Generalized differentiability conditions for Newton's method

Extended Newton’s Method for Mappings on Riemannian Manifolds With Values in a Cone

“…Another important result on Newton's method is Smale's point estimate theory (i.e., α-theory and γ-theory) in [2,32], where the notion of an approximate zero was introduced and the rules to judge an initial point to be an approximate zero were established, depending only on the information of the analytic nonlinear operator at initial point x 0 and a solution x * , respectively. There are a lot of works on the weakness and/or extension of the continuity made on the first derivatives of the operators, see for example, [10,11,14,15,37] and references therein. It should be noted that Wang introduced in [37] the notions of Lipschitz conditions with L-average to unify both Kantorovich's and Smale's criteria.…”

On Newton-like methods of “bounded deterioration” using recurrent functions

“…(d) We showed that our semilocal convergence analysis is finer at the general setting of [10]. All references in the literature [4,11,13,16,19,20] reduce to (2.79), which can always be replaced by (2.77). (e) A popular choice (but not the only one) for the triplet (b n , Δ, K 1 ) is given by…”

“…Using our new idea of recurrent functions, a combination of Lipschitz and center-Lipschitz conditions on the first Fréchet-derivative, instead of only Lipschitz conditions [10], we provided an analysis with the following advantages over the work in [4,10,11,13,16,19,20]: weaker sufficient convergence conditions, and larger convergence domain. Note that these advantages are obtained under the same computational cost as in [10], since in practice the computation of the Lipschitz constant K requires the computation of K 0 (see Sect.…”

“…The conditions in [4,11,13,16,19,20] are also weakened, since the methods there are special cases of the ones in [10]. Then, with these changes in Proposition 2.7, and Theorem 2.3, the conclusions will still hold in this setting.…”

“…We provide a convergence analysis based on our new idea of recurrent functions. In particular, using the majorant method [1][2][3][4][5][6], and more precise majorizing sequences than before [4,10,11,13,16,19,20], we provide under the same hypotheses and computational cost: in the semilocal case finer error bounds on the distances x n+1 − x n , x n − x , (n ≥ 0), and more precise information on the location of the solution x .…”