Local convergence of Newton's method in Banach space from the viewpoint of the majorant principle

Extended local convergence analysis of inexact Gauss-Newton method for singular systems of equations under weak conditions

“…The results in [6] improved the corresponding ones in [21,22,23,24,25,42,43]. In the present study, we improved the results in [6], since D 1 ⊂ U (x * , r) leading to an at least as tight function h λ,θ than the one used in [6] (see also the Examples).…”

“…The advantages of our analysis over earlier works such as [8,9,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43] are also shown under the same computational cost for the functions and constants involved. These advantages include: a large radius of convergence and more precise error estimates on the distances x n+1 − x * for each n = 0, 1, 2, .…”

“…and a clearer relationship between the majorant function (see (2.8) and the associated least squares problems (1.1)). These advantages are obtained because we use a center-type majorant condition (see (2.11)) for the computation of inverses involved which is more precise that the majorant condition used in [21,22,23,24,25,26,30,31,39,40,41,42,43]. Moreover, these advantages are obtained under the same computational cost, since as we will see in section 3 and section 4, the computation of the majorant function requires the computation of the center-majorant function.…”

“…. , (1.6) where x 0 ∈ D is an initial point and F (x n ) † denotes the Moore-Penrose inverse of the linear operator (of matrix) F (x n ) [1,12,14,15,17,18,20,21,36].…”

“…. Finally, the majorant functions (see ω and v) is not necessarily differentiable as in [21,26,30,31,39,40,41,42,43] but just differentiable. This is an improvement modification and extends the applicability of the method.…”