Volume 2013, Issue 1 367161

Research Article

Open Access

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

Jinsu He,

Corresponding Author

Jinsu He

[email protected]

orcid.org/0000-0002-3851-3181

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China zjnu.edu.cn

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Jinhua Wang,

Jinhua Wang

Department of Mathematics, Zhejiang University of Technology, Hangzhou 310032, China zjut.edu.cn

Department of Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan nsysu.edu.tw

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Jen-Chih Yao,

Jen-Chih Yao

Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan kmu.edu.tw

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

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Jinsu He,

Corresponding Author

Jinsu He

[email protected]

orcid.org/0000-0002-3851-3181

Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China zjnu.edu.cn

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Jinhua Wang,

Jinhua Wang

Department of Mathematics, Zhejiang University of Technology, Hangzhou 310032, China zjut.edu.cn

Department of Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan nsysu.edu.tw

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Jen-Chih Yao,

Jen-Chih Yao

Center for Fundamental Science, Kaohsiung Medical University, Kaohsiung 80702, Taiwan kmu.edu.tw

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

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First published: 13 November 2013

https://doi.org/10.1155/2013/367161

Citations: 1

Academic Editor: Antonio M. Peralta

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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 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 x₀ ∈ D,

()

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 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 [2], 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, [3–7] and references therein. In particular, Wang introduced in [6] 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 [8] for Newton’s method for vector fields on Riemannian manifolds while the extensions of the famous Smale α-theory and γ-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 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 [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 L-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 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 [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 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

()

then the sequence {x_n} 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 v ∈ 𝔤 such that

()

with a₀ = 0.081256… being the smallest positive root of the equation

. Clearly,

()

Note also that in general, there dose not exist v ∈ 𝔤 satisfying x₀ = x^*exp v 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 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 T_eG of G at e, equipped with Lie bracket [·, ·] : 𝔤 × 𝔤 → 𝔤.

In the sequel, we will make use of the left translation of the Lie group G. We define for each y ∈ G the left translation L_y : G → G by

()

The differential of L_y at z is denoted by

which clearly determines a linear isomorphism from T_zG to the tangent space T_(y·z)G. In particular, the differential

of L_y at e determines a linear isomorphism form 𝔤 to the tangent space T_yG. The exponential map exp : 𝔤 → G is certainly the most important construction associated with G and 𝔤 and is defined as follows. Given u ∈ 𝔤, let σ_u : ℝ → G be the one-parameter subgroup of G determined by the left invariant vector field

; that is, σ_u satisfies that

()

The value of the exponential map exp at u 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 0 ∈ 𝔤. In the case when G is Abelian, exp is also a homomorphism from 𝔤 to G; 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 u, v in a neighborhood of 0 ∈ 𝔤, where w is defined by

()

Let f : G → 𝔤 be a C¹-map and let x ∈ G. We use

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 Δ_x ∈ T_xG and any nontrivial smooth curve c : (−ε, ε) → G with c(0) = x and c^′(0) = Δ_x, one has

()

In particular,

()

Define the linear map df_x : 𝔤 → 𝔤 by

()

Then, by (13),

()

Also, in view of definition, we have that, for all t ≥ 0,

()

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, y ∈ G and define

()

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 x ∈ G and r > 0. We denoted the corresponding ball of radius r around x of G by C_r(x); 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 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

. The notion of Lipschitz condition in the inscribed sphere with the L 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 r > 0, x₀ ∈ G and let T be a mapping from G to ℒ(𝔤). Then T is said to satisfy the L-average Lipschitz condition on C_r(x₀) if

()

holds for any x ∈ C_r(x₀) and u ∈ 𝔤 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,

()

Write N(x^*, r): = x^*exp (B(0, r)). Clearly, N(x^*, r)⊆C_r(x^*). Let

be such that

()

Theorem 2. Let . Suppose that f(x^*) = 0 and 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 v ∈ 𝔤 such that y^* = x^*exp v and ∥v∥ < r. As L(·) is a positive function, it follows from [6] that the function ψ 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 x^* = y^*. Now,

()

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 [17], we define Newton’s method with initial point x₀ for f on a Lie group as follows:

()

Let r₀ > 0 and b > 0 be such that

()

Remark 3. (i) Since L(·) is a positive function, we always have b ≤ r₀. Indeed,

()

(ii) Consider . Indeed, recall from [6] that the function ψ 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 x₀ ∈ G is such that exists and satisfies the L-average Lipschitz condition on and that

()

Then the sequence {x_n} 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 exists.

Lemma 5. Let 0 < r ≤ r₀ and let x₀ ∈ C_r(x^*) be such that there exist j ≥ 1 and w₁, …, w_j ∈ 𝔤 satisfying

()

and

. Suppose that

satisfies the L-average Lipschitz condition on C_r(x^*). Then

exists,

()

Proof. It follows from [24, Lemma 2.1] that exists and (34) holds. Write y₀ = x^*, y_i = y_i−1 · exp w_i and for each i = 1, …, j. Thus, by (33), we have y_j = x₀ and so ρ_j = ρ(x^*, x₀). Fix i, one has from (17) that

()

which implies that

()

Since

satisfies the L-average Lipschitz condition on C_r(x^*), it follows that

()

where ρ₀ = 0. 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 L-average Lipschitz condition on . Suppose that ϱ(x^*, x₀)<(b/2). Then the sequence {x_n} 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 j ≥ 1 and w₁, …, w_j ∈ 𝔤 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 L(·) 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 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: satisfies the -average Lipschitz condition in . To do this, let be such that there exist v, v₁, …, v_l ∈ 𝔤 satisfying and . Since satisfies the L-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 {x_n} generated by Newton’s method (28) with initial point x₀ is well defined and converges to a zero y^* of f 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 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 satisfies the L-average Lipschitz condition on . Suppose that and ϱ(x^*, x₀)<(b/2). Then, the sequence {x_n} 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 {x_n} 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 , there exists u ∈ 𝔤 such that ∥u∥ ≤ r₀ and (x^*) ⁻¹y^* = exp u; that is, y^* = x^*exp u. Hence, y^* ∈ N(x^*, r₀): = x^*exp (B(0, r₀)). As r₀ ≤ r_u 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 satisfies the L-average Lipschitz condition on . Suppose that ϱ(x^*, x₀)<(b/2). Then, the sequence {x_n} generated by Newton’s method (28) with initial point x₀ is well defined and converges quadratically to x^*.

Proof. By Theorem 6, the sequence {x_n} 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 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 u ∈ 𝔤 such that y^* = x^* · exp u and ∥u∥ = ϱ(x^*, y^*) < r₀. Hence, y^* ∈ N(x^*, r₀): = x^*exp (B(0, r₀)). As by Remark 3(ii), Theorem 2 is applicable, and so y^* = x^*.

5. 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 L(·) is a constant, then the L-average Lipschitz condition is reduced to the classical Lipschitz condition.

Let r > 0, x₀ ∈ G and let T be a mapping from G to ℒ(𝔤). Then T is said to satisfy the L Lipschitz condition on C_r(x₀) if

()

holds for any u, u₀, …, u_k ∈ 𝔤 and x ∈ C_r(x₀) such that x = x₀exp u₀exp u₁ ⋯ exp u_k and ∥u∥ + ρ(x, x₀) < r, where

Hence, in the case when L(·) ≡ L, 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 0 < r ≤ (2/L), Suppose that 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 satisfies the L Lipschitz condition on C_1/L(x^*). Suppose that ϱ(x^*, x₀)<(1/4L). Then the sequence {x_n} 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 satisfies the L Lipschitz condition on C_1/L(x^*). Suppose that C_1/L(e)⊆exp (B(0, (1/L))) and ϱ(x^*, x₀)<(1/4L). Then, the sequence {x_n} 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 satisfies the L Lipschitz condition on C_1/L(x^*). Suppose that ϱ(x^*, x₀)<(1/4L). Then, the sequence {x_n} 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 : G → 𝔤 is a C^k-map. Define the map d^kf_x : 𝔤^k → 𝔤 by

()

for each (u₁, …, u_k) ∈ 𝔤^k. In particular,

()

Let 1 ≤ i ≤ k. Then, in view of the definition, one has

()

In particular, for fixed u₁, …, u_i−1, u_i+1, …, u_k ∈ 𝔤,

()

This implies that dⁱf_xu₁ ⋯ u_i−1u is linear with respect to u ∈ 𝔤 and so is d^kf_xu₁ ⋯ u_i−1uu_i+1 ⋯ u_k. Consequently, d^kf_x is a multilinear map from 𝔤^k to 𝔤 because 1 ≤ i ≤ k is arbitrary. Thus, we can define the norm of d^kf_x by

()

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

()

Thus, (17) is applied (with df · v in place of f(·) for each v ∈ 𝔤) to conclude the following formula:

()

The γ-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 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 γr ≤ 1.

Definition 13. Let x₀ ∈ G be such that exists. f is said to satisfy the γ-condition at x₀ on C_r(x₀) if, for any x ∈ C_r(x₀) with x = x₀ exp u₀ exp u₁ ⋯ exp u_k such that ,

()

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

()

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

Proposition 14. Suppose that f satisfies the γ-condition at x₀ on C_r(x₀). Then satisfies the L-average Lipschitz condition on C_r(x₀) with L being defined by (70).

In the case when L 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 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 . Suppose that . Then the sequence {x_n} generated by Newton’s method (28) with initial point x₀ is well defined and converges quadratically to a zero point y^* of f and .

Remark 17. Theorem 16 improves the corresponding results in [19, Corollary 4.1], where it was proved under the following assumption: there exists v ∈ 𝔤 such that x₀ = x^*exp v and ∥v∥ ≤ (a₀/γ) with a₀ = 0.081256… being the smallest positive root of the equation . Clearly, . Note also that in general, there dose not exist v ∈ 𝔤 satisfying x₀ = x^*exp v 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 f satisfies the γ-condition on . Suppose that and . Then, the sequence {x_n} 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 . Suppose that . Then, the sequence {x_n} 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 x ∈ G such that

exists, we define

()

Also we adopt the convention that γ(f, x) = ∞ if df_x is not invertible. Note that this definition is justified, and, in the case when df_x is invertible, γ(f, x) is finite by analyticity.

The following proposition is taken from [19].

Proposition 20. Let and let . Then f satisfies the -condition at x^* on C_r(x^*).

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

Theorem 21. Let . Then x^* is the unique zero point of f in N(x^*, r).

Theorem 22. Suppose that . Then the sequence {x_n} generated by Newton’s method (28) with initial point x₀ is well defined and converges quadratically to a zero point y^* of f and .

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

Corollary 23. Suppose that and . Then, the sequence {x_n} 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 . Then, the sequence {x_n} 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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Local Convergence of Newton’s Method on Lie Groups and Uniqueness Balls

Abstract

1. Introduction

2. Notions and Preliminaries

3. Uniqueness Ball of Zero Points of Mappings

4. Convergence Ball of Newton’s Method

5. Theorems under the Kantorovich Condition and the γ-Condition

6. Applications to Analytic Maps

Conflict of Interests

Acknowledgments

References

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

Abstract

1. Introduction

2. Notions and Preliminaries

3. Uniqueness Ball of Zero Points of Mappings

4. Convergence Ball of Newton’s Method

5. Theorems under the Kantorovich Condition and the γ-Condition

6. Applications to Analytic Maps

Conflict of Interests

Acknowledgments

References

Citing Literature

References

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