Unifying semilocal and local convergence of Newton's method on Banach space with a convergence structure

Applied Numerical Mathematics 115 (2017) 225–234

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Applied Numerical Mathematics www.elsevier.com/locate/apnum

Unifying semilocal and local convergence of Newton’s method on Banach space with a convergence structure Ioannis K. Argyros a , Ramandeep Behl b,∗ , S.S. Motsa b,c a

Cameron University, Department of Mathematics Sciences, Lawton, OK 73505, USA School of Mathematics, Statistics and Computer Sciences, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, Pietermaritzburg, South Africa c University of Swaziland, Mathematics Department, Private Bag 4, Kwaluseni, M201, Swaziland b

a r t i c l e

i n f o

Article history: Received 4 August 2015 Received in revised form 12 August 2016 Accepted 13 January 2017 Available online 26 January 2017 Keywords: Banach space with a convergence structure Semilocal convergence Local convergence Newton’s method

a b s t r a c t We present a semilocal and local convergence analysis of Newton’s method on a Banach space with a convergence structure to locate zeros of operators. P. Meyer introduced the concept of a Banach space with a convergence structure. Using this setting, he presented a finer semilocal convergence analysis for Newton’s method than in related studies using the real norm theory. In all these studies the operator involved as well as its Fréchet derivative is bounded above by the same bound-operator. In the present study, we introduce a second bound operator which is a special case of the bound-operator leading to tighter majorizing sequences for Newton’s method. Using this more flexible combination of bound-operators, we improve the results in the earlier studies. In the semilocal case, we obtain under the same or weaker sufficient convergence conditions more precise error bounds on the distances involved and in the local case not considered in the earlier studies, we obtain a larger radius of convergence. This way we expand the applicability of Newton’s method. Some numerical examples are also provided to show the superiority of the new results over the old results. © 2017 IMACS. Published by Elsevier B.V. All rights reserved.

1. Introduction In this study we are concerned with the problem of locating a locally unique zero x∗ of an operator G defined on a convex subset D of a Banach X with values in a Banach space Y . Our results will be presented for the operator F defined by

F (x) := J G (x0 + x) ,

(1.1)

where x0 is an initial point and J ∈ L (Y , X ) the space of bounded linear operators from Y into X . A lot of real life problems can be formulated like (1.1) using Mathematical Modeling [2,6,7,17,18,22,24–26]. The zeros of F can be found in closed form only in special cases. That is why most solution methods for these problems are usually iterative. There are mainly two types of convergence: semi-local and local convergence. The semi-local convergence case is based on the information around an initial point to find conditions ensuring the convergence of the iterative method; while the local one is based on the information around a solution, to find estimates of the radii of convergence balls [1–10,14–26].

*

Corresponding author. E-mail address: [email protected] (R. Behl).

http://dx.doi.org/10.1016/j.apnum.2017.01.008 0168-9274/© 2017 IMACS. Published by Elsevier B.V. All rights reserved.

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The most popular methods for approximating a zero of F are undoubtedly the so called Newton-like methods. Authors using real norm theory and Lipschitz-type conditions have provided sufficient convergence conditions for these methods. There is a plethora of local as well as semi-local convergence results in the case of the real norm theory [1,6–10,14–20,23, 26]. In these studies a Lipschitz-type bound is obtained which may be just a positive number or a non-negative function. These bounds are used in the construction of the majorizing sequences for Newton’s method. Tighter bounds lead to weaker sufficient convergence criteria and tighter error bounds on the distances xn − xn−1 , xn − x∗ . Looking at this direction P. Meyer [21,22] noticed that the properties of the operator F  can be analyzed more precisely in the setting of a Banach space with a convergence structure than in the real norm theory setting. In the present study using our new idea as already presented in the abstract of this study, we improve the results in [21,22] under the same computational cost. In particular, Semilocal case: The sufficient convergence criteria are weaker, the error estimates xn − xn−1 , xn − x∗  are tighter and the information on the location of the solution more precise. Therefore, if the old criteria are not satisfied, then new criteria may be satisfied. In the case when the old and new criteria are satisfied the new error bounds are at least as tight and the information on the location of the solution at least as precise (see Example 4.1 and Remark 4.2). Local case: The radius of convergence is larger and the error bounds are tighter leading to a wider choice of initial guesses and fewer iterations to achieve a desired error tolerance. Notice that the local case was not considered in [24,25] (see also the numerical examples). Therefore, we expand the applicability of Newton’s method in the setting of a Banach space with a convergence structure. Clearly, our idea can be used to improve the results in other settings such as the ones given by Gupta and Prashanth [11], Ferreira and Ferreira et al. [12,13]. In the future, we will provide the extensions of the works [11–13]. The rest of the paper is organized as follows. To make the paper as self continued as possible, we present some standard concepts on Banach spaces with a convergence structure in Section 2. The semilocal convergence analysis of Newton-like methods is presented in Section 3. The local convergence is given in section 4. Some examples are presented as practical applications. In particular, using the first semi local example (see Example 4.1), we show that the new results can apply to solve equations but the old ones cannot be applied. In the second local example (see section 5), we obtain a larger radius of convergence and tighter error bounds on the distance involved. This study is completed with a conclusion in section 6. 2. Banach spaces with convergence structure We present some results on Banach spaces with a convergence structure. More details can be found in [4–7,21,22] and the references there in. Definition 2.1. A triple ( X , V , E ) is a Banach space with convergence structure, if (i) ( X , ·) is a real Banach space. (ii) ( V , C , · V ) is a real Banach space which is partially ordered by the closed convex cone C . The norm · V is assumed to be monotone on C . (iii) E is a closed convex cone in X × V such that {0} × C ⊆ E ⊆ X × C . (iv) The operator /·/ : D → C

/x/ := inf {q ∈ C | (x, q) ∈ E } for each x ∈ Q , is well defined, where

Q := {x ∈ X |∃q ∈ E : (x, q) ∈ E } . (v) x ≤ /x/ V for each x ∈ Q . Notice that it follows by the definition of Q that Q + Q ⊆ Q and for each θ > 0, θ Q ⊆ Q . Define the set

U (a) := {x ∈ X | (x, a) ∈ E } . k Let us provide some examples when X = R   equipped with the max-norm [17,26]: k (a) V = R; E := (x, q) ∈ R × R| x∞ ≤ q .

    (c) V = Rk ; E := (x, q) ∈ Rk × Rk |0 ≤ x ≤ q .

(b) V = Rk ; E := (x, q) ∈ Rk × Rk | |x| ≤ q .

More cases can be found in [6–8,23,24]. Case (a) corresponds to the convergence analysis in a real Banach space; case (b) can be used for componentwise error analysis and case (c) may be used for monotone convergence analysis. The convergence analysis is considered in the space X × V . If (xn , qn ) ∈ E R is an increasing sequence, then:

(xn , qn ) ≤ (xn+m , qn+m ) ⇒ 0 ≤ (xn+m − xn , qn+m − qn ) .

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Moreover, if qn → q (n → ∞) then, we get: 0 ≤ (xn+m − xn , q − qn ). Hence, by (v) of Definition 2.1

xn+m − xn  ≤ q − qn  V → 0 (n → ∞). That is we conclude that {xn } is a complete sequence. Set qn = w 0 − w n , where { w n } ∈ C R is a decreasing sequence. Then, we have that

0 ≤ (xn+m − xn , w n − w n+m ) ≤ (xn+m − xn , w n ) . ∗ ∗ Furthermore,  j  if xn → x (n → ∞), then we deduce that /x − xn / ≤ w n . Let L X denote the space of multilinear, symmetric, bounded operators on a Banach space X , H : X j → X . Let also consider an ordered Banach space V :

 L+ V

j



     := L ∈ L V j |0 ≤ xi ⇒ 0 ≤ L x1 , x2 , ..., x j .

Let V L be an open subset of an ordered Banach space V . An operator L ∈ C 1 ( V L → V ) is defined to be order convex on an interval [a, b] ⊆ V L , if for each c , d ∈ [a, b], c ≤ d ⇒  L (d) − L  (c ) ∈ L + ( V ).





Definition 2.2. The set of bounds for an operator H ∈ L X j is defined by:





B ( H ) := L ∈ L + V

j



  

 | (xi , qi ) ∈ E ⇒ H x1 , ..., x j , L q1 , ..., q j ∈ E .







Lemma 2.3. Let H : [0, 1] → L X j and L : [0, 1] → L + V B ( H (t )) ⇒

1 0

L (t ) dt ∈ B



1 0



j



be continuous operators. Then, we have that for each t ∈ [0, 1] : L (t ) ∈

H (t ) dt .

Let T : Y → Y be an operator on a subset Y of a normed space. Denote by T n (x) the result of n-fold application of T . In particular in case of convergence, we write

T ∞ (x) := lim T n (x) . n→∞

Next, we define the right inverse: Definition 2.4. Let H ∈ L ( X ) and u ∈ X be given. Then,

H ∗ u := x∗ ⇔ x∗ ∈ T ∞ (0) , T (x) := ( I − H ) x + u ⇔ x∗ =

∞

( I − H ) j u,

j =0

provided that this limit exists. Finally, we need two auxiliary results on inequalities in normed spaces and the Banach perturbation Lemma: Lemma 2.5. Let L ∈ L + ( V ) and a, q ∈ C be given such that

Lq + a ≤ q and L n q → 0 (n → ∞). Then, the operator

( I − L )∗ : [0, a] → [0, a] is well defined and continuous. Lemma 2.6. Let H ∈ L ( X ), L ∈ B ( H ), u ∈ D and q ∈ C be given such that:

Lq + /u/ ≤ q and L n q → 0 (n → ∞). Then, the point given by x := ( I − H )∗ u is well defined, belongs in D and

/x/ ≤ ( I − L )∗ /u/ ≤ q.

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3. Semilocal convergence We present the semilocal convergence in this section to determine a zero x∗ of the operator (1.1) under certain conditions denoted by (A). Let X be a Banach space with convergence structure ( X , V , E ), where V = ( V , C , · V ), let operators F ∈ C 1 ( D → X ) with D ⊆ X , L 0 , L ∈ C 1 ( V L → V ) with V L ⊆ V and a point a ∈ C be such that the following conditions (A) hold:

( A 1 ) U (a) ⊆ D and [0, a] ⊆ V L . ( A 2 ) L 0 ≤ L and L 0 ≤ L  .   ( A 3 ) L 0 (0) ∈ B I − F  (0) , (− F (0) , L (0)) ∈ E .   ( A 4 ) L 0 (|x|) − L 0 (0) ∈ B F  (0) − F  (x) . 1 ( A5)

[ L  (/ y / + μ/ y − x/) − L  (/x/)]/ y − x/dμ.

0

( A 6 ) L (a) ≤ a. ( A 7 ) L  (a)n a → 0 (n → ∞). Next, we can show the following semilocal convergence result of Newton’s method using the preceding notation. Theorem 3.1. Suppose that the conditions (A) hold. Then (i) the sequences {xn }, {δn } defined by

x0 = 0, xn+1 := xn + F  (xn )∗ (xn ) (− F (xn )) ,

δ0 = 0, δn+1 := L (δn ) + L 0 (/xn /) γn , n = 0, 1, 2, . . . where and

γn := /xn+1 − xn /, are well defined, the sequence (xn , δn ) ∈ ( X × V )R remains in E R , for each n = 0, 1, 2, . . . , is monotone, δn ≤ b, for each n = 0, 1, 2, ...,

where b := L ∞ (0) is the smallest fixed point of L in [0, a]. (ii) The Newton sequence {xn } is well defined, it remains in U (a) for each n = 0, 1, 2, . . . , and converges to a unique zero x∗ of F in U (a). Proof. (i) We shall solve the equation





q = I − F  (xn ) q + (− F (xn )) , for each n = 0, 1, 2, ....

(1)

First notice that the conditions of Theorem 3.1 are satisfied with b replacing a. If n = 1 in (1) we get by ( A 2 ), ( A 3 ), ( A 5 ) and ( A 7 ) with q = b

L 0 (0) b + /− F (0)/ ≤ L (b) − L (0) + /− F (0)/ ≤ L (b) ≤ b. That is x1 is well defined and (x1 , b) ∈ E. We get the estimate





x1 = I − F  (0) x1 + (− F (0)) so,

/x1 / ≤ L 0 (0) /x1 / + L 0 (0) ≤ L 0 (0) /x1 / + L 0 (0) ≤ δ1 and by ( A 2 )

δ1 = L 0 (0) /x1 / + L (0) ≤ L  (0) (b) + L (0) ≤ L 0 (b) (b − 0) + L (0) ≤ L (b) − L (0) + L (0) = L (b) ≤ b. Suppose that the sequence is well defined and monotone for k = 1, 2, ..., n and δk ≤ b.

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Using the induction hypotheses and ( A 6 ) we get in turn that



/− F (xn )/ = − F (xn ) + F (xn−1 ) + F  (xn−1 ) (xn − xn−1 )

1 ≤



[ L  (/xn / + μγn−1 ) − L  (/xn−1 /)]γn−1 dμ

0

≤ L (/xn−1 / + γn−1 ) − L (/xn−1 /) − L  (/xn−1 /) γn−1 ≤ L (δn−1 + δn − δn−1 ) − L (δn−1 ) − L  (/xn−1 /) γn−1 



= L (δn ) − δn + L 0 (|xn−1 |)γn−1 − L (|xn−1 |)γn−1 . By ( A 2 )–( A 4 ) we have the estimate











(2) (3)



I − F  (xn ) ≤ I − F  (0) + F  (0) − F  (xn ) ≤ L 0 (0) + L 0 (/xn /) − L 0 (0)

= L 0 (/xn /) . Then, to solve the equation (1), let q = b − δn to obtain that

L 0 (/xn /) (b − δn ) + /− F (xn )/ + δn

≤ L 0 (δn ) (b − δn ) + L (δn ) + L 0 (|xn−1 |)γn−1 − L  (|xn−1 |)γn−1 ≤ L  (δn )(b − δn ) + L (δn ) ≤ L (b) − L (δn ) + L (δn ) ≤ L (b) ≤ b, since L 0 (|xn−1 |)γn−1 ≤ L  (|xn−1 |)γn−1 . That is xn+1 is well defined by Lemma 2.5 and

γn ≤ b − δn . Therefore, δn+1 is also well defined and we can have:



δn+1 ≤ L (δn ) + L 0 (δn ) (b − δn ) ≤ L (b) ≤ b. We also need to show the monotonicity of (xn , δn ) ≤ (xn+1 , δn+1 ):

γn + δn ≤ L 0 (/xn /) γn + /− F (xn )/ + δn + L 0 (|xn−1 |)γn−1 − L  (|xn−1 |)γn−1 ≤ L 0 (/xn /) γn + L (δn ) = δn+1 . The induction is complete and the statement (i) is shown. (ii) Using induction and the definition of sequence {δn } we get L n (0) ≤ δn ≤ b, which implies δn → b, since L n (0) → b. It follows from the discussion in Section 2 that sequence {xn } converges to some x∗ ∈ U (b) (since U (b) is a closed set). By letting n → ∞ in (3) we deduce that x∗ is a zero of F . Let y ∗ ∈ U (a) be a zero of F . Then as in [21] we get that





y ∗ − xn ≤ L n (a) (a) − L n (0) ,

so we conclude that x∗ = y ∗ .

2

Remark 3.2. Concerning a posteriori estimates, we can list a few. It follows from the proof of Theorem 3.1 that





x∗ − xn ≤ b − δn ≤ q − δn ,

where we can use for q any solution of M (q) q ≤ q. We can obtain more precise error estimates as in [22] by introducing monotone maps R n under the (A) conditions as follows:



∗

R n (q) := I − L 0 /xn /

S n (q) + γn ,

where

S n (q) := L (/xn / + q) − L (/xn /) − L  (/xn /) q. Notice that operator S n is monotone on the interval I n := [0, a − /xn /]. Suppose that there exists qn ∈ C such that /xn / + qn ≤ a and

S n (qn ) + L  (/xn /) (qn − γn ) ≤ qn − γn . It then follows that operator R n : [0, qn ] → [0, qn ] is well defined and monotone by Lemma 2.5 for each n = 0, 1, 2, . . . . A possible choice qn is a − δn . Indeed, this follows from the implications

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δn + γn ≤ δn+1 ⇒ L (a) − L (δn ) − L  (/xn /) γn ≤ a − δn − γn ⇒ L (a) − L (δn ) − L  (/xn /) γn ≤ a − δn − γn ⇒ S n (a − δn ) + L  (/xn /) (a − δn − γn ) ≤ a − δn − γn . The proofs of the next three results are omitted since they follow from the corresponding ones in [22] by simply using R n and S n instead of

 ∗  R n (q) = I − L  (/xn /)  S n (q) + cn ,  S n (q) = L (/xn / + q) − L (/xn /) − L  (/xn /) q, used in the preceding references. Let us also define the sequence {dn } by

d0 = 0, dn+1 = L (dn ) + L  (/xn /) cn .

(4)

Proposition 3.3. Suppose that there exists q ∈ I n such that R n (q) ≤ q. Then, the following hold

γn ≤ R n (q) =: q0 ≤ q and

R n+1 (q0 − γn ) ≤ q0 − γn . Proposition 3.4. Suppose that the (A) conditions hold. Moreover, suppose that there exist qn ∈ I n such that R n (qn ) ≤ qn . Then, the sequence { pn } defined by

pn = qn , pn+1 := R n ( pm ) − γn for m ≥ n leads to the estimate /x∗ − xn / ≤ pm . Proposition 3.5. Suppose that the (A) conditions hold. Then for any q ∈ I n satisfying R n (q) ≤ q we have that





x∗ − xn ≤ R n∞ (0) ≤ q.

The rest of the results in [22] can be generalized along the same framework. Remark 3.6. If L 0 = L and condition ( A 5 ) is replaced by the stronger condition

L  (/x/ + / y /) − L  (/x/) ∈ B ( F  (x) − F  (x + y )) then Theorem 3.1 reduces to Theorem 5 in [22]. Notice that Theorem 3.1 is weaker than Theorem 5 even if L 0 = L, since ( A 5 ) is weaker than ( A 5 ) . Condition ( A 5 ) implies ( A 4 ),

L0 ≤ L,

(5)

holds and LL can be arbitrary large [6,7]. In view of (5) and under the hypotheses of Theorem 3.1 and Theorem 5, a simple 0 inductive argument shows that

δn ≤ dn for each n = 0, 1, 2, . . . .

(6)

If L 0 < L, then strict inequality holds in (6) for n = 1, 2, . . . . Moreover, it follows from the proof of Theorem 3.1 that sequence {δ¯n } defined for each n = 0, 1, 2, . . . by

δ¯0 = 0, δ¯1 = L 0 (δ0 ) + L 0 (|x0 |)γ0 , δ¯n+1 = L (δ¯n ) + L 0 (|xn |)γn , n = 1, 2, . . .

(7)

can replace sequence {δn }. Furthermore, we have that

δ¯n ≤ δn ≤ dn for each n = 0, 1, 2, . . .

(8)

δ¯n < δn < dn for each n = 1, 2, . . . , if L 0 < L .

(9)

and

Finally, the estimates given in Remark 3.2, Propositions 3.3–3.5 are also improving the corresponding ones in [22]. Hence, the applicability of Newton’s method is extended under the same or weaker conditions than in [22].

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Example 3.7. Let us consider the case of a Banach space with a real norm  · . Suppose for simplicity that F  (0) = I and that there exist monotone maps ψ0 and ψ such that

 F  (x) − F  (x0 ) ≤ ψ0 (x − x0 ) and

 F  (x) − F  ( y ) ≤ ψ(x − y ) for each x, y ∈ U (a). Then, set

t L 0 (t ) =  F (0) +

s ds

0

and

L 0 (t ) =  F (0) +

1 2

ψ0 (t )dt 0

⎡ t ⎤  s t s ⎣ ds ψ0 (t )dt + ds ψ(t )dt ⎦ . 0

0

0

0

We have that L 0 (t ) ≤ L (t ) for each t ∈ [0, a], since ψ0 (t ) ≤ ψ(t ) for each t ∈ [0, a]. By estimating ψ0 (t ) ≤ ψ0 (a) = p 0 , and ψ(t ) ≤ ψ(a) = p, to satisfy ( A 5 ), we have to solve

1

 F (0) + ( p 0 + p )a2 ≤ a. 4

This is possible, if

h0 = ( p 0 + p ) F (0) ≤ 1.

(10)

Notice that

p0 ≤ p,

(11) p

holds in general and p can be arbitrary large [6,7]. If p 0 = p, condition (10) reduces to the famous for its simplicity and 0 clarity of Kantorovich hypothesis

h = 2p  F (0) ≤ 1,

(12)

for solving nonlinear equations [17]. However, if p 0 < p then

h ≤ 1 ⇒ h 0 ≤ 1,

(13)

but not vice versa. Hence, the sufficient convergence condition ( A 5 ) can be weakened in some interesting cases. Examples where (10) holds but (12) is violated can be found in [6–8]. 4. Local convergence The local convergence of the Newton’s method was not given in [22] on a Banach space with convergence structure. In this section, we present the local convergence of Newton’s method in a way analogous to the semilocal convergence case. We study the convergence of Newton’s method as defined in Theorem 3.1 as a fixed point problem of the form

yn = T¯ n ( yn ) := ( I − F  (xn )) yn − Q n ,

(14)

where

Q n = F (xn ) − F (x∗ ) − F  (xn )(xn − x∗ ) and F (x∗ ) = 0. Define also the sequences {αn } and {βn } by

α0 = /x0 − x∗ /, αn+1 := L (αn ) + L 0 (/xn /)βn , βn = /xn − x∗ /. The local convergence analysis of Newton’s method is given under the following conditions ( A ∗ ): Let X be a Banach space with convergence structure ( X , V , E ) where V = ( V , C ,  ·  v ), let operators F ∈ C 1 ( D → X ), with D ⊆ X , L 0 , L ∈ C 1 ( V L → V ) with V L ⊆ V , x∗ ∈ D with F (x∗ ) = 0 and a point a ∈ C be such that for each x, y ∈ D the following hold

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( A ∗1 ) U (a) ⊆ D and [0, a] ⊆ V L . ( A ∗2 ) L 0 ≤ L and L 0 ≤ L  . ( A ∗3 ) L 0 (0) ∈ B ( I − F  (0)), L 0 (0) ∈ B ( I − F  (x∗ )),

(− F (0), L 0 (0)) ∈ E .

( A ∗4 ) L 0 (/x∗ / + /x − x∗ /) − L 0 (|x∗ |) ∈ B ( F  (x∗ ) − F  (x)), 1

∗

( A5)

for each x ∈ U (a).

[ L  (/x∗ / + μ/x − x∗ /) − L  (/x/)]/x − x∗ /dμ.

0

∗

( A 6 ) L (a) ≤ a. ( A ∗7 ) L  (a)n a → 0(n → ∞). Then, sequences {xn } and {αn } are well defined, the sequence (xn , monotone,

αn ) ∈ ( X × V )R remains in E R for each n = 0, 1, 2, . . . is

αn+1 ≤ αn ≤ b, lim αn = 0, and lim xn = x∗ , n→∞

n→∞

L ∞ (0) is the smallest fixed point of L in [0, a].

where b =

Example 4.1. As in the case of Example 3.7 suppose that the exist monotone maps φ0 and φ such that

 F  (x) − F  (x∗ ) ≤ φ0 (x − x∗ ) ≤ φ0 (a) = N 0 and

 F  (x) − F  ( y ) ≤ φ(x − y ) ≤ φ(a) = N for each x, y ∈ V (a). Then, set

t L 0 (t ) = 2

s ds

0

φ0 (t )dt 0

and

⎡ L (t ) =

1 2

⎣2

t

s φ0 (t )dt +

ds 0

t

0

s ds

0

⎤ φ(t )dt ⎦ .

0

We have that L 0 (t ) ≤ L (t ) for each t ∈ [0, a]. Then, ( A ∗5 ) is satisfied if



2N 0 + N 2

 a ≤ 1.

That is we can choose

a=

2 2N 0 + N

.

The radius of convergence a improves the radius obtained independently by Traub [26] and Rheinboldt [25] given by

a¯ =

2 3N

.

Notice that

a¯ ≤ a, and

a¯ < a, if N 0 = N .

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Remark 4.2. (a) Let E = R and F  (x∗ ) = I . Then, we have that

a¯ = a=

2 3N

, 2

,

2N 0 + N

¯ αn2 N

αn+1 ≤ ξn+1 ≤

2(1 − N 0 αn ) N ξn2 2(1 − N ξn )

,

,

and a¯ a

→

1 3

as

N0 N

→ 0,

where the sequence {ξn } given in [25,26] is less precise than {αn }. Notice that a¯ is the radius of convergence for the Newton’s method obtained independently by Traub [26] and Rheinbold [25], where as a is the radius obtained by Argyros [6–8]. (b) The local results can be used for the 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 [6–8]. 5. Numerical examples In this section, we shall demonstrate the theoretical results which we have proposed in the section 2, by applying them on the following two numerical examples, for the local convergence case when E = R. Example 5.1. Let X = Y = R3 , D = U¯ (0, 1). Define F on D for v = (x, y , z) T by

 F ( v ) = e x − 1,

e−1 2

T y2 + y, z

.

(15)

Then the Fréchet-derivative is given by

⎡

ex  ⎣ F (v ) = 0 0

⎤

0 0 (e − 1) y + 1 0 ⎦ . 0 1

Notice that x∗ = (0, 0, 0), F  (x∗ ) = F  (x∗ )−1 = diag {1, 1, 1}, N 0 = e − 1 < N = e. Then, we get that

a¯ = 0.24525296 < a = 0.32497231. Example 5.2. Let X = Y = C [0, 1], the space of continuous functions defined on [0, 1] be and equipped with the max norm. Let D = U¯ (0, 1). Define function F on D by

1 xτ ϕ (τ )3 dτ ,

F (ϕ )(x) = ϕ (x) − 5

(16)

0

we have that



 F ϕ (ξ ) (x) = ξ(x) − 15 

1 xτ ϕ (τ )2 ξ(τ )dτ , for each ξ ∈ D . 0

Then, for x∗

= 0, we obtain that N 0 = 7.5, N = 15. Then, we get that

a¯ = 0.0444 · · · < a = 0.0666 . . . .

(17)

234

I.K. Argyros et al. / Applied Numerical Mathematics 115 (2017) 225–234

6. Conclusion In the present study, we used a more flexible choice of two bound L 0 and L operators for operator F  , instead of just one bound operator L used in earlier studies. Notice that in practice the computation of operator L involves the computation of operator L 0 as a special case. Moreover, L 0 is tighter than L leading to tighter majorizing sequences than in [20,21]. The benefits of the new approach have been explained previously. The technique introduced here can be used in settings other than the settings of a Banach space with convergence structure such as the setting [11–13]. The details of the extensions for the works in [11–13] shall be given in our future works. Another way of improving these results is to find a more precise domain than D or V (a) containing the Newton iterates. This will lead to even tighter operators L 0 and L. We shall also pursue this task in future works. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

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