Numerical Methods for Equations and its Applications

Improved convergence analysis for Newton-like methods

“…It follows from (2.18) and the Banach Lemma on invertible operators [2,4,14,19] that P −1 k+1 exists and…”

“…In the special case when b = 0, criterion (2.1) or criterion (2.7) reduce to the famous for its simplicity and clarity sufficient semilocal convergence criterion for Newton's method [2,4,14,19] defined for x −1 = x 0 and p n = q n for each n = 0, 1, 2, . .…”

“…We can simply set k 1 = k 0 and k 2 = k in (2.4) to obtain (2.5) However, in general k 0 ≤ k 1 and k ≤ k 2 (2.6) hold and k 1 k 0 , k k 2 can be arbitrarily large [2][3][4]. If we choose k 1 = k 2 , k 0 = k, then condition (2.4) reduces to a condition frequently used by many authors to study the convergence of Newton-like methods [2, 4, 7, 9-12, 15, 16, 19-24, 28, 30].…”

“…If T n = F (x n ) −1 , then (1.2) specializes to Newton's method [2,4], where as if T n = δF (x n , x n−1 ) −1 , δF being an approximation of the Fréchet-derivative F of the operator F [14], then (1.2) specializes to the secant method [2,4]. Other cases for T n are also possible [2,4,10]. It is well known that one obtains more efficient iterative algorithms if operator T n is kept piecewise constant.…”

“…Many Problems from Computational Sciences and other disciplines can be brought in the form of an equation like (1.1) using Mathematical Modelling [2,4,7,14,21,27]. The solutions of theses equations can be given in closed form only in special cased.…”