scholarly journals The Convergence Ball and Error Analysis of the Relaxed Secant Method

2017 ◽  
Vol 2017 ◽  
pp. 1-7
Author(s):  
Rongfei Lin ◽  
Qingbiao Wu ◽  
Minhong Chen ◽  
Lu Liu
Keyword(s):  
Nonlinear Equation ◽  
Error Analysis ◽  
Error Estimate ◽  
Secant Method ◽  
Speed Of Convergence ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Lipschitz Continuous ◽  
First Order ◽  
Nonlinear Equation Systems

A relaxed secant method is proposed. Radius estimate of the convergence ball of the relaxed secant method is attained for the nonlinear equation systems with Lipschitz continuous divided differences of first order. The error estimate is also established with matched convergence order. From the radius and error estimate, the relation between the radius and the speed of convergence is discussed with parameter. At last, some numerical examples are given.

scholarly journals On the Convergence Ball and Error Analysis of the Modified Secant Method

2018 ◽  
Vol 2018 ◽  
pp. 1-5
Author(s):  
Rongfei Lin ◽  
Qingbiao Wu ◽  
Minhong Chen ◽  
Xuemin Lei
Keyword(s):  
Error Estimate ◽  
Convergence Theorem ◽  
Nonlinear Operator ◽  
Secant Method ◽  
Convergence Properties ◽  
Fibonacci Series ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Iteration Methods ◽  
Local Convergence Theorem

We aim to study the convergence properties of a modification of secant iteration methods. We present a new local convergence theorem for the modified secant method, where the derivative of the nonlinear operator satisfies Lipchitz condition. We introduce the convergence ball and error estimate of the modified secant method, respectively. For that, we use a technique based on Fibonacci series. At last, some numerical examples are given.

scholarly journals Multiparameter extrapolation and deflation methods for solving equation systems

1984 ◽  
Vol 7 (4) ◽  
pp. 793-802 ◽  
Author(s):  
A. J. Hughes Hallett
Keyword(s):  
Applied Sciences ◽  
Nonlinear Equation ◽  
Newton Method ◽  
Single Parameter ◽  
Iterative Techniques ◽  
First Order ◽  
Convergence Results ◽  
Nonlinear Equation Systems

Most models in economics and the applied sciences are solved by first order iterative techniques, usually those based on the Gauss-Seidel algorithm. This paper examines the convergence of multiparameter extrapolations (accelerations) of first order iterations, as an improved approximation to the Newton method for solving arbitrary nonlinear equation systems. It generalises my earlier results on single parameter extrapolations. Richardson's generalised method and the deflation method for detecting successive solutions in nonlinear equation systems are also presented as multiparameter extrapolations of first order iterations. New convergence results are obtained for those methods.

scholarly journals A Discontinuous Finite Volume Method for the Darcy-Stokes Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Zhe Yin ◽  
Ziwen Jiang ◽  
Qiang Xu
Keyword(s):  
Error Estimate ◽  
Finite Volume Method ◽  
Finite Volume ◽  
Stokes Equations ◽  
Optimal Error Estimate ◽  
Optimal Error ◽  
Numerical Examples ◽  
First Order ◽  
Theoretical Predictions ◽  
Volume Method

This paper proposes a discontinuous finite volume method for the Darcy-Stokes equations. An optimal error estimate for the approximation of velocity is obtained in a mesh-dependent norm. First-orderL2-error estimates are derived for the approximations of both velocity and pressure. Some numerical examples verifying the theoretical predictions are presented.

The convergence ball and error analysis of the two-step Secant method

2017 ◽  
Vol 32 (4) ◽  
pp. 397-406 ◽  
Author(s):  
Rong-fei Lin ◽  
Qing-biao Wu ◽  
Min-hong Chen ◽  
Yasir Khan ◽  
Lu Liu
Keyword(s):  
Error Analysis ◽  
Secant Method ◽  
Convergence Ball

SOME HIGH ORDER ITERATIVE METHODS FOR NONLINEAR EQUATIONS BASED ON THE MODIFIED HOMOTOPY PERTURBATION METHODS

2010 ◽  
Vol 03 (03) ◽  
pp. 395-408
Author(s):  
Bilian Chen ◽  
Yajun Xie ◽  
Changfeng Ma
Keyword(s):  
Nonlinear Equation ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Perturbation Methods ◽  
Systems Of Nonlinear Equations ◽  
Convergence Criteria ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
Nonlinear Equation Systems ◽  
High Order Iterative Methods

In this paper, we present some efficient iterative methods for solving nonlinear equation (systems of nonlinear equations, respectively) by using modified homotopy perturbation methods. We also discuss the convergence criteria of the present methods. Some numerical examples are given to illustrate the performance and efficiency of the proposed methods.

scholarly journals Discontinuous Mixed Covolume Methods for Linear Parabolic Integrodifferential Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Ailing Zhu
Keyword(s):  
Error Analysis ◽  
Error Estimate ◽  
Optimal Order ◽  
Triangular Meshes ◽  
Element Space ◽  
Order Error ◽  
First Order ◽  
Fully Discrete ◽  
Mixed Element

The semidiscrete and fully discrete discontinuous mixed covolume schemes for the linear parabolic integrodifferential problems on triangular meshes are proposed. The error analysis of the semidiscrete and fully discrete discontinuous mixed covolume scheme is presented and the optimal order error estimate in discontinuousH(div)and first-order error estimate inL2are obtained with the lowest order Raviart-Thomas mixed element space.

scholarly journals A Novel Geometric Modification to the Newton-Secant Method to Achieve Convergence of Order1+2and Its Dynamics

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Gustavo Fernández-Torres
Keyword(s):  
Nonlinear Equation ◽  
Newton’S Method ◽  
Newton's Method ◽  
Secant Method ◽  
New Method ◽  
Convergence Order ◽  
Dynamical Analysis ◽  
Numerical Examples ◽  
Modified Method

A geometric modification to the Newton-Secant method to obtain the root of a nonlinear equation is described and analyzed. With the same number of evaluations, the modified method converges faster than Newton’s method and the convergence order of the new method is1+2≈2.4142. The numerical examples and the dynamical analysis show that the new method is robust and converges to the root in many cases where Newton’s method and other recently published methods fail.

Local convergence of deformed Euler–Halley-type methods in Banach space under weak conditions

2017 ◽  
Vol 10 (02) ◽  
pp. 1750086
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Error Estimates ◽  
Local Convergence ◽  
Fréchet Derivative ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Frechet Derivative ◽  
Local Convergence Analysis

We present a unified local convergence analysis for deformed Euler–Halley-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Euler, Halley and other high order methods. The convergence ball and error estimates are given for these methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second Fréchet derivative. Numerical examples are also provided in this study.

scholarly journals LOCAL CONVERGENCE OF JARRATT-TYPE METHODS WITH LESS COMPUTATION OF INVERSION UNDER WEAK CONDITIONS

2017 ◽  
Vol 22 (2) ◽  
pp. 228-236
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Nonlinear Equation ◽  
Convergence Analysis ◽  
Error Estimates ◽  
Local Convergence ◽  
Convergence Ball ◽  
Numerical Examples ◽  
Space Setting ◽  
Local Convergence Analysis ◽  
Methods Numerical

We present a local convergence analysis for Jarratt-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Earlier studies cannot be used to solve equations using such methods. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study.

scholarly journals Discontinuous Mixed Covolume Methods for Parabolic Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ailing Zhu ◽  
Ziwen Jiang
Keyword(s):  
Error Analysis ◽  
Error Estimate ◽  
Error Estimates ◽  
Parabolic Problems ◽  
Optimal Order ◽  
Order Error ◽  
First Order ◽  
Fully Discrete ◽  
Optimal Order Error Estimates ◽  
Backward Euler

We present the semidiscrete and the backward Euler fully discrete discontinuous mixed covolume schemes for parabolic problems on triangular meshes. We give the error analysis of the discontinuous mixed covolume schemes and obtain optimal order error estimates in discontinuousHdivand first-order error estimate inL2.

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