General convergence conditions of Newton’s method for m-Fréchet differentiable operators

We present a new semilocal convergence analysis of Newton's method on Riemannian manifolds with values in a cone in order to solve the inclusion problem. Using more precise majorizing sequences than in earlier studies such as [J. H. Wang, S. Huang and C. Li, Extended Newton's method for mappings on Riemannian manifolds with values in a cone, Taiwanese J. Math.13(2B) (2009) 633–656] and the concept of L-average Lipschitz condition we provide: weaker sufficient convergence conditions; tighter error analysis on the distances involved and an at least as precise information on the solutions. These advantages are obtained using the same parameters and functions. Applications include the celebrated Newton–Kantorovich theorem.

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An improved convergence analysis of Newton's method for twice Fréchet differentiable operators

Applicationes Mathematicae ◽

10.4064/am40-4-6 ◽

2013 ◽

Vol 40 (4) ◽

pp. 459-481

Author(s):

Ioannis K. Argyros ◽

Sanjay K. Khattri

Keyword(s):

Convergence Analysis ◽

Newton’S Method ◽

Newton's Method ◽

Fréchet Differentiable

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Extending the convergence domain of Newton’s method for twice Fréchet differentiable operators

Analysis and Applications ◽

10.1142/s0219530515500013 ◽

2016 ◽

Vol 14 (02) ◽

pp. 303-319

Author(s):

Ioannis K. Argyros ◽

Á. Alberto Magreñán

Keyword(s):

Convergence Analysis ◽

Newton’S Method ◽

Newton's Method ◽

Computational Cost ◽

Semilocal Convergence ◽

Convergence Criteria ◽

Fréchet Differentiable ◽

Local Convergence Analysis ◽

Appl Math ◽

Semilocal Convergence Theorem

We present a semi-local convergence analysis of Newton’s method in order to approximate a locally unique solution of a nonlinear equation in a Banach space setting. Using center-Lipschitz condition on the first and the second Fréchet derivatives, we provide under the same computational cost a new and more precise convergence analysis than in earlier studies by Huang [A note of Kantorovich theorem for Newton iteration, J. Comput. Appl. Math. 47 (1993) 211–217] and Gutiérrez [A new semilocal convergence theorem for Newton’s method, J. Comput. Appl. Math. 79 (1997) 131–145]. Numerical examples where the old convergence criteria cannot apply to solve nonlinear equations but the new convergence criteria are satisfied are also presented at the concluding section of this paper.

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On the quadratic convergence of Newton’s method under center-Lipschitz but not necessarily Lipschitz hypotheses

Mathematica Slovaca ◽

10.2478/s12175-013-0123-4 ◽

2013 ◽

Vol 63 (3) ◽

Cited By ~ 5

Author(s):

Ioannis Argyros ◽

Saïd Hilout

Keyword(s):

Banach Space ◽

Newton’S Method ◽

Newton's Method ◽

Quadratic Convergence ◽

Lipschitz Constant ◽

Semilocal Convergence ◽

Convergence Conditions ◽

Numerical Examples ◽

Convergence Results

AbstractWe provide new local and semilocal convergence results for Newton’s method in a Banach space. The sufficient convergence conditions do not include the Lipschitz constant usually associated with Newton’s method. Numerical examples demonstrating the expansion of Newton’s method are also provided in this study.

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