Convergence behavior of Gauss–Newton’s method and extensions of the Smale point estimate theory

Abstract

a b s t r a c tThe notions of Lipschitz conditions with L average are introduced to the study of convergence analysis of Gauss-Newton's method for singular systems of equations. Unified convergence criteria ensuring the convergence of Gauss-Newton's method for one kind of singular systems of equations with constant rank derivatives are established and unified estimates of radii of convergence balls are also obtained. Applications to some special cases such as the Kantorovich type conditions, γ -conditions and the Smale point estimate theory are provided and some important known results are extended and/or improved.

On the solution of systems of equations with constant rank derivatives

“…Let us compare our results with the corresponding ones in [19] (see also [22,Section 5.1]) which in turn have improved earlier results by Häubler [17]. We need to define scalar sequence {s n } by…”

“…In particular, we recommend the paper by Xu and Li [33], where (GNM) is studied under average Lipschitz conditions. Moreover, the paper by Li, Hu and Wang [22] is recommended, where (GNM) is studied using the Smale point estimate theory.…”

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, .…”

“…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.…”