Local Convergence of Newton ’ s Method on Lie Groups and Uniqueness Balls

and Applied Analysis 3 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 exp (w) = exp (u) ⋅ exp (V) , (10) for all u, V in a neighborhood of 0 ∈ g, where w is defined by


Introduction
In a vector space framework, when  is a differentiable operator from some domain  in a real or complex Banach space  to another , Newton's method is one of the most important methods for finding the approximation solution of the equation () = 0, which is formulated as follows: for any initial point  0 ∈ ,  +1 =   −   (  ) −1  (  ) ,  = 0, 1, . . . .
As is well known, one of the most important results on Newton's method is Kantorovich's theorem (cf.[1]), which provides a simple and clear criterion ensuring quadratic convergence of Newton's method under the mild condition that the second Fréchet derivative of  is bounded (or more generally, the first derivative is Lipschitz continuous) and the boundedness of ‖  () −1 ‖ on a proper open metric ball of the initial point  0 .Another important result on Newton's method is Smale's point estimate theory (i.e., -theory and -theory) in [2], where the notions of approximate zeros were introduced and the rules to judge an initial point  0 to be an approximate zero were established, depending on the information of the analytic nonlinear operator at this initial point and at a solution  * , 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, [3][4][5][6][7] and references therein.In particular, Wang introduced in [6] the notion of Lipschitz conditions with -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 [8] for Newton's method for vector fields on Riemannian manifolds while the extensions of the famous Smale -theory and theory in [2] to analytic vector fields and analytic mappings on Riemannian manifolds were done in [9].In the recent paper [10], the convergence criteria in [9] were improved by using the notion of the -condition for the vector fields and mappings on Riemannian manifolds.The radii of uniqueness balls of singular points of vector fields satisfying the conditions were estimated in [11], while the local behavior of Newton's method on Riemannian manifolds was studied in [12,13].Furthermore, in [14], Li and Wang extended the generalized -average Lipschitz condition (introduced in [6]) 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 [7] on Kantorovich's majorant method, Alvarez et al. introduced in [15] 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 [16], 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 [17] Newton's method, independent of affine connections on the Lie group, and showed the local quadratical convergence.Recently, Wang and Li [18] established Kantorovich's theorem (independent of the connection) for Newton's method on Lie group.More precisely, under the assumption that the differential of  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 [19].
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 -average Lipschitz condition.As applications, we obtain estimations of radius of convergence ball under the Kantorovich condition and the -condition, respectively.In particular, under the -condition, we get that (see Theorem 16) if ( * ) = 0 and then the sequence {  } generated by Newton's method (28) with initial point  0 is well defined and converges quadratically to a zero  * of .This improves the corresponding results in [19,Corollary 4.1], where it was proved under the following assumption: there exists V ∈ g such that with  0 = 0.081256 . . .being the smallest positive root of the equation Note also that in general, there dose not exist V ∈ g satisfying  0 =  * exp V because the exponential map is not surjective global, even if ( * ,  0 ) < (3 − 2 √ 2)/2.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 -condition are provided in Section 5.In the final section, we get the estimations of uniqueness ball and convergence ball under the assumption that  is analytic.

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 (, ⋅) 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 -dimensional.The symbol  designates the identity element of .Let g be the Lie algebra of the Lie group  which is the tangent space    of  at , equipped with Lie bracket [⋅, ⋅] : g × g → g.
In the sequel, we will make use of the left translation of the Lie group .We define for each  ∈  the left translation   :  →  by   () =  ⋅  for each  ∈ . ( The differential of   at  is denoted by (   )  which clearly determines a linear isomorphism from    to the tangent space  (⋅) .In particular, the differential (   )  of   at  determines a linear isomorphism form g to the tangent space   .The exponential map exp : g →  is certainly the most important construction associated with  and g and is defined as follows.Given  ∈ g, let   : R →  be the one-parameter subgroup of  determined by the left invariant vector field   :   → (   )  (); that is,   satisfies that The value of the exponential map exp at  is then defined by exp () =   (1) .
Moreover, we have that Note that the exponential map is not surjective in general.However, the exponential map is a diffeomorphism on an open neighborhood of 0 ∈ g.In the case when  is Abelian, exp is also a homomorphism from g to ; that is, 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 for all , V in a neighborhood of 0 ∈ g, where  is defined by Let  :  → g be a  1 -map and let  ∈ .We use    to denote the differential of  at .Then, by [22,Page 9] (the proof given there for a smooth mapping still works for a  1map), for each Δ  ∈    and any nontrivial smooth curve  : (−, ) →  with (0) =  and   (0) = Δ  , one has In particular, Define the linear map   : g → g by Then, by (13), Also, in view of definition, we have that, for all  ≥ 0,    ( ⋅ exp ()) =  ⋅exp()  for each  ∈ g, ( 16) 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  which plays a key role in the study.Let ,  ∈  and define where we adapt the convention that inf 0 = +∞.It is easy to verify that (⋅, ⋅) is a distance on  and that the topology induced by this distance is equivalent to the original one on .
Let  ∈  and  > 0. We denoted the corresponding ball of radius  around  of  by   (); that is, Let L(g) denote the set of all linear operators on g.Below, we will modify the notion of the Lipschitz condition with -average for mappings on Banach spaces to suit sections. Let holds for any  ∈   ( 0 ) and  ∈ g such that ‖‖ < −(,  0 ).
Proof.Let  * ∈ ( * , ) be another zero point of  in ( * , ).Then, there exists V ∈ g such that  * =  * exp V and ‖V‖ < .As (⋅) is a positive function, it follows from [6] that the function  defined by is strictly monotonically increasing.set Then, by (22), we get To complete the proof, it suffices to show that ‖V‖ ≤  ‖V‖ .
Granting this, one has that  * =  * .Now, where the third inequality holds because of ( 20) by selecting  =  0 =  * .Therefore, ( 26) is seen to hold and the proof is completed

Convergence Ball of Newton's Method
Following [17], we define Newton's method with initial point  0 for  on a Lie group as follows: Let  0 > 0 and  > 0 be such that Remark 3. (i) Since (⋅) is a positive function, we always have  ≤  0 .Indeed, (ii) Consider  0 ≤ r.Indeed, recall from [6] that the function  defined by is strictly monotonically increasing.Sine ( 0 ) ≤ 1 = (r), we get  0 ≤ r.
The following proposition plays a key role in this section, which is taken from [24].Proposition 4. Suppose that  0 ∈  is such that  −1  0 exists and  −1  0  satisfies the -average Lipschitz condition on   0 ( 0 ) and that Then the sequence {  } generated by Newton's method (28) with initial point  0 is well defined and converges to a zero point  * of  and ( * ,  0 ) <  0 .
The remainder of this section is devoted to an estimate of the convergence domain of Newton's method on  around a zero  * of .Below we will always assume that  * ∈  is such that  −1  * exists.
We make the following assumption throughout the remainder of the paper: Theorem 6 below gives an estimation of convergence ball of Newton's method.
Theorem 6 gives an estimate of the convergence domain for Newton's method.However, we do not know whether the limit  * of the sequence generated by Newton's method with initial point  0 from this domain is equal to the zero  * .The following corollary provides the convergence domain from which the sequence generated by Newton's method with initial point  0 converges to the zero  * .Recall that  designates the identity element of .
Recall that in the special case when  is a compact connected Lie group,  has a bi-invariant Riemannian metric (cf.[22, page 46]).Below, we assume that  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  be a compact connected Lie group and endowed with a bi-invariant Riemannian metric.Suppose that  −1  *  satisfies the -average Lipschitz condition on   0 ( * ).Suppose that ( * ,  0 ) < (/2).Then, the sequence {  } generated by Newton's method (28) with initial point  0 is well defined and converges quadratically to  * .Proof.By Theorem 6, the sequence {  } generated by Newton's method (28) with initial point  0 is well defined and converges to a zero, say  * , of  with ( * ,  * ) <  0 .Clearly, there is a minimizing geodesic  connecting  * −1 ⋅  * and .Since  is a compact connected Lie group and endowed with a bi-invariant Riemannian metric, it follows from [20, page 224] that  is also a one-parameter subgroup of .

Theorems under the Kantorovich Condition and the 𝛾-Condition
This section is devoted to the study of some applications of the results obtained in the preceding sections.At first, if (⋅) is a constant, then the -average Lipschitz condition is reduced to the classical Lipschitz condition.
Furthermore, by Corollaries 7 and 8, one has the following results.
Let  be a positive integer, and assume further that  :  → g is a   -map.Define the map     : g  → g by      1 ⋅ ⋅ ⋅   = (     ⋅ ⋅ ⋅  1  ( ⋅ exp     ⋅ ⋅ ⋅ exp  1  1 )) for each ( 1 , . . .,   ) ∈ g  .In particular, For the remainder of the paper, we always assume that  is a  2 -map from  to g.Then taking  = 2, we have  2   V = ( ⋅ V)   for any , V ∈ g and each  ∈ .
be a positive nondecreasing integrable function on [0, ], where  is a positive number large enough such that ∫  0 ( − )() ≥ .The notion of Lipschitz condition in the inscribed sphere with the  average for operators on Banach spaces was first introduced in[23]by Wang for the study of Smale's point estimate theory.