Error Bounds for Newton’s Method Under the Kantorovich Assumptions

1986 ◽  
pp. 197-208 ◽  
Author(s):  
Tetsuro Yamamoto
Keyword(s):  
Error Bounds ◽  
Newton’S Method ◽  
Newton's Method

A CONTINUATION METHOD AND ITS CONVERGENCE FOR SOLVING NONLINEAR EQUATIONS IN BANACH SPACES

2013 ◽  
Vol 10 (04) ◽  
pp. 1350021 ◽  
Author(s):  
M. PRASHANTH ◽  
D. K. GUPTA
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Nonlinear Equations ◽  
Error Bounds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Continuation Method ◽  
Continuity Condition ◽  
Majorizing Sequences ◽  
Chebyshev’S Method

A continuation method is a parameter based iterative method establishing a continuous connection between two given functions/operators and used for solving nonlinear equations in Banach spaces. The semilocal convergence of a continuation method combining Chebyshev's method and Convex acceleration of Newton's method for solving nonlinear equations in Banach spaces is established in [J. A. Ezquerro, J. M. Gutiérrez and M. A. Hernández [1997] J. Appl. Math. Comput.85: 181–199] using majorizing sequences under the assumption that the second Frechet derivative satisfies the Lipschitz continuity condition. The aim of this paper is to use recurrence relations instead of majorizing sequences to establish the convergence analysis of such a method. This leads to a simpler approach with improved results. An existence–uniqueness theorem is given. Also, a closed form of error bounds is derived in terms of a real parameter α ∈ [0, 1]. Four numerical examples are worked out to demonstrate the efficacy of our convergence analysis. On comparing the existence and uniqueness region and error bounds for the solution obtained by our analysis with those obtained by using majorizing sequences, it is found that our analysis gives better results in three examples, whereas in one example it gives the same results. Further, we have observed that for particular values of the α, our analysis reduces to those for Chebyshev's method (α = 0) and Convex acceleration of Newton's method (α = 1) respectively with improved results.

scholarly journals Error bounds for the modified Newton's method

1976 ◽  
Vol 14 (3) ◽  
pp. 427-433 ◽  
Author(s):  
A.L. Andrew
Keyword(s):  
Error Bounds ◽  
Newton’S Method ◽  
Convergence Theorem ◽  
Newton's Method ◽  
Modified Newton’S Method ◽  
Strengthened Form

A strengthened form of the Kantorovich convergence theorem for the modified Newton's method is proved. The result is compared with previously known results.

scholarly journals On Newton's method and nondiscrete mathematical induction

1988 ◽  
Vol 38 (1) ◽  
pp. 131-140 ◽  
Author(s):  
Ioannis K. Argyros
Keyword(s):  
Lipschitz Condition ◽  
Error Bounds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Mathematical Induction ◽  
Fréchet Derivative ◽  
Hölder Continuous ◽  
Frechet Derivative ◽  
Nondiscrete Mathematical Induction ◽  
Sharp Error

The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton's method. We assume only that the operator has Hölder continuous derivatives. In the case when the Fréchet-derivative of the operator satisfies a Lipschitz condition, our results reduce to the ones obtained by Ptak and Potra in 1972.

scholarly journals Semilocal analysis of equations with smooth operators

1981 ◽  
Vol 4 (3) ◽  
pp. 553-563 ◽  
Author(s):  
George J. Miel
Keyword(s):  
Recent Work ◽  
Error Bounds ◽  
Newton’S Method ◽  
Newton's Method ◽  
Nonlinear Operator ◽  
Operator Equations ◽  
Nonlinear Operator Equations ◽  
Kantorovich Theorem ◽  
Refined Version

Recent work on semilocal analysis of nonlinear operator equations is informally reviewed. A refined version of the Kantorovich theorem for Newton's method, with new error bounds, is presented. Related topics are briefly surveyed.

Optimal Power Flow Using Newton’s Method

2012 ◽  
Vol 3 (2) ◽  
pp. 167-169
Author(s):  
F.M.PATEL F.M.PATEL ◽  
◽  
N. B. PANCHAL N. B. PANCHAL
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Power Flow ◽  
Optimal Power Flow ◽  
Optimal Power

An Efficient Sixth-Order Convergent Newton-Type Iterative Method for Nonlinear Equations

2012 ◽  
Vol 220-223 ◽  
pp. 2585-2588
Author(s):  
Zhong Yong Hu ◽  
Fang Liang ◽  
Lian Zhong Li ◽  
Rui Chen
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Efficiency Index ◽  
Sixth Order ◽  
Type Method ◽  
Second Derivatives ◽  
Better Than

In this paper, we present a modified sixth order convergent Newton-type method for solving nonlinear equations. It is free from second derivatives, and requires three evaluations of the functions and two evaluations of derivatives per iteration. Hence the efficiency index of the presented method is 1.43097 which is better than that of classical Newton’s method 1.41421. Several results are given to illustrate the advantage and efficiency the algorithm.

Sign in / Sign up

Export Citation Format

Share Document