A new approach for finding weaker conditions for the convergence of Newton's method

Abstract

Abstract. The Newton-Kantorovich hypothesis (15) has been used for a long time as a sufficient condition for convergence of Newton's method to a locally unique solution of a nonlinear equation in a Banach space setting. Recently in [3], [4] we showed that this hypothesis can always be replaced by a condition weaker in general (see (18), (19) or (20)) whose verification requires the same computational cost. Moreover, finer error bounds and at least as precise information on the location of the solution can be obtained this way. Here we show that we can further weaken conditions (18)- (20) A large number of problems in applied mathematics and also in engineering are solved by finding solutions of certain equations. For example, dynamical systems are mathematically modeled by difference or differential equations, and their solutions usually represent states of the systems. For the sake of simplicity, assume that a time-invariant system is driven by the equationẋ = G(x) (for some suitable operator G), where x is the state. Then the equilibrium states are determined by solving equation (1). Similar equations are used in the case of discrete systems. The unknowns of engineering equations can be functions (difference, differential, and integral equations), 2000 Mathematics Subject Classification: 65H10, 65G99, 47H17, 49M15.

Expanding the applicability of inexact Newton methods using restricted convergence domains

“…(3.5) Then (3.4)⇒(3.5) by (2.4). Examples where (1.6) or (2.11) hold as strict inequalities and therefore the advantages (A) hold can be found in [1,2,3,4].…”

Extended convergence of a sixth order scheme for solving equations under ω–continuity conditions

“…But this is achieved only in special occasions. That is why iterative schemes are developed generating sequences converging to x * under suitable convergence criteria [1,2,3,4,5,6,7,8,9,10,11,12,13].…”

Extending the applicability of Newton's and Secant methods under regular smoothness

“…A convergence analysis for both methods has been provided under various assumptions by many authors. A survey of such results can be found in [3], [9], and the references there (see, also [1], [2], [4]- [8], [10]- [18]). In the excellent works by Galperin [6], [7], the concept of regular smoothness was introduced, which became a viable framework for the study of the convergence of iterative procedures such as Newton's method, and Secant method.…”

“…It follows from (2.37), and the Banach lemma of invertible operators [11], [2], [4], [5], that F ′ 0 (x n ) −1 ∈ L(Y, X), so that (2.27), and (2.28) hold. Note that in [7], less precise estimates were obtained with ω replacing ω 0 in estimates (2.36), and (2.37).…”