scholarly journals On High Order Barycentric Root-Finding Methods

2016 ◽  
Vol 17 (3) ◽  
pp. 321
Author(s):  
Mario Meireles Graça ◽  
Pedro Miguel Lima
Keyword(s):  
Real Function ◽  
Simple Root ◽  
High Order ◽  
Convergence Order ◽  
Numerical Examples ◽  
Root Finding ◽  
Iterative Maps

To approximate a simple root of a real function f we construct a family of iterative maps, which we call Newton-barycentric functions, and analyse their convergence order. The performance of the resulting methods is illustrated by means of numerical examples. 

scholarly journals High-order compact finite volume scheme for the 2D multi-term time fractional sub-diffusion equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Baojin Su ◽  
Ziwen Jiang
Keyword(s):  
Diffusion Equation ◽  
Finite Volume ◽  
High Order ◽  
Convergence Order ◽  
Finite Volume Scheme ◽  
Interpolation Operator ◽  
Step Size ◽  
Numerical Examples ◽  
Volume Scheme ◽  
The Difference

AbstractBased on an L1 interpolation operator, a new high-order compact finite volume scheme is derived for the 2D multi-term time fractional sub-diffusion equation. It is shown that the difference scheme is unconditionally convergent and stable in $L_{\infty }$ L ∞ -norm. The convergence order is $O(\tau ^{2-\alpha }+h_{1}^{4}+h_{2}^{4})$ O ( τ 2 − α + h 1 4 + h 2 4 ) , where τ is the temporal step size and $h_{1}$ h 1 is the spatial step size in one direction, $h_{2}$ h 2 is the spatial step size in another direction. Two numerical examples are implemented, testifying to their efficiency and confirming their convergence order.

scholarly journals On an Aitken-Steffensen-Newton type method

2018 ◽  
Vol 34 (1) ◽  
pp. 85-92
Author(s):  
ION PAVALOIU ◽  
Keyword(s):  
Local Convergence ◽  
Convergence Result ◽  
Convergence Order ◽  
Monotone Convergence ◽  
Type Method ◽  
Numerical Examples ◽  
Newton Iterations ◽  
Convergence Orders

We consider an Aitken-Steffensen type method in which the nodes are controlled by Newton and two-step Newton iterations. We prove a local convergence result showing the q-convergence order 7 of the iterations. Under certain supplementary conditions, we obtain monotone convergence of the iterations, providing an alternative to the usual ball attraction theorems. Numerical examples show that this method may, in some cases, have larger (possibly sided) convergence domains than other methods with similar convergence orders.

scholarly journals A high order immersed finite element method for parabolic interface problems

2019 ◽  
Vol 29 ◽  
pp. 01007
Author(s):  
Derrick Jones ◽  
Xu Zhang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
High Order ◽  
Interface Problems ◽  
Numerical Schemes ◽  
Numerical Examples ◽  
Immersed Finite Element Method ◽  
Immersed Finite Element ◽  
Time Marching ◽  
Element Method

We present a high order immersed finite element (IFE) method for solving 1D parabolic interface problems. These methods allow the solution mesh to be independent of the interface. Time marching schemes including Backward-Eulerand Crank-Nicolson methods are implemented to fully discretize the system. Numerical examples are provided to test the performance of our numerical schemes.

scholarly journals Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 260 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Higher Order ◽  
High Order ◽  
Convergence Order ◽  
Eighth Order ◽  
First Derivative ◽  
Ball Convergence ◽  
Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

scholarly journals A fourth order method for finding a simple root of univariate function

2015 ◽  
Vol 34 (2) ◽  
pp. 197-211
Author(s):  
D. Sbibih ◽  
Abdelhafid Serghini ◽  
A. Tijini ◽  
A. Zidna
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Simple Root ◽  
Newton's Method ◽  
Order Method ◽  
Numerical Examples ◽  
Quadratic Interpolation ◽  
Univariate Function ◽  
Fourth Order Method

In this paper, we describe an iterative method for approximating asimple zero $z$ of a real defined function. This method is aessentially based on the idea to extend Newton's method to be theinverse quadratic interpolation. We prove that for a sufficientlysmooth function $f$ in a neighborhood of $z$ the order of theconvergence is quartic. Using Mathematica with its high precisioncompatibility, we present some numerical examples to confirm thetheoretical results and to compare our method with the others givenin the literature.

scholarly journals Improved Convergence Analysis of Gauss-Newton-Secant Method for Solving Nonlinear Least Squares Problems

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 99 ◽  
Author(s):  
Ioannis Argyros ◽  
Stepan Shakhno ◽  
Yurii Shunkin
Keyword(s):  
Least Squares ◽  
Convergence Analysis ◽  
Difference Method ◽  
Divided Difference ◽  
Nonlinear Least Squares ◽  
Convergence Order ◽  
Least Squares Problems ◽  
Convergence Radius ◽  
Numerical Examples ◽  
Theoretical Results

We study an iterative differential-difference method for solving nonlinear least squares problems, which uses, instead of the Jacobian, the sum of derivative of differentiable parts of operator and divided difference of nondifferentiable parts. Moreover, we introduce a method that uses the derivative of differentiable parts instead of the Jacobian. Results that establish the conditions of convergence, radius and the convergence order of the proposed methods in earlier work are presented. The numerical examples illustrate the theoretical results.

scholarly journals A High-Order Compact 2D FDFD Method for Waveguides

2012 ◽  
Vol 2012 ◽  
pp. 1-6
Author(s):  
Gang Zheng ◽  
Bing-Zhong Wang
Keyword(s):  
Boundary Condition ◽  
Finite Difference ◽  
Surface Impedance ◽  
Transverse Field ◽  
High Order ◽  
Impedance Boundary Condition ◽  
Two Dimensional ◽  
Dispersion Characteristics ◽  
Impedance Boundary ◽  
Numerical Examples

A high-order compact two-dimensional finite-difference frequency-domain (2D FDFD) method is proposed for the analysis of the dispersion characteristics of waveguides. A surface impedance boundary condition (SIBC) for the high-order compact 2D FDFD method is also given to model lossy metal waveguides. Four transverse field components are involved in the final eigenequation. Numerical examples are given, which show that this high-order compact 2D FDFD method is more efficient than the low-order compact 2D FDFD method and has a less storage cost.

scholarly journals A family of two-point methods with memory for solving nonlinear equations

2011 ◽  
Vol 5 (2) ◽  
pp. 298-317 ◽  
Author(s):  
Miodrag Petkovic ◽  
Jovana Dzunic ◽  
Ljiljana Petkovic
Keyword(s):  
Nonlinear Equations ◽  
Free Parameter ◽  
Convergence Order ◽  
Optimal Order ◽  
Additional Function ◽  
Numerical Examples ◽  
Derivative Free ◽  
With Memory ◽  
Theoretical Results ◽  
High Computational Efficiency

An efficient family of two-point derivative free methods with memory for solving nonlinear equations is presented. It is proved that the convergence order of the proposed family is increased from 4 to at least 2 + ?6 ? 4.45, 5, 1/2 (5 + ?33) ? 5.37 and 6, depending on the accelerating technique. The increase of convergence order is attained using a suitable accelerating technique by varying a free parameter in each iteration. The improvement of convergence rate is achieved without any additional function evaluations meaning that the proposed methods with memory are very efficient. Moreover, the presented methods are more efficient than all existing methods known in literature in the class of two-point methods and three-point methods of optimal order eight. Numerical examples and the comparison with the existing two-point methods are included to confirm theoretical results and high computational efficiency. 2010 Mathematics Subject Classification. 65H05

scholarly journals Families of optimal multipoint methods for solving nonlinear equations: A survey

2010 ◽  
Vol 4 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Miodrag Petkovic ◽  
Ljiljana Petkovic
Keyword(s):  
Nonlinear Equations ◽  
Computational Efficiency ◽  
Rapid Development ◽  
Convergence Order ◽  
Important Class ◽  
Iterative Root ◽  
Numerical Examples ◽  
Multipoint Methods ◽  
The 1960S ◽  
Order Four

Multipoint iterative root-solvers belong to the class of the most powerful methods for solving nonlinear equations since they overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. Although the construction of these methods has occurred in the 1960s, their rapid development have started in the first decade of the 21-st century. The most important class of multipoint methods are optimal methods which attain the convergence order 2n using n + 1 function evaluations per iteration. In this paper we give a review of optimal multipoint methods of the order four (n = 2), eight (n = 3) and higher (n > 3), some of which being proposed by the authors. All of them possess as high as possible computational efficiency in the sense of the Kung-Traub hypothesis (1974). Numerical examples are included to demonstrate a very fast convergence of the presented optimal multipoint methods.

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