scholarly journals A convergence analysis of the iteratively regularized Gauss–Newton method under the Lipschitz condition

2008 ◽  
Vol 24 (4) ◽  
pp. 045002 ◽  
Author(s):  
Qinian Jin
Keyword(s):  
Convergence Analysis ◽  
Newton Method ◽  
Lipschitz Condition

ON THE SEMILOCAL CONVERGENCE OF NEWTON'S METHOD FOR SECTIONS ON RIEMANNIAN MANIFOLDS

2014 ◽  
Vol 07 (01) ◽  
pp. 1450007
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Convergence Analysis ◽  
Lipschitz Condition ◽  
Riemannian Manifolds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Geometric Model ◽  
Semilocal Convergence ◽  
Convergence Result ◽  
Numer Anal ◽  
The One

We present a semilocal convergence analysis of Newton's method for sections on Riemannian manifolds. Using the notion of a 2-piece L-average Lipschitz condition introduced in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] in combination with the weaker center 2-piece L1-average Lipschitz condition given by us in this paper, we provide a tighter convergence analysis than the one given in [C. Li and J. H. Wang, Newton's method for sections on Riemannian manifolds: Generalized covariant α-theory, J. Complexity24 (2008) 423–451] which in turn has improved the works in earlier studies such as [R. L. Adler, J. P. Dedieu, J. Y. Margulies, M. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numer. Anal.22 (2002) 359–390; F. Alvarez, J. Bolte and J. Munier, A unifying local convergence result for Newton's method in Riemannian manifolds, Found. Comput. Math.8 (2008) 197–226; J. P. Dedieu, P. Priouret and G. Malajovich, Newton's method on Riemannian manifolds: Covariant α-theory, IMA J. Numer. Anal.23 (2003) 395–419].

scholarly journals On Local Convergence Analysis of Inexact Newton Method for Singular Systems of Equations under Majorant Condition

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Fangqin Zhou
Keyword(s):  
Convergence Analysis ◽  
Newton Method ◽  
Local Convergence ◽  
Singular Systems ◽  
Inexact Newton Method ◽  
Systems Of Equations ◽  
Convergence Ball ◽  
Special Cases ◽  
Majorant Condition ◽  
Local Convergence Analysis

We present a local convergence analysis of inexact Newton method for solving singular systems of equations. Under the hypothesis that the derivative of the function associated with the singular systems satisfies a majorant condition, we obtain that the method is well defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the singular systems. It also allows us to obtain an estimate of convergence ball for inexact Newton method and some important special cases.

SIMPLE YET EFFICIENT ALGORITHM FOR SOLVING NONLINEAR EQUATIONS

2010 ◽  
Vol 01 (04) ◽  
pp. 509-522
Author(s):  
SANJAY KUMAR KHATTRI ◽  
MUHAMMAD ASLAM NOOR
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Newton Method ◽  
Efficient Algorithm ◽  
Iterative Process ◽  
Computational Results ◽  
Numerical Work ◽  
Software Packages ◽  
Practical Algorithm ◽  
Convergence Orders

In this work, we develop a simple yet robust and highly practical algorithm for constructing iterative methods of higher convergence orders. The algorithm can be easily implemented in software packages for achieving desired convergence orders. Convergence analysis shows that the algorithm can develop methods of various convergence orders which is also supported through the numerical work. The algorithm is shown to converge even if the derivative of the function vanishes during the iterative process. Computational results ascertain that the developed algorithm is efficient and demonstrate equal or better performance as compared with other well known methods and the classical Newton method.

Derivative-Free Optimal Iterative Methods

2010 ◽  
Vol 10 (4) ◽  
pp. 368-375 ◽  
Author(s):  
S.K. Khattri ◽  
R.P. Agarwal
Keyword(s):  
Convergence Analysis ◽  
Iterative Methods ◽  
Newton Method ◽  
Fourth Order ◽  
Free Parameter ◽  
Convergence Order ◽  
Derivative Free ◽  
The Family ◽  
Optimal Value

AbstractIn this study, we develop an optimal family of derivative-free iterative methods. Convergence analysis shows that the methods are fourth order convergent, which is also verified numerically. The methods require three functional evaluations during each iteration. Though the methods are independent of derivatives, computa- tional results demonstrate that the family of methods are efficient and demonstrate equal or better performance as compared with many well-known methods and the clas- sical Newton method. Through optimization we derive an optimal value for the free parameter and implement it adaptively, which enhances the convergence order without increasing functional evaluations.

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