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Convergence criteria of Newton’s method on Lie groups

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Abstract

In the present paper, we study Newton’s method on Lie groups (independent of affine connections) for finding zeros of a mapping f from a Lie group to its Lie algebra. Under a generalized L-average Lipschitz condition of the differential of f, we establish a unified convergence criterion of Newton’s method. As applications, we get the convergence criteria under the Kantorovich’s condition and the γ-condition, respectively. Moreover, applications to optimization problems are also provided.

MSC:65H10, 65D99.

1 Introduction

Newton’s method is one of the most important methods for finding the approximation solution of the equation f(x)=0, where f is an operator from some domain D in a real or complex Banach space X to another Y. As is well known, one of the most important results on Newton’s method is Kantorovich’s theorem (cf. [1]). Under the mild condition that the second Fréchet derivative of F is bounded (or more general, the first derivative is Lipschitz continuous) on a proper open metric ball of the initial point x 0 , Kantorovich’s theorem provides a simple and clear criterion ensuring the quadratic convergence of Newton’s method. 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 x 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 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, [37] and references therein. In particular, Zabrejko-Nguen parametrized in [7] the classical Lipschitz continuity. 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’s theorem was given in [8] for Newton’s method for vector fields on Riemannian manifolds while the extensions of the famous Smale’s 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 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 established 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 solving 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 the 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 establish a unified convergence criterion for Newton’s method (independent of the connection) on Lie groups under a generalized L-average Lipschitz condition. As applications, we get the convergence criteria under the Kantorovich’s condition and the γ-condition, respectively. Hence, our results extend the corresponding results in [18] and [19], respectively. Moreover, applications to optimization problems are also provided.

The remainder of the paper is organized as follows. Some preliminary results and notions are given in Section 2, while the main results about a unified convergence criterion are presented in Section 3. In Section 4, applications to optimization problems are explored. Theorems under the Kantorovich’s condition and the γ-condition are provided in the final section.

2 Notions and preliminaries

Most of the notions and notations which are used in the present paper are standard; see, for example, [20, 21]. The 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 shall 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 e G of G at e, equipped with Lie bracket [,]:G×GG.

In the sequel we make use of the left translation of the Lie group G. We define, for each yG, the left translation L y :GG by

L y (z)=yzfor each zG.
(2.1)

The differential of L y at z is denoted by ( L y ) z , which clearly determines a linear isomorphism from T z G to the tangent space T ( y z ) G. In particular, the differential ( L y ) e of L y at e determines a linear isomorphism from G to the tangent space T y G. The exponential map exp:GG is certainly the most important construction associated to G and G, and is defined as follows. Given uG, let σ u :RG be a one-parameter subgroup of G determined by the left invariant vector field X u :y ( L y ) e (u); i.e., σ u satisfies that

σ u (0)=eand σ u (t)= X u ( σ u ( t ) ) = ( L σ u ( t ) ) e (u)for each tR.
(2.2)

The value of the exponential map exp at u is then defined by

exp(u)= σ u (1).

Moreover, we have that

exp(tu)= σ t u (1)= σ u (t)for each tR and uG
(2.3)

and

exp(t+s)u=exp(tu)exp(su)for any t,sR and uG.
(2.4)

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, i.e.,

exp(u+v)=exp(u)exp(v)for all u,vG.
(2.5)

In the non-abelian case, exp is not a homomorphism and, by the Baker-Campbell-Hausdorff (BCH) formula (cf. [[21], p.114]), (2.5) must be replaced by

exp(w)=exp(u)exp(v)
(2.6)

for all u,v in a neighborhood of 0G, where w is defined by

w:=u+v+ 1 2 [u,v]+ 1 12 ( [ u , [ u , v ] ] + [ v , [ v , u ] ] ) +.
(2.7)

Let f:GG be a C 1 -map and let xG. We use f x to denote the differential of f at x. Then, by [[22], p.9] (the proof given there for a smooth mapping still works for a C 1 -map), for each x T x G and any nontrivial smooth curve c:(ε,ε)G with c(0)=x and c (0)= x , one has that

f x x = ( d d t ( f c ) ( t ) ) t = 0 .
(2.8)

In particular,

f x x = ( d d t f ( x exp ( t ( L x 1 ) x x ) ) ) t = 0 for each  x T x G.
(2.9)

Define the linear map d f x :GG by

d f x u= ( d d t f ( x exp ( t u ) ) ) t = 0 for each uG.
(2.10)

Then, by (2.9),

d f x = f x ( L x ) e .
(2.11)

Also, in view of the definition, we have that for all t0,

d d t f ( x exp ( t u ) ) =d f x exp ( t u ) ufor each uG
(2.12)

and

f ( x exp ( t u ) ) f(x)= 0 t d f x exp ( s u ) udsfor each uG.
(2.13)

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

ϱ(x,y):=inf { i = 1 k u i | there exist  k 1  and  u 1 , , u k G  such that y = x exp u 1 exp u k } ,
(2.14)

where we adapt the convention that inf=+. It is easy to verify that ϱ(,) is a distance on G and the topology induced by this distance is equivalent to the original one on G.

Let xG and r>0. We denote the corresponding ball of radius r around x of G by C r (x), that is,

C r (x):= { y G | ϱ ( x , y ) < r } .

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 0 R (Rs)L(s)dsR. The notion of Lipschitz condition in the inscribed sphere with the L average for operators from Banach spaces to Banach spaces was first introduced in [23] by Wang for the study of Smale’s point estimate theory.

Definition 2.1 Let r>0, x 0 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 ( x 0 ) if

T ( x exp u ) T ( x ) ρ ( x 0 , x ) ρ ( x 0 , x ) + u L(s)ds
(2.15)

holds for any u, u 0 ,, u k G and x C r ( x 0 ) such that x= x 0 exp u 0 exp u 1 exp u k and u+ρ(x, x 0 )<r, where ρ(x, x 0 ):= i = 0 k u i .

The majorizing function h defined in the following, which was first introduced and studied by Wang (cf. [23]), is a powerful tool in our study. Let r 0 >0 and b>0 be such that

0 r 0 L(s)ds=1andb= 0 r 0 L(s)sds.
(2.16)

For β>0, define the majorizing function h by

h(t)=βt+ 0 t L(s)(ts)dsfor each 0tR.
(2.17)

Some useful properties are described in the following propositions, see [23].

Proposition 2.1 The function h is monotonic decreasing on [0, r 0 ] and monotonic increasing on [ r 0 ,R]. Moreover, if βb, h has a unique zero respectively in [0, r 0 ] and [ r 0 ,R], which are denoted by r 1 and r 2 .

Let { t n } denote the sequence generated by Newton’s method with initial value t 0 =0 for h, that is,

t n + 1 = t n h ( t n ) 1 h( t n )for each n=0,1,.
(2.18)

Proposition 2.2 Suppose that βb. Then the sequence { t n } generated by (2.18) is monotonic increasing and convergent to r 1 .

The following lemma will be useful in the proof of the main theorem.

Lemma 2.1 Let 0<r r 0 and let x 0 G be such that d f x 0 1 exists. Suppose that d f x 0 1 df satisfies the L-average Lipschitz condition on C r ( x 0 ). Let x C r ( x 0 ) be such that there exist k1 and u 0 ,, u k G satisfying x= x 0 exp u 0 exp u k and ρ(x, x 0 ):= i = 0 k u i <r. Then d f x 1 exists and

d f x 1 d f x 0 1 1 0 ρ ( x , x 0 ) L ( s ) d s .
(2.19)

Proof Write y 0 = x 0 and y i + 1 = y i exp u i for each i=0,,k. Since (2.15) holds with T=d f x 0 1 df, one has that

d f x 0 1 ( d f y i exp u i d f y i ) ρ ( y i , x 0 ) ρ ( y i + 1 , x 0 ) L(s)dsfor each 0ik.
(2.20)

Noting that y k + 1 =x, we have that

Thus the conclusion follows from the Banach lemma and the proof is complete. □

3 Convergence criteria

Following [17], we define Newton’s method with initial point x 0 for f on a Lie group as follows:

x n + 1 = x n exp ( d f x n 1 f ( x n ) ) for each n=0,1,.
(3.1)

Recall that f:GG is a C 1 -mapping. In the remainder of this section, we always assume that x 0 G is such that d f x 0 1 exists and set β:=d f x 0 1 f( x 0 ). Let r 0 and b given by (2.16), and r 1 be given by Proposition 2.1.

Theorem 3.1 Suppose that d f x 0 1 df satisfies the L-average Lipschitz condition on C r 1 ( x 0 ) and that

β= d f x 0 1 f ( x 0 ) b.
(3.2)

Then the sequence { x n } generated by Newton’s method (3.1) with initial point x 0 is well defined and converges to a zero x of f. Moreover, the following assertions hold for each n=0,1, :

ϱ( x n + 1 , x n ) d f x n 1 f ( x n ) t n + 1 t n ;
(3.3)
ϱ ( x n , x ) r 1 t n .
(3.4)

Proof Write v n =d f x n 1 f( x n ) for each n=0,1, . Below we shall show that each v n is well defined and

ϱ( x n + 1 , x n ) v n t n + 1 t n
(3.5)

holds for each n=0,1, . Granting this, one sees that the sequence { x n } generated by Newton’s method (3.1) with initial point x 0 is well defined and converges to a zero x of f, because, by (3.1),

x n + 1 = x n exp v n for each n=0,1,.

Furthermore, assertions (3.3) and (3.4) hold for each n and the proof of the theorem is completed.

Note that v 0 is well defined by assumption and x 1 = x 0 exp v 0 . Hence, ϱ( x 1 , x 0 ) v 0 . Since v 0 =d f x 0 1 (f( x 0 ))=β= t 1 t 0 , it follows that (3.5) is true for n=0. We now proceed by mathematical induction on n. For this purpose, assume that v n is well defined and (3.5) holds for each nk1. Then

i = 0 k 1 v i t k t 0 = t k < r 1 and x k = x 0 exp v 0 exp v k 1 .
(3.6)

Thus, we use Lemma 2.1 to conclude that d f x k 1 exists and

d f x k 1 d f x 0 1 1 0 t k L ( s ) d s = h ( t k ) 1 .
(3.7)

Therefore, v k is well defined. Observe that

f ( x k ) = f ( x k ) f ( x k 1 ) d f x k 1 v k 1 = 0 1 d f x k 1 exp ( t v k 1 ) v k 1 d t d f x k 1 v k 1 = 0 1 [ d f x k 1 exp ( t v k 1 ) d f x k 1 ] v k 1 d t ,

where the second equality is valid because of (2.13). Therefore, applying (2.15), one has that

d f x 0 1 f ( x k ) 0 1 d f x 0 1 [ d f x k 1 exp ( t v k 1 ) d f x k 1 ] v k 1 d t 0 1 ρ ( x k 1 , x 0 ) ρ ( x k 1 , x 0 ) + t v k 1 L ( s ) d s v k 1 d t 0 1 t k 1 t k 1 + t ( t k t k 1 ) L ( s ) d s ( t k t k 1 ) d t = t k 1 t k L ( s ) ( t k s ) d s = h ( t k ) h ( t k 1 ) h ( t k 1 ) ( t k t k 1 ) = h ( t k ) ,
(3.8)

where the first equality holds because h( t k 1 )+ h ( t k 1 )( t k t k 1 )=0. Combining this with (3.7) yields that

v k = d f x k 1 f ( x k ) d f x k 1 d f x 0 d f x 0 1 f ( x k ) h ( t k ) 1 h ( t k ) = t k + 1 t k .
(3.9)

Since x k + 1 = x k exp v k , we have ϱ( x k + 1 , x k ) v k . This together with (3.9) gives that (3.5) holds for n=k, which completes the proof of the theorem. □

4 Applications to optimization problems

Let ϕ:GR be a C 2 -map. Consider the following optimization problem:

min x G ϕ(x).
(4.1)

Newton’s method for solving (4.1) was presented in [16], where local quadratical convergence result was established for a smooth function ϕ.

Let XG. Following [16], we use X ˜ to denote the left invariant vector field associated with X defined by

X ˜ (x)= ( L x ) e Xfor each xG,

and X ˜ ϕ the Lie derivative of ϕ with respect to the left invariant vector field X ˜ , that is, for each xG,

( X ˜ ϕ)(x)= d d t | t = 0 ϕ(xexptX).
(4.2)

Let { X 1 ,, X n } be an orthonormal basis of G. According to [[24], p.356] (see also [16]), gradϕ is a vector field on G defined by

gradϕ(x)=( X ˜ 1 ,, X ˜ n ) ( X ˜ 1 ϕ ( x ) , , X ˜ n ϕ ( x ) ) T = j = 1 n X ˜ j ϕ(x) X ˜ j for each xG.
(4.3)

Then Newton’s method with initial point x 0 G considered in [16] can be written in a coordinate-free form as follows.

Algorithm 4.1 Find X k G such that X ˜ k = ( L x ) e X k and

gradϕ( x k )+grad ( X ˜ k ϕ ) ( x k )=0;

Set x k + 1 = x k exp X k ;

Set kk+1 and repeat.

Let f:GG be a mapping defined by

f(x)= ( L x ) e 1 gradϕ(x)for each xG.
(4.4)

Define the linear operator H x ϕ:GG for each xG by

( H x ϕ)X= ( L x ) e 1 grad( X ˜ ϕ)(x)for each XG.
(4.5)

Then H ( ) ϕ defines a mapping from G to L(G). The following proposition gives the equivalence between d f x and H x ϕ. The following proposition was given in [18].

Proposition 4.1 Let f() and H ( ) ϕ be defined respectively by (4.4) and (4.5). Then

d f x = H x ϕfor each xG.
(4.6)

Remark 4.1 One can easily see from Proposition 4.1 that, with the same initial point, the sequence generated by Algorithm 4.1 for ϕ coincides with the one generated by Newton’s method (3.1) for f defined by (4.4).

Let x 0 G be such that ( H x 0 ϕ ) 1 exists, and let β ϕ := ( H x 0 ϕ ) 1 ( L x 0 ) e 1 gradϕ( x 0 ). Recall that r 0 and b are given by (2.16), and r 1 is given by Proposition 2.1. Then the main theorem of this section is as follows.

Theorem 4.1 Suppose that

β ϕ = ( H x 0 ϕ ) 1 ( L x 0 ) e 1 grad ϕ ( x 0 ) b,
(4.7)

and that ( H x 0 ϕ ) 1 ( H ( ) ϕ) satisfies the L-average Lipschitz condition on C r 1 ( x 0 ). Then the sequence generated by Algorithm 4.1 with initial point x 0 is well defined and converges to a critical point x of ϕ: gradϕ( x )=0.

Furthermore, if H x 0 ϕ is additionally positive definite and the following Lipschitz condition is satisfied:

( H x 0 ϕ ) 1 H x exp u ϕ H x ϕ ρ ( x , x 0 ) ρ ( x , x 0 ) + u L ( s ) d s for  x G  and  u G  with  ρ ( x 0 , x ) + u < r 1 .
(4.8)

Then x is a local solution of (4.1).

Proof Recall that f is defined by (4.4). Then by Proposition 4.1, d f x = H x ϕ for each xG. Hence, by assumptions, d f x 0 1 df satisfies the L-average Lipschitz condition on C r 1 ( x 0 ) and condition (3.2) is satisfied because β ϕ b. Thus, Theorem 3.1 is applicable; hence the sequence generated by Newton’s method for f with initial point x 0 is well defined and converges to a zero x of f. Consequently, by Remark 4.1, one sees that the first assertion holds.

To prove the second assertion, we assume that H x 0 ϕ is additionally positive definite and the Lipschitz condition (4.8) is satisfied. It is sufficient to prove that H x ϕ is positive definite. Let λ and λ 0 be the minimum eigenvalues of H x ϕ and H x 0 ϕ, respectively. Then λ 0 >0. We have to show that λ >0. To do this, let { x n } be the sequence generated by Algorithm 4.1 and write v n =d f x n 1 f( x n ) for each n=0,1, . Then, by Remark 4.1,

x n + 1 = x n exp( v n )for each n=0,1,,
(4.9)

and by Theorem 3.1,

v n t n + 1 t n for each n=0,1,.
(4.10)

Therefore, for each n=0,1, ,

H x 0 ϕ 1 ( H x n + 1 ϕ H x 0 ϕ ) = H x 0 ϕ 1 ( H x n exp ( v n ) ϕ H x 0 ϕ ) = j = 0 n H x 0 ϕ 1 H x j exp ( v n ) ϕ H x j ϕ j = 0 n ρ ( x j , x 0 ) ρ ( x j , x 0 ) + v n L ( s ) d s 0 t k L ( s ) d s < 0 r 0 L ( s ) d s = 1
(4.11)

thanks to (4.8)-(4.10). Since

| λ λ 0 1 | = 1 λ 0 | min v G , v = 1 ( H x ϕ ) v , v min v G , v = 1 ( H x 0 ϕ ) v , v | H x 0 ϕ 1 H x ϕ H x 0 ϕ ,

it follows that

| λ λ 0 1| lim n H x 0 ϕ 1 H x n + 1 ϕ H x 0 ϕ<1

thanks to (4.11). This implies that λ >0 and completes the proof. □

5 Theorems under the Kantorovich’s condition and the γ-condition

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

Let r>0, x 0 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 ( x 0 ) if

T ( x exp u ) T ( x ) Lu

holds for any u, u 0 ,, u k G and x C r ( x 0 ) such that x= x 0 exp u 0 exp u 1 exp u k and u+ρ(x, x 0 )<r, where ρ(x, x 0 )= i = 0 k u i .

Let β>0 and L>0. The quadratic majorizing function h is reduced to

h(t)= L 2 t 2 t+βfor each t0.

Let { t n } denote the sequence generated by Newton’s method with initial value t 0 =0 for h, that is,

t n + 1 = t n h ( t n ) 1 h( t n )for each n=0,1,.

Assume that λ:=Lβ 1 2 . Then h has two zeros r 1 and r 2 :

r 1 = 1 1 2 λ L and r 2 = 1 + 1 2 λ L ;
(5.1)

moreover, { t n } is monotonic increasing and convergent to r 1 , and satisfies that

r 1 t n = q 2 n 1 j = 0 2 n 1 q j r 1 for each n=0,1,,

where

q= 1 1 2 λ 1 + 1 2 λ .

Recall that f:GG is a C 1 -mapping. As in the previous section, we always assume that x 0 G is such that d f x 0 1 exists and set β:=d f x 0 1 f( x 0 ). Then, by Theorem 3.1, we obtain the following results, which were given in [18].

Theorem 5.1 Suppose that d f x 0 1 df satisfies the L-Lipschitz condition on C r 1 ( x 0 ) and that λ=Lβ 1 2 . Then the sequence { x n } generated by Newton’s method (3.1) with initial point x 0 is well defined and converges to a zero x of f. Moreover, the following assertions hold for each n=0,1, :

ϱ ( x n + 1 , x n ) d f x n 1 f ( x n ) t n + 1 t n ; ϱ ( x n , x ) q 2 n 1 j = 0 2 n 1 q j r 1 .

Let x 0 G be such that ( H x 0 ϕ ) 1 exists, and let β ϕ = ( H x 0 ϕ ) 1 ( L x 0 ) e 1 gradϕ( x 0 ). Recall that r 1 is defined by (5.1). Then, by Theorem 4.1, we get the following results, which were given in [18].

Theorem 5.2 Suppose that λ=L β ϕ 1 2 , and that ( H x 0 ϕ ) 1 ( H ( ) ϕ) satisfies the L-Lipschitz condition on C r 1 ( x 0 ). Then the sequence generated by Algorithm 4.1 with initial point x 0 is well defined and converges to a critical point x of ϕ: gradϕ( x )=0.

Furthermore, if H x 0 ϕ is additionally positive definite and the following Lipschitz condition is satisfied:

( H x 0 ϕ ) 1 H x exp u ϕ H x ϕLufor xG and uG with ϱ( x 0 ,x)+u< r 1 .

Then x is a local solution of (4.1).

Let k be a positive integer and assume further that f:GG is a C k -map. Define the map d k f x : G k G by

d k f x u 1 u k = ( k t k t 1 f ( x exp t k u k exp t 1 u 1 ) ) t k = = t 1 = 0

for each ( u 1 ,, u k ) G k . In particular,

d k f x u k = ( d k d t k f ( x exp t u ) ) t = 0 for each uG.

Let 1ik. Then, in view of the definition, one has that

d k f x u 1 u k = d k i ( d i f ( u 1 u i ) ) x u i + 1 u k for each ( u 1 ,, u k ) G k .

In particular, for fixed u 1 ,, u i 1 , u i + 1 ,, u k G,

d i f x u 1 u i 1 =d ( d i 1 f ( u 1 u i 1 ) ) x ().

This implies that d i f x u 1 u i 1 u is linear with respect to uG and so is d k f x u 1 u i 1 u u i + 1 u k . Consequently, d k f x is a multilinear map from G k to G because 1ik is arbitrary. Thus we can define the norm of d k f x by

d k f x =sup { d k f x u 1 u 2 u k : ( u 1 , , u k ) G k  with each  u j = 1 } .

For the remainder of the paper, we always assume that f is a C 2 -map from G to G. Then, taking i=2, we have

d 2 f z vu=d ( d f v ) z ufor any u,vG and each zG.

Thus, (2.13) is applied (with d f v in place of f() for each vG) to conclude the following formula:

d f x exp ( t u ) d f x = 0 t d 2 f x exp ( s u ) udsfor each uG and tR.
(5.2)

The γ-conditions for nonlinear operators in Banach spaces were first introduced and explored by Wang [25, 26] to study Smale’s point estimate theory, which was extended in [19] for a map f from a Lie group to its Lie algebra in view of the map d 2 f as given in Definition 5.1 below. Let r>0 and γ>0 be such that γr1.

Definition 5.1 Let x 0 G be such that d f x 0 1 exists. f is said to satisfy the γ-condition at x 0 on C r ( x 0 ) if, for any x C r ( x 0 ) with x= x 0 exp u 0 exp u 1 exp u k such that ρ(x, x 0 ):= i = 0 k u i <r,

d f x 0 1 d 2 f x 2 γ ( 1 γ ρ ( x , x 0 ) ) 3 .

As shown in Proposition 5.3, if f is analytic at x 0 , then f satisfies the γ-condition at x 0 .

Let γ>0 and let L be the function defined by

L(s)= 2 γ ( 1 γ s ) 3 for each 0<s< 1 γ .
(5.3)

The following proposition shows that the γ-condition implies the L-average Lipschitz condition.

Proposition 5.1 Suppose that f satisfies the γ-condition at x 0 on C r ( x 0 ). Then d f x 0 1 df satisfies the L-average Lipschitz condition on C r ( x 0 ) with L defined by (5.3).

Proof Let x C r ( x 0 ) and let u, u 0 ,, u k G be such that x= x 0 exp u 0 exp u 1 exp u k and i = 0 k u i +u<r. Write ρ(x, x 0 ):= i = 0 k u i . Observe from (5.2) that

d f x exp u d f x = 0 1 d 2 f x exp ( s u ) uds.

Combining this with the assumption yields that

d f x 0 1 ( d f x exp u d f x ) 0 1 d f x 0 1 d 2 f x exp ( s u ) u d s 0 1 2 γ ( 1 γ ( ρ ( x , x 0 ) + s u ) ) 3 u d s = ρ ( x , x 0 ) ρ ( x , x 0 ) + u 2 γ ( 1 γ t ) 3 d t .

Hence, d f x 0 1 df satisfies the L-average Lipschitz condition on C r ( x 0 ) with L defined by (5.3). □

Corresponding to the function L defined by (5.3), r 0 and b in (2.16) are r 0 =(1 2 2 ) 1 γ and b=(32 2 ) 1 γ , and the majorizing function given in (2.17) reduces to

h(t)=βt+ γ t 2 1 γ t for each 0tR.

Hence the condition βb is equivalent to α=γβ32 2 . Let { t n } denote the sequence generated by Newton’s method with the initial value t 0 =0 for h. Then the following proposition was proved in [27], see also [10] and [6].

Proposition 5.2 Assume that α=γβ32 2 . Then the zeros of h are

r 1 = 1 + α ( 1 + α ) 2 8 α 4 γ , r 2 = 1 + α + ( 1 + α ) 2 8 α 4 γ

and

β r 1 ( 1 + 1 2 ) β ( 1 1 2 ) 1 γ r 2 1 2 γ .

Moreover, the following assertions hold:

t n + 1 t n = ( 1 μ 2 n ) ( 1 + α ) 2 8 α 2 α ( 1 ν μ 2 n 1 ) ( 1 ν μ 2 n + 1 1 ) ν μ 2 n 1 β μ 2 n 1 βfor each n=0,1,,

where

μ= 1 α ( 1 + α ) 2 8 α 1 α + ( 1 + α ) 2 8 α andν= 1 + α ( 1 + α ) 2 8 α 1 + α + ( 1 + α ) 2 8 α .
(5.4)

Recall that x 0 G is such that d f x 0 1 exists, and let β:=d f x 0 1 f( x 0 ). Then, by Theorem 3.1 and Proposition 5.2, we get the following results, which were given in [19].

Theorem 5.3 Suppose that

α:=βγ32 2

and that f satisfies the γ-condition at x 0 on C r 1 ( x 0 ). Then Newton’s method (3.1) with initial point x 0 is well defined, and the generated sequence { x n } converges to a zero x of f. Moreover, if α<32 2 , then for each n=0,1, ,

ϱ( x n + 1 , x n ) ν 2 n 1 β,

where ν is given by (5.4).

Below, we always assume that f is analytic on G. For xG such that d f x 1 exists, we define

γ x :=γ(f,x)= sup i 2 d f x 1 d i f x i ! 1 i 1 .

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

The following proposition is taken from [19].

Proposition 5.3 Let γ x 0 :=γ(f, x 0 ) and let r= 2 2 2 γ x 0 . Then f satisfies the γ x 0 -condition at x 0 on C r ( x 0 ).

Thus, by Theorem 5.3 and Proposition 5.3, we get the following corollary, which was given in [19].

Corollary 5.1 Suppose that

α:=β γ x 0 32 2 .

Then Newton’s method (3.1) with initial point x 0 is well defined and the generated sequence { x n } converges to a zero x of f. Moreover, if α<32 2 , then for each n=0,1, ,

ϱ( x n + 1 , x n ) ν 2 n 1 β,

where ν is given by (5.4).

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Acknowledgements

The research of the second author was partially supported by the National Natural Science Foundation of China (grant 11001241; 11371325) and by Zhejiang Provincial Natural Science Foundation of China (grant LY13A010011). The research of the third author was partially supported by a grant from NSC of Taiwan (NSC 102-2115-M-037-002-MY3).

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He, J., Wang, J. & Yao, J. Convergence criteria of Newton’s method on Lie groups. Fixed Point Theory Appl 2013, 293 (2013) doi:10.1186/1687-1812-2013-293

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Keywords

  • Newton’s method
  • Lie group
  • L-average Lipschitz condition