Convergence analysis for a deformed Newton's method with third-order in Banach space under γ-condition

We present a new modified Newton's method with third-order convergence and compare it with the Jarratt method, which is of fourth-order. Based on this new method, we obtain a family of Newton-type methods, which converge cubically. Numerical examples show that the presented method can compete with Newton's method and other known third-order modifications of Newton's method.

Download Full-text

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

Asian-European Journal of Mathematics ◽

10.1142/s1793557114500077 ◽

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].

Download Full-text

UNA NUEVA FORMA DEL TEOREMA DE KANTOROVICH PARA EL ME´ TODO DE NEWTON

Revista TECNIA ◽

10.21754/tecnia.v23i1.69 ◽

2017 ◽

Vol 23 (1) ◽

pp. 79

Author(s):

Leopoldo Paredes Soria ◽

Pedro Canales García

Keyword(s):

Banach Space ◽

Linear Operator ◽

Unique Solution ◽

Newton’S Method ◽

Convergence Theorem ◽

Newton's Method ◽

Convergence Condition ◽

Palabras Clave ◽

Lipschitz Conditions

Una nueva forma de convergencia de tipo Kantorovich para el me´todo de Newton es establecido para aproximarse localmente a una solucio´n u´nica de la ecuacio´n F (x) = 0 definido sobre un espacio de Banach. Se asume que el operador F es dos veces diferenciable Fre´chet, y que Fr, F rr satisface las condiciones de Lipschitz. Nuestra condicio´n de convergencia difiere de los me´todos conocidos y por lo tanto tiene un valor teo´rico y pra´ctico Palabras clave.-Operador lineal, Diferenciable Fre´chet, Sucesio´n convergente, Unicidad. ABSTRACTA new Kantorovich-type convergence theorem for Newton’s method is established for approximating a locally unique solution of an equation F (x) = 0 defined on a Banach space. It is assumed that the operator F is twice Fre´chet differentiable, and that Fr, F rr satisfy Lipschitz conditions. Our convergence condition differs from earlier ones and therefore it has theoretical and practical value. Keywords.-Linear operator, Differentiable Fre´chet, Convergent succession, Uniqueness.

Download Full-text

Unified Local Convergence for Newton’s Method and Uniqueness of the Solution of Equations under Generalized Conditions in a Banach Space

Mathematics ◽

10.3390/math7050463 ◽

2019 ◽

Vol 7 (5) ◽

pp. 463 ◽

Cited By ~ 2

Author(s):

Ioannis K. Argyros ◽

Ángel Alberto Magreñán ◽

Lara Orcos ◽

Íñigo Sarría

Keyword(s):

Banach Space ◽

Newton’S Method ◽

Newton's Method ◽

Convergence Region ◽

Precise Information ◽

Solution Of Equations ◽

Lipschitz Constants ◽

Uniqueness Of The Solution ◽

Generalized Newton ◽

Theoretical Results

Under the hypotheses that a function and its Fréchet derivative satisfy some generalized Newton–Mysovskii conditions, precise estimates on the radii of the convergence balls of Newton’s method, and of the uniqueness ball for the solution of the equations, are given for Banach space-valued operators. Some of the existing results are improved with the advantages of larger convergence region, tighter error estimates on the distances involved, and at-least-as-precise information on the location of the solution. These advantages are obtained using the same functions and Lipschitz constants as in earlier studies. Numerical examples are used to test the theoretical results.

Download Full-text

Convergence of Newton's method and uniqueness of the solution of equations in Banach space

IMA Journal of Numerical Analysis ◽

10.1093/imanum/20.1.123 ◽

2000 ◽

Vol 20 (1) ◽

pp. 123-134 ◽

Cited By ~ 96

Author(s):

X Wang

Keyword(s):

Banach Space ◽

Newton’S Method ◽

Newton's Method ◽

Solution Of Equations ◽

Uniqueness Of The Solution

Download Full-text

Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equations

Geodesy and Cartography ◽

10.2478/v10277-012-0013-x ◽

2011 ◽

Vol 60 (2) ◽

pp. 145-159 ◽

Cited By ~ 20

Author(s):

Marcin Ligas ◽

Piotr Banasik

Keyword(s):

Nonlinear Equations ◽

Newton’S Method ◽

Newton's Method ◽

System Of Nonlinear Equations ◽

Order Convergence ◽

Ellipsoidal Height ◽

Third Order ◽

The Third ◽

Geodetic Coordinates ◽

Modified Newton’S Method

Conversion between Cartesian and geodetic coordinates on a rotational ellipsoid by solving a system of nonlinear equationsA new method to transform from Cartesian to geodetic coordinates is presented. It is based on the solution of a system of nonlinear equations with respect to the coordinates of the point projected onto the ellipsoid along the normal. Newton's method and a modification of Newton's method were applied to give third-order convergence. The method developed was compared to some well known iterative techniques. All methods were tested on three ellipsoidal height ranges: namely, (-10 - 10 km) (terrestrial), (20 - 1000 km), and (1000 - 36000 km) (satellite). One iteration of the presented method, implemented with the third-order convergence modified Newton's method, is necessary to obtain a satisfactory level of accuracy for the geodetic latitude (σφ < 0.0004") and height (σh< 10-6km, i.e. less than a millimetre) for all the heights tested. The method is slightly slower than the method of Fukushima (2006) and Fukushima's (1999) fast implementation of Bowring's (1976) method.

Download Full-text