Complexity of Bezout's theorem IV: probability of success; extensions
SIAM Journal on Numerical Analysis
A new semilocal convergence theorem for Newton's method
Journal of Computational and Applied Mathematics
Complexity and real computation
Complexity and real computation
Convergence and Complexity of Newton Iteration for Operator Equations
Journal of the ACM (JACM)
Newton's method for overdetermined systems of equations
Mathematics of Computation
The Newton method for operators with Hölder continuous first derivative
Journal of Optimization Theory and Applications
Newton's method for analytic systems of equations with constant rank derivatives
Journal of Complexity
On the Newton-Kantorovich hypothesis for solving equations
Journal of Computational and Applied Mathematics
Kantorovich's type theorems for systems of equations with constant rank derivatives
Journal of Computational and Applied Mathematics
On a class of Newton-like methods for solving nonlinear equations
Journal of Computational and Applied Mathematics
Computational Theory of Iterative Methods, Volume 15
Computational Theory of Iterative Methods, Volume 15
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The famous for its simplicity and clarity Newton---Kantorovich hypothesis of Newton's method has been used for a long time as the sufficient convergence condition for solving nonlinear equations. Recently, in the elegant study by Hu et al. (J Comput Appl Math 219:110---122, 2008), a Kantorovich-type convergence analysis for the Gauss---Newton method (GNM) was given improving earlier results by Häubler (Numer Math 48:119---125, 1986), and extending some results by Argyros (Adv Nonlinear Var Inequal 8:93---99, 2005, 2007) to hold for systems of equations with constant rank derivatives. In this study, we use our new idea of recurrent functions to extend the applicability of (GNM) by replacing existing conditions by weaker ones. Finally, we provide numerical examples to solve equations in cases not covered before (Häubler, Numer Math 48:119---125, 1986; Hu et al., J Comput Appl Math 219:110---122, 2008; Kontorovich and Akilov 2004).