Expanding the applicability of Inexact Newton Methods under Smale’s (α, γ)-theory

Applied Mathematics and Computation 224 (2013) 224–237

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Expanding the applicability of Inexact Newton Methods underSmale’s (a, c)-theory

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.08.050

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

Ioannis K. Argyros a, Sanjay K. Khattri b, Saïd Hilout c,⇑a Cameron University, Department of Mathematical Sciences, Lawton, OK 73505, USAb Stord Haugesund University college, Department of Engineering, Bjørnsonsgt 45, N-5528, Haugesund, Norwayc Poitiers University, Laboratoire de Mathématiques et Applications, Bd. Pierre et Marie Curie, Téléport 2, B.P. 30179, 86962 Futuroscope Chasseneuil Cedex, France

a r t i c l e i n f o

Keywords:Inexact Newton MethodBanach spacec-ConditionSmale’s a-theoryFréchet-derivative

a b s t r a c t

We present a new semilocal convergence analysis for an Inexact Newton Method (INM)using Smale’s (a; c)-theory. Our approach is based on the concept of center-c0-condition.Developed sufficient convergence conditions are weaker and the error estimates are tighterthan those proposed in earlier studies such as Shen and Li [30,31], Guo [22], Smale [33–35],Morini [26], Argyros [2,8,9] and Argyros and Hilout [12]. Numerical examples illustratingthe theoretical results are also provided 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 xH of nonlinear equation

FðxÞ ¼ 0 ð1Þ

where F is a twice-Fréchet continuously differentiable operator defined on a nonempty convex subset D of a Banach space Xwith values in a Banach space Y. Many problems from computational sciences and other disciplines can be expressed in theform of Eq. (1) using Mathematical Modelling (cf. [1,4,5,7,10,11]). The solution of these equations can rarely be found inclosed form. That is why the most solution methods for these equations are iterative. The study about convergence analysisof iterative procedures are usually divided into two categories: semilocal and local convergence analysis. The semilocal con-vergence analysis derive convergence criteria from the information around an initial point while the local analysis find esti-mates of the radii of convergence balls from the information around a solution. There is ample literature on local as well assemilocal convergence results under various conditions. Interested readers are referred to (cf. [1–42] and references therein).

The most popular iterative method for solving Eq. (1) is undoubtedly the Newton’s method defined by

xnþ1 ¼ xn � F0ðxnÞ�1FðxnÞ for each n ¼ 0;1;2; . . . ð2Þ

where x0 is an initial point. And, the most famous among the semilocal convergence results is the Kantorovich’s theorem forsolving nonlinear equations. This theorem provides a simple and transparent convergence criterion for operators withbounded second derivatives F00 or the Lipschitz continuous first derivatives (cf. [15,25]). Another important theorem wasinaugurated by Professor Smale at the International Conference of Mathematicians (cf. [32–34]). In this theorem, the conceptof an approximate zero was proposed and the convergence criteria were provided to determine an approximate zero for ana-lytic functions depending on the information at the initial point. Wang et al. [36–40] generalized Smale’s result by introduc-ing the c-condition (see Definition 2.3). According to (2), Newton’s method requires to solve equation

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I.K. Argyros et al. / Applied Mathematics and Computation 224 (2013) 224–237 225

F0ðxnÞwn ¼ �FðxnÞ for each n ¼ 0;1;2; . . . ð3Þ

However, in practice it may not be feasible to solve the preceeding equation exactly especially when F0ðxnÞ is too large anddense. That is why, as a suitable alternative to solving Eq. (3) [27,28,42], linear iterative methods have been extensivelydeveloped to approximate the solution of (3). These methods are called Inexact Newton Methods (INM) and are definedby Algorithm 1 [42].

Algorithm 1. (Inexact Newton Method).

For n ¼ 0 and a given initial point x0 until convergence, do

Step 1: Given the residual control zn and iteration xn, find the step wn satisfying

F0ðxnÞwn ¼ �FðxnÞ þ zn

Step 2: Set xnþ1 ¼ xn þwn

Step 3: Set nþ 1 instead n and go to Step 1.

Note that fzng 2 Y and in general depends on fxng. There are many convergence results for the (INM). We refer the reader to(cf. [14,15,2,3,22,26,27,29–31,42] and references therein). These results depend either on the Kantorovich or the Smale (a; c)-theory and residual controls mainly of the form

kF 0ðx0Þ�1znk 6 gnkF 0ðx0Þ�1FðxnÞk1þj ð4Þ

for each n ¼ 0;1;2; . . . and 0 6 j 6 1. In the present paper, we are motivated by optimization considerations and the recentwork by Shen et al. [30,31] which improved and extended some classical results on Smale’s (a; c)-theory. In particular, weexpand the applicability of (INM) by using the center c0-condition (defined in the Definition 2.1) and the center Lipschitzcondition (see (76)). These conditions are employed for computing the upper bound of kF 0ðxnÞ�1Fðx0Þk instead of usingthe less precise and more expensive c-condition and Lipschitz condition. This idea has already been used to provide weakersufficient convergence conditions and a tighter convergence analysis for Newton-type methods using the Kantorovich or theSmale’s (a; c)-theory (cf. [2–15]).

The rest of the paper is organized as follows. In Section 2, we provide the semilocal convergence analysis of the (INM)whereas Section 3 presents the applications and numerical examples.

2. Semilocal convergence of Inexact Newton Methods

Let Uðx; rÞ and Uðx; rÞ be the open and closed ball, respectively, in X with center x and radius r > 0. Let LðY;XÞ be the spaceof bounded linear operators from Y into X.

Definition 2.1. Let c0 > 0 and let 0 < r 6 1=c0 be such that Uðx0; rÞ#D. The operator F is said to satisfy thecenter-c0-condition at x0 2 Uðx0; rÞ if

kF 0ðx0Þ�1ðF 0ðxÞ � F0ðx0ÞÞk 61

ð1� c0kx� x0kÞ2� 1 ð5Þ

for each x on Uðx0; rÞ. We need the following Banach lemma on invertible operators (cf. [15,25,27]).

Lemma 2.2. Let 0 < r 6 1=c0 be such that Uðx0; rÞ#D. Suppose that F satisfies the center-c0-condition (5) at x0 on Uðx0; rÞ. If

kx� x0k 6 1� 1ffiffiffi2p

� �1c0;

then F0ðxÞ�1 2 LðY;XÞ and satisfies

kF 0ðxÞ�1F0ðx0Þk 6 2� 1

ð1� c0kx� x0kÞ2

!�1

: ð6Þ

Proof. We have by (5) and the choice of x that

kF 0ðx0Þ�1ðF 0ðxÞ � F0ðx0ÞÞk 61

ð1� c0kx� x0kÞ2� 1 < 1: ð7Þ

226 I.K. Argyros et al. / Applied Mathematics and Computation 224 (2013) 224–237

It follows from (7) and the Banach lemma on invertible operators that F0ðxÞ�1 2 LðY;XÞ so that (7) is satisfied. The proof ofLemma 2.2 is complete. h

The following definition and related Lemma are taken from (cf. [30–34,36–40]).

Definition 2.3. Let c > 0 and let 0 < r 6 1=c be such that Uðx0; rÞ#D. The operator F is said to satisfy the c-condition at x0

on Uðx0; rÞ if

kF0ðx0Þ�1F00ðxÞk 6 2cð1� ckx� x0kÞ3

ð8Þ

for each x on Uðx0; rÞ.

Lemma 2.4. Let 0 < r 6 1=c be such that Uðx0; rÞ#D. Suppose that F satisfies the c-condition at x0 on Uðx0; rÞ. If

kx� x0k 6 1� 1ffiffiffi2p

� �1c;

then F0ðxÞ�1 2 LðY;XÞ and satisfies

kF0ðxÞ�1F0ðx0Þk 6 2� 1

1� ckx� x0kð Þ2

!�1

: ð9Þ

Remark 2.5. It follows from (8) that

kF0ðxÞ�1ðF 0ðxÞ � F 0ðx0ÞÞk 61

ð1� ckx� x0kÞ2� 1 < 1: ð10Þ

Clearly, we have that

c0 6 c ð11Þ

and c=c0 can be arbitrarily large [4,15]. In case c0 < c, estimate (6) is tighter than (9). This observation leads to a tighter con-vergence analysis for (INM) (see Remark 2.10).

Let c0; c; k and c be positive constants. The majorizing function / defined by

/ðtÞ ¼ k� t þ cc t2

1� c tfor each 0 6 t 6

1c

ð12Þ

was introduced by Wang et al. [36–40] for c ¼ 1 in connection to the Smale’s a-theory [32] in order to compute approximatezeros. Function / was also used by Shen et al. [30,31] to study the convergence of Algorithm 1. Function /0ðtÞ has uniquepositive zero in ð0;1=cÞ. Let us denote such a zero by l. We have

l ¼ 1�ffiffiffiffiffiffiffiffiffiffiffi

cc þ 1

r� �1c: ð13Þ

In the present study, we introduce function /0 defined by

/0ðtÞ ¼ k� t þ cc0 t2

1� c0 tfor each 0 6 t 6

1c0: ð14Þ

Note that if c0 ¼ c then /0ðtÞ ¼ /ðtÞ for each 0 6 t 6 1=c. Consider the iteration ftng given in [31] and defined by

tnþ1 ¼ tn �/ðtnÞ/0ðtnÞ

for each n ¼ 0;1;2; . . . ð15Þ

In order to make the study as self-contained as possible and to use these auxiliary results for our main semilocal results, weneed the following two results which can be found in [30,31].

Lemma 2.6. If

kc 6 1þ 2c � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficðc þ 1Þ

p; ð16Þ

then the function /, defined by (12), has two zeros

tHH

tH

)¼

1þ kc�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ kcÞ2 � 4ðc þ 1Þkc

q2ðc þ 1Þc

ð17Þ


satisfying

k 6 tH6 l 6 tHH;

where l is given in Eq. (13).

Lemma 2.7. Let tH be defined by (17) and ftng be the Newton sequence generated by (15). Let (16) hold. Then for each n P 1 thefollowing assertions hold

tH � tn

tH � tn�16 q2n�1

;tnþ1 � tn

tn � tn�16 q2n�1

and/ðtnÞ

/ðtn�1Þ6 q2n�1

; ð18Þ

where

q ¼1� kc�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ kcÞ2 � 4ðc þ 1Þkc

q1� kcþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ kcÞ2 � 4ðc þ 1Þkc

q :

Let the residuals fzng be satisfying (4) (with j ¼ 1) and furthermore let g ¼ supnP0gn < 1. Consequently for n P 0, the se-quence fxng is well defined and

kF 0ðx0Þ�1znk 6 gnkF 0ðx0Þ�1FðxnÞk26 gkF 0ðx0Þ�1FðxnÞk2

: ð19Þ

Let us define

b ¼ kF0ðx0Þ�1Fðx0Þk: ð20Þ

Without loss of generality we may assume that the starting point x0 is not a zero of F . As a result b is well defined and b > 0.We set

k ¼ ð1þ ffiffiffigp Þb; d ¼ 1� ck

1þ c0kand c ¼ cðdÞ :¼

ffiffiffi2p

gð1þ ffiffiffigp Þdcð1� ffiffiffigp Þ2 þ 1þ ffiffiffi

gp

: ð21Þ

The proof is essentially given in [31] with the exception of the first estimate in A2 (see also (31)) where weaker (5) is usedinstead of more expensive (8) for the computation of kF 0ðxnÞ�1Fðx0Þk. Moreover, estimates (30), (31), (40)–(43) are alsoneeded for the justification of the introduction of the tighter than ftng majorizing sequences introduced in Remark 2.10and after. Furthermore, the major improvement on the selection of parameter c, defined in (21), is justified in the proof.

Note that 0 6 d 6 1 if ck < 1. We need the following auxiliary lemma.

Lemma 2.8. Let fxng be the sequence generated by Algorithm 1. Suppose that the function F satisfy the center-c0-condition (5),the c-condition (8) with r ¼ tH and the inequality

ð1þ ffiffiffigp Þbc 6 1þ 2c � 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficð1þ cÞ

pð22Þ

is satisfied. Then the following assertions hold:

(A1) tH6 1� 1=

ffiffiffi2p� �

1=c or 1=ð1� c tHÞ 6ffiffiffi2p

.(A2) If n P 1 such that kxn � x0k 6 tn then

kF 0ðxnÞ�1F0ðx0Þk 6 �/00ðtnÞ�16 �/0ðtnÞ�1

:

(A3) Let k ¼ 1;2; . . .. If
ffiffiffigp kF0ðx0Þ�1Fðxn�1Þk 6 1 ð23Þ
and

kxn � xn�1k 6 tn � tn�1 ð24Þ

for each 1 6 n 6 k, then the following assertions hold

ð1þ ffiffiffigp ÞkF0ðx0Þ�1FðxkÞk 6 /ðtkÞ; ð25Þ

ffiffiffigp kF0ðx0Þ�1FðxkÞk 6 1; ð26Þ

kF 0ðx0Þ�1FðxkÞkkF 0ðx0Þ�1Fðxk�1Þk

6/ðtkÞ

/ðtk�1Þð27Þ


and

kxkþ1 � xkk 6tkþ1 � tk

tk � tk�1kxk � xk�1k: ð28Þ

Proof. Since k is defined by (21) thus the conditions (22) and (16) are similar. As a result of Lemmas 2.6 and 2.7 are appli-cable. Therefore ftng is strictly increasing and it satisfies

tn 6 tH6 l ð29Þ

for each n P 0. From the Eq. (21), using the fact c P 1, we obtain 1�ffiffiffiffiffiffiffiffiffi

cð1þcÞ

q6 1� 1ffiffi

2p . Thus assertion (A1) follows from Lem-

ma 2.6, (13), (29)the preceding inequality. To show the assertion (A2), let us presume that kxn � x0k 6 tn is valid for n P 1.Thus the Eq. (13), the inequality (29) and the assertion (A1) yield

kxn � x0k 6 tn < tH6 1� 1ffiffiffi

2p

� �1c:

Hence Lemma 2.4 can be applied to obtain

kF0ðxnÞ�1F0ðx0Þk 6 2� 1

ð1� c0kxn � x0kÞ2

!�1

6 2� 1

ð1� c0tnÞ2

!�1

: ð30Þ

Through simple algebraic manipulations we see that

�/00ðtnÞ � 2� 1

ð1� c0tnÞ2

!¼ c � 1� c � 1

ð1� c0tnÞ26 0:

It follows from (30) that

kF0ðxnÞ�1F0ðx0Þk 6 �/00ðtnÞ�16 �/0ðtnÞ�1

: ð31Þ

Accordingly the assertion (A2) holds. To prove the assertion (A3), let k ¼ 1;2; . . . and suppose that (23) and (24) hold for each1 6 n 6 k. We may write

xssk�1 ¼ xk�1 þ ssðxk � xk�1Þ for any 0 6 s; s 6 1: ð32Þ

Now applying Algorithm 1, we have that

FðxkÞ ¼ FðxkÞ � Fðxk�1Þ � F0ðxk�1Þðxk � xk�1Þ þ zk�1 ð33Þ

¼Z 1

0

Z 1

0F00ðxss

k�1Þsdsdsðxk � xk�1Þ2 þ zk�1: ð34Þ

Hence

kF0ðx0Þ�1FðxkÞk 6 F0ðx0Þ�1Z 1

0

Z 1

0F00ðxss

k�1Þsdsdsðxk � xk�1Þ2��

��þ kF0ðx0Þ�1zk�1k ¼ I1 þ I2: ð35Þ

To estimate the integral I1. From (24) and (29) we write

kxssk�1 � x0k 6

Xk�1

n¼1

kxn � xn�1k þ sskxk � xk�1k 6 sstk þ ð1� ssÞtk�1 < tH ð36Þ

and which in turn yields

kxk�1 � x0k 6 tk�1 < tH and kxk � x0k 6 tk < tH: ð37Þ

Using the c-condition (8) in I1, we get that

I1 6

Z 1

0

Z 1

0

2cð1� ckxss

k�1 � x0kÞ3sdsdskxk � xk�1k2

6

Z 1

0

Z 1

0

2cð1� ckxk�1 � x0k � csskxk � xk�1kÞ3

sdsdskxk � xk�1k2

¼ ckxk � xk�1k2

ð1� ckxk�1 � x0k � ckxk � xk�1kÞð1� ckxk�1 � x0kÞ2: ð38Þ

From (24) (for n ¼ k), (37) and (38), we have that

I1 6cðtk � tk�1Þ2

ð1� ctkÞð1� ctk�1Þ2kxk � xk�1k

tk � tk�1

� �2

: ð39Þ


Now we will estimate I2. We can write

F0ðx0Þ�1F0ðxk�1Þ ¼ 1þ F0ðx0Þ�1ðF 0ðxk�1Þ � F0ðx0ÞÞ:

It follows from the center-c0-condition (5) that

kF 0ðx0Þ�1F0ðxk�1Þk 6 1þZ 1

0

2c0

ð1� c0skxk�1 � x0kÞ3dskxk�1 � x0k ¼

1

ð1� c0kxk�1 � x0kÞ2: ð40Þ

By Algorithm 1, we obtain that

kF 0ðx0Þ�1F0ðxk�1Þðxk � xk�1ÞkP kF 0ðx0Þ�1Fðxk�1Þk � kF 0ðx0Þ�1zk�1k:

Thus from (19) and (23) (for n ¼ k), we have that

kF 0ðx0Þ�1F0ðxk�1Þðxk � xk�1ÞkP kF 0ðx0Þ�1Fðxk�1Þk � gkF 0ðx0Þ�1Fðxk�1Þk2 P ð1� ffiffiffigp ÞkF 0ðx0Þ�1Fðxk�1Þk:

Thus the above inequality furnishes

kF 0ðx0Þ�1Fðxk�1Þk 6kF 0ðx0Þ�1F0ðxk�1Þkkxk � xk�1k

1� ffiffiffigp 6kxk � xk�1k

ð1� ffiffiffigp Þð1� c0kxk�1 � x0kÞ2: ð41Þ

In the preceding expression the last inequality holds because of (40). Thus by (19), (37) and (41)

I2 6 gkF 0ðx0Þ�1Fðxk�1Þk26

gkxk � xk�1k2

ð1� ffiffiffigp Þ2ð1� c0kxk�1 � x0kÞ46

gðtk � tk�1Þ2

ð1� ffiffiffigp Þ2ð1� c0tk�1Þ4kxk � xk�1k

tk � tk�1

� �2

: ð42Þ

The observation tk�1 < tk < tH, the inequality (42) and the assertion (A1) yield

I2 6gðtk � tk�1Þ2

ð1� ffiffiffigp Þ2ð1� c0tHÞð1� c0tkÞð1� c0tk�1Þ2kxk � xk�1k

tk � tk�1

� �2

6

ffiffiffi2p

gðtk � tk�1Þ2

ð1� ffiffiffigp Þ2ð1� c0tkÞð1� c0tk�1Þ2kxk � xk�1k

tk � tk�1

� �2

6

ffiffiffi2p

gðtk � tk�1Þ2ð1� ctkÞð1� ctk�1Þ2

ð1� ffiffiffigp Þ2ð1� c0tkÞð1� c0tk�1Þ2ð1� ctkÞð1� ctk�1Þ2kxk � xk�1k

tk � tk�1

� �2

: ð43Þ

Note that

1� ctk

1� c0tk6

1� ctk�1

1� c0tk�16

1� ck1� c0k

¼ d;

since ð1� c tÞ=ð1� c0 tÞ is a decreasing function on ð0;1=cÞ. Thus from (43), we get that

I2 6

ffiffiffi2p

dgðtk � tk�1Þ2

ð1� ffiffiffigp Þ2ð1� ctkÞð1� ctk�1Þ2kxk � xk�1k

tk � tk�1

� �2

: ð44Þ

Consequently, combining (35), (39) and (43) give

kF 0ðx0Þ�1FðxkÞk26 I1 þ I2 6 1þ dg

ffiffiffi2p

cð1� ffiffiffigp Þ2 !

cðtk � tk�1Þ2


tk � tk�1

� �2

:

Since from (21) we know c ¼ dffiffi2p

gð1þ ffiffigp Þcð1� ffiffigp Þ2 þ 1þ ffiffiffigp . Thus the preceding inequality yields

ð1þ ffiffiffigp ÞkF0ðx0Þ�1FðxkÞk2

6ccðtk � tk�1Þ2


tk � tk�1

� �2

: ð45Þ

By Eq. (12) and (15), we have

ccðtk � tk�1Þ2

ð1� ctkÞð1� ctk�1Þ2¼ /ðtkÞ � /ðtk�1Þ � /0ðtk�1Þðtk � tk�1Þ ¼ /ðtkÞ:

From (45), we obtain


6 /ðtkÞkxk � xk�1k

tk � tk�1

� �2

ð46Þ


and from (24) (with n ¼ k) and (25). Moreover, since / is strictly decreasing on ½0;l�, it follows from (25) that


6 /ðtkÞ 6 /ðt0Þ ¼ k:

This implies that
ffiffiffigp kF0ðx0Þ�1FðxkÞk 6
ffiffiffigp1þ ffiffiffigp k ¼ ffiffiffi

gp

b ¼ ffiffiffigp kF0ðx0Þ�1Fðx0Þk

using the definition of k and b. Hence, (26) holds with (23) (with n ¼ 1). Thus to complete the proof of the lemma, we need toshow that (27) and (28) hold. For this purpose, we have from Algorithm 1 that

xnþ1 � xn ¼ F0ðx0Þ�1F0ðx0Þ �F0ðx0Þ�1FðxnÞ þ F0ðx0Þ�1zn

� �:

By (23)ffiffiffigp kF0ðx0Þ�1Fðxk�1Þk 6 1 (with n ¼ k), it follows from (19) that

kF0ðx0Þ�1zk�1k 6 gkF 0ðx0Þ�1Fðxk�1Þk26 gkF 0ðx0Þ�1Fðxk�1Þk:

Consequently we get that

kxk � xk�1k 6 kF 0ðxk�1Þ�1F0ðx0Þk kF 0ðx0Þ�1Fðxk�1Þk þ kF 0ðx0Þ�1zk�1k� �

6 ð1þ ffiffiffigp ÞkF 0ðxk�1Þ�1F0ðx0Þk � kF 0ðx0Þ�1Fðxk�1Þk ð47Þ

Similarly, we also have that (observing thatffiffiffigp kF0ðx0Þ�1FðxkÞk 6 1 by (27))

kxkþ1 � xkk 6 ð1þffiffiffigp ÞkF 0ðxkÞ�1F0ðx0Þk � kF 0ðx0Þ�1FðxkÞk: ð48Þ

By (46) and (47), we get that


¼ ð1þffiffiffigp ÞkF 0ðx0Þ�1FðxkÞk

ð1þ ffiffiffigp ÞkF 0ðx0Þ�1Fðxk�1Þk

6/ðtkÞ

ð1þ ffiffiffigp ÞkF0ðx0Þ�1Fðxk�1Þkkxk � xk�1k

tk � tk�1

� �2

/ðtmÞF 0ðxk�1Þ�1F0ðx0Þkkxk � xk�1kðtk � tk�1Þ2

6 /ðtmÞkF 0ðxk�1Þ�1F0ðx0Þk

tk � tk�1usingð24Þ: ð49Þ

Similarly by (46), (49) and (24) (with n ¼ k), we obtain

kxk � xk�1k 6 /ðtkÞkF 0ðxkÞ�1F0ðx0Þkkxk � xk�1k

tk � tk�1: ð50Þ

Furthermore, from assertion (A2) and (37), we get that

kF0ðxk�1Þ�1F0ðx0Þk 6 �/0ðtk�1Þ�1 and kF 0ðxkÞ�1F0ðx0Þk 6 �/0ðtkÞ�1:

From (49), (50) and the above inequalities, we get that


6/ðtkÞ

�/0ðtk�1Þðtk � tk�1Þ¼ /ðtkÞ

/ðtk�1Þ

and

kxkþ1 � xkk 6�/ðtkÞ/0ðtkÞ�1

tk � tk�1kxk � xk�1k ¼

tkþ1 � tk

tk � tk�1kxk � xk�1k:

Thus (27) and (28) hold. The proof is complete. h

We have only added the center-c0-condition to bring in our new second result in Lemma 2.8. Now we present an impor-tant theorem from [31] and for the proof of the next result we refer the reader to [31], our Lemma 2.2, (30) and (31).

Theorem 2.9. Let us assume that F fulfill the c0-condition (5) and the c-condition (8) with r ¼ tH. Let us further assume that

b 6min1ffiffiffigp ;

1þ 2c � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficðc þ 1Þ

pcð1þ ffiffiffigp Þ

( ): ð51Þ

Let fxng be a sequence generated by Algorithm 1. Then fxng converges to a solution xH of (1) and the following assertions hold foreach n P 1

kxn � xHk 6 q2n�1kxn�1 � xHk; kxnþ1 � xnk 6 q2n�1kxn � xn�1k; ð52Þ


kF 0ðx0Þ�1FðxnÞk 6 q2n�1kF 0ðx0Þ�1Fðxn�1Þk; ð53Þ

where q is given in Lemma 2.7.

Remark 2.10.

(R1) If c0 ¼ c, Lemma 2.6 and the Theorem 2.9 reduce, respectively, to the Lemma 3.2 and the Theorem 2.7 in [31] which inturn reduces to the classical ones proposed by Smale et al. [34,35]. Otherwise, i.e. if c0 < c, our results constitute animprovement, since (see (21)) then

c < d :¼ cð1Þ: ð54Þ

In particular (22) provides a wider convergence domain, since d was used instead of c to obtain the above result by Shen andLi [31]. Moreover, the new error analysis is tighter in this case. Furthermore

cd! P :¼ P0

d< 1 as

cc0!1; ð55Þ

since d! 1� ck as cc0!1, where

P0 ¼ffiffiffi2p

g ð1þ ffiffiffigp Þð1� ckÞc ð1� ffiffiffigp Þ2 þ 1þ ffiffiffi

gp

:

Estimate (55) shows by at most how much the value of parameter c is improved.(R2) The proofs of the same results show that the sequences frng and fsng defined by

r0 ¼ 0; r1 ¼ k; r2 ¼ r1 þc0ðr1 � r0Þ2

2� 1ð1�c0r1Þ2

� �ð1� cr1Þð1� cr0Þ2

rnþ2 ¼ rnþ1 þc0ðrnþ1 � rnÞ2

2� 1ð1�c0rnþ1Þ2

� �ð1� crnþ1Þð1� crnÞ2

;

ð56Þ

and

s0 ¼ 0; s1 ¼ k

snþ2 ¼ snþ1 þcðsnþ1 � snÞ2

2� 1ð1�c0snþ1Þ2

� �ð1� csnþ1Þð1� csnÞ2

:ð57Þ

are tighter majorizing sequences for fxng than ftng. Indeed, under sufficient convergence condition (22) the following asser-tions hold

rn 6 sn 6 tn; ð58Þ

rnþ1 � rn 6 snþ1 � sn 6 tnþ1 � tn ð59Þ

and

rH ¼ limn!1

rn ¼6 sH6 tH: ð60Þ

Estimates (59) and (60) hold as strict inequalities if c0 < c for n > 1. In view of (59)–(61), we are wondering if frng and fsngconverge under weaker conditions than (22). Next we present such conditions for frng which is a tighter majorizing se-quence for fxng than fsng if c0 < c.

Proposition 2.11. Suppose that the operator F satisfies the center-c0-condition at x0 on Uðx0;1=c0Þ (5) and the c-condition at x0

on Uðx0;1=cÞ (8). Furthermore let

rn <

1c for each n ¼ 0;1;2; . . . ;

1� 1ffiffi2p

� �1c0

if 1� 1ffiffi2p 6

c0c

8<: ð61Þ

and

Uðx0; rHÞ#D: ð62Þ

Then the following assertions hold


(A1) The sequence frng generated by (56) is increasing and converges to its unique least upper bound rH which satisfies

k 6 rH6 rHH ¼min

1c; 1� 1ffiffiffi

2p

� �1c0

� : ð63Þ

(A2) Sequence fxng generated by Algorithm 1 is well defined, remains in Uðx0; rHÞ for all n P 0 and converges to a solutionxH 2 Uðx0; rHÞ of the equation FðxÞ ¼ 0. Moreover, the following estimates hold

kxnþ1 � xnk 6 rnþ1 � rn ð64Þ

and

kxn � xHk 6 rH � rn: ð65Þ

Proof. In view of the proofs of Lemma 2.6 and Theorem 2.9, we only need to show (A1). It follows from (56) and (61) thatsequence frng is increasing, bounded from above by rII and as such it converges to its unique least upper bound denoted byrH. The proof of the proposition is complete. h

Remark 2.12.

(R1) Limit point rH can be replaced by rHH and which is given in closed form in (63).(R2) Sequence frng depends on the initial data c0; c and k. Therefore, sufficient condition (61) can be checked to see if it is

satisfied. Since in case of convergence rN ¼ rnþN for a finite integer N and each n ¼ 0;1;2; . . . Clearly (61) is weaker suf-ficient convergent condition than (22). Next we find more sufficient convergent conditions for frng that can replace(61) in the Proposition 2.11.

Case (R2ð1Þ): First we propose one such set of conditions. Let there exists a minimum integer N P 1 such that

r0 6 r1 6 � � � rN < rHH ð66Þ

and (22) is satisfied with k replaced by kN ¼ rN � rN�1. That is

ðrN � rN�1Þc 6 1þ 2c � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficðc þ 1Þ

p: ð67Þ

Note that k1 ¼ k and (67) reduces to (22). Clearly (67) is at least as weak as (22) if rN � rN�1 6 r1 � r0 and weaker ifrN � rN�1 < r1 � r0.

Case (R2ð2Þ): Motivated by estimates

2� 1

ð1� c0rnþ1Þ2P 2� 1

ð1� c0gÞand

11� crn

6

ffiffiffi2p

; ð68Þ

we consider sequence fvng defined as

v0 ¼ 0; v1 ¼ k; v2 ¼ v1 þK0ðv1 � v0Þ2

2ð1� cv1Þvnþ2 ¼ vnþ1 þ

Kðvnþ1 � vnÞ2

2ð1� cvnþ1Þfor each n ¼ 0;1;2; . . . ; ð69Þ

where

K0 ¼2c0

1� 12ð1�c0gÞ

2

h i and K ¼ 2c

1� 12ð1�c0gÞ

2

h i : ð70Þ

Kantorovich type sufficient convergence conditions for sequences of the type (67) have been given by us in [4,10,13,15].Next, in particular, we present some relevant ones. Sufficient convergent condition

h ¼ 18

4cþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficKþ 8c2

qþ

ffiffiffiffiffiffifficK

p� �k 6

12

ð71Þ

[13] can replace (61) in the Proposition 2.11.Let us state one more result for the sequence rn (see [13, Lemma 2.4]).

Lemma 2.13. Suppose that there exists a minimum integer N > 1 such that iterates ri (i ¼ 0;1;2; . . . ;N � 1) given by (56) are welldefined

ri < riþ1 <1c

for each i ¼ 0;1;2; . . . ;N � 2 ð72Þ

and


rN 61c

1� ð1� crN�1Þn½ �; ð73Þ

where

n ¼ 2c

KþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK2 þ 8cK

q :

Then the following assertions hold:

crN < 1; rNþ1 61c

1� ð1� crNÞn½ �;

nN�1 ¼KðrNþ1 � rNÞ2ð1� crNþ1Þ

6 n 6 1� cðrNþ1 � rNÞ1� crN

;

sequence frng given by (56) is well defined, increasing, bounded from above by

bN ¼ rN�1 þ1

1� nðrN � rN�1Þ ð74Þ

and converges to its unique least upper bound b which satisfies b 2 ½g; bN�. Moreover, the following estimates hold

0 < rNþ1 � rNþn�1 6 nn�1ðrNþ1 � rNÞ for each n ¼ 1;2; . . . ð75Þ

Clearly (61) is weaker than (71) or (72) and (73) which in turn can be weaker than (22). A direct comparison between frng and allthe sequences introduced above is not practical (see also Section 3 for numerical examples).

Remark 2.14. Let us introduce center-Lipschitz condition

kF 0ðx0Þ�1ðF 0ðxÞ � F0ðx0ÞÞk 6 L0kx� x0k ð76Þ

for each x in D as an alternative to (2.1) or (2.4) for the computation of the upper bounds on kF 0ðxÞ�1F0ðx0Þk. Then, we haveinstead of (2.2) or (2.5) that

kF 0ðxÞ�1F0ðx0Þk 61

1� L0kx� x0kð77Þ

for each x on Uðx0;1=L0Þ. This modification leads to majorizing sequence s1n defined by

s10 ¼ 0; s1

1 ¼ k; s1nþ2 ¼ s1

nþ1 þcðs1

nþ1 � s1nÞ

ð1� L0s1nþ1Þð1� cs1

nþ1Þð1� cs1nÞ

2 : ð78Þ

Then fs1ng can replace ftng or fsng in Theorem 2.9. Noting that

11� cs1

k

6

ffiffiffi2p

(see also the numerical examples in the next section) we arrive at the less tight majorizing sequence ft1ng

t10 ¼ 0; t1

1 ¼ k; t1nþ2 ¼ t1

nþ1 þnðt1

nþ1 � t1nÞ

2

2ð1� c0t1nþ1Þ

ð79Þ

where n ¼ 4ffiffiffi2p

c. Then, the sufficient convergence condition is given by (cf. [13,15])

H ¼ 18

nþ 4L0 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin2 þ 8L0n

q� �k 6

12: ð80Þ

Similar comments to the Remark 2.10 can follow in this new setting.Summing up we conclude that in practice we shall test existing sufficient convergence conditions to see if any are satis-

fied. Then, we choose the tightest majorizing sequence for the computation of the upper bounds on the distances kxnþ1 � xnkand kxn � xHk for each n ¼ 0;1;2; . . .

3. Special cases and numerical examples

Remark 3.1. If gn :¼ g ¼ 0 then Algorithm 1 reduces to Newton’s method (NM). Moreover if c0 ¼ c we have that


b ¼ k; c ¼ 1 and tH ¼1þ kc�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ kcÞ2 � 8kc

q4c

:

Thus Theorem 2.9 reduces to the following corollary which was obtained by Wang et al. [37].

Corollary 3.2. Suppose that F satisfies the c-condition (8) with r ¼ tH. Furthermore suppose that

kc 6 3� 2ffiffiffi2p

: ð81Þ

Let fxng be the sequence generated by the Newton’s method. Then fxng converges to a solution xH of (1) and the assertions 52,53hold for each n P 1 and with q as defined in Theorem 2.9.

Remark 3.3. One classic and important type of examples satisfying the c-condition (8) is the analytic functions. As in [32],we define

c :¼ supnP2

F0ðx0Þ�1F nðx0Þn!

��

1=ðn�1Þ

: ð82Þ

Then 0 6 c < þ1. Then the following lemma is known (cf. [37]).

Lemma 3.4. Let 0 < r < 1=c be such that Uðx0; rÞ#D. Then the analytic operator F satisfies the c-condition (8) at x0 on Uðx0; rÞwith c defined by (82).

The following corollary is obvious.

Corollary 3.5 [31]. Suppose that the operator F is analytic and the condition (51) holds. Furthermore let fxng be a sequencegenerated by Algorithm 1. Then fxng converges to a solution xH of (1) and the 52,53 hold assertions hold for each n P 1.

Remark 3.6. Suppose that the nonlinear operator D# X�!Y has second continuous Fréchet derivative. Let x0 2 D be suchthat F0ðx0Þ�1 exists. Smale introduced the concept of an approximate zero for analyzing the computational complexity ofNewton’s method for nonlinear operators in Banach spaces (cf. [35]). However later it was discovered that the proposed ideacannot capture the quadratic convergent nature of the Newton’s method and it is not suitable for studying the computationalcomplexity. To remove these drawbacks, Smale in [32] proposed two modifications. The first modification is based upon anapproximate zero in the sense of kxn � xn�1kwhile the second modification is based upon an approximate zero in the sense ofkxn � xIk. Smale [32], also introduced the following criterion

a :¼ ck

for examining x0 as an approximate zero of F . Here, b is defined by (20). Wang et al. in [32] introduced the following def-inition of an approximate zero for Newton’s method.

Definition 3.7. Let eðxnÞ be some measurement of the approximation degree between xn and xH. Let x0 2 D be such that thesequence fxng generated by the Newton’s method is well defined and the assertion

eðxnÞ 612

� �2n�1

eðxn�1Þ ð83Þ

holds for each n P 1. Then x0 is called an approximate zero of F in the sense of eðxnÞ (cf. [37]).In the next definition, the notion of approximate zero is defined for (INM).

Definition 3.8. Let x0 2 D be such that the sequence fxng generated by Algorithm 1 is well defined and satisfies thedefinition (83). The x0 is called an approximate zero of F in the sense of eðxnÞ.

By Theorem 2.9, the following theorem can be deduced to provide criterion for the approximate zero for (INM). For thespecial case of c ¼ cð1Þ, the proof was given in [30].

Theorem 3.9. Suppose that F satisfies the center-c0-condition (5) and the c-condition (8) at x0 with r ¼ tH. Furthermore supposethat

k 6 min1ffiffiffigp ;

4þ 9c � 3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficð9c þ 8Þ

p4c ð1þ ffiffiffigp Þ

( ): ð84Þ


Let fxng be the sequence generated by Algorithm 1. Then fxng converges to a solution xH of (1) and x0 is an approximate zero of F inthe sense of kxn � xHk; kxnþ1 � xnk and kF 0ðx0Þ�1FðxnÞk.

Proof. Since there is no difference between the proof of Theorems 3.9 and 2.9. Hence, proof is being omitted. h

In the special case when gn ¼ 0, Algorithm 1 reduces to (NM). Thus we obtain the following corollary for (NM). For theproof, see [37].

Corollary 3.10. Let F satisfies c-condition (8) with

r ¼1þ kc�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1þ kcÞ2 � 8kc

q4c

:

Suppose that the following assertion hold

kc 613� 3

ffiffiffiffiffiffi17p

4:

Let fxng be the sequence generated by the (NM). Then fxng converges to a solution xH of (1) and x0 is an approximate zero of F inthe sense of kxn � xHk; kxnþ1 � xnk and kF 0ðx0Þ�1FðxnÞk.

From Theorem 3.9 and Lemma 3.4 the following corollary is obvious.

Corollary 3.11. Let the operator F be analytic and that (83) holds. Furthermore let the sequence fxng be generated by Algorithm 1.Then fxng converges to a solution xH of (1) and x0 is an approximate zero of the operator F in the sense of kxn � xHk; kxnþ1 � xnkand kF 0ðx0Þ�1FðxnÞk.

Next we consider some simple numerical examples to illustrate our theoretical results.

Example 3.12. Let X ¼ Y ¼ R2; x0 ¼ ½1;0�T;D ¼ Uðx0;1� NÞ for N 2 ð0;1Þ. Let us define function F on D as follows

FðxÞ ¼ Fðn1; n2Þ ¼n3

1 � n2 � N

n1 þ 3n2 �ffiffiffiffiN3p

" #: ð85Þ

Jacobian of FðxÞ is

F0ðxÞ ¼ n21 �1

1 3

" #:

Using (85) we see that (8) is satisfied for c ¼ 2� N. We also have that k ¼ ð1� NÞ=3.Case ILet N ¼ 0:6255. Then we notice that the condition (81) is not satisfied. Since

Fig. 1. Comparison between conditions (81) and (80).

Table 1Comparison among the sequences frng; fsng; ftng; fs1

ng and ft1ng. Part I.

n rn sn tn t1n s1

n

0 0:000000eþ 00 0:000000eþ 00 0:000000eþ 00 0:000000eþ 00 0:000000eþ 001 1:000000e� 01 1:000000e� 01 1:000000e� 01 1:000000e� 01 1:000000e� 012 1:152215e� 01 1:195920e� 01 1:220125e� 01 1:409005e� 01 2:940588e� 013 1:157271e� 01 1:206972e� 01 1:236963e� 01 1:480721e� 01 1:961109eþ 004 1:157277e� 01 1:207009e� 01 1:237070e� 01 1:482945e� 01 3:005236eþ 005 1:157277e� 01 1:207009e� 01 1:237070e� 01 1:482947e� 01 3:038136eþ 006 1:157277e� 01 1:207009e� 01 1:237070e� 01 1:482947e� 01 3:038422eþ 007 1:157277e� 01 1:207009e� 01 1:237070e� 01 1:482947e� 01 3:038425eþ 008 1:157277e� 01 1:207009e� 01 1:237070e� 01 1:482947e� 01 3:038425eþ 009 1:157277e� 01 1:207009e� 01 1:237070e� 01 1:482947e� 01 3:038425eþ 00

Table 2Comparison among the sequences frng; fsng; ftng; fs1

ng and ft1ng. Part II.

n rnþ1 � rn snþ1 � sn tnþ1 � tn t1nþ1 � t1

n s1nþ1 � s1

n

0 1:000000e� 01 1:000000e� 01 1:000000e� 01 1:000000e� 01 1:000000e� 011 1:522150e� 02 1:959203e� 02 2:201246e� 02 4:090050e� 02 1:940588e� 012 5:055776e� 04 1:105150e� 03 1:683820e� 03 7:171583e� 03 1:667050eþ 003 5:856846e� 07 3:755605e� 06 1:069600e� 05 2:223674e� 04 1:044127eþ 004 7:872070e� 13 4:351969e� 11 4:338887e� 10 2:138444e� 07 3:289973e� 025 1:422132e� 24 5:843899e� 21 7:140132e� 19 1:977660e� 13 2:866045e� 046 4:641326e� 48 1:053747e� 40 1:933579e� 36 1:691450e� 25 2:424286e� 067 4:943634e� 95 3:426132e� 80 1:417992e� 71 1:237297e� 49 2:050096e� 088 5:608605e� 189 3:621924e� 159 7:626002e� 142 6:620700e� 98 1:733659e� 109 7:218912e� 377 4:047712e� 317 2:205685e� 282 1:895674e� 194 1:466065e� 12


kc ¼ 0:171583417 > 3� 2ffiffiffi2p¼ 0:171572875:

Hence there is no guarantee that (NM) starting from x0 ¼ 1 will converge to xH ¼ffiffiffiffiN3p¼ 0:85521599 (cf. [30–40]). However,

our results can apply. Indeed using (76) we have that

L0 ¼ 3� N; n ¼ 2ffiffiffi2pð2� NÞ:

Hence, (80) is satisfied. Since

H ¼ 0:3560521103 < 0:5;

we conclude that iteration fs1ng given in (78) converges to xH.

Fig. 1 presents a comparison between (81) and (80) after normalization.Case IILet N ¼ 0:7. It can be seen that the condition (81) holds since

kc ¼ 0:1300000000 6 3� 2ffiffiffi2p

:

From the condition (80), we get that

H ¼ 0:2735840806 < 0:5

Thus the condition (80) also holds. Using (85), we have that (5) is satisfied if

F0ðx0Þ�1 F0ðxÞ � F 0ðx0Þð Þ�� ¼ kx2 � x2

0k <1

ð1� c0kx� x0kÞ2� 1

or if

kðx� x0Þðx� x0 þ 2x0Þk <1

ð1� c0kx� x0kÞ2� 1

or

c0kx� x0k2 � ð1� c0Þ2kx� x0k � 2ð1� c0ÞP 0;

which is certainly satisfied for say c0 ¼ 1:01. Note that c0 < 1:3 ¼ c. Let us now compare the sequences frng; fsng; ftng; fs1ng

and ft1ng. Tables 1 and 2 compare the sequences. In the Table 2, we observe that the sequence frng is converging the fastest

and certainly satisfies the conditions of the Proposition 2.11, Since


1c¼ 0:7692307; 1� 1ffiffiffi

2p

� �1c0¼ 0:2899932

and

1� 1ffiffiffi2p ¼ :29289321 < 0:776923:

4. Conclusion

A convergence analysis of Inexact Newton Method is provided for approximating a locally unique solution of a nonlinearequation in a Banach space setting. Using Smale’s ða; cÞ-theory in combination with our new idea of center-c0-condition, wepresented a new convergence analysis with larger convergence domain and weaker sufficient convergence conditions thanin earlier studies such as (cf. [2,3,15,16,22,30–40]). Numerical examples illustrating the theoretical results are also providedin this study.

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Expanding the applicability of Inexact Newton Methods under Smale’s (α, γ)-theory

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