scholarly journals On the Superlinear Convergence of Newton’s Method on Riemannian Manifolds

2017 ◽  
Vol 173 (3) ◽  
pp. 828-843 ◽  
Author(s):  
Teles A. Fernandes ◽  
Orizon P. Ferreira ◽  
Jinyun Yuan
Keyword(s):  
Riemannian Manifolds ◽  
Newton’S Method ◽  
Superlinear Convergence ◽  
Newton's Method

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 Kantorovich's Theorem on Newton's Method in Riemannian Manifolds

2002 ◽  
Vol 18 (1) ◽  
pp. 304-329 ◽  
Author(s):  
O.P. Ferreira ◽  
B.F. Svaiter
Keyword(s):  
Riemannian Manifolds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Kantorovich’S Theorem

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

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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