On the Secant method

Journal of Complexity 29 (2013) 454–471

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On the Secant method I.K. Argyros a , S.K. Khattri b,∗ a

Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

b

Department of Engineering, Stord Haugesund University College, Norway

article

info

Article history: Received 3 December 2012 Accepted 26 March 2013 Available online 8 April 2013 Keywords: Secant method Newton’s method Banach space Semilocal convergence Majorizing sequence Divided difference

abstract We present a new semilocal convergence analysis for the Secant method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Our analysis is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been ubiquitously employed in other studies such as Amat et al. (2004) [2], Bosarge and Falb (1969) [9], Dennis (1971) [10], Ezquerro et al. (2010) [11], Hernández et al. (2005, 2000) [13,12], Kantorovich and Akilov (1982) [14], Laasonen (1969) [15], Ortega and Rheinboldt (1970) [16], Parida and Gupta (2007) [17], Potra (1982, 1984–1985, 1985) [18–20], Proinov (2009, 2010) [21,22], Schmidt (1978) [23], Wolfe (1978) [24] and Yamamoto (1987) [25] for computing the inverses of the linear operators. We also provide lower and upper bounds on the limit point of the majorizing sequences for the Secant method. Under the same computational cost, our error analysis is tighter than that proposed in earlier studies. Numerical examples illustrating the theoretical results are also given in this study. © 2013 Elsevier Inc. All rights reserved.

1. Introduction In this study, we are concerned with the problem of approximating a locally unique solution x⋆ of nonlinear equation

F ( x) = 0 ,

(1.1)

where F is a continuous operator defined on a nonempty convex subset D of a Banach space X with values in a Banach space Y. Many problems from computational sciences and other disciplines can

∗

Corresponding author. E-mail addresses: [email protected] (I.K. Argyros), [email protected] (S.K. Khattri).

0885-064X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jco.2013.04.001

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be expressed in the form of Eq. (1.1) using Mathematical Modeling [5,6,10,12,16]. The solution of these equations can rarely be found in closed form. That is why the most solution methods for these equations are iterative. The study about convergence analysis of iterative procedures is usually divided into two categories: semilocal and local convergence analysis. The semilocal convergence analysis derives convergence criteria from the information around an initial point while the local analysis finds estimates of the radii of convergence balls from the information around a solution. The Secant method – also known under the names of Regular falsi or the method of chords – is one of the most used iterative methods for solving nonlinear equations. According to A.N. Ostrowski this method is known since the time of early Italian algebraists. It is well known that in the case of scalar equations, the Secant method is better than Newton’s method from the point of view of efficiency index [7,6,14,16,18–23]. The Secant method was extended for the solution of Eq. (1.1) in Banach spaces by J.W. Schmidt [23]. They have used the notion of a divided difference of a nonlinear operator. In later studies, it was observed that it is possible to use the more general concept of consistent approximation of a derivative [7,6,16] instead of the notion of divided differences. In this case the Secant method can be described by the recurrence scheme xn+1 = xn − δ F (xn−1 , xn )−1 F (xn ),

for each n = 0, 1, 2, 3, . . .

(1.2)

where x−1 , x0 ∈ D are initial points and δ F (x, y) ∈ L(X, Y) (x, y ∈ D) is a consistent approximation of the Fréchet derivative of F . L(X, Y) denotes the space of bounded linear operators from X into Y. The Secant method is an alternative to the well-known Newton’s method −1

xn+1 = xn − F ′ (xn )

F (xn ),

for each n = 0, 1, 2, 3, . . .

where x0 ∈ D is an initial point. The following conditions are commonly associated with the semilocal convergence of the Secant method (1.2): C1 : F is a nonlinear operator defined on a convex subset D of a Banach space X with values in a Banach space Y, C2 : x−1 and x0 are two points belonging to the interior D0 of D and satisfying the inequality

∥ x 0 − x −1 ∥ ≤ c , C3 : F is Fréchet-differentiable on D0 and there exists a linear operator δ F : D0 × D0 → L(X, Y) such that A = δ F (x−1 , x0 ) is invertible, its inverse A−1 is bounded,

∥A−1 F (x0 )∥ ≤ η, ∥A−1 (δ F (x, y) − F ′ (z ))∥ ≤ ℓ (∥x − z ∥ + ∥y − z ∥) for all x, y, z ∈ D, U (x0 , r ) = {x ∈ X : ∥x − x0 ∥ ≤ r } ⊆ D0 , for some r > 0 depending on ℓ, c , η and

 ℓ c + 2 ℓ η ≤ 1.

(1.3)

√

Under condition (1.3), the R-order of convergence for the Secant method is (1 + 5)/2. Several researchers, such as Amat et al. [2,1], Bosarge et al. [9], Dennis [10], Potra [18–20], Hernández et al. [9–11,13,12], Argyros [3–5,7,8,6], Parida et al. [17] and others [23,25,24], have used the conditions (C1 )–(C3 ) to provide sufficient convergence criteria for the Secant method. In many situations, the sufficient convergence criteria (1.3) may fail. For example, consider X = Y = R, x0 = 1, D = [p, 2 − p] for p ∈ (0, 1) and c = 0.15. Define function F on D by

F (x) = x3 − p. For p = 0.7252 (see the computations in Section 4), we get

η = 0.102934888,

l = 1.474985421

and lc + 2



lη = 1.000548677 > 1.

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√

Hence, there is no guarantee that the Secant method converges to x⋆ = 3 p. In this work, we are motivated by optimization considerations and the preceding observation where criterion (1.3) is not satisfied. Our approach is based on the weaker center-Lipschitz concept instead of the stronger Lipschitz condition which has been frequently used in earlier studies such as [2,1,9–11,13,12,14–16, 18–23,25,24] for computing the inverses of the linear operators involved. In this study, we provide lower and upper bounds on the limit point of the majorizing sequence for the Secant method. Under the same computational cost, the developed analysis give tighter error bounds than provided in earlier work. Hence, this way we expand the applicability of the Secant method. The rest of the paper is organized as follows. In Section 2, we study the convergence of scalar sequences that will be shown to be majorizing for the Secant method. Semilocal convergence of the Secant method is developed in Sections 3 and 4. Numerical examples illustrating the theoretical results are given in Section 5. 2. Majorizing sequences for the Secant method In this section, we shall first study some scalar sequences which are related to the Secant method. Let there be parameters c ≥ 0, η > 0, l0 > 0 and l > 0 with l0 ≤ l. Define the scalar sequence {αn } by

α−1 = 0,

α0 = c , α1 = c + η, (2.1) l(αn+1 − αn−1 )(αn+1 − αn ) for each n = 0, 1, 2, . . . . αn+2 = αn+1 + 1 − l0 (αn+1 + αn − c ) Special cases of the sequence {αn } have been used as a majorizing sequence for the Secant method by several authors, for example the case l0 = l was studied in [2,1,9–11,13,12,14–23,25,24] where as the case l0 ≤ l was studied by us in [3–5,7,8,6]. Moreover, sufficient convergence criteria have been presented for {αn } (and consequently for the Secant method) using different techniques such as majorizing functions, recurrent functions and recurrent relations. Here, we unify and extend these techniques. Moreover, we provide computable lower and upper bounds on the limit points of these sequences. In order for us to achieve these goals, we employ a technique that modifies sequence {αn }. Let L0 =

l0 1 + l0 c

and

L=

l 1 + l0 c

.

(2.2)

Then, sequence {αn } can be written as

α−1 = 0, αn+2

α0 = c , α1 = c + η, L(αn+1 − αn−1 )(αn+1 − αn ) for each n = 0, 1, 2, . . . . = αn+1 + 1 − L0 (αn+1 + αn )

(2.3)

Moreover, let L = b L0

for some b ≥ 1

(2.4)

and

βn = L0 αn .

(2.5)

Then, we can define sequence {βn } by

β−1 = 0, βn+2

β0 = L0 c , β1 = L0 (c + η), b (βn+1 − βn−1 )(βn+1 − βn ) = βn+1 + for each n = 0, 1, 2, . . . . 1 − (βn+1 + βn )

(2.6)

Furthermore, let

γn =

1 2

− βn .

(2.7)

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Then, sequence {γn } is defined as

γ−1 =

1 2

,

1

γ0 =

2

− L0 c ,

γ1 =

1 2

− L0 (c + η),

b (γn+1 − γn−1 )(γn+1 − γn )

γn+2 = γn+1 −

γn+1 + γn

(2.8) for each n = 0, 1, 2, . . . .

Finally, let

δn = 1 −

γn . γn−1

(2.9)

Then, we can define the sequence {δn } by

γ1 , γ0 δn + δn+1 (1 − δn ) = b δn+1 for each n = 0, 1, 2, . . . . (1 − δn )(1 − δn+1 )(2 − δn+1 )

δ0 = 1 − δn+2

γ0 , γ−1

δ1 = 1 −

(2.10)

Next, we study the convergence of these sequences starting from {δn }. Lemma 2.1. Let δ1 > 0, δ2 > 0 and b ≥ 1 be given parameters. Suppose that 0 < δ2 ≤ δ1 ≤ a :=

2

√

b2 + 4 b

b+2+

.

(2.11)

Let {δn } be the scalar sequence defined by (2.10). Then, the following assertions hold: A1 : if

δ1 = δ2

(2.12)

δn = a < 1 for each n = 1, 2, 3, . . . .

(2.13)

δ2 < δ1 < a

(2.14)

then,

A2 : if

then, sequence {δn } is decreasing and converges to 0. Proof. We first notice the polynomial g defined by g (t ) = t 3 − (b + 4)t 2 + (2b + 5)t − 2 = (t − 2)(t − a)(t − a+ ),

(2.15)

where a < 2 < a+

(2.16)

and

√ a+ =

b2 + 4b

b+2+ 2

.

(2.17)

Moreover, function g is increasing on (0, 1), since

 g (t ) = 3(t − 1) t − ′

2b + 5 3

 > 0.

(2.18)

Using (2.10) and δ2 ≤ δ1 , we deduce that δ3 > 0. We shall show that

δ3 ≤ δ2 .

(2.19)

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In view of (2.10) for n = 1, it suffices to show that

(1 − δ1 )δ22 − (1 − δ1 )(b + 3)δ2 + 2 − (b + 2)δ1 ≥ 0.

(2.20)

If we consider the left hand side of (2.20) to be a quadratic polynomial with respect to δ2 , then the discriminant is

  ∆ = (1 − δ1 ) (3 + b)2 (1 − δ1 ) + 4b > 0.

(2.21)

Then, (2.20) shall be true if

δ2 ≤ δ−

(2.22)

where

(1 − δ1 )(b + 3) − δ− = 2(1 − δ1 )

√ ∆

.

(2.23)

It follows from δ2 ≤ δ1 that (2.22) is true, if δ1 ≤ δ− or if g (δ1 ) ≤ 0 which is true, since δ1 ≤ a. Hence, so far we have shown that 0 < δ3 ≤ δ2 .

(2.24)

Let us assume that 0 < δk+1 ≤ δk

for each k = 1, 2, 3, . . . , n.

(2.25)

Then, we must show 0 < δk+2 ≤ δk+1 .

(2.26)

It follows from (2.10), δk < 1 and δk+1 < 1 that δk+2 > 0. In view of (2.10) the right hand side inequality in (2.26) is true if

δk+2 = b δk+1

δk + δk+1 (1 − δk ) ≤ δk+1 (1 − δk )(1 − δk+1 )(2 − δk+1 )

(2.27)

or g (δk ) ≤ 0,

(2.28)

which is true, since δk ≤ δ1 ≤ a for each k = 1, 2, 3 . . . , n + 1. The induction for (2.26) is complete. If δ1 = δ2 = a, then it follows from (2.10) for n = 1 that δ3 = a and δn = a for n = 4, 5, . . . , which shows (2.13). If δ2 < δ1 , then sequence {δn } is decreasing, bounded from below by 0 and as such it converges to its unique largest lower bound δ . We then have from (2.10) that

δ = bδ

(2 − δ)δ H⇒ δ = a or δ = 0. (1 − δ)(1 − δ)(2 − δ)

But δ ≤ δ1 < a. Hence, we conclude that δ = 0. The proof of the lemma is complete.

(2.29) 

Lemma 2.2. Under the hypothesis (2.14) further suppose that l0 c < 1.

(2.30)

Then, the sequence {γn } is decreasingly convergent and sequences {αn } and {βn } are increasingly convergent. Proof. Using (2.2), (2.9) and (2.30), we get that

γn = (1 − δn )γn−1 = · · · = (1 − δn ) · · · (1 − δ1 )γ0   1 = (1 − δn ) · · · (1 − δ1 ) − L0 c > 0 2

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and

γn < γn−1 (since δn < 1). Hence, sequence {γn } converges to its unique largest lower bound denoted by γ ⋆ . We also have that βn = 1/2 − γn < 1/2, thus the sequence {βn } is increasing, bounded from above by 1/2 and as such it converges to its unique least upper bound denoted by β ⋆ . Then, in view of (2.5) sequence {αn } is also increasing, bounded from above by 1/(2L0 ) = (1 + l0 c )/(2l0 ) and as such it converges to its unique least upper bound denoted by α ⋆ . The proof of the lemma is complete.  Lemma 2.3. Suppose that (2.11), (2.12) and (2.30) hold. Then, the following assertions hold for each n = 1, 2, . . .

δ n = a, γn = (1 − a)n γ0 , βn =

1

αn =

1

γ ⋆ = lim γn = 0, n→∞

β ⋆ = lim βn =

− (1 − a)n γ0 ,

2

n→∞

1 2

,

and



1 2

L0

 α ⋆ = lim αn =

− (1 − a) γ0 , n

n→∞

1 2L0

=

1 + l0 c 2l0

.

Remarks 2.4. The sufficient convergence criteria (2.11), (2.12) and (2.30) determine the smallness of η for the convergence of sequence {αn }. These criteria can be simplified and furthermore these criteria can also be expressed in terms of the initial data, l0 , l, c and η. We have that δ1 ≤ a is satisfied, if

δ1 =

2l0 η 1 − l0 c

≤ a.

We also have that δ0 = 2l0 c /(1 + l0 c ). Then, δ2 ≤ δ1 , if

δ2 = bδ1

δ0 + δ1 (1 − δ0 ) ≤ δ1 (1 − δ0 )(1 − δ1 )(2 − δ1 )

or 2(l0 η)2 − (b + 3)(1 − l0 c )l0 η + (1 − l0 c )[(1 + b)(1 − l0 c ) − b] ≥ 0.

(2.31)

If we see the left hand side of (2.31) as a quadratic polynomial in l0 η, then its discriminant ∆1 is given by

∆1 = [(b − 1)2 (1 − l0 c ) + 8b](1 − l0 c ) > 0. Hence, the quadratic polynomial has two distinct roots denoted by ρ1 and ρ2 with ρ1 < ρ2 . Note that if l0 c < 1/(b + 1), then ((b + 1)(1 − l0 c ) − b)(1 − l0 c ) > 0. Hence, we have that 0 < ρ1 < ρ2 . However, if l0 c > 1/(1 + b), then ρ1 < 0 < ρ2 . We must also have that

δ1 ≤ a.

(2.32)

Therefore, using (2.31) and (2.32) we deduce that {αn } converges if the following criteria are satisfied: l0 ≤ l c<

η≤

     

1 l0 (b + 1) 1 l0

 min

(1 − l0 c )a 2

1  (C )    , ρ1  

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or if l0 ≤ l

ρ2 ≤

(1 − l0 c )a 2

1


1

        

 l0 (b + 1) l0     ρ2 (1 − l0 c )a    ≤η≤ l0

1

(C ).

2l0

Hence, we obtain the following result as a consequence of the preceding lemmas and Remarks 2.4. 1

Corollary 2.5. Suppose that the (C1 ) and (C ) criteria hold. Then, sequence {αn } defined in (2.1) is non decreasing and converges to

α⋆ =

β ⋆ (1 + l0 c ) l0

,

(2.33)

where β ⋆ is defined in Lemmas 2.2 and 2.3. We need an auxiliary result for a certain function related to the sequence {αn }. Lemma 2.6. Let the function φ : [0, 1)2 −→ R be defined by

φ(s, t ) =

t + s( 1 − t ) . (1 − t )(1 − s)(2 − s)

(2.34)

Then, function φ is increasing in both variables. Proof. Let s ∈ [0, 1) be fixed. Then, using (2.34) we get

∂φ 1 = > 0 for each t ∈ [0, 1). ∂t (1 − t )2 (1 − s)(2 − s)

(2.35)

Moreover, we have that for fixed t ∈ [0, 1)

ψ(s, t ) + 2 ∂φ = ∂s (1 − t )(1 − s)2 (2 − s)2

(2.36)

ψ(s, t ) = (t − 1)s2 − 2st + t .

(2.37)

where

It follows from (2.37) that for each (s, t ) ∈ [0, 1)2

− 1 ≤ ψ(s, t ) ≤ t .

(2.38)

Indeed, the right hand side of (2.38) reduces to showing that 2(s(t − 1) − t ) ≤ 0 which is true. The left hand side inequality in (2.38) reduces to showing that t (s − 1)2 + 1 − s2 ≥ 0 which is also true. Then, in view of (2.36) and (2.38) we deduce that

∂φ > 0. ∂s That completes the proof of lemma.

(2.39) 

Next, we present lower and upper bounds on the limit point α ⋆ .

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Lemma 2.7. Suppose that the (C1 ) conditions hold but last inequality in (C1 ) is strict. Then, the following assertion holds b11 ≤ α ⋆ ≤ b12

(2.40)

where b11

=

b12 =

1 + l0 c l0 1 + l0 c

δ⋆ = −





δ1 δ2 − exp −2 + 2 2 − δ1 2 − δ2   1

1

l0

2



r =b



(2.41)

δ2 δ1 + 1−r

1 − δ1

,

− exp(δ ⋆ ) ,



1





 + ln

2(1 + l0 c )

 ,

1 − l0 c

δ1 + δ2 (1 − δ1 ) . (1 − δ1 )(1 − δ2 )(2 − δ2 )

Proof. Using (2.10) and (2.14), we have that 0 < δ3 < δ2 < δ1 . Let us assume that 0 < δk+1 < δk < · · · < δ1 . Then, it follows from the Lemma 2.6, the induction hypotheses and (2.41) that

δk+2 = δk+1 b

δk + δk+1 (1 − δk ) < r δk+1 < r 2 δk ≤ · · · ≤ r k−1 δ3 ≤ r k δ2 . (1 − δk )(1 − δk+1 )(2 − δk+1 )

We have that ∞ 

γ ⋆ = lim γn = n→∞

(1 − δn )γ0 .

i =1

This is equivalent to

 ln

1



∞ 

=

γ⋆

 ln

1

 + ln

1 − δn

n=1



2(1 + l0 c )

 ,

1 − l0 c

recalling that γ0 = (1 − l0 c )/(2(1 + l0 c )). We shall use the following bounds for ln t , t > 1:

 2

t −1 t +1

 ≤ ln t ≤

t2 − 1 2t

.

First we shall find an upper bound for ln(1/γ ⋆ ). We have that

 ln

1

γ⋆

 ≤

∞  δn (2 − δn ) n=1

≤

≤

2(1 − δn )

1

∞ 

1 − δ1 n=1 1 1 − δ1

 + ln 

δn + ln



2(1 + l0 c )

,

1 − l0 c 2(1 + l0 c )

 ,

1 − l0 c

 (δ1 + δ2 + δ3 + · · ·) + ln

2(1 + l0 c ) 1 − l0 c

 ,

    2(1 + l0 c ) 2 n ≤ δ1 + δ2 + r δ2 + r δ2 + · · · + r δ2 + · · · + ln , 1 − δ1 1 − l0 c 1

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   2(1 + l0 c ) δ1 + δ2 (1 + r + r + · · · + r + · · ·) + ln ≤ , 1 − δ1 1 − l0 c   1  δ2 2(1 + l0 c )  ≤ δ1 + = −δ ⋆ . + ln 1 − δ1 1−r 1 − l0 c 1



2

n

1 ⋆ As β ⋆ = 1/2 − γ ⋆ and α ⋆ = L− 0 β , we obtain the upper bound in (2.40). Moreover, in order to obtain ⋆ the lower bound for ln(1/γ ), we have that

 ln

1

γ⋆

 ≥2

∞  n =1

  δn δ1 δ2 >2 + 2 − δn 2 − δ1 2 − δ2

which implies the lower bound in (2.40). The proof of the lemma is complete.



Remarks 2.8. R1 : Let us introduce the notation c N = αN −1 − αN −2 ,

ηN = αN − αN −1

where N ≥ 1 is an integer. Note that c 1 = α0 − α−1 = c and η1 = α1 − α0 = η. The results can be weakened even further as follows. Consider the criteria (CN⋆ ) for N > 1:

 1 (C1 ) or (C ) with c , η replaced by c N , ηN , respectively,  (CN⋆ ). α−1 < α0 < α1 < · · · < αN < αN +1 ,   l0 (αN + αN +1 ) < 1 + l0 c . Then, all the results obtained above hold with c , η, δ1 , δ2 , ρ1 , ρ2 , b11 , b12 replaced, respectively, by c N , ηN , δN , δN +1 , ρ1N , ρ2N , bN1 , bN2 . R2 : Clearly, if l0 (αn + αn+1 ) < 1 + l0 c

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

(2.42)

then, it follows from (2.1) that sequence {αn } is increasing, bounded from above by (1 + l0 c )/(2l0 ) and as such it converges to its unique least upper bound α ⋆ . Criterion (2.42) is the weakest of all criteria for the convergence of sequence {αn }. Clearly all the convergence criteria presented above imply (2.42). Define the criteria for N ≥ 1:

(IN ) =

 (CN⋆ ), (2.42)

if criteria (CN ⋆ ) fail.

(2.43)

3. Semilocal convergence of the Secant method-I In this section, we present the semilocal convergence analysis of the Secant method using {αn } (defined by (2.1)) as a majorizing sequence. Let U (x, R) stand for an open ball centered at x ∈ X with radius R > 0. Let U (x, R) denote its closure. We shall study the Secant method for triplets (F , x−1 , x0 ) belonging to the class K = K (l0 , l, η, c ) defined as follows. Definition 3.1. Let ℓ0 , ℓ, η, c be constants satisfying the hypotheses (IN ) for some fixed integer N ≥ 1. A triplet (F , x−1 , x0 ) belongs to the class K (ℓ0 , ℓ, η, c ) if: D1 : F is a nonlinear operator defined on a convex subset D of a Banach space X with values in a Banach space Y, D2 : x−1 and x0 are two points belonging to the interior D0 of D and satisfying the inequality

∥ x 0 − x −1 ∥ ≤ c ,

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D3 : F is Fréchet-differentiable on D0 and there exists an operator δ F : D0 × D0 → L(X, Y) such that A−1 = δ F (x−1 , x0 )−1 ∈ L(Y, X), for all x, y, z ∈ D the following hold

∥A−1 F (x0 )∥ ≤ η, ∥A−1 (δ F (x, y) − F ′ (z ))∥ ≤ ℓ (∥x − z ∥ + ∥y − z ∥) and

∥A−1 (δ F (x, y) − F ′ (x0 ))∥ ≤ ℓ0 (∥x − x0 ∥ + ∥y − x0 ∥), D4 :

U (x0 , α ⋆ ) ⊆ Dc = {x ∈ D : F is continuous at x} ⊆ D where α ⋆ is given in Lemma 2.1. Next, we present the semilocal convergence result for the Secant method. Theorem 3.2. If (F , x−1 , x0 ) ∈ K (l, l0 , η, c ) then, the sequence {xn } (n ≥ −1) generated by the Secant method is well defined, remains in U (x0 , α ⋆ − c ) for each n = 0, 1, 2, . . . and converges to a unique solution x⋆ ∈ U (x0 , α ⋆ − c ) of equation F (x) = 0. Moreover, the following assertions hold for each n = 0, 1, 2, . . .

∥xn − xn−1 ∥ ≤ αn − αn−1 ,

(3.1)

∥xn − x⋆ ∥ ≤ α ⋆ − αn

(3.2)

and

where {αn } (n ≥ 0) is given by (2.1). Furthermore, if there exists R such that U (x0 , R) ⊆ D,

R ≥ α⋆ − c

and

l0 α ⋆ − c + R + ∥A−1 (F ′ (x0 ) − A)∥ < 1,





(3.3)

then, the solution x⋆ is unique in U (x0 , R). Proof. First, we show that L = δ F (xk , xk+1 ) is invertible for xk , xk+1 ∈ U (x0 , α ⋆ − c ). By (D2 ) and (D3 ), we have

∥I − A−1 L∥ = ∥A−1 (L − A)∥ ≤ ∥A−1 (L − F ′ (x0 ))∥ + ∥A−1 (F ′ (x0 ) − A)∥ ≤ ℓ0 (∥xk − x0 ∥ + ∥xk+1 − x0 ∥ + ∥x0 − x−1 ∥) ≤ ℓ0 (αk − α0 + αk+1 − α0 + c ) < 1 (by (2.11)).

(3.4)

Using the Banach Lemma on invertible operators [5,14] and (2.39), we deduce that L is invertible and

 −1 ∥L−1 A∥ ≤ 1 − ℓ0 (αk+1 − αk − α0 ) .

(3.5)

By (D3 ), we have

∥A−1 (F ′ (u) − F ′ (v))∥ ≤ 2ℓ∥u − v∥,

u, v ∈ D 0 .

(3.6)

We can write the identity

F (x) − F (y) =

1



F ′ (y + t (x − y)) dt (x − y)

(3.7)

0

then, for all x, y, u, v ∈ D0 , we obtain

∥A−1 (F (x) − F (y) − F ′ (u)(x − y))∥ ≤ ℓ (∥x − u∥ + ∥y − u∥) ∥x − y∥

(3.8)

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and

∥A−1 (F (x) − F (y) − δ F (u, v) (x − y))∥ ≤ ℓ (∥x − v∥ + ∥y − v∥ + ∥u − v∥)∥x − y∥. (3.9) By a continuity argument (3.6)–(3.9) remain valid if x and/or y belong to Dc . Next, we show (3.1). If (3.1) holds for all n ≤ k and if {xn } (n ≥ 0) is well defined for n = 0, 1, 2, . . . , k, then

∥xn − x0 ∥ ≤ αn − α0 < α ⋆ − α0 ,

n ≤ k.

(3.10)

That is (1.2) is well defined for n = k + 1. For n = −1 and n = 0, (3.1) reduces to ∥x−1 − x0 ∥ ≤ c and ∥x0 − x1 ∥ ≤ η. Suppose (3.1) holds for n = −1, 0, 1, . . . , k (k ≥ 0). By (3.5), (3.9), and

F (xk+1 ) = F (xk+1 ) − F (xk ) − δ F (xk−1 , xk ) (xk+1 − xk )

(3.11)

we obtain in turn the following estimates

∥A−1 F (xk+1 )∥ ≤ ℓ (∥xk+1 − xk ∥ + ∥xk − xk−1 ∥)∥xk+1 − xk ∥

(3.12)

and

∥xk+2 − xk+1 ∥ = ∥δ F (xk , xk+1 )−1 F (xk+1 )∥ ≤ ∥δ F (xk , xk+1 )−1 A∥ ∥A−1 F (xk+1 )∥ ℓ (∥xk+1 − xk ∥ + ∥xk − xk−1 ∥) ≤ ∥xk+1 − xk ∥ 1 − ℓ0 (∥xk+1 − x0 ∥ + ∥xk − x0 ∥ + c ) ℓ (αk+1 − αk + αk − αk−1 ) ≤ (αk+1 − αk ) 1 − ℓ0 (αk+1 − α0 + αk − α0 + α0 − α−1 ) = αk+2 − αk+1 . The induction for (3.1) is complete. It follows from (3.1) and Lemma 2.1 that {xn } (n ≥ −1) is a complete sequence in a Banach space X and as such it converges to some x⋆ ∈ U (x0 , α ⋆ − c ) (since U (x0 , α ⋆ − c ) is a closed set). By letting k → ∞ in (3.12), we obtain F (x⋆ ) = 0. Moreover, estimate (3.2) follows from (3.1) by using standard majoration techniques [5,6,14,16]. Finally, for showing the uniqueness in U (x0 , R), let y⋆ ∈ U (x0 , R) be a solution (1.1). Set 1

 M=

F ′ (y⋆ + t (y⋆ − x⋆ )) dt .

0

Using (D3 ) and (3.3) we get in turn that

  ∥A−1 (A − M)∥ = ℓ0 ∥y⋆ − x0 ∥ + ∥x⋆ − x0 ∥ + ∥A−1 (F ′ (x0 ) − A)∥   ≤ ℓ0 (t ⋆ − α0 ) + R + ∥A−1 (F ′ (x0 ) − A)∥ < 1.

(3.13)

It follows from (3.13) and the Banach lemma on invertible operators that M −1 exists on U (x0 , α ⋆ ). Using the identity:

F (x⋆ ) − F (y⋆ ) = M (x⋆ − y⋆ ) ⋆

(3.14)

⋆

we deduce that x = y . The proof of Theorem is complete.



Remarks 3.3. R1 : In the uniqueness part (see (2.38)), we can replace A−1 (F ′ (x0 ) − A) by the less tight l0 c, since by (D3 )



 −1 ′  A (F (x0 ) − A) ≤ l0 ∥x0 − x−1 ∥ ≤ l0 c .



(3.15)

In fact according to (3.4), majorizing sequence {αn } can be replaced by the following iteration, which is at least as tight as the sequence defined by (2.4), t1 = 0, t1 = c + η,

t0 = c , tn+2 = tn+1 +

l(tn+1 − tn−1 )(tn+1 − tn ) 1 − (L + l0 (tn+1 + tn − 2c ))

.

(3.16)

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465

This is also a majorizing sequence for {xn } (converging under the same hypotheses to some α ⋆ ≤ α ⋆ ). R2 : A more popular hypothesis used, instead of the first inequality in (D3 ), is

  −1 A (δ F (x, y) − δ F (u, v)) ≤ l (∥x − u∥ + ∥y − v∥)

(3.17)

for all x, y, u, v ∈ D. Note that (3.17) implies both hypotheses in (D3 ) but not necessarily vice versa. Note also that

ℓ0 ≤ ℓ,

ℓ ≤ ℓ,

and l/l0 , l/l can be arbitrarily large [2,1,3–5,7]. R3 : Let us define the majorizing sequence {wn }, used in [2,1,9–11,13,12,14–16,18–23,25,24] (under the condition (1.3))

w−1 = 0, wn+2

w0 = c , w1 = c + η, ⋆ l (wn+1 − wn−1 ) (wn+1 − wn ) = wn+1 + 1 − l⋆ (wn+1 − w0 + wn )

(3.18)

where l⋆ =



l l

if the first inequality inD3 is used if (3.17) is used.

Note that in general l0 ≤ l

(3.19)

holds and l/l0 can be arbitrarily large. In the case, l0 = l then αn = wn (n ≥ −1). Otherwise, a simple inductive argument shows that

αn ≤ wn ,

αn+1 − αn ≤ wn+1 − wn ,

(3.20)

and 0 ≤ α ⋆ − α n ≤ w ⋆ − wn ,

w ⋆ = lim wn . n−→∞

(3.21)

Note also that strict inequality holds in (3.20) for each n = 1, 2, 3, . . . , if l0 < l. R4 : It follows from (D3 ) and (D4 ) that there exist constants κ, κ0 , κ1 such that

    −1  A δ F (x1 , x0 ) − δ F (x0 , x−1 )  ≤ κ∥x1 − x0 ∥,    −1 ′  A (F (x0 ) − A) ≤ κ0 ∥x0 − x−1 ∥

(3.22) (3.23)

and

    −1  A δ F (x0 , x1 ) − F ′ (x0 )  ≤ κ1 ∥x1 − x0 ∥, where x1 = x0 − δ F (x0 , x−1 )−1 F (x0 ). Clearly

κ ≤ l,

κ0 ≤ l0 and κ1 ≤ l0

hold. Then, in view of the proof of Theorem 3.2, we obtain the estimates

    δ F (x0 , x−1 )−1 A ≤

1 1 − (κ1 η + κ0 c )

and

∥x 2 − x 1 ∥ ≤

κ(c + η)η . 1 − (κ1 η + κ0 c )

(3.24)

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Hence, the above justify the introduction of sequence {rn } defined by r−1 = 0,

r0 = c ,

r1 = c + η,

r2 = r1 +

l(rn+1 − rn−1 )(rn+1 − rn )

rn+2 = rn+1 +

1 − l0 (rn+1 + rn − c )

κ(c + η)η 1 − (κ1 η + κ0 c )

,

,

for each n = 1, 2, . . . . Sequence {rn } is a tighter majorizing sequence for {xn } than {αn } if

κ < l or κ0 < l0 or κ1 < l0 . Moreover, we have in this case that rn < αn ,

rn+1 − rn < αn+1 − αn

and r ⋆ = lim rn ≤ α ⋆ . n→∞

Note that the verification of constants κ, κ0 , κ1 requires only computations at the initial data. Clearly, tighter than {wn } sequences {αn } and {rn } converge under condition (1.3). In practice, we shall test the old convergence criterion (i.e. (1.3)) and if it is satisfied we shall use {αn } or {rn } instead of {wn } for the computation of the error bounds. If both the old and the new convergence criteria hold then again we shall use {αn } or {rn } for the computation of the error bounds. Finally if criterion (1.3) fails, then we shall test the (IN ) criteria. 4. Semilocal convergence of the Secant method-II In this section, we present a semilocal convergence analysis of the Secant method using a different majorizing sequence. Indeed, let there be parameters c ≥ 0, η > 0, l0 > 0 and l > 0 with l0 ≤ l. Define the scalar sequence {αn } by

α−1 = 0, αn+2

α0 = c , α1 = c + η, l(αn+1 − αn−1 )(αn+1 − αn ) = αn+1 + 1 − l0 (αn+1 + αn − 2c )

(4.1)

for each n = 0, 1, 2, . . . . We shall show that the new sequence is also majorizing for the Secant method. The study of the convergence of the new sequence {αn } is given in an analogous way but in this case L0 =

l0

and L =

1 + 2l0 c

l 1 + 2l0 c

.

(4.2)

Then, in an identical way as in Section 2 the results can be rewritten with the changes (due to (4.2))

γ0 = b11

=

b12 =

1 2(1 + 2l0 c ) 1 + 2l0 c l0 1 + 2l0 c

δ =−

α⋆ = 

1 + 2l0 c 2l0

,

δ1 = 2l0 η,

(4.3)

 ,

(4.4)

δ1 δ2 − exp −2 + 2 2 − δ1 2 − δ2  

l0

 ⋆



,

1

1 2

1 1 − δ1





− exp(δ ⋆ ) , δ2 δ1 + 1−r



(4.5)

 + ln 2(1 + 2l0 c ) .

(4.6)

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467

Notice also that (2.30) is dropped, since

γn = (1 − δn ) · · · (1 − δ1 )γ0 > 0 and condition (2.11) reduces to the new (C1 ) condition given by

(C1 )

   η ≤ min  

2(1 − lc )

l + 3l0 +

lc < 1.



1



 , , 2 (l + 3l0 )2 − 8l20 (1 − lc ) l + 2l0 + l + 4l0 l

Finally notice that the last condition in (CN ⋆ ) is given by l0 (αN + αN +1 ) < 1 + 2l0 c and the condition (2.42) is replaced by l0 (αn + αn+1 ) < 1 + 2l0 c for each n = 0, 1, 2, . . . with the above changes. Let us define the class K = K (l0 , l, η, c ) as follows. Definition 4.1. Let l0 , l, η, c be constants satisfying the hypotheses (IN ) as defined above in this section for some fixed integer N ≥ 1. A triplet (F , x−1 , x0 ) belongs to the class K (l0 , l, η, c ) if: D1 : F is a nonlinear operator defined on a convex subset D of a Banach space X with values in a Banach space Y, D2 : x−1 and x0 are two points belonging to the interior D0 of D and satisfying the inequality

∥ x 0 − x −1 ∥ ≤ c , D3 : F is Fréchet-differentiable on D0 and there exist operators δ F : D0 × D0 −→ L(X, Y) such that A−1 = δ F (x−1 , x0 )−1 ∈ L(Y, X), for all x, y, x ∈ D the following hold

∥A−1 F (x0 )∥ ≤ η, ∥F ′ (x0 )−1 (δ F (x, y) − F ′ (z ))∥ ≤ l(∥x − z ∥ + ∥y − z ∥) and

∥F ′ (x0 )−1 (δ F (x, y) − F ′ (x0 ))∥ ≤ l0 (∥x − x0 ∥ + ∥y − x0 ∥). D4 : U (x0 , α ⋆ ) ⊆ Dc ⊆ D.

Then, in an analogous way to Theorem 3.2 we arrive at the semilocal result for the Secant method. Theorem 4.2. If (F , x−1 , x0 ) ∈ K then, the sequence {xn } (n ≥ −1) generated by the Secant method is well defined, remains in U (x0 , α ⋆ − c ) for each n = 0, 1, 2, . . . and converges to a unique solution x⋆ ∈ U (x0 , α ⋆ − c ) of equation F (x) = 0. Moreover, the following assertions hold for each n = 0, 1, 2, . . .

∥xn − xn−1 ∥ ≤ αn − αn−1 and

∥xn − x⋆ ∥ ≤ α ⋆ − αn where {αn } (n ≥ 0) is given by (4.1). Furthermore, if there exists R such that U (x0 , R) ⊆ D, R ≥ α ⋆ − c and l0 (α ⋆ − c + R) < 1 then, the solution x⋆ is unique in U (x0 , R). Remarks 4.3. R1 : The hypotheses in the class K differ from the hypotheses in the class K . Notice also that the new (C1 ) hypotheses are simpler than the corresponding ones of Section 2. R2 : Remarks similar to the ones following Theorem 3.2 can now follow in this setting. R3 : Notice that condition (1.3) is also the sufficient convergence condition used in the earlier studies mentioned in the introduction when A−1 is replaced by F ′ (x0 )−1 in the condition (C3 ) (see the Introduction). In practice, we shall use the most suitable between the classes K and K .

468

I.K. Argyros, S.K. Khattri / Journal of Complexity 29 (2013) 454–471 Table 1 Comparison between the sequences {αn } and {wn }. n

αn

wn

0 1 2 3 4 5 6 7 8 9

0.000000 × 10 1.500000 × 10−01 2.529349 × 10−01 3.117477 × 10−01 3.443003 × 10−01 3.586517 × 10−01 3.627660 × 10−01 3.632848 × 10−01 3.633018 × 10−01 3.633018 × 10−01 +00

αn+1 − αn

0.000000 × 10 1.500000 × 10−01 2.529349 × 10−01 3.141902 × 10−01 3.527472 × 10−01 3.766455 × 10−01 3.917863 × 10−01 4.017136 × 10−01 4.089325 × 10−01 4.160761 × 10−01 +00

wn+1 − wn

1.500000 × 10 1.029349 × 10−01 5.881284 × 10−02 3.255262 × 10−02 1.435138 × 10−02 4.114289 × 10−03 5.188264 × 10−04 1.691342 × 10−05 6.398041 × 10−08 7.670581 × 10−12 −01

1.500000 × 10−01 1.029349 × 10−01 6.125530 × 10−02 3.855705 × 10−02 2.389829 × 10−02 1.514075 × 10−02 9.927280 × 10−03 7.218911 × 10−03 7.143669 × 10−03 3.461348 × 10−02

5. Numerical examples Here, we illustrate the theoretical results with some numerical examples. Example 5.1. Let us return back to the example discussed in the introduction section. Let D = U (x0 , 1 − p). A consistent approximation to the Fréchet derivative is

δ F (x, y) = x2 + xy + y2 for all x, y ∈ D. Then, we have

|δ F (x, y) − F ′ (z )| = = = ≤

|x2 + xy + y2 − 3z 2 | |x2 − z 2 + xy − yz + yz − z 2 + y2 − z 2 | |(x + z + y)(x − z ) + (y − z )(y + 2z )| |x + z + y| · |x − z | + |y + 2z | · |y − z |,

(5.1)

but

|x + z + y| = |(x − x0 ) + (y − x0 ) + (z − x0 ) + 3x0 | ≤ |x − x0 | + |y − x0 | + |z − x0 | + 3|x0 | ≤ 3(1 − p) + 3 = 3(2 − p)

(5.2)

and for y = z

|y + 2z | ≤ 3(2 − p).

(5.3)

From Definition 3.1 and (5.1)–(5.3), we obtain l = 3(2 − p)|A−1 |,

η = (1 − p)|A−1 |.

By considering z = x0 , we get l0 = (5 − 2p)|A−1 |. We may notice that l0 < l. Choosing x−1 = 0.85 and p = 0.7352, we get

|A−1 | = 0.388726919, l0 = 1.372050533, l = 1.474985422, η = 0.1029348882, a = 0.369632444, b = 1.075022665 and

δ1 = 0.355661586 > δ2 = 0.315378771. Hence, the new convergence criteria are satisfied. A comparison between sequences {αn } and {wn } is presented in Table 1. In the sequence {wn }, l⋆ = l.

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469

Table 2 Comparison between the sequences {αn } and {wn }. n

αn

wn

0 1 2 3 4 5 6 7 8 9

0.000000 × 10 1.500000 × 10−01 2.528183 × 10−01 3.115038 × 10−01 3.438848 × 10−01 3.580631 × 10−01 3.620718 × 10−01 3.625653 × 10−01 3.625808 × 10−01 3.625808 × 10−01

+00

αn+ 1 − αn

0.000000 × 10 1.500000 × 10−01 2.528183 × 10−01 3.139361 × 10−01 3.522689 × 10−01 3.758115 × 10−01 3.903709 × 10−01 3.993253 × 10−01 4.048031 × 10−01 4.080964 × 10−01

+00

wn+1 − wn

1.500000 × 10 1.028183 × 10−01 5.868551 × 10−02 3.238104 × 10−02 1.417823 × 10−02 4.008790 × 10−03 4.934508 × 10−04 1.547500 × 10−05 5.503991 × 10−08 5.974296 × 10−12

−01

1.500000 × 10−01 1.028183 × 10−01 6.111779 × 10−02 3.833284 × 10−02 2.354261 × 10−02 1.455935 × 10−02 8.954449 × 10−03 5.477805 × 10−03 3.293327 × 10−03 1.896154 × 10−03

In Table 1, we observe that the assertions (3.20) hold. Note also that sequence {wn } converges although criterion (1.3) fails. From (2.40) and (2.41), we obtain b11 = 0.04729011896 ≤ α ⋆ ≤ b12 = 0.4264774759. Example 5.2. Let us choose p = 0.7355 in Example 4.1, then (1.3) is satisfied, since 0.9999622526 < 1.0 as well as the new convergence criteria. Then, we can compare majorizing sequences {αn } and {wn } with c = 0.15, l⋆ = l = 1.474635567, l0 = 1.371817297, η = 0.1028182701, a = 0.369643879 and b = 1.074950411. In Table 2, we observe that assertions (3.20) and (3.21) hold. From (2.40) and (2.41), we obtain b11 = 0.04647459627 ≤ α ⋆ ≤ b12 = 0.4257511671. Example 5.3. Let X = Y = R and x0 = 0. Define function F on D = X by

F (x) = d0 x + d1 + d2 sin(exp(d3 x)) where di , i = 0, 1, 2, 3 are given parameters. Let us also define divided difference as

δ F (x, y) =

1



F ′ (y + t (x − y))dt , 0

=

d2 (sin(ed3 x ) − sin(ed3 y ))

(x − y)

+ d0

and

F ′ (z ) = d0 + d2 d3 ed3 z cos(ed3 z ). Then, for d3 sufficiently large and d2 sufficiently small, l/l0 can be arbitrarily large. Example 5.4. Let X = Y = C[0, 1] be equipped with the max-norm. Consider the following nonlinear boundary value problem u′′ = −u3 − γ u2 , u(0) = 0, u(1) = 1.



It is well known that the above nonlinear differential equation can be formulated as the integral equation u(s) = s +

1



Q (s, t )(u3 (t ) + γ u2 (t ))dt 0

(5.4)

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where Q is the Green function Q (s, t ) =



t (1 − s), s(1 − t ),

t ≤ s, s < t.

We observe that 1

 0≤s≤1

1

|Q (s, t )|dt =

max

8

0

.

We may notice that the equivalent problem (5.4) is in the form (1.1). Here, the operator F : D −→ Y is given as 1



[F (x)](s) = x(s) − s −

Q (s, t )(x3 (t ) + γ x2 (t ))dt . 0

Set x0 (s) = s and D = U (x0 , R0 ). It is easy to verify that U (x0 , R0 ) ⊂ U (0, R0 + 1) since ∥x0 ∥ = 1. Hence, if 2γ < 5, we can choose

η=

1+γ

(5 − 2γ )(1 − l0 c )

,

l=

γ + 6R0 + 3 2γ + 3R0 + 6 and l0 = . 8(5 − 2γ )(1 − l0 c ) 16(5 − 2γ )(1 − l0 c )

Note that l > l0 . The Fréchet derivative of the operator F is given by

[F ′ (x)y](s) = y(s) − 3



1

Q (s, t )x2 (t )y(t )dt − 2γ

0



1

Q (s, t )x(t )y(t )dt .

0

Then, we have that

[(I − F ′ (x0 ))(y)](s) = 3

1



Q (s, t )x20 (t )y(t )dt + 2γ 0

1



Q (s, t )x0 (t )y(t )dt . 0

Hence, if 2γ < 5, then

∥I − F ′ (x0 )∥ ≤ 2(γ − 2) < 1. It follows that F ′ (x0 )−1 exists and

∥F ′ (x0 )−1 ∥ ≤

1 5 − 2γ

.

We also have that ∥F (x0 )∥ ≤ 1 + γ . Define

δ F (x, y) =



1

F ′ (y + t (x − y))dt .

0

Choosing x−1 (s) such that ∥x−1 − x0 ∥ ≤ c and l0 c < 1, then we have

∥δ F (x−1 , x0 )−1 F ′ (x0 )∥ ≤

1 1 − l0 c

.

Notice also that

∥δ F (x−1 , x0 )−1 F (x0 )∥ ≤ ∥δ F (x−1 , x0 )−1 F ′ (x0 )∥ ∥F ′ (x0 )−1 F (x0 )∥ 1 1+γ ≤ = η. (1 − l0 c ) (5 − 2γ ) Hence, the choices for η, l0 and l are justified. Furthermore, let us compare the condition (1.3) and the new condition (C1 ). We choose γ = 0.5, R0 = 1.0 and c = 1.0. Thus, η = 0.4651451253, l = 0.3682398909 and l0 = 0.1938137822. The condition (1.3) yields 1.195971708 ≤ 1

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whereas the new condition (C1 ) yields

(C1 )



0.4651451253 ≤ min(0.7045246916, 0.5847273982), 0.3682398909 < 1.

Thus the new condition (C1 ) holds where the condition (1.3) fails. More examples where l0 < l can be found in [5,8,6]. Conclusions We have presented new sufficient convergence criteria of the Secant method for approximating a locally unique solution of a nonlinear equation in a Banach space. To perform the semilocal convergence analysis, we employ an easier to study version of the majorizing sequence, Lipschitz and center-Lipschitz conditions on the divided difference operator. The new error analysis is tighter even under the old convergence criteria. Our error bounds are more precise than provided in earlier studies, and under our convergence hypotheses we can examine cases where earlier conditions are not satisfied [10,13,15,17,18,20,23,24]. We have also provided some computable upper and lower bounds on the limit points of the majorizing sequences for the Secant method. Numerical examples illustrating theoretical results are also given in this study. References [1] S. Amat, S. Busquier, V. Candela, A class of quasi-Newton generalized Steffensen methods on Banach spaces, J. Comput. Appl. Math. 149 (2002) 397–406. [2] S. Amat, S. Busquier, M. Negra, Adaptive approximation of nonlinear operators, Numer. Funct. Anal. Optim. 25 (2004) 397–405. [3] I.K. Argyros, A unifying local–semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004) 374–397. [4] I.K. Argyros, New sufficient convergence conditions for the Secant method, Chechoslovak Math. J. 55 (2005) 175–187. [5] I.K. Argyros, in: K. Chui, L. Wuytack (Eds.), Computational Theory of Iterative Methods, in: Studies in Computational Mathematics, vol. 15, Elsevier, New York, USA, 2007. [6] I.K. Argyros, Y.J. Cho, S. Hilout, Numerical Methods for Equations and its Applications, CRC Press, Taylor and Francis, New York, 2012. [7] I.K. Argyros, S. Hilout, Convergence conditions for Secant-type methods, Chechoslovak Math. J. 60 (2010) 253–272. [8] I.K. Argyros, S. Hilout, Weaker conditions for the convergence of Newton’s method, J. Complexity 28 (2012) 364–387. [9] W.E. Bosarge, P.L. Falb, A multipoint method of third order, J. Optim. Theory Appl. 4 (1969) 156–166. [10] J.E. Dennis, Toward a unified convergence theory for Newton-like methods, in: L.B. Rall (Ed.), Nonlinear Functional Analysis and Applications, Academic Press, New York, 1971, pp. 425–472. [11] J.A. Ezquerro, J.M. Gutiérrez, M.A. Hernández, N. Romero, M.J. Rubio, The Newton method: from Newton to Kantorovich, Gac. R. Soc. Mat. Esp. 13 (2010) 53–76 (in Spanish). [12] M.A. Hernández, M.J. Rubio, J.A. Ezquerro, Secant-like methods for solving nonlinear integral equations of the Hammerstein type, J. Comput. Appl. Math. 115 (2000) 245–254. [13] M.A. Hernández, M.J. Rubio, J.A. Ezquerro, Solving a special case of conservative problems by Secant-like method, Appl. Math. Comput. 169 (2005) 926–942. [14] L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982. [15] P. Laasonen, Ein überquadratisch konvergenter iterativer algorithmus, Ann. Acad. Sci. Fenn. Ser. I 450 (1969) 1–10. [16] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. [17] P.K. Parida, D.K. Gupta, Recurrence relations for a Newton-like method in Banach spaces, J. Comput. Appl. Math. 206 (2007) 873–887. [18] F.A. Potra, On the convergence of a class of Newton-like methods, in: Iterative Solution of Nonlinear Systems of Equations (Oberwolfach, 1982), in: Lecture Notes in Math., vol. 953, Springer, Berlin, New York, 1982, pp. 125–137. [19] F.A. Potra, On an iterative algorithm of order 1.839 · · · for solving nonlinear operator equations, Numer. Funct. Anal. Optim. 7 (1) (1985) 75–106. [20] F.A. Potra, Sharp error bounds for a class of Newton-like methods, Libertas Math. 5 (1985) 71–84. [21] P.D. Proinov, General local convergence theory for a class of iterative processes and its applications to Newton’s method, J. Complexity 25 (2009) 38–62. [22] P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton–Kantorovich type theorems, J. Complexity 26 (2010) 3–42. [23] J.W. Schmidt, Untere Fehlerschranken fur Regula–Falsi Verhafren, Period. Math. Hungar. 9 (1978) 241–247. [24] M.A. Wolfe, Extended iterative methods for the solution of operator equations, Numer. Math. 31 (1978) 153–174. [25] T. Yamamoto, A convergence theorem for Newton-like methods in Banach spaces, Numer. Math. 51 (1987) 545–557.