2006)
17. Wang, X, Kou, J, Gu, C: Semilocal convergence of a class of modified super-Halley methods in Banach spaces.
J. Optim. Theory Appl. 153, 779-793 (2012)
18. Hernández, MA, Salanova, MA: Sufficient conditions for semilocal convergence of a fourth order multipoint iterative
method for solving equations in Banach spaces. Southwest J. Pure Appl. Math. 1, 29-40 (1999)
19. Wu, Q, Zhao, Y: A Newton-Kantorovich convergence theorem for inverse-free Jarratt method in Banach space. Appl.
Math. Comput. 179, 39-46 (2006)
20. Kou, J, Li, Y, Wang, X: A variant of super-Halley method with accelerated fourth-order convergence. Appl. Math.
Comput. 186, 535-539 (2007)
21. Zheng, L, Gu, C: Fourth-order convergence theorem by using majorizing functions for super-Halley method in
Banach spaces. Int. J. Comput. Math. 90, 423-434 (2013)
22. Powell, MJD: On the convergence of trust region algorithms for unconstrained minimization without derivatives.
Comput. Optim. Appl. 53, 527-555 (2012)