Saïd Hilout

Weaker conditions for the convergence of Newton's method

Newtons method is often used for solving nonlinear equations. In this paper, we show that Newtons method converges under weaker convergence criteria than those given in earlier studies, such as Argyros (2004) [2, p. 387], Argyros and Hilout (2010)[11, p. 12], Argyros et al. (2011) [12, p. 26], Ortega and Rheinboldt (1970) [26, p. 421], Potra and Ptak (1984) [36, p. 22]. These new results are illustrated by several numerical examples, for which the older convergence criteria do not hold but for which our weaker convergence criteria are satisfied.

Computational methods in nonlinear analysis : efficient algorithms, fixed point theory and applications

Kantorovich Theory for Newton-Like Methods Holder Conditions and Newton-Type Methods Regular Smoothness Conditions for Iterative Methods Fixed Point Theory and Iterative Methods Mathematical Programming Fixed Point Theory for Set-Valued Mapping Special Convergence Conditions Recurrent Functions and Newton-Like Methods Recurrent Functions and Special Iterative Methods.

Extending the Newton-Kantorovich hypothesis for solving equations

The famous Newton-Kantorovich hypothesis (Kantorovich and Akilov, 1982 [3], Argyros, 2007 [2], Argyros and Hilout, 2009 [7]) has been used for a long time as a sufficient condition for the convergence of Newtons method to a solution of an equation in connection with the Lipschitz continuity of the Frechet-derivative of the operator involved. Here, using Lipschitz and center-Lipschitz conditions, and our new idea of recurrent functions, we show that the Newton-Kantorovich hypothesis can be weakened, under the same information. Moreover, the error bounds are tighter than the corresponding ones given by the dominating Newton-Kantorovich theorem (Argyros, 1998 [1]; [2,7]; Ezquerro and Hernandez, 2002 [11]; [3]; Proinov 2009, 2010 [16,17]). Numerical examples including a nonlinear integral equation of Chandrasekhar-type (Chandrasekhar, 1960 [9]), as well as a two boundary value problem with a Greens kernel (Argyros, 2007 [2]) are also provided in this study.

Improved local convergence of Newton's method under weak majorant condition

We provide a local convergence analysis for Newtons method under a weak majorant condition in a Banach space setting. Our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than before [14]. Special cases and numerical examples are also provided in this study.

Extending the applicability of the Gauss---Newton method under average Lipschitz---type conditions

We extend the applicability of the Gauss–Newton method for solving singular systems of equations under the notions of average Lipschitz–type conditions introduced recently in Li et al. (J Complex 26(3):268–295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss–Newton method than in Li et al. (J Complex 26(3):268–295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110–122, 2008; Li et al. Comput Math Appl 47:1057–1067, 2004; Wang Math Comput 68(255):169–186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268–295, 2010). Applications to some special cases of Kantorovich–type conditions are also provided in this study.

On the local convergence of fast two-step Newton-like methods for solving nonlinear equations

A local convergence analysis is presented for a fast two-step Newton-like method (TSNLM) for solving nonlinear equations in a Banach space setting. The TSNLM unifies earlier methods such as Newtons, Secant, Newton-like, Chebyshev-Secant, Chebyshev-Newton, Steffensen, Stirlings and other single or multistep methods. Numerical examples and a comparative study of these methods validating our theoretical results are also given in the concluding section of this paper.

Improved generalized differentiability conditions for Newton-like methods

We provide a semilocal convergence analysis for Newton-like methods using the @w-versions of the famous Newton-Kantorovich theorem (Argyros (2004) [1], Argyros (2007) [3], Kantorovich and Akilov (1982) [13]). In the special case of Newtons method, our results have the following advantages over the corresponding ones (Ezquerro and Hernaandez (2002) [10], Proinov (2010) [17]) under the same information and computational cost: finer error estimates on the distances involved; at least as precise information on the location of the solution, and weaker sufficient convergence conditions. Numerical examples, involving a Chandrasekhar-type nonlinear integral equation as well as a differential equation with Greens kernel are provided in this study.

A convergence analysis for directional two-step Newton methods

A semilocal convergence analysis for directional two-step Newton methods in a Hilbert space setting is provided in this study. Two different techniques are used to generate the sufficient convergence results, as well as the corresponding error bounds. The first technique uses our new idea of recurrent functions, whereas the second uses recurrent sequences. We also compare the results of the two techniques.

On an improved convergence analysis of Newton's method

We present a local as well as a semi-local convergence analysis of Newtons method for solving nonlinear equations in a Banach space setting. The new approach leads in the local case to larger convergence radius than before (Argyros and Hilout, 2013, 2012 [5,6] and Rheinboldt, 1977 [17]). In the semi-local case, we obtain weaker sufficient convergence conditions; tighter error bounds distances involved and a more precise information on the location of the solution than in earlier studies such as Argyros (2007) [2], Argyros and Hilout (2013, 2012) [5,6] and Ortega and Rheinboldt (1970) [13]. Upper and lower bounds on the limit points of the majorizing sequences are also provided in this study. These advantages are obtained under the same computational cost as in the earlier stated studies. Finally, the numerical examples illustrate the theoretical results.

Majorizing sequences for iterative methods

We provide convergence results for very general majorizing sequences of iterative methods. Using our new concept of recurrent functions, we unify the semilocal convergence analysis of Newton-type methods (NTM) under more general Lipschitz-type conditions. We present two very general majorizing sequences and we extend the applicability of (NTM) using the same information before Chen and Yamamoto (1989) [13], Deuflhard (2004) [16], Kantorovich and Akilov (1982) [19], Miel (1979) [20], Miel (1980) [21] and Rheinboldt (1968) [30]. Applications, special cases and examples are also provided in this study to justify the theoretical results of our new approach.