scholarly journals An Improved Secant Algorithm of Variable Order to Solve Nonlinear Equations Based on the Disassociation of Numerical Approximations and Iterative Progression

2020 ◽  
Vol 12 (6) ◽  
pp. 50
Author(s):  
Christian Vanhille
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Secant Method ◽  
Variable Order ◽  
Discretization Step ◽  
Newton Raphson ◽  
Analytic Expression ◽  
The One ◽  
Derivatives Of ◽  
Raphson Method

We propose an iterative method to evaluate the roots of nonlinear equations. This Secant-based technique approximates the derivatives of the function numerically through a constant discretization step h disassociated from the iterative progression. The algorithm is developed, implemented, and tested. Its order of convergence is found to be h-dependent. The results obtained corroborate the theoretical deductions and evidence its excellent behavior. For infinitesimal h-values, the algorithm accelerates the convergence of the Secant method to order 2 (the one of the Newton-Raphson method) with no need for analytic expression of derivatives (the advantage of the Secant method).

scholarly journals Numerical Solution of Nonlinear Equations in Maple

2021 ◽  
Vol 8 (4) ◽  
Author(s):  
Qani Yalda
Keyword(s):  
Numerical Methods ◽  
Numerical Solution ◽  
Numerical Method ◽  
Nonlinear Equations ◽  
Secant Method ◽  
Bisection Method ◽  
Real Roots ◽  
Newton Raphson ◽  
Root Analysis ◽  
Raphson Method

The main purpose of this paper is to obtain the real roots of an expression using the Numerical method, bisection method, Newton's method and secant method. Root analysis is calculated using specific, precise starting points and numerical methods and is represented by Maple. In this research, we used Maple software to analyze the roots of nonlinear equations by special methods, and by showing geometric diagrams, we examined the relevant examples. In this process, the Newton-Raphson method, the algorithm for root access, is fully illustrated by Maple. Also, the secant method and the bisection method were demonstrated by Maple by solving examples and drawing graphs related to each method.

scholarly journals A New Second-Order Iteration Method for Solving Nonlinear Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
Shin Min Kang ◽  
Arif Rafiq ◽  
Young Chel Kwun
Keyword(s):  
Nonlinear Equations ◽  
Iteration Method ◽  
Nonlinear Operator ◽  
Efficiency Index ◽  
Second Order ◽  
First Order ◽  
Newton Raphson ◽  
Second Order Convergence ◽  
Derivatives Of ◽  
Raphson Method

We establish a new second-order iteration method for solving nonlinear equations. The efficiency index of the method is 1.4142 which is the same as the Newton-Raphson method. By using some examples, the efficiency of the method is also discussed. It is worth to note that (i) our method is performing very well in comparison to the fixed point method and the method discussed in Babolian and Biazar (2002) and (ii) our method is so simple to apply in comparison to the method discussed in Babolian and Biazar (2002) and involves only first-order derivative but showing second-order convergence and this is not the case in Babolian and Biazar (2002), where the method requires the computations of higher-order derivatives of the nonlinear operator involved in the functional equation.

scholarly journals On the Modification of Newton-Secant Method in Solving Nonlinear Equations for Multiple Zeros of Trigonometric Function

CAUCHY ◽  
2021 ◽  
Vol 7 (1) ◽  
pp. 84-96
Author(s):  
Juhari Juhari
Keyword(s):  
Mathematical Model ◽  
Nonlinear Equation ◽  
Nonlinear Equations ◽  
Trigonometric Function ◽  
Secant Method ◽  
Second Step ◽  
Initial Values ◽  
Newton Raphson ◽  
Modified Formula ◽  
Raphson Method

This study discusses the analysis of the modification of Newton-Secant method and solving nonlinear equations having a multiplicity of  by using a modified Newton-Secant method. A nonlinear equation that has a multiplicity   is an equation that has more than one root. The first step is to analyze the modification of the Newton-Secant method, namely to construct a mathematical model of the Newton-Secant method using the concept of the Newton method and the concept of the Secant method. The second step is to construct a modified mathematical model of the Newton-Secant method by adding the parameter . After obtaining the modified formula for the Newton-Secant method, then applying the method to solve a nonlinear equations that have a multiplicity . In this case, it is applied to the nonlinear equation which has a multiplicity of . The solution is done by selecting two different initial values, namely  and . Furthermore, to determine the effectivity of this method, the researcher compared the result with the Newton-Raphson method, the Secant method, and the Newton-Secant method that has not been modified. The obtained results from the analysis of modification of Newton-Secant method is an iteration formula of the modified Newton-Secant method. And for the result of  using a modified Newton-Secant method with two different initial values, the root of  is obtained approximately, namely  with less than iterations. whereas when using the Newton-Raphson method, the Secant method, and the Newton-Secant method, the root  is also approximated, namely  with more than  iterations. Based on the problem to find the root of the nonlinear equation  it can be concluded that the modified Newton-Secant method is more effective than the Newton-Raphson method, the Secant method, and the Newton-Secant method that has not been modified

scholarly journals Fractional Newton–Raphson Method Accelerated with Aitken’s Method

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 47
Author(s):  
A. Torres-Hernandez ◽  
F. Brambila-Paz ◽  
U. Iturrarán-Viveros ◽  
R. Caballero-Cruz
Keyword(s):  
Fractional Derivative ◽  
Iterative Methods ◽  
Fractional Integral ◽  
Order Of Convergence ◽  
Fractional Operators ◽  
Speed Of Convergence ◽  
Newton Raphson ◽  
Raphson Method

In the following paper, we present a way to accelerate the speed of convergence of the fractional Newton–Raphson (F N–R) method, which seems to have an order of convergence at least linearly for the case in which the order α of the derivative is different from one. A simplified way of constructing the Riemann–Liouville (R–L) fractional operators, fractional integral and fractional derivative is presented along with examples of its application on different functions. Furthermore, an introduction to Aitken’s method is made and it is explained why it has the ability to accelerate the convergence of the iterative methods, in order to finally present the results that were obtained when implementing Aitken’s method in the F N–R method, where it is shown that F N–R with Aitken’s method converges faster than the simple F N–R.

scholarly journals Two Higher Order Iterative Methods for Solving Nonlinear Equations

2018 ◽  
Vol 14 (1) ◽  
pp. 179-187
Author(s):  
Jivandhar Jnawali ◽  
Chet Raj Bhatta
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Higher Order ◽  
Secant Method ◽  
Subject Classification ◽  
Numerical Examples ◽  
Ams Subject Classification

 The main purpose of this paper is to derive two higher order iterative methods for solving nonlinear equations as variants of Mir, Ayub and Rafiq method. These methods are free from higher order derivatives. We obtain these methods by amalgamating Mir, Ayub and Rafiq method with standard secant method and modified secant method given by Amat and Busquier. The order of convergence of new variants are four and six. Also, numerical examples are given to compare the performance of newly introduced methods with the similar existing methods. 2010 AMS Subject Classification: 65H05 Journal of the Institute of Engineering, 2018, 14(1): 179-187

ONE-STEP INVERSE APPROACH BASED ON QUASI-CONJUGATE-GRADIENT METHOD

2012 ◽  
Vol 11 (02) ◽  
pp. 165-172
Author(s):  
JINGXIN NA ◽  
WEI CHEN ◽  
HAIPENG LIU
Keyword(s):  
Conjugate Gradient Method ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Inverse Method ◽  
Inverse Approach ◽  
One Step ◽  
Unbalance Force ◽  
Newton Raphson ◽  
The One ◽  
Raphson Method

For the one-step inverse method, an iteration method based on a quasi-conjugate-gradient method is proposed to replace the Newton–Raphson method. It commences from the physical meaning of elemental unbalance force. It does not need to solve the system of finite element equations. It not only inherits the advantage of conjugate gradient method but also avoids non-convergence of the solving process. Finally, the validity of the algorithm proposed is proved by comparing the simulation results obtained by the method in this paper with those obtained through the module of one-step inverse method in Dynaform and practical drawn parts.

HASIL PERBANDINGAN METODE IMPROVED NEWTON-RAPHSON BERBASIS DEKOMPOSISI ADOMIAN DAN BEBERAPA METODE KLASIK PADA MASALAH PERSAMAAN NON-LINIER

2018 ◽  
Vol 1 (1) ◽  
pp. 39
Author(s):  
Indah Jumawanti ◽  
Sutrisno Sutrisno ◽  
Bayu Surarso
Keyword(s):  
Adomian Decomposition Method ◽  
Secant Method ◽  
Computational Time ◽  
Equation Solving ◽  
Adomian Decomposition ◽  
Newton Raphson ◽  
Iteration Number ◽  
The Difference ◽  
Raphson Method ◽  
Better Than

In this paper, we work with ten nonlinear equations to compare a new method in nonlinear equation solving, Improved Newton-Raphson based on Adomian Decomposition method (INR-ADM) that consisting of two types called INR-ADM 1 and INR-ADM 2. The difference between INR-ADM 1 and INR-ADM 2 is on the iteration formula form. From our results, it was showed that INR-ADM 1 and INR-ADM 2 are not always better than classic Newton-Raphson method in term of the iteration number. However, if INR-ADM 1 and INR-ADM 2 are compared to Regula False method and Secant method, they are always better i.e. they had fewer number of iteration. The INR-ADM 1 and INR-ADM 2 had shorter computational time than Regula False method. Furthermore, the computational time of INR-ADM 1 and INR-ADM 2 cannot be claimed that they had shorter or longer if they are compared to Newton-Raphson method and Secant method.

Application of the Newton-Raphson Method in a Voice Localization System

2014 ◽  
Vol 513-517 ◽  
pp. 4435-4438
Author(s):  
Ming Yu Tong ◽  
Di Jian Xu ◽  
Jin Liang Shi ◽  
Yan Shi
Keyword(s):  
Nonlinear Equations ◽  
Jacobian Matrix ◽  
Initial Solution ◽  
Test Results ◽  
Localization System ◽  
Voice Source ◽  
Solution Of Equations ◽  
Newton Raphson ◽  
The Voice ◽  
Raphson Method

In a voice localization system, the nonlinear equation of voice sources coordinates were established according the accepted information, so the algorithm for solving nonlinear equations is the key problem to the voice source localization syetem. In this paper, the Newton - Raphson method (N-R) is applied to solve the nonlinear equations, by setting the initial solution of equations,then calculating the unbalance vector and Jacobian matrix (J) so as to attain corrected vetor, after modification and iteration the initial solution,until meet the precision numerical solution. Test results show that, application of N-R method in voice localization system have advantange of less number of iterations, saving chip resources and high precision, can meet the precision requirements of voice localization system.

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