Abstract and Applied Analysis

Volume 2013 (2013), Article ID 367161, 9 pages

http://dx.doi.org/10.1155/2013/367161

Research Article

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

Jinsu He,¹ Jinhua Wang,2,3 and Jen-Chih Yao4,5

¹Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

²Department of Mathematics, Zhejiang University of Technology, Hangzhou 310032, China

³Department of Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan

⁴Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan

⁵Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 20 August 2013; Accepted 1 October 2013

Academic Editor: Antonio M. Peralta

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.

Abstract.

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 -condition, respectively. In particular, under the -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 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 , which is formulated as follows: for any initial point ,

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 on a proper open metric ball of the initial point . 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 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, [3–7] 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 and

then the sequence generated by Newton’s method (28) with initial point 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 such that

with being the smallest positive root of the equation . Clearly,

Note also that in general, there dose not exist satisfying because the exponential map is not surjective global, even if . 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.

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 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 be the Lie algebra of the Lie group which is the tangent space of at , equipped with Lie bracket .

In the sequel, we will make use of the left translation of the Lie group . We define for each the left translation by

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 to the tangent space . The exponential map is certainly the most important construction associated with and and is defined as follows. Given , let be the one-parameter subgroup of determined by the left invariant vector field ; that is, satisfies that

The value of the exponential map at is then defined by

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 . In the case when is Abelian, is also a homomorphism from to ; that is,

In the non-Abelian case, is not a homomorphism and, by the Baker-Campbell-Hausdorff (BCH) formula (cf. [21, page 114]), (9) must be replaced by

for all in a neighborhood of , where is defined by

Let be a -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 -map), for each and any nontrivial smooth curve with and , one has

In particular,

Define the linear map by

Then, by (13),

Also, in view of definition, we have that, for all ,

For the remainder of the present paper, we always assume that is an inner product on and is the associated norm on . We now introduce the following distance on which plays a key role in the study. Let and define

where we adapt the convention that . 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 . We denoted the corresponding ball of radius around of by ; that is,

Let denote the set of all linear operators on . Below, we will modify the notion of the Lipschitz condition with -average for mappings on Banach spaces to suit sections. Let be a positive nondecreasing integrable function on , where is a positive number large enough such that . 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.

Definition 1. Let ,   and let be a mapping from to . Then is said to satisfy the -average Lipschitz condition on if

holds for any and such that .

3. Uniqueness Ball of Zero Points of Mappings

This section is devoted to the study of uniqueness ball of zero points of mappings. Let . We use to denote the open ball at with radius on ; that is,

Write . Clearly, . Let be such that

Theorem 2. Let . Suppose that and satisfies the -average Lipschitz condition in . Then is the unique zero point of in .

Proof. Let be another zero point of in . Then, there exists such that and . 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

Granting this, one has that . Now,

where the third inequality holds because of (20) by selecting . Therefore, (26) is seen to hold and the proof is completed

4. Convergence Ball of Newton’s Method

Following [17], we define Newton’s method with initial point for on a Lie group as follows:

Let and be such that

Remark 3. (i) Since is a positive function, we always have . Indeed,

(ii) Consider . Indeed, recall from [6] that the function defined by

is strictly monotonically increasing. Sine , we get .

The following proposition plays a key role in this section, which is taken from [24].

Proposition 4. Suppose that is such that exists and satisfies the -average Lipschitz condition on and that

Then the sequence generated by Newton’s method (28) with initial point is well defined and converges to a zero point of and .

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 exists.

Lemma 5. Let and let be such that there exist and satisfying

and . Suppose that satisfies the -average Lipschitz condition on . Then exists,

Proof. It follows from [24, Lemma 2.1] that exists and (34) holds. Write ,  and for each . Thus, by (33), we have and so . Fix , one has from (17) that

which implies that

Since satisfies the -average Lipschitz condition on , it follows that

where . Noting that , we have from (37) and (38) that

Write . Then,

and so

This, together with (39), yields that

Combining this with (34) implies that

which completes the proof of the lemma.

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. Suppose that satisfies the -average Lipschitz condition on . Suppose that . Then the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to a zero point of and .

Proof. Since , there exist and satisfying

and , where the last inequality holds because of Remark 3(i). By Lemma 5, exists and

Write

Let be such that

This gives that

Hence,

As is a nondecreasing and positive integrable function, one has

Therefore,

Below, we will show that

To do this, by (46), it remains to show that

that is,

which always holds because 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: satisfies the -average Lipschitz condition in . To do this, let be such that there exist satisfying and . Since satisfies the -average Lipschitz condition in and

thanks to (51), we obtain that

Combining this with (34) yields that

Hence, satisfies the -average Lipschitz condition in . Thus, we apply Proposition 4 to conclude that the sequence generated by Newton’s method (28) with initial point is well defined and converges to a zero of and . And

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 of the sequence generated by Newton’s method with initial point 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 converges to the zero . Recall that designates the identity element of .

Corollary 7. Suppose that satisfies the -average Lipschitz condition on . Suppose that and . Then, the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to .

Proof. Since , we apply Theorem 6 to conclude that the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to a zero point of and ; that is, . Since , there exists such that and ; that is, . Hence, . As by Remark 3(ii), Theorem 2 is applicable, and so .

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 satisfies the -average Lipschitz condition on . Suppose that . Then, the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to .

Proof. By Theorem 6, the sequence generated by Newton’s method (28) with initial point is well defined and converges to a zero, say , of with . Clearly, there is a minimizing geodesic connecting 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 . Consequently, there exists such that and . Hence, . As by Remark 3(ii), Theorem 2 is applicable, and so .

5. Theorems under the Kantorovich Condition and the -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.

Let ,   and let be a mapping from to . Then is said to satisfy the Lipschitz condition on if

holds for any and such that and , where .

Hence, in the case when , we obtain from (22) and (29) that

Thus, by Theorems 2 and 6, we have the following results, where Theorem 10 has been given in [18].

Theorem 9. Let , Suppose that satisfies the Lipschitz condition in . Then is the unique zero point of in .

Theorem 10. Suppose that satisfies the Lipschitz condition on . Suppose that . Then the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to a zero of and .

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

Corollary 11. Suppose that satisfies the Lipschitz condition on . Suppose that and . Then, the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to .

Corollary 12. Let be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that satisfies the Lipschitz condition on . Suppose that . Then, the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to .

Let be a positive integer, and assume further that is a -map. Define the map by

for each . In particular,

Let . Then, in view of the definition, one has

In particular, for fixed , ,

This implies that is linear with respect to and so is . Consequently, is a multilinear map from to because is arbitrary. Thus, we can define the norm of by

For the remainder of the paper, we always assume that is a -map from to . Then taking , we have

Thus, (17) is applied (with in place of for each ) to conclude the following formula:

The -condition for nonlinear operators in Banach spaces was first introduced by Wang and Han [25], and explored further by Wang [26] to study Smale’s point estimate theory, which has been extended in [19] for a map from a Lie group to its Lie algebra in view of the map as given in Definition 13 below. Let and be such that .

Definition 13. Let be such that exists. is said to satisfy the -condition at on if, for any with such that ,

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

Let and let be the function defined by

The following proposition shows that the -condition implies the -average Lipschitz condition, which is taken from [24].

Proposition 14. Suppose that satisfies the -condition at on . Then satisfies the -average Lipschitz condition on with being defined by (70).

In the case when is given by (70), we have from (22) and (29) that

Thus, by Theorems 2 and 6, we have the following results.

Theorem 15. Let . Suppose that satisfies the -condition in . Then is the unique zero point of in .

Theorem 16. Suppose that satisfies the -condition on . Suppose that . Then the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to a zero point of and .

Remark 17. Theorem 16 improves the corresponding results in [19, Corollary 4.1], where it was proved under the following assumption: there exists such that and with being the smallest positive root of the equation . Clearly, . Note also that in general, there dose not exist satisfying because the exponential map is not surjective global, even if . 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 satisfies the -condition on . Suppose that and . Then, the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to .

Corollary 19. Let be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that satisfies the -condition on . Suppose that . Then, the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to .

6. Applications to Analytic Maps

Throughout this section, we always assume that is analytic on . For such that exists, we define

Also we adopt the convention that if is not invertible. Note that this definition is justified, and, in the case when is invertible, is finite by analyticity.

The following proposition is taken from [19].

Proposition 20. Let and let . Then satisfies the -condition at on .

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

Theorem 21. Let . Then is the unique zero point of in .

Theorem 22. Suppose that . Then the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to a zero point of and .

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

Corollary 23. Suppose that and . Then, the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to .

Corollary 24. Let be a compact connected Lie group and endowed with a bi-invariant Riemannian metric. Suppose that . Then, the sequence generated by Newton’s method (28) with initial point is well defined and converges quadratically to .

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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