Convergence criterion of Newton's method for singular systems with constant rank derivatives

Abstract

The present paper is concerned with the convergence problem of Newton's method to solve singular systems of equations with constant rank derivatives. Under the hypothesis that the derivatives satisfy a type of weak Lipschitz condition, a convergence criterion based on the information around the initial point is established for Newton's method for singular systems of equations with constant rank derivatives. Applications to two special and important cases: the classical Lipschitz condition and the Smale's assumption, are provided; the latter, in particular, extends and improves the corresponding result due to Dedieu and Kim in [J.P. Dedieu, M. Kim, Newton's method for analytic systems of equations with constant rank derivatives, J. Complexity 18 (2002) 187-209].

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

“…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, .…”

“…Moreover, if taking λ = 0 (in this case λ n = 0 and r n = 0) in Theorem 3.1, we obtain the local convergence of Newton's method for solving the singular systems, which has been studied by Dedieu and Kim in [17] for analytic singular systems with constant rank derivatives and Li, Xu in [39] and Wang in [38] for some special singular systems with constant rank derivatives. (c) If g(t) < h λ,θ (t) then the improvements as mentioned in the Introduction of this study we obtained (see also the discussion above and below Definition 2.6) If F (x) is full column rank for every x ∈ U (x * , r), then we have…”

“…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.…”

“…. 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.…”

Convergence Analysis of General Version of Gauss-Type Proximal Point Method for Metrically Regular Mappings

“…A huge number of contributions have been studied on semilocal analysis for the Gauss-Newton method (cf. [4] [13]- [16]). In [4], Rashid et al have given a semilocal convergence analysis for the classical Gauss-type proximal point method.…”