A Convergence Analysis of the Iterated Lavrentiev Regularization Method under a Lipschitz Condition

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].

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Regularized Mixed Variational-Like Inequalities

Journal of Applied Mathematics ◽

10.1155/2012/863450 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14

Author(s):

Muhammad Aslam Noor ◽

Khalida Inayat Noor ◽

Saira Zainab ◽

Eisa Al-Said

Keyword(s):

Convergence Analysis ◽

Iterative Methods ◽

Regularization Method ◽

Auxiliary Principle ◽

Iterative Regularization ◽

Auxiliary Principle Technique ◽

Iterative Schemes ◽

Special Cases

We use auxiliary principle technique coupled with iterative regularization method to suggest and analyze some new iterative methods for solving mixed variational-like inequalities. The convergence analysis of these new iterative schemes is considered under some suitable conditions. Some special cases are also discussed. Our method of proofs is very simple as compared with other methods. Our results represent a significant refinement of the previously known results.

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EXTENDING THE APPLICABILITY OF NEWTON'S METHOD ON RIEMANNIAN MANIFOLDS WITH VALUES IN A CONE

Asian-European Journal of Mathematics ◽

10.1142/s1793557113500411 ◽

2013 ◽

Vol 06 (03) ◽

pp. 1350041

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Error Analysis ◽

Convergence Analysis ◽

Lipschitz Condition ◽

Riemannian Manifolds ◽

Newton’S Method ◽

Newton's Method ◽

Semilocal Convergence ◽

Inclusion Problem ◽

Convergence Conditions ◽

Precise Information

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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