scholarly journals Recursive Elucidation of Polynomial Congruences Using Root-Finding Numerical Techniques

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
M. Khalid Mahmood ◽  
Farooq Ahmad
Keyword(s):  
Iterative Methods ◽  
Logarithmic Scale ◽  
Numerical Techniques ◽  
Root Finding ◽  
Polynomial Congruences ◽  
Polynomial Congruence

In this paper we put forward a family of algorithms for lifting solutions of a polynomial congruencemod pto polynomial congruencemod pk. For this purpose, root-finding iterative methods are employed for solving polynomial congruences of the formaxn≡b(mod pk),k≥1,wherea,b,andn>0are integers which are not divisible by an odd primep. It is shown that the algorithms suggested in this paper drastically reduce the complexity for such computations to a logarithmic scale. The efficacy of the proposed technique for solving negative exponent equations of the formax-n≡b(mod pk)has also been addressed.

scholarly journals Blended Root Finding Algorithm Outperforms Bisection and Regula Falsi Algorithms

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1118
Author(s):  
Sabharwal
Keyword(s):  
Computational Complexity ◽  
Empirical Evidence ◽  
Analytic Solution ◽  
Fundamental Problem ◽  
Numerical Techniques ◽  
Physical Sciences ◽  
Root Finding ◽  
Numerical Computing ◽  
Newton Raphson ◽  
Raphson Algorithm

Finding the roots of an equation is a fundamental problem in various fields, including numerical computing, social and physical sciences. Numerical techniques are used when an analytic solution is not available. There is not a single algorithm that works best for every function. We designed and implemented a new algorithm that is a dynamic blend of the bisection and regula falsi algorithms. The implementation results validate that the new algorithm outperforms both bisection and regula falsi algorithms. It is also observed that the new algorithm outperforms the secant algorithm and the Newton–Raphson algorithm because the new algorithm requires fewer computational iterations and is guaranteed to find a root. The theoretical and empirical evidence shows that the average computational complexity of the new algorithm is considerably less than that of the classical algorithms.

scholarly journals Modelling of Planetary Gravitational Microlensing Events

2021 ◽  
Author(s):  
◽  
Michael L. J. Miller
Keyword(s):  
Extrasolar Planets ◽  
Photometric Data ◽  
Analytic Method ◽  
Numerical Techniques ◽  
Root Finding ◽  
Gravitational Microlensing ◽  
Single Lens ◽  
Limb Darkening ◽  
Polynomial Root Finding ◽  
Speed And Accuracy

<p>This thesis describes and develops procedures for the generation of theoretical lightcurves that can be used to model gravitational microlensing events that involve multiple lenses. Of particular interest are the cases involving a single lens star with one or more orbiting planets, as this has proven to be an effective way of detecting extrasolar planets. Although there is an analytical expression for microlensing lightcurves produced by single lensing body, the generation of model lightcurves for more than one lensing body requires the use of numerical techniques. The method developed here, known as the semi-analytic method, involves the analytical rearrangement of the relatively simple ‘lens equation’ to produce a high-order complex lens polynomial. Root-finding algorithms are then used to obtain the roots of this ‘lens polynomial’ in order to locate the positions of the images and calculate their magnifications. By running example microlensing events through the root-finding algorithms, both the speed and accuracy of the Laguerre and Jenkins-Traub algorithms were investigated. It was discovered that, in order to correctly identify the image positions, a method involving solutions of several ‘lens polynomials’ corresponding to different coordinate origins needed to be invoked. Multipole and polygon approximations were also developed to include finite source and limb darkening effects. The semi-analytical method and the appropriate numerical techniques were incorporated into a C++ modelling code at VUW (Victoria University of Wellington) known as mlens2. The effectiveness of the semi-analytic method was demonstrated using mlens2 to generate theoretical lightcurves for the microlensing events MOA-2009-BLG-319 and OGLE-2006-BLG-109. By comparing these theoretical lightcurves with the observed photometric data and the published models, it was demonstrated that the semi-analytic method described in this thesis is a robust and efficient method for discovering extrasolar planets.</p>

scholarly journals A Family of Multiple-Root Finding Iterative Methods Based on Weight Functions

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2194
Author(s):  
Francisco I. Chicharro ◽  
Rafael A. Contreras ◽  
Neus Garrido
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Numerical Test ◽  
Multiple Root ◽  
Initial Guess ◽  
Dynamical Analysis ◽  
Weight Functions ◽  
Root Finding ◽  
Wide Range ◽  
The Family

A straightforward family of one-point multiple-root iterative methods is introduced. The family is generated using the technique of weight functions. The order of convergence of the family is determined in its convergence analysis, which shows the constraints that the weight function must satisfy to achieve order three. In this sense, a family of iterative methods can be obtained with a suitable design of the weight function. That is, an iterative algorithm that depends on one or more parameters is designed. This family of iterative methods, starting with proper initial estimations, generates a sequence of approximations to the solution of a problem. A dynamical analysis is also included in the manuscript to study the long-term behavior of the family depending on the parameter value and the initial guess considered. This analysis reveals the good properties of the family for a wide range of values of the parameter. In addition, a numerical test on academic and engineering multiple-root functions is performed.

scholarly journals An Optimal Eighth-Order Family of Iterative Methods for Multiple Roots

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 672 ◽  
Author(s):  
Saima Akram ◽  
Fiza Zafar ◽  
Nusrat Yasmin
Keyword(s):  
Weight Function ◽  
Iterative Methods ◽  
Multiple Root ◽  
Multiple Roots ◽  
Order Convergence ◽  
Eighth Order ◽  
Root Finding ◽  
New Family ◽  
Two Parameters ◽  
Theoretical Results

In this paper, we introduce a new family of efficient and optimal iterative methods for finding multiple roots of nonlinear equations with known multiplicity ( m ≥ 1 ) . We use the weight function approach involving one and two parameters to develop the new family. A comprehensive convergence analysis is studied to demonstrate the optimal eighth-order convergence of the suggested scheme. Finally, numerical and dynamical tests are presented, which validates the theoretical results formulated in this paper and illustrates that the suggested family is efficient among the domain of multiple root finding methods.

scholarly journals Higher-Order Root-Finding Algorithms and Their Basins of Attraction

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Complex Polynomials ◽  
Convergence Criteria ◽  
Root Finding ◽  
Variational Iteration ◽  
Order Root ◽  
Iteration Technique

In this paper, we proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable. We derive these algorithms with the help of variational iteration technique. We discuss the convergence criteria of these newly developed algorithms. The dominance of the proposed algorithms is illustrated by solving several test examples and comparing them with other well-known existing iterative methods in the literature. In the end, we present the basins of attraction using some complex polynomials of different degrees to observe the fractal behavior and dynamical aspects of the proposed algorithms.

scholarly journals Rootfinding : An Iterative Tool to Solve Different Problems

2015 ◽  
Vol 5 ◽  
pp. 121-125
Author(s):  
Iswarmani Adhikari
Keyword(s):  
Numerical Analysis ◽  
Iterative Methods ◽  
Linear Equations ◽  
Secant Method ◽  
Root Finding ◽  
Non Linear ◽  
Iteration Methods ◽  
False Position ◽  
The Common ◽  
Non Linear Equations

The aim of this paper is to apply the iteration methods for the solution of non-linear equations. Among the various root finding techniques, two of the common iterative methods Regula-falsi (false position) and the Secant method are used in two different problems to show the applications of numerical analysis in different fields. The Himalayan Physics Vol. 5, No. 5, Nov. 2014 Page: 121-125

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