Newton s method on Riemannian manifolds: Smale s point estimate theory under the γ-condition

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1 IMA Journal of Numerical Analysis 6) 6, 8 5 doi:.93/imanum/dri39 Advance Access publication on October 3, 5 Newton s method on Riemannian manifolds: Smale s point estimate theory under the γ-condition CHONG LI AND JINHUA WANG Department of Mathematics, Zhejiang University, Hangzhou 37, People s Republic of China [Received on 9 November 4; revised on August 5] The γ-conditions for vector fields on Riemannian manifolds are introduced. The γ-theory and the α-theory for Newton s method on Riemannian manifolds are established under the γ-conditions. Applications to analytic vector fields are provided and the results due to Dedieu et al. 3, IMA J. Numer. Anal., 3, ) are improved. Keywords: Newton s method; Riemannian manifold; vector field; Smale s point estimate theory; the γ-condition.. Introduction Newton s method and its variants are among the most efficient methods known for solving systems of non-linear equations when the functions involved are continuously differentiable. Besides its practical applications, Newton s method is also a powerful theoretical tool. Therefore, it has been studied and used extensively. One of the famous results on Newton s method is the well-known Kantorovich theorem cf. Kantorovich & Akilov, 98) which guarantees convergence of Newton s sequence to a solution under very mild conditions. Another important result concerning Newton s method is Smale s point estimate theory cf. Blum et al., 997, Smale, 98, 986 and 997). Newton s method has been extended to finding numerically zeros of vector fields on Riemannian manifolds, see, e.g. Edelman et al., 998; Gabay, 98; Smith, 993, 994; Udriste, 994. Recent research has focused on extensions of the Kantorovich theorem and Smale s point estimate theory, see Ferreira & Svaiter, ; Dedieu et al., 3. Here we are particularly interested in the work due to Dedieu et al. 3). Let X be an analytic vector field on an analytic Riemannian manifold M. Let p M be such that DX p) exists, and define Write γx, p) = sup DX p) Dk X p) k! k k βx, p) = DX p) X p) and αx, p) = βx, p)γ X, p). p. cli@zju.edu.cn wjh@zjut.edu.cn c The author 5. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

2 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 9 Let α = denote the unique root of the equation u = ψu) in the interval, ), where ψu) = 4u + u for each u, ). Set σ = ) k = k and s = σ + σα ) ψσα ) + σ σα ) = Then the main results in Dedieu et al. 3) are as follows. For the definition of the geometrical constant K p and of other undefined notations in the sequel, we refer to Dedieu et al. 3), see also Sections 3 and 6. THEOREM. Suppose that X p ) = and let p M.If dp, p ) min r p, + K p K p + 4K p + γx, p ) then Newton s method with the initial point p is well-defined for all n, and ) n dp n, p ) dp, p ), n =,,,... THEOREM. Let p M be such that βx, p ) s r p and αx, p )<α..) Then Newton s method with the initial point p is well-defined for all n and converges to a zero p of X. Moreover, ) n dp n+, p n ) βx, p ). THEOREM.3 Suppose that X p ) =, and let p M.If { } dp, p û )<min ŝ r p, γx, p,.) ) where û = is the smallest positive root of the equation = α ψû ) and ŝ = = , then Newton s method with the initial point p is well-defined for all s s + û ψû ) n, and ) n dp n, p ) σ dp, p ). The γ-conditions for non-linear operators in Banach spaces were first introduced and explored by Wang & Han 997) for the study of Smale s point estimate theory. The purpose of the present paper is to extend the notion of γ-conditions to the case of vector fields on Riemannian manifolds and to establish the γ-theory and the α-theory of Newton s method on Riemannian manifolds under the γ-conditions. In particular, when the results obtained in the present paper are applied to the special case when the, û

3 3 C. LI AND J. WANG vector field X is analytic, Theorem. becomes a direct consequence, while Theorems. and.3 are improved in such a way that the criteria.) and.) in Theorems. and.3 are, respectively, replaced by the weaker conditions.3) and.4) below: β )r p and α = βγ ) 4 and { } dp, p ū )<min t r p, γx, p,.4) ) where ū = is the smallest positive root of the equation ū ψū ) = while t =. Notions and preliminaries We begin with some basic notions and notations. Most of them are standard, see, e.g. Boothby, 986; DoCarmo, 99; Lang, 995. Let M be a real complete m-dimensional Riemannian manifold. Let p M and let T p M denote the tangent space at p to M. Let, be the scalar product on T p M with the associated norm p, where the subscript p is sometimes omitted. For any two distinct elements p, q M, let c: [, ] M be a piecewise smooth curve connecting p and q. Then the arc-length of c is defined by lc) := c t) dt, and the Riemannian distance from p to q by dp, q) := inf c lc), where the infimum is taken over all piecewise smooth curves c: [, ] M connecting p and q. Thus, M, d) is a complete metric space by the Hopf Rinow theorem cf. Boothby, 986; DoCarmo, 99; Lang, 995). For a finite-dimensional space or a Riemannian manifold Z, let B Z p, r) and B Z p, r) denote, respectively, the open metric ball and the closed metric ball at p with radius r, i.e. B Z p, r) ={q Z: dp, q) <r}, B Z p, r) ={q Z: dp, q) r}. In particular, we write, respectively, Bp, r) and Bp, r) for B M p, r) and B M p, r) in the case when M is a Riemannian manifold. Noting that M is complete, the exponential map at p, i.e. exp p : T p M M, is well-defined on T p M. Furthermore, the radius of injectivity of the exponential map at p is denoted by r p. Thus, exp p is a one-to-one mapping from B Tp M, r p ) to Bp, r p ). The following proposition gives the relationship of the radii r p and r q, see Dedieu et al., 3, Lemma 4.4. PROPOSITION. Let p, q M. Then r p dp, q) r q. Recall that a geodesic in M connecting p and q is called a minimizing geodesic if its arc-length equals its Riemannian distance between p and q. Note that there is at least one minimizing geodesic connecting p and q. In particular, the curve c:[, ] M is a minimizing geodesic connecting p and q if, and only if, there exists a vector v T p M such that v =dp, q), q = exp p v) and ct) = exp p tv) for each t [, ]. Let denote the Levi Civita connection on M. For any two vector fields X and Y on M, the covariant derivative of X with respect to Y is denoted by Y X. Define the linear map

4 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 3 DX p): T p M T p M by DX p)w) = Y X p) w T p M, where Y is a vector field satisfying Y p) = w. Then the value DX p)w) of DX p) at w depends only on the tangent vector w = Y p) T p M since is tensorial in Y. Let c: R M be a C curve and let P c,, denote the parallel transport along c, which is defined by P c,cb),ca) v) = V cb)) a, b R and v T ca) M, where V is the unique C vector field satisfying c t)v = and V ca)) = v. Then, for any a, b R, P c,cb),ca) is an isometry from T ca) M to T cb) M. Note that, for any a, b, b, b R, P c,cb ),cb ) P c,cb ),ca) = P c,cb ),ca) and P c,cb),ca) = P c,ca),cb). In particular, we write P q,p for P c,q,p in the case when c is a minimizing geodesic connecting p and q. Let X be a C vector field on M and let p M. Following Ferreira & Svaiter ), Newton s method with the initial point p for X is defined as follows. p n+ = exp pn DX p n ) X p n )), n =,,,....) The γ-conditions for operators in Banach spaces were first presented by Wang & Han 997) for the study of Smale s point estimate theory. In the following, we extend this notion to the case of vector fields on a Riemannian manifold M. Let k be a positive integer. We first define the notion of kth covariant derivatives. DEFINITION. Let {Y,...,Y k } be a finite sequence of vector fields on M. Then, the kth covariant derivative of X with respect to {Y,...,Y k } is denoted by k X and is defined inductively by {Y i } i= k k {Y i } i= k X = Yk k {Y i } i= k DEFINITION. Let p M and v,...,v k ) T p M) k. Let {Y,...,Y k } be a finite sequence of vector fields on M such that Y i p) = v i for each i =,...,k. Then, the value of the kth covariant derivative of X with respect to {Y,...,Y k } at p is denoted by ) X. D k X p)v v v k = k {Y i } k i= X p). Note that D k X p)v v v k only depends on the k-tuple of vectors v,...,v k ) since the covariant derivative is tensorial in each Y i. Clearly, by Definition., the kth covariant derivative D k X p) at a point p is a k-multilinear map from T p M) k to T p M. We define the norm of D k X p) by D k X p) p = sup D k X p)v v v k p,.) where the supremum is taken over all k-tuples of vectors v,...,v k ) T p M) k each with v j =. Let r > and γ> be such that γ r. Also let k be a positive integer. Throughout the paper, we always assume that X is a C vector field on M.

5 3 C. LI AND J. WANG DEFINITION.3 Let q M be such that DX q ) exists. X is said to satisfy the k-piece γ-condition at q in Bq, r), if DX q ) P q,q P q,q P qk,q k D γ X q k ) γ ) 3.3) k dq i, q i ) i= holds for any k points q, q,...,q k Bq, r) satisfying k i= dq i, q i )<r. REMARK. i) The k + )-piece γ-condition at q implies the k-piece γ-condition at q in Bq, r). ii) Let b denote the bound of DX q ) P q,q P q,q P qk,qd X q) on Bq, r). Then it is easy to see that X satisfies the k-piece b -condition at q in Bq, r). It follows that γ b if γ is the minimum of γ> such that X satisfies the k-piece γ-condition at q in Bq, r). The following two lemmas will be used later. The first one is stated in Ferreira & Svaiter, p. 38) while the second one is its consequence. LEMMA. Let c: [, ] M be a geodesic and Z a C vector field on M. Then P c,c),ct) Zct)) = Zc)) + t P c,c),cs) DZcs))c s)) ds. LEMMA. Let c: [, ] M be a geodesic and Y a C vector field on M. Then P c,c),ct) DX ct))y ct)) = DX c))y c)) + In particular, P c,c),ct) DX ct))c t) = DX c))c ) + t t P c,c),cs) D X cs))y cs))c s)) ds..4) P c,c),cs) D X cs))c s)) ) ds..5) Proof. Clearly,.5) is a direct consequence of.4). Thus, we only need to show.4). Let Z = Y X. Then Z is a C vector field on M and Lemma. is applicable. Hence, P c,c),ct) Zct)) = Zc)) + t P c,c),cs) DZcs))c s)) ds. Since DZcs))c s) = DDX cs))y cs)))c s) = D X cs))y cs))c s),.4) follows and the proof is complete. Finally, we state a lemma, which will play a key role in this paper. This lemma is true for the general case although it is stated and proved for the special case when k. For simplicity, we use the function ψ defined by [ ) ψu) := 4u + u, u,..6) Note that ψ is strictly monotonic decreasing on [, ).

6 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 33 LEMMA.3 Let r < γ and let k. Let q M be such that DX q ) exists. Suppose that X satisfies the k-piece γ-condition at q in Bq, r). Then, for each point q Bq, r), DX q) exists, and for any k points q, q,...,q k in Bq, r) satisfying k i= dq i, q i+ )<r, DX q) P q,qm P q,q P q,q DX q ) γ k ψ ) dq i, q i+ ) i= γ k i= dq i, q i+ ) ),.7) where q k = q. Proof. We only prove the lemma for the case when k = because the proof for the case when k = and for the general case) is similar. By the Banach lemma, to complete the proof, it is sufficient to show that DX q ) P q,q P q,qdx q)p q,q P q,q I Tq M + γdq, q ) + dq, q))) <.8) because P q,q and P q,q are isometries, where I Tq M is the identity on T q M. To verify.8), let v T q M. Let v T q M and v T q M be such that the curve c t) := exp q tv ), t [, ], is a minimizing geodesic connecting q and q and that the curve c t) := exp q tv ), t [, ], is a minimizing geodesic connecting q and q. Note that there exist vector fields Y and Y such that Y c )) = v, D c t)y c t)) =, Y c )) = P c,q,q v and D c t)y c t)) =. Then we apply Lemma. to conclude that DX q ) P c,q,q DX q )Y c )) DX q )Y c )) ) and = DX q ) P c,q,c s)d X c s))y c s))c s) ds.9) DX q ) P c,q,q Pc,q,qDX q)y c )) DX p)y c )) ) = DX q ) P c,q,q P c,q,c s)d X c s))y c s))c s) ds..) Hence, in view of.3) with k =,, respectively), we have that DX q ) P c,q,q DX q )Y c )) DX q )Y c )) ) DX q ) P c,q,c s)d X c s)) Y c s)) c s) ds = γ γ s v ) 3 v v ds.)

7 34 C. LI AND J. WANG and DX q ) P c,q,q Pc,q,qDX q)y c )) DX p)y c )) ) DX q ) P c,q,q P c,q,c s)d X c s)) Y c s)) c s) ds γ γ v +s v )) 3 v v ds..) Since ) DX q ) P q,q P q,qdx q)p q,q P q,q I Tq M v = DX q ) [ P c,q,q Pc,q,qDX q)y c )) DX q )Y c )) ] + DX q ) [ P c,q,q DX q )Y c )) DX q )Y c )) ],.3) it follows from.) and.) that ) DX q ) P q,q P q,qdx q)p q,q P q,q I Tq M v = γ γ v +s v )) 3 v v ds + + γdq, q ) + dq, q))) γ γ s v ) 3 v v ds ) v,.4) where the last equality holds because v =dq, q ) and v =dq, q). Asv T q M is arbitrary,.8) follows. 3. Generalized γ-theory Recall that X is a C vector field on M. Let p M be such that DX p ) exists. The approach for the generalized γ-theory in this section depends on the geometrical number K p while the approach independent of K p will be considered in Section 5. The K p is related to the sectional curvature at p M and is defined by K p = sup dexp q w), exp q v)), 3.) w v q where the supremum is taken over all q Bp, r p ), and v,w v B Tq M, r p ) with w v, see Dedieu et al., 3. REMARK 3. ) K p measures how fast the geodesics spread apart in M. In particular, if w = or more generally if w and v are on the same line through, then dexp q w), exp q v)) = w v q. This means that K p. ) In the case when M has non-negative sectional curvature, the geodesics spread apart less than the rays cf. Dedieu et al., 3) so that dexp q w), exp q v)) w v q

8 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 35 and consequently K p =. Examples of manifolds with non-negative sectional curvature are given in Dedieu et al. 3). The main theorem of this section gives an estimate of the radius of the convergence ball around the zero of X for Newton s method. Recall that ψ is defined by ψu) = 4u + u for each u [, ). THEOREM 3. Let K p + 4 K p r c = + 8K p + 8 and r = min{r p, r c }. 3.) 4γ Suppose that X p ) = and that X satisfies the one-piece γ-condition at p in Bp, r).ifdp, p )< r, then Newton s method.) with the initial point p is well-defined, and for n =,,,..., where Proof. By 3.), γ r c = dp n, p ) λ n dp, p ), 3.3) λ = K p γ dp, p ) ψγdp, p )) K p + 4 K p + 8K p <. 3.4) < K p + 4 K p + 3) 4 Then γ dp, p )<γr c < it follows that λ = K p γ dp, p ) ψγdp, p )) < K p γ r c ψγr c ) <. 3.5). Since the function ψ is strictly monotonic decreasing on [, ), =. 3.6) Below we will show that 3.3) holds for each n =,,... by induction. Clearly, it is trivial in the case when n =. Now assume that 3.3) holds for n. Note that, for each n =,,..., 3.3) implies p n Bp, r). Then, by 3.5), we have dp n, p )<r c <. γ Hence, Lemma.3 is applicable with k = ). It follows that DX p n ) exists and DX p n ) P pn,p DX p ) γ dp n, p )) ψγdp n, p. 3.7) )) Thus, p n+ is well-defined. Consequently, to complete the proof, it remains to verify that 3.3) holds for n +. To do this, let v T p M be such that p n = exp p v) and v =dp n, p ). We claim that DX p n ) X p n ) P pn,p v) γ dp n, p ) ψγdp n, p )). 3.8)

9 36 C. LI AND J. WANG In fact, since the curve ct) := exp p tv),t [, ], is the minimizing geodesic connecting p and p n, by Lemma., we have that Also, by Lemma., P c,p,p n X p n ) X p ) = DX p n )P c,pn,p v P c,p n,cτ)dx cτ))c τ) = Hence, the two equalities above imply that DX p n ) X p n ) P pn,p v) P c,p,cτ)dx cτ))c τ) dτ. 3.9) τ P c,pn,cs)d X cs))c s)) ds. 3.) = DX p n ) P c,pn,p Pc,p,p n X p n ) X p ) ) + P c,pn,p v = DX p n ) DX pn )P c,pn,p v P c,p n,cτ)dx cτ))c τ) ) dτ = DX p n ) P c,pn,p τ P c,p,cs)d X cs))c s)) ds dτ. 3.) Consequently, by 3.), 3.7) and.3) with k = ), we obtain that DX p n ) X p n ) P pn,p v) DX p n ) P c,pn,p DX p ) DX p ) P c,p,cs)d X cs))c s)) ds dτ γ dp n, p )) ψγdp n, p )) = γ dp n, p ) ψγdp n, p )). τ τ γ γ sdp n, p )) 3 dp n, p ) ds dτ 3.) This shows that 3.8) holds and hence DX p n ) X p n ) P pn,p v) λdpn, p ) r p. 3.3) On the other hand, since Pc,pn,p v = v =dpn, p )<r p, 3.4) in view of the definition of K p and 3.8), one gets that d exp pn DX p n ) X p n )), exp pn Pc,pn,p v)) K p γ dp n, p ) ψγdp n, p )). 3.5)

10 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 37 As p n+ = exp pn DX p n ) X p n )) and p = exp pn Pc,pn,p v), 3.5) means that dp n+, p ) K p γ dp n, p ) ψγdp n, p )) K p γ λ n ) dp, p ) ψγdp, p )) = λ n+ dp, p ). Therefore, 3.3) holds for n Generalized α-theory The majorizing function h, which is due to Wang 999) and Wang & Han 99), will play a key role in this section. Let β>and γ>. Define ht) = β t + γ t γ t, for each t < γ. 4.) Let {t n } denote the sequence generated by Newton s method with the initial value t = for h, i.e. t n+ = t n h t n ) ht n ), for each n =,,... 4.) Then we have the following proposition which was proved in Wang 999) and Wang & Han 99). PROPOSITION 4. Suppose that α = γβ 3. Then the zeros of h are r = + α + α) 8α, r = + α + + α) 8α 4γ 4γ 4.3) and they satisfy Moreover, and where and t n+ t n = β r + ) β ) γ r γ. 4.4) t n = µn µ n η r 4.5) µ n ) + α) 8α α ηµ n ) ηµ n+ )ηµn β, n =,,..., 4.6) µ = α + α) 8α α + + α) 8α 4.7) η = + α + α) 8α + α + + α) 8α. 4.8) Lemma 4. was shown in Wang 999) and Wang & Han 99). However, here we give a direct and simpler proof of this lemma.

11 38 C. LI AND J. WANG LEMMA 4. Suppose that α<3. Then µ n ) + α) 8α α ηµ n ) )η, ηµ n+ n =,, ) Proof. Let a n = µ n ) + α) 8α α ηµ n ) ηµ n+ )η. Since <η< and ηµ >, one has that a n = µn ηµ )µ n µn a n ηµ n+ µ n µ n+ µ n. µ n Hence, a n a n a = t t = β and 4.9) follows. Recall that X is a C vector field. In the remainder of this section, let p M be such that DX p ) exists, and define β = DX p ) X p ), α = γβ. THEOREM 4. Let β )r p and α = βγ 3. Suppose that X satisfies the two-piece γ-condition at p in Bp, r ). Then Newton s method.) with the initial point p is well-defined and the generated sequence {p n } converges to a zero p of X in Bp, r ). Moreover, dp n+, p n ) µ n ) + α) 8α α ηµ n ) ηµ n+ )ηµn dp, p ), 4.) for all n =,,,..., where µ and η are given by 4.7) and 4.8), respectively. Proof. Recall from.) that p n+ = exp pn DXp n ) X p n )), n =,,... 4.) Let We will use induction to prove that v n = DXp n ) X p n ). 4.) dp n+, p n ) = v n t n+ t n 4.3) holds for each n =,,...Since t =, t = β, v =β<r p and p = exp p v ), dp, p ) = v t t.

12 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 39 Therefore, the result is clear for the case when n =. Assume that dp n+, p n ) = v n t n+ t n, n =,,...,k. 4.4) Then, we have k k dp k, p ) dp n+, p n ) t n+ t n ) = t k < r <. 4.5) γ n= n= Hence, DX p k ) exists by Lemma.3 and p k+ is well-defined. Furthermore, by.7) with k = ), we have that DX p k ) P pk,p k P pk,p DX p ) γdp k, p ) + dp k, p k ))). 4.6) ψγdp k, p ) + dp k, p k ))) Since u) ψu) = h ), it follows that u γ DX p k ) P pk,p k P pk,p DX p ) h dp k, p ) + dp k, p k )) h t k ) 4.7) because h t) is monotonic increasing on [, γ ). Define the curve ct) := exp pk tv k ), t [, ]. By 4.4), dp k, p k ) = v k ; hence, c is the minimizing geodesic connecting p k and p k. Using Lemma., we obtain that P c,pk,p k X p k ) = P c,pk,p k X p k ) X p k ) DX p k )v k = Pc,pk,cτ)DX cτ))c ) τ) DX p k )v k dτ. 4.8) By Lemma., it follows that P c,pk,cτ)dx cτ))c τ) DX p k )v k = γ τ P c,pk,cs)d X cs))c s)) ds. 4.9) Note that h u) = ; hence by 4.8), 4.9) and.3) with k = ), we get that γ u) 3 DX p ) P p,p k P c,pk,p k X p k ) τ DX p ) P p,p k P c,pk,cs)d X cs)) c s) ds dτ τ τ h t k + s v k ) v k ds dτ h t k + st k t k ))t k t k ) ds dτ = ht k ) ht k ) h t k )t k t k ) = ht k ), 4.)

13 4 C. LI AND J. WANG where the last equality holds because ht k ) h t k )t k t k ) = by 4.) with n = k). Therefore, 4.7) and 4.) imply that DX p k ) X p k ) DX p k ) P c,pk,p k P pk,p DX p ) DX p ) P p,p k P c,pk,p k X p k ) h t k ) ht k ) = t k+ t k. Hence, in view of 4.), v k = DX p k ) X p k ) t k+ t k. 4.) As β< )r p, it follows from 4.4) that r r p. Thus, 4.5) and 4.) yield that This, together with Proposition., implies that v k +dp k, p ) t k+ < r r p. v k r p dp k, p ) r pk. Since p k+ = exp pk v k ), it follows from the definition of r pk that dp k+, p k ) = v k. Hence, it is seen that 4.3) holds for n = k thanks to 4.). Combining 4.3) and 4.6), we get 4.) and complete the proof. By 4.9), we arrive at the following corollary. COROLLARY 4. Let β )r p and α<3. Suppose that X satisfies the two-piece γ-condition at p in Bp, r ). Then Newton s method.) with the initial point p is well-defined and the generated sequence {p n } converges to a zero p of X in Bp, r ). Moreover, dp n+, p n ) µ n dp, p ), n =,,..., where µ is defined by 4.7). 5. Alternative formulation of the generalized γ-theorem This section will provide an alternative formulation of the generalized γ-theorem, which is independent of the geometric number K p. Recall that X is a C vector field on M and that p M is such that DX p ) exists. Recall from.6) that the function ψ is defined by ψu) = 4u + u, [ ) u,. The following lemma estimates the value of the quantity DX p ) X p ), which will be used in the proof of the main theorem of this section.

14 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 4 LEMMA 5. Let r < γ and let p Bp, r). Suppose that X satisfies the one-piece γ-condition at p in Bp, r). Then DX p ) exists and where u = γ dp, p ). Proof. By Lemma.3, DXp ) exists and DX p ) P p,p DX p ) DX p ) X p ) u ψu) dp, p ), 5.) u). 5.) ψu) Below, we will show that DX p ) P p,p X p ) dp, p ) u). 5.3) Granting this, by 5.), we have that DXp ) X p ) DX p ) P p,p DX p ) DX p ) P p,p X p ) u ψu) dp, p ) and so 5.) is seen to hold. To verify 5.3), let c: [, ] M be a minimizing geodesic connecting p and p. Then there exists v T p M such that v =dp, p ) and ct) = exp p tv) for each t [, ]. Observe that P c,p,p X p ) = P c,p,p X p ) X p ) DX p )v + DX p )v = = τ P c,p,cτ)dx cτ))c τ) dτ DX p )v + DX p )v P c,p,cs)d X cs))c s) ds dτ + DX p )v, 5.4) where the second equality holds because of Lemma. while the third equality is valid because of Lemma.. Thus, by.3) with k = ), τ DX p ) P c,p,p X p ) DX p ) P c,p,cs)d X cs))c s) ds dτ + v τ = dp, p ) u) γ sγ v ) 3 v ds dτ + v and hence 5.3) holds. Let u = be the smallest positive root of the equation u ψu ) = )

15 4 C. LI AND J. WANG Also, let Then t = + u = ) ψu ) u t =. 5.7) ψu ) t Recall that u = γ dp, p ) and β = DX p ) X p ). Furthermore, set γ = γ ψu) u) and ᾱ = β γ. THEOREM 5. Let { r = min t r p, u } γ and { r = min r p, }. γ Suppose that X p ) = and that X satisfies the three-piece γ-condition at p in Bp, r). If dp, p )<r, then Newton s method.) with the initial point p is well-defined and dp n, p ) σ µ) n dp, p ), where µ = ᾱ +ᾱ) 8ᾱ ᾱ + +ᾱ) 8ᾱ 5.8) and σ = n µ) n. 5.9) Proof. By Lemma 5., DX p ) exists and β = DX p ) X p ) u ψu) dp, p ) = u u ψu) γ. 5.) Then ᾱ = β γ u ψu) < u ψu ) = 3 5.) because the function u u is strictly monotonic increasing on [, ) ψu). Let r = +ᾱ +ᾱ) 8ᾱ. 5.) 4 γ Then, by 4.4), β r + ) β ) γ 5.3)

16 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 43 thanks to 5.). Since dp, p ) t r p and the function u ψu) u on [, ), by 5.), we have that is strictly monotonic increasing thanks to 5.7). Hence, by 5.3), By Proposition., Therefore, Thus, 5.4) implies that β u t t )r p = ) t )r p 5.4) ψu ) t r t )r p. 5.5) r p r p + dp, p ) r p + t r p. 5.6) r p r p t. 5.7) β )r p. 5.8) Then, in order to ensure that Corollary 4. is applicable, we have to show the following assertion: there exists ˆr r such that X satisfies the two-piece γ -condition at p in Bp, ˆr). For this purpose, let { ˆr = r dp, p ) = min r p, } dp, p ). 5.9) γ We claim that ˆr is the number desired. First, we have that ˆr r. 5.) In fact, if r = r p, then if r = γ, then ˆr = γ ˆr = r p dp, p ) t )r p r ; u ) γ ψu) u) γ ) = γ r. Therefore, 5.) is proved. Next, we have that X satisfies the two-piece γ -condition at p in Bp, ˆr). Indeed, for any two points p, q Bp, ˆr) with dp, p) + dp, q) <ˆr 5.) since X satisfies the three-piece γ-condition at p in Bp, r) and dp, p ) + dp, p) + dp, q) < r, we obtain that DX p ) P p,p P p,p P p,q D X q) γ γdp, p ) + dp, p) + dp, q))) 3. 5.)

17 44 C. LI AND J. WANG Consequently, using Lemma.3 with k = ) and 5.), we conclude that DX p ) P p,p P p,q D X p) = DX p ) P p,p DX p ) DX p ) P p,p P p,p P p,q D X q) u) ψu) γ γdp, p ) + dp, p) + dp, q))) 3 γ u) 3 = ψu) u) u γdp, p) + dp, q))) 3 γ = γ u dp, p) + dp, q)) = ) 3 γ γ ψu) u) dp, p) + dp, q)) γ γdp, p) + dp, q))) 3 because <ψu)< for all u, ). Therefore, X satisfies the two-piece γ -condition at p in Bp, ˆr) and the assertion holds. Thus, we apply Corollary 4. to conclude that the sequence {p n } generated by Newton s method.) with the initial point p converges to a zero q of X in Bp, r ) and dp n+, p n ) µ) n dp, p ), n =,,, ) To complete the proof, it remains to verify that p = q. To this end, let v T p M be such that q = exp p v) and v =dp, q ). Then the curve c defined by ct) = exp p tv),t [, ], is a minimizing geodesic connecting p and q.asc t) = P c,ct),p v, it follows from Lemma. that ) DX p ) P c,p,ct)dx ct))p c,ct),p dt v = DX p ) [P c,p,q X q ) X p )] ) 3 =. 5.4) We claim that DX p ) P c,p,ct)dx ct))p c,ct),p dt is invertible. Granting this, 5.4) implies that v = and so p = q. Let v T p M and let Y be the unique vector field such that Y c)) = v and D c t)y ct)) =. By Lemma., one has that = DX p ) [P c,p,ct)dx ct))p c,ct),p DX p )] v dt t Note that, by 5.9) and 5.), DX p ) P c,p,cs)d X cs))y cs))c s) ds dt. 5.5) dq, p ) dq, p ) + dp, p ) r + dp, p ) ˆr + dp, p ) = r. 5.6)

18 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 45 This implies that dcs), p )< r for each s, ). It follows that, for each s, ), DX p ) P c,p,cs)d X cs))y cs))c γ s) γ sdp, q )) 3 v dp, q ) 5.7) since X satisfies the three-piece γ-condition and therefore the one-piece γ-condition) at p in Bp, r). Consequently, by 5.5) and 5.7), DX p ) [P c,p,ct)dx ct))p c,ct),p DX p )] v dt t γ γ sdp, q )) 3 v dp, q ) ds dt γ < γ sdp, q )) 3 v dp, q ) ds dt = γ dp, q )) ) v v, 5.8) where the last inequality follows from 5.6) and the fact that r γ. Hence, DX p ) [P c,p,ct)dx ct))p c,ct),p X p )]dt <. By the Banach lemma, the claim holds and the proof is complete. 6. Application to analytic vector fields Throughout this section, we shall always assume that M is an analytic complete m-dimensional Riemannian manifold. Let p M. Recall from Boothby 986) and DoCarmo 99) that a vector field X is said to be analytic at p if there exists a local coordinate system U, {x i }) of p and m analytic functions X i : U R, i =,,...,m, such that X U = m i= X i x i. Then the vector field X is analytic on M if it is analytic at each point of M. In the remainder of this section, we assume that X is analytic on M. Let p M be such that DX p) exists. Following Dedieu et al. 3), we define γx, p) = sup DX p) Dk X p) k! k k p. 6.) Also we adopt the convention that γx, p) = if DX p) is not invertible. Note that this definition is justified and in the case when DX p) is invertible, by analyticity, γx, p) is finite. The following Taylor formula for vector fields can be found in Dedieu et al. 3).

19 46 C. LI AND J. WANG LEMMA 6. Let r = min { r p, γx,p)}. Let q Bp, r) and v Tp M be such that q = exp p v). Then ) X q) = P q,p k! Dk X p)v k. Taking the lth covariant derivative in Lemma 6. gives the following corollary. COROLLARY 6. Under the same hypotheses as in Lemma 6., for any l, we have ) D l X q) = P q,p k! Dk+l X p)v k Pp,q l, k= k= where P l p,q stands for the map from T q M) l to T p M) l defined by P l p,q v,...,v l ) = P p,q v,...,p p,q v l ) v,...,v l ) T q M) l. The following two lemmas were given in Dedieu et al. 3). Let q M be such that DX q ) exists. LEMMA 6. Let r < and let k be a positive integer. Then l= k + l)! r l = k!l! r) k+. LEMMA 6.3 Let u = γx, q )dp, q ).Ifdp, q )<min { r q, γx,q )}, then γx, p) γx, q ) u)ψu). 6.) The following lemma shows that an analytic vector field satisfies the three-piece γ-condition at q in Bq, r), where γ = γx, q ) and r = min { r q, γx,q )}. LEMMA 6.4 Let < r min { r q, γx,q )}. Then X satisfies the three-piece γ-condition at q in Bq, r). Proof. Let p, p, q Bq, r) be such that Set { ˆr = min r p, dq, p ) + dp, p) + dp, q) <r. 6.3) } {, ˆr = min r p, γx, p) } {, ˆr 3 = min r q, γx, p ) }. 6.4) γx, q ) Below, we claim that q Bp, ˆr ), p Bp, ˆr ), p Bq, ˆr 3 ). 6.5) We only show that q Bp, ˆr ) since the proofs for p Bp, ˆr ) and p Bq, ˆr 3 ) are similar. As dq, p ) + dp, p) + dp, q) <r r q, it follows from Proposition. that dp, q) r q dq, p) r p. 6.6)

20 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 47 Write u = γx, q )dp, q ). Since p Bq, r), Lemma 6.3 is applicable. It follows that γx, p) γx, q ) u)ψu). 6.7) By a simple calculation, we see that u)ψu) γx, q ) γx, q ) dp, q ). Then, by 6.7) and 6.3), dp, q) γx, p). 6.8) This, together with 6.6), implies that q Bp, ˆr ); hence, our claim holds. Thus, by 6.5), Corollary 6. is applicable to conclude that DX q ) P q,p P p,p P p,q D X q) = DX q ) P q,p P p,p = DX q ) P q,p = l= l! j= j! k= l= l! l= j= l! Dl+ X p)v l 3 P p,q j! Dl+ j+ X p )v j Pl+ p,p vl 3 P p,q k! DX q ) D l+ j+k+ X q )v k j+ Pl+ q,p v j Pl+ p,p vl 3 P p,q, 6.9) where v T q M, v T p M and v 3 T p M satisfy that p = exp q v ), p = exp p v ) and q = exp p v 3 ), respectively. Since DX q ) D l+ j+k+ X q ) γx, q ) l+ j+k+, l + j + k + )! one has from 6.9) that DX q ) P q,p P p,p P p,q D X q) l + j + )! l + j + k + )! l! j! k!l + j + )! γx, q ) l+ j+k+ v k v j v 3 l. 6.) l= j= k= Using Lemma 6. to calculate the quantity on the right-hand side of the inequality 6.), we get that DX q ) P q,p P p,p P p,q D γx, q ) X q) γx, q ) v + v + v 3 )) 3. 6.) Since v =dq, p ), v =dp, p) and v 3 =dp, q), it follows from 6.) that DX q ) P q,p P p,p P p,q D γx, q ) X q) γx, q )dq, p ) + dp, p) + dp, q))) 3. Hence, X satisfies the three-piece γ-condition at q in Bq, r) and the proof is complete.

21 48 C. LI AND J. WANG Then, by Theorem 3., we have the following corollary which was obtained in Dedieu et al. 3) with a different technique. COROLLARY 6. Let p M be such that DX p ) exists. Suppose X p ) = and let p M.If dp, p ) min r p, + K p K p + 4K p + γx, p ), then Newton s method.) with the initial point p is well-defined, and dp n, p ) ) n dp, p ), n =,,,... 6.) Proof. Write + K p K p δ = + 4K p +. Let γ = γx, p ). Then p Bp, r c ) because γ δ < r c, where r c is defined by 3.). As r c < γ, it follows from Lemma 6.4 that X satisfies the one-piece γ-condition at p in Bp, r) with r = min{r p, r c }. Hence, Theorem 3. is applicable to conclude that Newton s method.) is welldefined for p, and 3.3) holds for λ defined by 3.4). As K p γ dp, p ) λ = 4γ dp, p ) + γ dp, p )) K p δ 4δ + δ =, 6.) holds by 3.3). The proof is complete. Similarly, by Corollary 4. and Theorem 5., we also have the following two corollaries, which improve the corresponding results due to Dedieu et al. 3). Recall that p M is such that DX p ) exists and also that β = DX p ) X p ) and α = βγ, where γ = γx, p ). Let r = + α + α) 8α. 4γ Let ū = be the smallest positive root of the equation COROLLARY 6.3 If ū ψū ) = ) 4 β )r p and α , 4 then Newton s method.) with the initial point p is well-defined and the generated sequence {p n } converges to a zero p of X in Bp, r ). Moreover, dp n+, p n ) ) n dp, p ).

22 NEWTON S METHOD ON RIEMANNIAN MANIFOLDS 49 Proof. As β )r p, it follows from 4.4) that r min{r p, γx,p )}. Thus, by Lemma 6.4, X satisfies the two-piece γ-condition at p in Bp, r ) with γ = γx, p ). Note also that α Corollary 4. is applicable to conclude that < 3. dp n+, p n ) µ n dp, p ), where µ = α + α) 8α α + + α) 8α. Therefore, we have that µ because µ increases as α does on [, 3 3 ] 7 4 and the value of µ at α = is. COROLLARY 6.4 Let p M be such that DX p ) exists. Suppose X p ) =. Let p M and t be given by 5.6). If { } dp, p u )<min t r p, γx, p, ) then Newton s method.) with the initial point p is well-defined and where σ = ) n. n Proof. Let dp n, p ) σ γ = γx, p ), γ = ) n dp, p ), γ ψu) u) and ᾱ = β γ, where u = γ dp, p ). By Lemma 6.4, X satisfies the three-piece γ-condition at p in Bp, r) with r = min{r p, γ }. Since ū determined by 6.3) is less than u given by 5.5), Theorem 5. is applicable. Thus, we have that Newton s method.) with the initial point p is well-defined and dp n+, p n ) µ n dp, p ), where By Lemma 5., we have that µ = ᾱ +ᾱ) 8ᾱ ᾱ + +ᾱ) 8ᾱ. β = DX p ) X p ) u ψu) dp, p ).

23 5 C. LI AND J. WANG It follows that ᾱ u ψu) ū ψū ) = ) u ψu) because the function u is strictly monotonic increasing on [, ). Hence, we have that µ and the proof is complete. 7. Conclusion We have established the γ-theory and the α-theory under the γ-conditions. In particular, when these results are applied to analytic vector fields, some results due to Dedieu et al. 3) are improved. In addition, we should remark that the issue on mappings from manifolds to m-dimensional spaces can be addressed in almost the same way, so we do not elaborate further on this here. Acknowledgements We wish to thank the referees for their valuable comments and suggestions. This work was supported in part by the National Natural Science Foundation of China grant 75) and Program for New Century Excellent Talents in University. REFERENCES BLUM, L., CUCKER, F., SHUB, M.& SMALE, S. 997) Complexity and Real Computation. New York: Springer. BOOTHBY, W. M. 986) An Introduction to Differentiable Manifolds and Riemannian Geometry, nd edn. New York: Academic Press, Inc. DEDIEU, J.P.,PRIOURET, P.&MALAJOVICH, G. 3) Newton s method on Riemannian manifolds: covariant alpha theory. IMA J. Numer. Anal., 3, DOCARMO, M. P. 99) Riemannian Geometry. Boston, MA: Birkhauser. EDELMAN, A., ARIAS, T. A.& SMITH, T. 998) The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl.,, FERREIRA, O. P.& SVAITER, B. F. ) Kantorovich s theorem on Newton s method in Riemannian manifolds. J. Complex., 8, GABAY, D. 98) Minimizing a differentiable function over a differential manifold. J. Optim. Theory Appl., 37, KANTOROVICH, L.V.&AKILOV, G. P. 98) Functional Analysis. Oxford: Pergamon. LANG, S. 995) Differential and Riemannian Manifolds, GTM 6. New York: Springer. SMALE, S. 98) The fundamental theorem of algebra and complexity theory. Bull. Am. Math. Soc., 4, 36. SMALE, S. 986) Newton s method estimates from data at one point. The Merging of Disciplines: New Directions in Pure, Applied and Computational Mathematics R. Ewing, K. Gross & C. Martin eds). New York: Springer, pp SMALE, S. 997) Complexity theory and numerical analysis. Acta Numer., 6, SMITH, S. T. 993) Geometric optimization method for adaptive filtering. Ph.D. Thesis, Harvard University, Cambridge, MA. SMITH, S. T. 994) Optimization techniques on Riemannian manifolds. Fields Institute Communications, vol. 3. A. Bloch ed.) Providence, RI: American Mathematical Society, pp UDRISTE, C. 994) Convex Functions and Optimization Methods on Riemannian Manifolds. Dordrecht: Kluwer Academic.

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