NEWTON’S METHOD IN RIEMANNIAN MANIFOLDS

. Using more precise majorizing sequences than before [1], [8], and under the same computational cost, we provide a ﬁner semilocal convergence analysis of Newton’s method in Riemannian manifolds with the following advantages: larger convergence domain, ﬁner error bounds on the distances involved, and a more precise information on the location of the singularity of the vector ﬁeld.


INTRODUCTION
We refer the reader to [1], [5], [7] for some of the concepts introduced but not detailed here.
Let X be a C 1 vector field defined on a connected, complete and finite-dimensional Riemannian manifold (M, g).In this study we are concerned with the following problem: (1. 1) find p * ∈ M such that X(p * ) = o ∈ T p * M.
A point p * satisfying (1.1) is called a singularity of X.
The most popular method for generating a sequence {p n } (n ≥ 0) approximating p * is undoubtedly Newton's method, described here as follows: Assume there exists an initial guess p 0 ∈ M such that the covariant X (p 0 ) of X at p 0 given by (1.2) is invertible, at p = p 0 , for each pair of continuously differentiable vector fields X, Y where the vector field ∇ y X stands for the covariant derivative of X with respect to Y .Define the Riemannian-Newton method by where exp p : T p M → M is the exponential map at p.
A survey of local as well as semilocal convergence results for method (1.3) can be found in [1]- [10] and the references there.
Here we are motivated by optimization considerations and the recent excellent study [1] of the Riemannian analogue of the property used by Zabreijko and Nguen [10].
In particular we show: Under weaker hypotheses and the same computational cost finer error bounds on the distances d(p n+1 , p n ), d(p n , p * ) (n ≥ 0) and a more precise information on the location of the solution p * are obtained.
All the above advantages are achieved because we use more precise error estimates on the distances involved than in [1] along the lines of our relevant works for Newton's method for solving nonlinear equations on Banach spaces [2]- [6].
In Section 2 we cover the local whereas in Section 3 we study the semilocal convergence of method (1.3).

LOCAL CONVERGENCE ANALYSIS OF METHOD (1.3)
We need the definition [1], [5]: We can now introduce a Lipschitz as well as a center-Lipschitz-type continuity of X : Let R > 0. We suppose there exist continuous and nondecreasing functions where Without loss of generality we assume 0 (r) > 0, (r) > 0 on (0, R].
Let us consider the assumption: the function w given by (2.6) has a unique zero r * in [0, R] with We showed in [2] (see also [6], [7]): (a) the vector field X admits a unique singularity (b) Sequence {p n } (n ≥ 0) generated by method (3) is well defined, p n ∈ U (p 0 , t n ) for all n ≥ 0, and lim n→∞ p n = p * .(c) The following estimates hold true for all n ≥ 0: and Proof.Uses the more accurate (i.e. the one really needed) condition (2.1) instead of condition (2.2) used in [1] for the computation of the inverses X (p n ) −1 (n ≥ 0).The rest of the proof is identical to [1] and is omitted.scalar {t n } (n ≥ 0) given by (2.17) is also a majorizing sequence for {p n } (n ≥ 0).Sufficient convergence conditions for sequence (1.27) which are weaker than (2.10) have already been given in [3]- [7].Note that See also Remark 3.3.

SEMILOCAL CONVERGENCE ANALYSIS OF METHOD (1.3)
An extension of the Newton-Kantorovich theorem to finite-dimensional and complete Riemannian manifolds has been given by Ferreira and Svaiter in [7] and has been improved by us in [4].Moreover the results in [7] were shown to be special cases of Theorem 3.1 in [1].Here we weaken these results using L-Lipschitz and L 0 -center Lipschitz conditions for tensors.Definition 3.1.A (1, k)-tensor T on M is said to be: L 0 -center Lipschitz continuous on a subset S of M , if for all geodesic curve γ : [0, 1] → M with endpoints in S and p 0 ∈ M (3.1) and L-Lipschitz continuous on S, if We can show the following improvement of Theorem 5.1 in [1] for the semilocal convergence of method (1.3) (see [3, page 387, Case 3, for δ = δ 0 ]): Theorem 3.2.Under hypotheses (3.1) and (3.2) on S = U (x 0 , R) further suppose there exists p 0 ∈ M such that X (p 0 ) ∈ GL(T p 0 M ).Set: where L 0 ≤ L holds in general and L L 0 can be arbitrarily large [3]- [7].If L 0 = L holds then our theorem reduces to Theorem 5.1 in [1].Otherwise it is an improvement.Indeed, the condition corresponding to (3.3) in [1] is given by but not vice versa.Moreover the corresponding majorizing sequence is given by (3.13) z 0 = 0, z n+1 = z n − w 1 (zn) w 1 (zn) , where ) holding as strict inequalities for n ≥ 1 if L 0 < L. Finally note that in [3] we provided sufficient convergence conditions for iteration (3.6) that are even weaker than (3.3).
All the above justify the advantages of our approach already stated in the introduction of the paper.These ideas can be used to improve the rest of the results stated in [1].However we leave the details for the motivated reader.
r) denote the class of all the piecewise geodesic curves c : [0, T ] → M which satisfy: (a) c(0) = p 0 and the length of c is no greater than r; (b) there exists τ ∈ (0, T ) such that c [0,τ ] is a minimizing geodesic and c [τ,T ] is a geodesic.
Remark 2.5.(a)If 0 (r) = (r) our Theorem 2.4 reduces to Theorem 3.1 in[1].Otherwise (see(2.4)) it is an improvement over it as already shown in Proposition 2.3.Note also that the rest of the bounds obtained in Theorem 3.1 in[1] (see parts (iv)-(v) of Theorem 3.1) hold true with {t n } replacing sequence {r n } there but we decided not to include those bounds here to avoid repetitions.(b) Theorem 2.4 remains valid for a C 1 vector field X :D ⊂ M → T M which is defined only on an open subset D of M provided U (p 0 , R) ⊆ D [1], [5],[7].(c) It follows from the proof of the theorem that the sharper (than {t n }) scalar sequence {s n } generated by (3.6) is monotonically increasing, bounded above by b η and converges to s * ∈ [η, b η].(b) Sequence {p n } (n ≥ 0) generated by method (1.3) is well defined, remains in U (p 0 , s * ) for all n ≥ 0 and converges to a unique singularity of X in U (p 0 , s * ).(c) The following error bounds hold true for all n ≥ 0: (3.7) d(p n+1 , p n ) ≤ s n+1 − s n , and (3.8) d(p n , p * ) ≤ s * − p n .(d) If there exists R 1 ∈ [s * , R] such that (3.9) aL 0 (s * + R 1 ) ≤ 2,then p * is unique in U (p 0 , R 1 ).Proof.As the proof in Theorem 5.1 in [1, p. 18] but we use condition (3.1) for the computation of the upper bounds of X (p n ) −1 (n ≥ 0) which is really needed instead of (3.2) used in[1].Remark 3.3.(a) As in Remark 2.2 (3.10)