AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 367161 10.1155/2013/367161 367161 Research Article Local Convergence of Newton’s Method on Lie Groups and Uniqueness Balls http://orcid.org/0000-0002-3851-3181 He Jinsu 1 Wang Jinhua 2, 3 Yao Jen-Chih 4, 5 Peralta Antonio M. 1 Department of Mathematics Zhejiang Normal University Jinhua 321004 China zjnu.edu.cn 2 Department of Mathematics Zhejiang University of Technology Hangzhou 310032 China zjut.edu.cn 3 Department of Mathematics National Sun Yat-sen University Kaohsiung 804 Taiwan nsysu.edu.tw 4 Center for Fundamental Science Kaohsiung Medical University Kaohsiung 80702 Taiwan kmu.edu.tw 5 Department of Mathematics King Abdulaziz University P.O. Box 80203, Jeddah 21589 Saudi Arabia kau.edu.sa 2013 13 11 2013 2013 20 08 2013 01 10 2013 2013 Copyright © 2013 Jinsu He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An estimation of uniqueness ball of a zero point of a mapping on Lie group is established. Furthermore, we obtain a unified estimation of radius of convergence ball of Newton’s method on Lie groups under a generalized L-average Lipschitz condition. As applications, we get estimations of radius of convergence ball under the Kantorovich condition and the γ-condition, respectively. In particular, under the γ-condition, our results improve the corresponding results in (Li et al. 2009, Corollary 4.1) as showed in Remark 17. Finally, applications to analytical mappings are also given.

1. Introduction

In a vector space framework, when f is a differentiable operator from some domain D in a real or complex Banach space X to another Y, Newton’s method is one of the most important methods for finding the approximation solution of the equation f(x)=0, which is formulated as follows: for any initial point x0D, (1)  xn+1=xn-f(xn)-1f(xn),n=0,1,.

As is well known, one of the most important results on Newton’s method is Kantorovich’s theorem (cf. ), which provides a simple and clear criterion ensuring quadratic convergence of Newton’s method under the mild condition that the second Fréchet derivative of f is bounded (or more generally, the first derivative is Lipschitz continuous) and the boundedness of f(x)-1 on a proper open metric ball of the initial point x0. Another important result on Newton’s method is Smale’s point estimate theory (i.e., α-theory and γ-theory) in , where the notions of approximate zeros were introduced and the rules to judge an initial point x0 to be an approximate zero were established, depending on the information of the analytic nonlinear operator at this initial point and at a solution x*, respectively. There are a lot of works on the weakness and/or the extension of the Lipschitz continuity made on the mappings; see, for example,  and references therein. In particular, Wang introduced in  the notion of Lipschitz conditions with L-average to unify both Kantorovich’s and Smale’s criteria.

In a Riemannian manifold framework, an analogue of the well-known Kantorovich theorem was given in  for Newton’s method for vector fields on Riemannian manifolds while the extensions of the famous Smale α-theory and γ-theory in  to analytic vector fields and analytic mappings on Riemannian manifolds were done in . In the recent paper , the convergence criteria in  were improved by using the notion of the γ-condition for the vector fields and mappings on Riemannian manifolds. The radii of uniqueness balls of singular points of vector fields satisfying the γ-conditions were estimated in , while the local behavior of Newton’s method on Riemannian manifolds was studied in [12, 13]. Furthermore, in , Li and Wang extended the generalized L-average Lipschitz condition (introduced in ) to Riemannian manifolds and established a unified convergence criterion of Newton’s method on Riemannian manifolds. Similarly, inspired by the previous work of Zabrejko and Nguen in  on Kantorovich’s majorant method, Alvarez et al. introduced in  a Lipschitz-type radial function for the covariant derivative of vector fields and mappings on Riemannian manifolds and gave a unified convergence criterion of Newton’s method on Riemannian manifolds.

Note also that Mahony used one-parameter subgroups of a Lie group to develop a version of Newton’s method on an arbitrary Lie group in , where the algorithm presented is independent of affine connections on the Lie group. This means that Newton’s method on Lie groups is different from the one defined on Riemannian manifolds. On the other hand, motivated by looking for approaches to solv ordinary differential equations on Lie groups, Owren and Welfert also studied in  Newton’s method, independent of affine connections on the Lie group, and showed the local quadratical convergence. Recently, Wang and Li  established Kantorovich’s theorem (independent of the connection) for Newton’s method on Lie group. More precisely, under the assumption that the differential of f satisfies the Lipschitz condition around the initial point (which is in terms of one-parameter semigroups and independent of the metric), the convergence criterion of Newton’s method is presented. Extensions of Smale’s point estimate theory for Newton’s method on Lie groups were given in .

The purpose of the present paper is to continue the study of Newton’s method on Lie groups. At first, we give an estimation of uniqueness ball of a zero point of a mapping on a Lie group. Second, we establish a unified estimation of radius of convergence ball of Newton’s method on Lie groups under a generalized L-average Lipschitz condition. As applications, we obtain estimations of radius of convergence ball under the Kantorovich condition and the γ-condition, respectively. In particular, under the γ-condition, we get that (see Theorem 16) if f(x*)=0 and (2)ϱ(x*,x0)<3-222γ, then the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to a zero y* of f. This improves the corresponding results in [19, Corollary 4.1], where it was proved under the following assumption: there exists v𝔤 such that (3)x0=x*expv,va0γ, with a0=0.081256 being the smallest positive root of the equation a0/(1-4a0+2a02)2=3-22. Clearly, (4)a0γ<3-222γ. Note also that in general, there dose not exist v𝔤 satisfying x0=x*expv because the exponential map is not surjective global, even if ϱ(x*,x0)<(3-22)/2γ. In view of this, our results somewhat improve the corresponding results in [19, Corollary 4.1].

The remainder of the paper is organized as follows. Some preliminary results and notions are given in Section 2, while the estimation of uniqueness ball is presented in Section 3. In Section 4, the main results about estimations of convergence ball are explored. Theorems under the Kantorovich condition and the γ-condition are provided in Section 5. In the final section, we get the estimations of uniqueness ball and convergence ball under the assumption that f is analytic.

2. Notions and Preliminaries

Most of the notions and notations which are used in the present paper are standard; see, for example, [20, 21]. A Lie group (G,·) is a Hausdorff topological group with countable bases which also has the structure of an analytic manifold such that the group product and the inversion are analytic operations in the differentiable structure given on the manifold. The dimension of a Lie group is that of the underlying manifold, and we will always assume that it is m-dimensional. The symbol e designates the identity element of G. Let 𝔤 be the Lie algebra of the Lie group G which is the tangent space TeG of G at e, equipped with Lie bracket [·,·]:𝔤×𝔤𝔤.

In the sequel, we will make use of the left translation of the Lie group G. We define for each yG the left translation Ly:GG by (5)Ly(z)=y·zfor  each  zG. The differential of Ly at z is denoted by (Ly)z which clearly determines a linear isomorphism from TzG to the tangent space T(y·z)G. In particular, the differential (Ly)e of Ly at e determines a linear isomorphism form 𝔤 to the tangent space TyG. The exponential map exp:𝔤G is certainly the most important construction associated with G and 𝔤 and is defined as follows. Given u𝔤, let σu:G be the one-parameter subgroup of G determined by the left invariant vector field Xu:y(Ly)e(u); that is, σu satisfies that (6)σu(0)=e,σu(t)=Xu(σu(t))=(Lσu(t))e(u)for  each  t. The value of the exponential map exp at u is then defined by (7)exp(u)=σu(1). Moreover, we have that (8)exp(tu)=σtu(1)=σu(t)for  each  t,u𝔤,exp(t+s)u=exp(tu)·exp(su)for  any  t,s,u𝔤. Note that the exponential map is not surjective in general. However, the exponential map is a diffeomorphism on an open neighborhood of 0𝔤. In the case when G is Abelian, exp is also a homomorphism from 𝔤 to G; that is, (9)exp(u+v)=exp(u)·exp(v),u,v𝔤. In the non-Abelian case, exp is not a homomorphism and, by the Baker-Campbell-Hausdorff (BCH) formula (cf. [21, page 114]), (9) must be replaced by (10)exp(w)=exp(u)·exp(v), for all u,v in a neighborhood of 0𝔤, where w is defined by (11)w:=u+v+12[u,v]+112([u,[u,v]]+[v,[v,u]])+.

Let f:G𝔤 be a C1-map and let xG. We use fx to denote the differential of f at x. Then, by [22, Page 9] (the proof given there for a smooth mapping still works for a C1-map), for each ΔxTxG and any nontrivial smooth curve c:(-ε,ε)G with c(0)=x and c(0)=Δx, one has (12)fxΔx=(ddt(fc)(t))t=0. In particular, (13)fxΔx=(ddtf(x·exp(t(Lx-1)xΔx)))t=0  ((Lx-1)xΔx)foreachΔxTxG. Define the linear map dfx:𝔤𝔤 by (14)dfxu=(ddtf(x·exp(tu)))t=0for  each  u𝔤. Then, by (13), (15)dfx=fx(Lx)e. Also, in view of definition, we have that, for all t0, (16)ddtf(x·exp(tu))=dfx·exp(tu)ufor  each  u𝔤,(17)f(x·exp(tu))-f(x)=0tdfx·exp(su)udsfor  each  u𝔤.

For the remainder of the present paper, we always assume that ·,· is an inner product on 𝔤 and · is the associated norm on 𝔤. We now introduce the following distance on G which plays a key role in the study. Let x,yG and define (18)ϱ(x,y)inf{i=1kuithere  exist  k1  and  u1,,uk𝔤  such  that  y=x·expu1expuki=1kuithere  exist  k1  and  u1}, where we adapt the convention that inf=+. It is easy to verify that ϱ(·,·) is a distance on G and that the topology induced by this distance is equivalent to the original one on G.

Let xG and r>0. We denoted the corresponding ball of radius r around x of G by Cr(x); that is, (19)Cr(x):={yGϱ(x,y)<r}. Let (𝔤) denote the set of all linear operators on 𝔤. Below, we will modify the notion of the Lipschitz condition with L-average for mappings on Banach spaces to suit sections. Let L be a positive nondecreasing integrable function on [0,R], where R is a positive number large enough such that 0R(R-s)L(s)dsR. The notion of Lipschitz condition in the inscribed sphere with the L average for operators on Banach spaces was first introduced in  by Wang for the study of Smale’s point estimate theory.

Definition 1.

Let r>0,  x0G and let T be a mapping from G to (𝔤). Then T is said to satisfy the L-average Lipschitz condition on Cr(x0) if (20)T(x·expu)-T(x)ϱ(x0,x)ϱ(x0,x)+uL(s)ds holds for any xCr(x0) and u𝔤 such that u<r-ϱ(x,x0).

3. Uniqueness Ball of Zero Points of Mappings

This section is devoted to the study of uniqueness ball of zero points of mappings. Let r>0. We use B(0,r) to denote the open ball at 0 with radius r on 𝔤; that is, (21)B(0,r):={v𝔤v<r}. Write N(x*,r):=x*exp(B(0,r)). Clearly, N(x*,r)Cr(x*). Let r^>0 be such that (22)1r^0r^L(u)(r^-u)du=1.

Theorem 2.

Let 0<rr^. Suppose that f(x*)=0 and dfx*-1df satisfies the L-average Lipschitz condition in N(x*,r). Then x* is the unique zero point of f in N(x*,r).

Proof.

Let y*N(x*,r) be another zero point of f in N(x*,r). Then, there exists v𝔤 such that y*=x*expv and v<r. As L(·) is a positive function, it follows from  that the function ψ defined by (23)ψ(t)=1t0tL(s)(t-s)ds,t(0,r^] is strictly monotonically increasing. set (24)λ:=1v0vL(s)(v-s)ds. Then, by (22), we get (25)λ<1r^0r^L(s)(r^-s)ds=1. To complete the proof, it suffices to show that (26)vλv. Granting this, one has that x*=y*. Now, (27)v=-dfx*-1(f(y*)-f(x*))+v-dfx*-101dfx*exp(sv)vds+v01dfx*-1(dfx*exp(sv)-df(x*))vds010svL(t)dtvds=0vL(t)(v-t)dt=λv, where the third inequality holds because of (20) by selecting x=x0=x*. Therefore, (26) is seen to hold and the proof is completed

4. Convergence Ball of Newton’s Method

Following , we define Newton’s method with initial point x0 for f on a Lie group as follows: (28)xn+1=xn·exp(-dfxn-1f(xn))for  each  n=0,1,.

Let r0>0 and b>0 be such that (29)0r0L(s)ds=1,b=0r0L(s)sds.

Remark 3.

(i) Since L(·) is a positive function, we always have br0. Indeed, (30)b-r0=0r0L(s)sds-r00r0L(s)ds=0r0L(s)(s-r0)ds0.

(ii) Consider r0r^. Indeed, recall from  that the function ψ defined by (31)ψ(t)=1t0tL(s)(t-s)ds,t(0,r^] is strictly monotonically increasing. Sine ψ(r0)1=ψ(r^), we get r0r^.

The following proposition plays a key role in this section, which is taken from .

Proposition 4.

Suppose that x0G is such that dfx0-1 exists and dfx0-1df satisfies the L-average Lipschitz condition on Cr0(x0) and that (32)β:=dfx0-1f(x0)b. Then the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges to a zero point x* of f and ϱ(x*,x0)<r0.

The remainder of this section is devoted to an estimate of the convergence domain of Newton’s method on G around a zero x* of f. Below we will always assume that x*G is such that dfx*-1 exists.

Lemma 5.

Let 0<rr0 and let x0Cr(x*) be such that there exist j1 and w1,,wj𝔤 satisfying (33)x0=x*·expw1expwj, and ρ(x*,x0):=i=1jwi<r. Suppose that dfx*-1df satisfies the L-average Lipschitz condition on Cr(x*). Then dfx0-1 exists, (34)dfx0-1dfx*11-0ρ(x*,x0)L(s)ds,(35)dfx0-1f(x0)0ρ(x*,x0)L(s)(ρ(x*,x0)-s)ds+ρ(x*,x0)1-0ρ(x*,x0)L(s)ds.

Proof.

It follows from [24, Lemma 2.1] that dfx0-1 exists and (34) holds. Write y0=x*,  yi=yi-1·expwi and ρi:=l=1iwl for each i=1,,j. Thus, by (33), we have yj=x0 and so ρj=ρ(x*,x0). Fix i, one has from (17) that (36)f(yi)-f(yi-1)=01dfyi-1·exp(τwi)widτ, which implies that (37)dfx*-1(f(yi)-f(yi-1))=01dfx*-1(dfyi-1·exp(τwi)-dfx*)widτ+wi. Since dfx*-1df satisfies the L-average Lipschitz condition on Cr(x*), it follows that (38)dfx*-1(dfyi-1·exp(τwi)-dfx*)dfx*-1(dfyi-1·exp(τwi)-dfyi-1)+l=1i-1dfx*-1(dfyl-dfyl-1)ρi-1ρi-1+τwiL(s)ds+l=1i-1ρl-1ρlL(s)ds0ρi-1+τwiL(s)ds, where ρ0=0. Noting that f(x0)=i=1j(f(yi)-f(yi-1)), we have from (37) and (38) that (39)dfx*-1f(x0)i=1j(01dfx*-1(dfyi-1·exp(τwi)-dfx*)×widτ+wi01dfx*-1(dfyi-1·exp(τwi)-dfx*))i=1j(010ρi-1+τwiL(s)dswidτ+wi). Write F(t):=0tL(s)ds. Then, (40)010ρi-1+τwiL(s)dswidτ=01F(ρi-1+τwi)widτ=ρi-1ρiF(t)dt, and so (41)i=1j01dfx*-1(dfyi-1·exp(τwi)-dfx*)widτi=1jρi-1ρiF(t)dt=0ρ(x*,x0)F(t)dt=0ρ(x*,x0)L(s)(ρ(x*,x0)-s)ds. This, together with (39), yields that (42)dfx*-1f(x0)0ρ(x*,x0)L(s)(ρ(x*,x0)-s)ds+ρ(x*,x0). Combining this with (34) implies that (43)dfx0-1f(x0)dfx0-1dfx*dfx*-1f(x0)0ρ(x*,x0)L(s)(ρ(x*,x0)-s)ds+ρ(x*,x0)1-0ρ(x*,x0)L(s)ds, which completes the proof of the lemma.

We make the following assumption throughout the remainder of the paper: (44)x*Gsuch  that  f(x*)=0,dfx*-1  exists. Theorem 6 below gives an estimation of convergence ball of Newton’s method.

Theorem 6.

Suppose that dfx*-1df satisfies the L-average Lipschitz condition on Cr0(x*). Suppose that ϱ(x*,x0)<(b/2). Then the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to a zero point y* of f and ϱ(y*,x*)<r0.

Proof.

Since ϱ(x*,x0)<(b/2), there exist j1 and w1,,wj𝔤 satisfying (45)x0=x*·expw1expwj, and ρ(x*,x0):=i=1jwi<(b/2)r0, where the last inequality holds because of Remark 3(i). By Lemma 5, dfx0-1 exists and (46)β:=dfx0-1f(x0)0ρ(x*,x0)L(s)(ρ(x*,x0)-s)ds+ρ(x*,x0)1-0ρ(x*,x0)L(s)ds. Write (47)L-(s):=L(s+ρ(x*,x0))1-0ρ(x*,x0)L(s)ds. Let r-0,b- be such that (48)0r-0L-(s)ds=1,b-=0r-0L-(s)sds. This gives that (49)0r-0L(s+ρ(x*,x0))ds=1-0ρ(x*,x0)L(s)ds=0r0L(s)ds-0ρ(x*,x0)L(s)ds=ρ(x*,x0)r0L(s)ds. Hence, (50)ρ(x*,x0)r-0+ρ(x*,x0)L(s)ds=ρ(x*,x0)r0L(s)ds. As L(·) is a nondecreasing and positive integrable function, one has (51)r-0+ρ(x*,x0)=r0. Therefore, (52)b-=0r-0L-(s)sds=0r0-ρ(x*,x0)L-(s)sds=ρ(x*,x0)r0L(s)(s-ρ(x*,x0))ds1-0ρ(x*,x0)L(s)ds. Below, we will show that (53)β=dfx0-1f(x0)b-. To do this, by (46), it remains to show that (54)0ρ(x*,x0)L(s)(ρ(x*,x0)-s)ds+ρ(x*,x0)ρ(x*,x0)r0L(s)(s-ρ(x*,x0))ds; that is, (55)ρ(x*,x0)0r0L(s)(s-ρ(x*,x0))ds=0r0L(s)sds-ρ(x*,x0)0r0L(s)ds=b-ρ(x*,x0), which always holds because ρ(x*,x0)(b/2) by assumption. Hence, (53) is seen to hold.

Then in order to ensure that Proposition 4 is applicable, we have to show the following assertion: dfx0-1df satisfies the L--average Lipschitz condition in Cr-0(x0). To do this, let xCr-0(x0) be such that there exist v,v1,,vl𝔤 satisfying x=x0expv1expvl,ρ(x0,x):=i=1lvi and v+ρ(x0,x)<r-0. Since dfx*-1df satisfies the L-average Lipschitz condition in Cr0(x0) and (56)ρ(x*,x0)+v+ρ(x0,x)<ρ(x*,x0)+r-0=r0 thanks to (51), we obtain that (57)dfx*-1(dfxexpv-df(x))ρ(x*,x0)+ρ(x0,x)ρ(x*,x0)+ρ(x0,x)+vL(s)ds. Combining this with (34) yields that (58)dfx0-1(dfxexpv-df(x))dfx0-1dfx*dfx*-1(dfxexpu-df(x))11-0ρ(x*,x0)L(s)dsρ(x*,x0)+ρ(x0,x)ρ(x*,x0)+ρ(x0,x)+vL(s)ds=ρ(x0,x)ρ(x0,x)+vL(s+ρ(x*,x0))1-0ρ(x*,x0)L(s)dsds=ρ(x0,x)ρ(x0,x)+vL-(s)ds. Hence, dfx0-1df satisfies the L--average Lipschitz condition in Cr-0(x0). Thus, we apply Proposition 4 to conclude that the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges to a zero y* of f and ϱ(y*,x0)<r-0. And (59)ϱ(y*,x*)ϱ(y*,x0)+ϱ(x*,x0)ρ(x*,x0)+r-0=r0. The proof of the theorem is completed.

Theorem 6 gives an estimate of the convergence domain for Newton’s method. However, we do not know whether the limit y* of the sequence generated by Newton’s method with initial point x0 from this domain is equal to the zero x*. The following corollary provides the convergence domain from which the sequence generated by Newton’s method with initial point x0 converges to the zero x*. Recall that e designates the identity element of G.

Corollary 7.

Suppose that dfx*-1df satisfies the L-average Lipschitz condition on Cr0(x*). Suppose that Cr0(e)exp(B(0,r0)) and ϱ(x*,x0)<(b/2). Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.

Proof.

Since ϱ(x*,x0)<(b/2), we apply Theorem 6 to conclude that the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to a zero point y* of f and ϱ(y*,x*)<r0; that is, ϱ((x*)-1y*,e)<r0. Since Cr0(e)exp(B(0,r0)), there exists u𝔤 such that ur0 and (x*)-1y*=expu; that is, y*=x*expu. Hence, y*N(x*,r0):=x*exp(B(0,r0)). As r0ru by Remark 3(ii), Theorem 2 is applicable, and so y*=x*.

Recall that in the special case when G is a compact connected Lie group, G has a bi-invariant Riemannian metric (cf. [22, page 46]). Below, we assume that G is a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Therefore, an estimate of the convergence domain with the same property as in Corollary 7 is described in the following corollary.

Corollary 8.

Let G be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that dfx*-1df satisfies the L-average Lipschitz condition on Cr0(x*). Suppose that ϱ(x*,x0)<(b/2). Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.

Proof.

By Theorem 6, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges to a zero, say y*, of f with ϱ(x*,y*)<r0. Clearly, there is a minimizing geodesic c connecting x*-1·y* and e. Since G is a compact connected Lie group and endowed with a bi-invariant Riemannian metric, it follows from [20, page 224] that c is also a one-parameter subgroup of G. Consequently, there exists u𝔤 such that y*=x*·expu and u=ϱ(x*,y*)<r0. Hence, y*N(x*,r0):=x*exp(B(0,r0)). As r0r^ by Remark 3(ii), Theorem 2 is applicable, and so y*=x*.

5. Theorems under the Kantorovich Condition and the <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M353"><mml:mrow><mml:mi>γ</mml:mi></mml:mrow></mml:math></inline-formula>-Condition

This section is devoted to the study of some applications of the results obtained in the preceding sections. At first, if L(·) is a constant, then the L-average Lipschitz condition is reduced to the classical Lipschitz condition.

Let r>0,  x0G and let T be a mapping from G to (𝔤). Then T is said to satisfy the L Lipschitz condition on Cr(x0) if (60)T(x·expu)-T(x)Lu holds for any u,u0,,uk𝔤 and xCr(x0) such that x=x0expu0expu1expuk and u+ρ(x,x0)<r, where ρ(x,x0)=i=0kui.

Hence, in the case when L(·)L, we obtain from (22) and (29) that (61)r0=1L,b=12L,r^=2L. Thus, by Theorems 2 and 6, we have the following results, where Theorem 10 has been given in .

Theorem 9.

Let 0<r(2/L), Suppose that dfx*-1df satisfies the L Lipschitz condition in N(x*,r). Then x* is the unique zero point of f in N(x*,r).

Theorem 10.

Suppose that dfx*-1df satisfies the L Lipschitz condition on C1/L(x*). Suppose that ϱ(x*,x0)<(1/4L). Then the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to a zero y* of f and ϱ(y*,x*)<(1/L).

Furthermore, by Corollaries 7 and 8, one has the following results.

Corollary 11.

Suppose that dfx*-1df satisfies the L Lipschitz condition on C1/L(x*). Suppose that C1/L(e)exp(B(0,(1/L))) and ϱ(x*,x0)<(1/4L). Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.

Corollary 12.

Let G be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that dfx*-1df satisfies the L Lipschitz condition on C1/L(x*). Suppose that ϱ(x*,x0)<(1/4L). Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.

Let k be a positive integer, and assume further that f:G𝔤 is a Ck-map. Define the map dkfx:𝔤k𝔤 by (62)dkfxu1uk=(ktkt1f(x·exptkukexpt1u1))tk==t1=0, for each (u1,,uk)𝔤k. In particular, (63)dkfxuk=(dkdtkf(x·exptu))t=0for  each  u𝔤. Let 1ik. Then, in view of the definition, one has (64)dkfxu1uk=dk-i(dif·(u1ui))xui+1ukd11k-ifor  each  (u1,,uk)𝔤k. In particular, for fixed u1,,ui-1, ui+1,,uk𝔤, (65)difxu1ui-1(u)=d(di-1f(u1ui-1))(u)=d(di-1f(u1ui-1))(u)(u)for  each  u𝔤. This implies that difxu1ui-1u is linear with respect to u𝔤 and so is dkfxu1ui-1uui+1uk. Consequently, dkfx is a multilinear map from 𝔤k to 𝔤 because 1ik is arbitrary. Thus, we can define the norm of dkfx by (66)dkfx:=sup{ujdkfxu1u2uk(u1,,uk)𝔤kwith  each  uj=1dkfxu1u2ukuj}.

For the remainder of the paper, we always assume that f is a C2-map from G to 𝔤. Then taking i=2, we have (67)d2fzvu=d(df·v)zufor  any  u,v𝔤  and  each  zG. Thus, (17) is applied (with df·v in place of f(·) for each v𝔤) to conclude the following formula: (68)dfx·exp(tu)-dfx=0td2fx·exp(su)uds  dfx·exp(tu)-dfxfor  eachu  𝔤,t.

The γ-condition for nonlinear operators in Banach spaces was first introduced by Wang and Han , and explored further by Wang  to study Smale’s point estimate theory, which has been extended in  for a map f from a Lie group to its Lie algebra in view of the map d2f as given in Definition 13 below. Let r>0 and γ>0 be such that γr1.

Definition 13.

Let x0G be such that dfx0-1 exists. f is said to satisfy the γ-condition at x0 on Cr(x0) if, for any xCr(x0) with x=x0  expu0expu1expuk such that ρ(x,x0):=i=0kui<r, (69)dfx0-1d2fx2γ(1-γρ(x,x0))3.

As shown in Proposition 20, if f is analytic at x0, then f satisfies the γ-condition at x0.

Let γ>0 and let L be the function defined by (70)L(s)=2γ(1-γs)3for  each  0<s<1γ. The following proposition shows that the γ-condition implies the L-average Lipschitz condition, which is taken from .

Proposition 14.

Suppose that f satisfies the γ-condition at x0 on Cr(x0). Then dfx0-1df satisfies the L-average Lipschitz condition on Cr(x0) with L being defined by (70).

In the case when L is given by (70), we have from (22) and (29) that (71)r0=2-22γ,b=3-22γ,r^=12γ. Thus, by Theorems 2 and 6, we have the following results.

Theorem 15.

Let 0<r(1/2γ). Suppose that f satisfies the γ-condition in N(x*,r):=x*exp(B(0,r)). Then x* is the unique zero point of f in N(x*,r).

Theorem 16.

Suppose that f satisfies the γ-condition on C(2-2)/2γ(x*). Suppose that ϱ(x*,x0)<(3-22)/2γ. Then the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to a zero point y* of f and ϱ(y*,x*)<(2-2)/2γ.

Remark 17.

Theorem 16 improves the corresponding results in [19, Corollary 4.1], where it was proved under the following assumption: there exists v𝔤 such that x0=x*expv and v(a0/γ) with a0=0.081256 being the smallest positive root of the equation a0/(1-4a0+2a02)2=3-22. Clearly, (a0/γ)<(3-22)/2γ. Note also that in general, there dose not exist v𝔤 satisfying x0=x*expv because the exponential map is not surjective global, even if ϱ(x*,x0)<(3-22)/2γ. In view of this, our results somewhat improves the corresponding results in [19, Corollary 4.1].

Moreover, we get the following two corollaries from Corollaries 7 and 8.

Corollary 18.

Suppose that f satisfies the γ-condition on C(2-2)/2γ(x*). Suppose that C(2-2)/2γ(e)exp(B(0,(2-2)/2γ)) and ϱ(x*,x0)<(3-22)/2γ. Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.

Corollary 19.

Let G be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that f satisfies the γ-condition on C(2-2)/2γ(x*). Suppose that ϱ(x*,x0)<(3-22)/2γ. Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.

6. Applications to Analytic Maps

Throughout this section, we always assume that f is analytic on G. For xG such that dfx-1 exists, we define (72)γx:=γ(f,x)=supi2dfx-1difxi!1/(i-1). Also we adopt the convention that γ(f,x)= if dfx is not invertible. Note that this definition is justified, and, in the case when dfx is invertible, γ(f,x) is finite by analyticity.

The following proposition is taken from .

Proposition 20.

Let γx*:=γ(f,x*) and let r=(2-2)/2γx*. Then f satisfies the γx*-condition at x* on Cr(x*).

Thus, by Theorems 15 and 16 and Proposition 20, we have the following results.

Theorem 21.

Let 0<r(1/2γx*). Then x* is the unique zero point of f in N(x*,r).

Theorem 22.

Suppose that ϱ(x*,x0)<(3-22)/2γx*. Then the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to a zero point y* of f and ϱ(y*,x*)<(2-2)/2γx*.

Moreover, we get the following two corollaries from Corollaries 7 and 8 and Proposition 20.

Corollary 23.

Suppose that C(2-2)/2γx*(e)exp(B(0,(2-2)/2γx*)) and ϱ(x*,x0)<((3-22)/2γx*). Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.

Corollary 24.

Let G be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that ϱ(x*,x0)<(3-22)/2γx*. Then, the sequence {xn} generated by Newton’s method (28) with initial point x0 is well defined and converges quadratically to x*.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research of the second author was partially supported by the National Natural Science Foundation of China (Grant nos. 11001241 and 11371325) and by Zhejiang Provincial Natural Science Foundation of China (Grant no. LY13A010011). The research of the third author was partially supported by a Grant from NSC of Taiwan (NSC 102-2221-E-037-004-MY3).

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