Characterizing Mesh Independent Quadratic Convergence of Newton's Method for a Class of Elliptic Problems

2012 ◽  
Vol 44 (3) ◽  
pp. 1279-1303 ◽  
Author(s):  
János Karátson
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Elliptic Problems

scholarly journals Quadratic Convergence of Smoothing Newton's Method for 0/1 Loss Optimization

2021 ◽  
Vol 31 (4) ◽  
pp. 3184-3211
Author(s):  
Shenglong Zhou ◽  
Lili Pan ◽  
Naihua Xiu ◽  
Hou-Duo Qi
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence

A New SPH Iterative Method for Solving Nonlinear Equations

2019 ◽  
Vol 17 (01) ◽  
pp. 1843005 ◽  
Author(s):  
Rahmatjan Imin ◽  
Ahmatjan Iminjan
Keyword(s):  
Iterative Method ◽  
Nonlinear Equations ◽  
Basic Principle ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Taylor Series Expansion ◽  
Numerical Examples ◽  
First Order ◽  
Kernel Approximation

In this paper, based on the basic principle of the SPH method’s kernel approximation, a new kernel approximation was constructed to compute first-order derivative through Taylor series expansion. Derivative in Newton’s method was replaced to propose a new SPH iterative method for solving nonlinear equations. The advantage of this method is that it does not require any evaluation of derivatives, which overcame the shortcoming of Newton’s method. Quadratic convergence of new method was proved and a variety of numerical examples were given to illustrate that the method has the same computational efficiency as Newton’s method.

scholarly journals A Picard-Type Iterative Scheme for Fredholm Integral Equations of the Second Kind

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 83
Author(s):  
José M. Gutiérrez ◽  
Miguel Á. Hernández-Verón
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Integral Equations ◽  
Nonlinear Equations ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Iterative Scheme ◽  
Fredholm Integral Equations

In this work, we present an application of Newton’s method for solving nonlinear equations in Banach spaces to a particular problem: the approximation of the inverse operators that appear in the solution of Fredholm integral equations. Therefore, we construct an iterative method with quadratic convergence that does not use either derivatives or inverse operators. Consequently, this new procedure is especially useful for solving non-homogeneous Fredholm integral equations of the first kind. We combine this method with a technique to find the solution of Fredholm integral equations with separable kernels to obtain a procedure that allows us to approach the solution when the kernel is non-separable.

scholarly journals Newton's method for the eigenvalue problem of a symmetric matrix

2021 ◽  
Vol 25 (2(36)) ◽  
pp. 75-82
Author(s):  
V. V. Verbitskyi ◽  
A. G. Huk
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Symmetric Matrix ◽  
Simple Eigenvalue ◽  
Normalization Condition ◽  
Nonlinear System Of Equations ◽  
Computational Costs ◽  
The Matrix ◽  
Eigenvalue And Eigenvector

Newton's method for calculating the eigenvalue and the corresponding eigenvector of a symmetric real matrix is considered. The nonlinear system of equations solved by Newton's method consists of an equation that determines the eigenvalue and eigenvector of the matrix and the normalization condition for the eigenvector. The method allows someone to simultaneously calculate the eigenvalue and the corresponding eigenvector. Initial approximations for the eigenvalue and the corresponding eigenvector can be found by the power method or by the reverse iteration with shift. A simple proof of the convergence of Newton's method in a neighborhood of a simple eigenvalue is proposed. It is shown that the method has a quadratic convergence rate. In terms of computational costs per iteration, Newton's method is comparable to the reverse iteration method with the Rayleigh ratio. Unlike reverse iteration, Newton's method allows to compute the eigenpair with better accuracy.

On the quadratic convergence of Newton’s method under center-Lipschitz but not necessarily Lipschitz hypotheses

2013 ◽  
Vol 63 (3) ◽  
Author(s):  
Ioannis Argyros ◽  
Saïd Hilout
Keyword(s):  
Banach Space ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Lipschitz Constant ◽  
Semilocal Convergence ◽  
Convergence Conditions ◽  
Numerical Examples ◽  
Convergence Results

AbstractWe provide new local and semilocal convergence results for Newton’s method in a Banach space. The sufficient convergence conditions do not include the Lipschitz constant usually associated with Newton’s method. Numerical examples demonstrating the expansion of Newton’s method are also provided in this study.

scholarly journals A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Ababu Teklemariam Tiruneh ◽  
W. N. Ndlela ◽  
S. J. Nkambule
Keyword(s):  
Newton Method ◽  
Newton’S Method ◽  
Newton's Method ◽  
Quadratic Convergence ◽  
Computational Effort ◽  
Secant Methods ◽  
Iterative Solutions ◽  
Solutions Of Equations ◽  
Modified Formula

An iterative formula based on Newton’s method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method has super-quadratic convergence of order 2.414 (i.e., ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number, or offshooting away to another root further from the desired domain or offshooting to an invalid domain where the function may not be defined. In addition when the derivative at the iteration point is zero, Newton method stalls. In most of these cases, hybrids of several methods such as Newton, bisection, and secant methods are suggested as substitute methods and Newton method is essentially blended with other methods or altogether abandoned. This paper argues that a solution is still possible in most of these cases by the application of Newton method alone without resorting to other methods and with the same computational effort (two functional evaluations per iteration) like the traditional Newton method. In addition, the proposed modified formula based on Newton method has better convergence characteristics than the traditional Newton method.

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
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