scholarly journals Derivative Free Fourth Order Solvers of Equations with Applications in Applied Disciplines

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 586 ◽  
Author(s):  
Ramandeep Behl ◽  
Ioannis K. Argyros ◽  
Fouad Othman Mallawi ◽  
J. A. Tenreiro Machado
Keyword(s):  
Fourth Order ◽  
Error Bounds ◽  
Initial Guess ◽  
Numerical Examples ◽  
Know How ◽  
First Order ◽  
Derivative Free ◽  
Lipschitz Constants ◽  
Fourth Derivative ◽  
Complex Valued

This paper develops efficient equation solvers for real- and complex-valued functions. An earlier study by Lee and Kim, used the Taylor-type expansions and hypotheses on higher than first order derivatives, but no derivatives appeared in the suggested method. However, we have many cases where the calculations of the fourth derivative are expensive, or the result is unbounded, or even does not exist. We only use the first order derivative of function Ω in the proposed convergence analysis. Hence, we expand the utilization of the earlier scheme, and we study the computable radii of convergence and error bounds based on the Lipschitz constants. Furthermore, the range of starting points is also explored to know how close the initial guess should be considered for assuring convergence. Several numerical examples where earlier studies cannot be applied illustrate the new technique.

scholarly journals Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1242
Author(s):  
Ramandeep Behl ◽  
Sonia Bhalla ◽  
Eulalia Martínez ◽  
Majed Aali Alsulami
Keyword(s):  
Iterative Methods ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Real Life ◽  
Multiple Roots ◽  
Computational Results ◽  
Nonlinear Functions ◽  
First Order ◽  
Life Problems ◽  
Derivative Free

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots (m≥2). In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods.

scholarly journals A Family of Derivative-Free Methods with High Order of Convergence and Its Application to Nonsmooth Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Alicia Cordero ◽  
José L. Hueso ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Fourth Order ◽  
Order Of Convergence ◽  
Nonsmooth Equations ◽  
Order Convergence ◽  
Linear Combinations ◽  
Numerical Examples ◽  
Derivative Free ◽  
Derivative Free Methods ◽  
Seventh Order ◽  
The Given

A family of derivative-free methods of seventh-order convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to 1.6266. Also, numerical examples are used to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other derivative-free methods, including some optimal fourth-order ones, in the sense of Kung-Traub’s conjecture.

scholarly journals Ball convergence theorem for a Steffensen-type third-order method

2017 ◽  
Vol 51 (1) ◽  
pp. 1-14
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Nonlinear Equation ◽  
Error Bounds ◽  
Order Method ◽  
Third Order ◽  
Numerical Examples ◽  
First Derivative ◽  
Fourth Derivative ◽  
Ball Convergence ◽  
Computable Error Bounds ◽  
Local Convergence Analysis

We present a local convergence analysis for a family of Steffensen-type third-order methods in order to approximate a solution of a nonlinear equation. We use hypothesis up to the first derivative in contrast to earlier studies such as [2, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28] using hypotheses up to the fourth derivative. This way the applicability of these methods is extended under weaker hypothesis. Moreover the radius of convergence and computable error bounds on the distances involved are also given in this study. Numerical examples are also presented in this study.

scholarly journals Novel Computational Iterative Methods with Optimal Order for Nonlinear Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
F. Soleymani
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Nonlinear Equations ◽  
Fourth Order ◽  
General Class ◽  
Analytical Study ◽  
Order Of Convergence ◽  
Optimal Order ◽  
Numerical Examples ◽  
First Order

This paper contributes a very general class of two-point iterative methods without memory for solving nonlinear equations. The class of methods is developed using weight function approach. Per iteration, each method of the class includes two evaluations of the function and one of its first-order derivative. The analytical study of the main theorem is presented in detail to show the fourth order of convergence. Furthermore, it is discussed that many of the existing fourth-order methods without memory are members from this developed class. Finally, numerical examples are taken into account to manifest the accuracy of the derived methods.

Compact Finite Difference Scheme for the Fourth-Order Fractional Subdiffusion System

2014 ◽  
Vol 6 (4) ◽  
pp. 419-435 ◽  
Author(s):  
Seakweng Vong ◽  
Zhibo Wang
Keyword(s):  
Difference Scheme ◽  
Fourth Order ◽  
High Order ◽  
Compact Difference Scheme ◽  
Compact Finite Difference Scheme ◽  
Numerical Examples ◽  
First Order ◽  
Compact Difference ◽  
High Order Convergence ◽  
Theoretical Results

AbstractIn this paper, we study a high-order compact difference scheme for the fourth-order fractional subdiffusion system. We consider the situation in which the unknown function and its first-order derivative are given at the boundary. The scheme is shown to have high order convergence. Numerical examples are given to verify the theoretical results.

scholarly journals New error bounds for linear complementarity problems of Σ-SDD matrices and SB-matrices

2019 ◽  
Vol 17 (1) ◽  
pp. 1599-1614
Author(s):  
Zhiwu Hou ◽  
Xia Jing ◽  
Lei Gao
Keyword(s):  
Linear Algebra ◽  
Error Bound ◽  
Linear Complementarity Problem ◽  
Error Bounds ◽  
Complementarity Problems ◽  
Linear Complementarity Problems ◽  
Linear Complementarity ◽  
Numerical Examples ◽  
Numer Algor ◽  
Better Than

Abstract A new error bound for the linear complementarity problem (LCP) of Σ-SDD matrices is given, which depends only on the entries of the involved matrices. Numerical examples are given to show that the new bound is better than that provided by García-Esnaola and Peña [Linear Algebra Appl., 2013, 438, 1339–1446] in some cases. Based on the obtained results, we also give an error bound for the LCP of SB-matrices. It is proved that the new bound is sharper than that provided by Dai et al. [Numer. Algor., 2012, 61, 121–139] under certain assumptions.

The Wave Theory of the Electron

1928 ◽  
Vol 24 (4) ◽  
pp. 501-505 ◽  
Author(s):  
J. M. Whittaker
Keyword(s):  
Wave Function ◽  
Differential Equations ◽  
Fourth Order ◽  
Commutative Algebra ◽  
Wave Theory ◽  
Wave Functions ◽  
Linear Differential Equations ◽  
Wave Mechanics ◽  
First Order ◽  
Linear Differential

In two recent papers Dirac has shown how the “duplexity” phenomena of the atom can be accounted for without recourse to the hypothesis of the spinning electron. The investigation is carried out by the methods of non-commutative algebra, the wave function ψ being a matrix of the fourth order. An alternative presentation of the theory, using the methods of wave mechanics, has been given by Darwin. The four-rowed matrix ψ is replaced by four wave functions ψ1, ψ2, ψ3, ψ4 satisfying four linear differential equations of the first order. These functions are related to one particular direction, and the work can only be given invariance of form at the expense of much additional complication, the four wave functions being replaced by sixteen.

Incommensurately modulated structure of TaGe0.354Te2: application of crenel functions

1996 ◽  
Vol 52 (1) ◽  
pp. 100-109 ◽  
Author(s):  
F. Boucher ◽  
M. Evain ◽  
V. Petříček
Keyword(s):  
Detailed Analysis ◽  
Fourth Order ◽  
X Ray Diffraction ◽  
Third Order ◽  
Modulated Structure ◽  
X Ray ◽  
First Order ◽  
Germanium Telluride ◽  
Orthorhombic Cell ◽  
Orthogonalization Procedure

The incommensurately modulated structure of tantalum germanium telluride, TaGe0.354Te2, was determined by single-crystal X-ray diffraction. The dimensions of the basic orthorhombic cell are a = 6.4394 (5), b = 14.025 (2), c = 3.8456 (5) Å, V = 347.3 (1) Å3 and Z = 4. The (3 + 1)-dimensional superspace group is Pnma(00γ)s00, γ = 0.3544 (3). Refinements on 1641 reflections with I ≥ 3σ(I) converged to R = 0.065 and 0.044 for 526 main reflections and R = 0.061, 0.12, 0.28 and 0.32 for 782 first-order, 237 second-order, 37 third-order and 59 fourth-order satellites, respectively. Since the structure exhibits a strong occupational modulation of both Ta and Ge atoms, along with important displacive modulation waves, crenel functions were used in the refinement in combination with an orthogonalization procedure. Such an approach is shown to be the most convenient and to give reliable coordinations and distances. A detailed analysis of some Te...Te distances is performed, in connection with already known commensurately and incommensurately modulated MAx Te2 structures.

An Accelerated Iterative Method with Third-Order Convergence

2012 ◽  
Vol 220-223 ◽  
pp. 2658-2661
Author(s):  
Zhong Yong Hu ◽  
Liang Fang ◽  
Lian Zhong Li
Keyword(s):  
Iterative Method ◽  
Fourth Order ◽  
Newton’S Method ◽  
Newton's Method ◽  
New Method ◽  
Order Convergence ◽  
Third Order ◽  
Numerical Examples ◽  
Modified Newton’S Method ◽  
Jarratt Method

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.

scholarly journals Hexagonal Grid Computation of the Derivatives of the Solution to the Heat Equation by Using Fourth-Order Accurate Two-Stage Implicit Methods

2021 ◽  
Vol 5 (4) ◽  
pp. 203
Author(s):  
Suzan Cival Buranay ◽  
Nouman Arshad ◽  
Ahmed Hersi Matan
Keyword(s):  
Heat Equation ◽  
Fourth Order ◽  
Boundary Value ◽  
Time Variable ◽  
Implicit Methods ◽  
First Order ◽  
Mixed Derivatives ◽  
Hexagonal Grids ◽  
Spatial Derivatives ◽  
Derivatives Of

We give fourth-order accurate implicit methods for the computation of the first-order spatial derivatives and second-order mixed derivatives involving the time derivative of the solution of first type boundary value problem of two dimensional heat equation. The methods are constructed based on two stages: At the first stage of the methods, the solution and its derivative with respect to time variable are approximated by using the implicit scheme in Buranay and Arshad in 2020. Therefore, Oh4+τ of convergence on constructed hexagonal grids is obtained that the step sizes in the space variables x1, x2 and in time variable are indicated by h, 32h and τ, respectively. Special difference boundary value problems on hexagonal grids are constructed at the second stages to approximate the first order spatial derivatives and the second order mixed derivatives of the solution. Further, Oh4+τ order of uniform convergence of these schemes are shown for r=ωτh2≥116,ω>0. Additionally, the methods are applied on two sample problems.

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