scholarly journals A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1914
Author(s):  
José Ignacio Extreminana-Aldana ◽  
José Manuel Gutiérrez-Jiménez ◽  
Luis Javier Hernández-Paricio ◽  
María Teresa Rivas-Rodríguéz
Keyword(s):  
Graphical Representation ◽  
Polynomial Equation ◽  
Mathematical Community ◽  
Basins Of Attraction ◽  
Point Of View ◽  
Multiple Roots ◽  
Graphic Method ◽  
Hopf Fibration ◽  
Polynomial Equations ◽  
Homogeneous Coordinates

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of self-maps of the Hopf fibration. As an application, our algorithms can be used to check the existence of multiple roots for polynomial equations as well as to give a graphical representation of the union of the basins of attraction of simple roots and the union of the basins of multiple roots. Finally, we would like to highlight that all the algorithms developed in this work have been implemented in Julia, a programming language with increasing use in the mathematical community.

BASIN CONFIGURATION OF A SIX-DIMENSIONAL MODEL OF AN ELECTRIC POWER SYSTEM

2002 ◽  
Vol 12 (06) ◽  
pp. 1333-1356 ◽  
Author(s):  
YOSHISUKE UEDA ◽  
HIROYUKI AMANO ◽  
RALPH H. ABRAHAM ◽  
H. BRUCE STEWART
Keyword(s):  
Power Systems ◽  
Power System ◽  
Initial Conditions ◽  
Electrical Power ◽  
Practical Importance ◽  
Intervention Strategy ◽  
Basins Of Attraction ◽  
Electric Power System ◽  
Point Of View ◽  
Electrical Power System

As part of an ongoing project on the stability of massively complex electrical power systems, we discuss the global geometric structure of contacts among the basins of attraction of a six-dimensional dynamical system. This system represents a simple model of an electrical power system involving three machines and an infinite bus. Apart from the possible occurrence of attractors representing pathological states, the contacts between the basins have a practical importance, from the point of view of the operation of a real electrical power system. With the aid of a global map of basins, one could hope to design an intervention strategy to boot the power system back into its normal state. Our method involves taking two-dimensional sections of the six-dimensional state space, and then determining the basins directly by numerical simulation from a dense grid of initial conditions. The relations among all the basins are given for a specific numerical example, that is, choosing particular values for the parameters in our model.

THE KINEMATIC SYNTHESIS OF STEERING MECHANISMS

2000 ◽  
Vol 24 (3-4) ◽  
pp. 453-476 ◽  
Author(s):  
Jin Yao ◽  
Jorge Angeles
Keyword(s):  
Polynomial Equation ◽  
Global Optimum ◽  
Polynomial Equations ◽  
Kinematic Synthesis ◽  
Local Optima ◽  
Speed Reduction ◽  
Design Variables ◽  
Transmission Quality ◽  
Four Bar Linkage ◽  
Maximum Transmission

We propose a computational-kinematics approach based on elimination procedures to synthesize a steering four-bar linkage. In this regard, we aim at minimizing the root-mean square error of the synthesized linkage in meeting the steering condition over a number of linkage configurations within the linkage range of motion. A minimization problem is thus formulated, whose normality conditions lead to two polynomial equations in two unknown design variables. Upon eliminating one of these two variables, a monovariate polynomial equation is obtained, whose roots yield all locally-optimum linkages. From these roots, the global optimum, as well as unfeasible local optima, are readily identified. The global optimum, however, turns out to be impractical because of the large differences in its link lengths, which we refer to as dimensional unbalance. To cope with this drawback, we use a kinematically-equivalent focal mechanism, i.e., a six-bar linkage with an input-output function identical to that of the four-bar linkage. Given that the synthesized linkage requires a rotational input, as opposed to most existing steering linkages, which require a translational input, we propose a spherical four-bar linkage to drive the steering linkage. The spherical linkage is synthesized so as to yield a speed reduction as close as possible to 2:1 and to have a maximum transmission quality.

scholarly journals Generating Optimal Eighth Order Methods for Computing Multiple Roots

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1947
Author(s):  
Deepak Kumar ◽  
Sunil Kumar ◽  
Janak Raj Sharma ◽  
Matteo d’Amore
Keyword(s):  
Real Life ◽  
Nonlinear Function ◽  
Basins Of Attraction ◽  
Computational Time ◽  
Multiple Roots ◽  
Time Stability ◽  
Academic Problems ◽  
Eighth Order ◽  
Multiple Zeros ◽  
New Family

There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of proposed methods is demonstrated on some real life and academic problems that illustrate the efficient convergence behavior. It is shown that the newly developed schemes are able to compete with other methods in terms of numerical error, convergence and computational time. Stability is also demonstrated by means of a pictorial tool, namely, basins of attraction that have the fractal-like shapes along the borders through which basins are symmetric.

Dynamics of a Piecewise-Linear Oscillator With Limited Power Supply

2009 ◽  
Author(s):  
Silvio L. T. de Souza ◽  
Ibereˆ L. Caldas ◽  
Jose´ M. Balthazar ◽  
Reyolando M. L. R. F. Brasil
Keyword(s):  
Power Supply ◽  
Piecewise Linear ◽  
Basins Of Attraction ◽  
Point Of View ◽  
Restoring Force ◽  
Rich Variety ◽  
Coexistence Of Attractors ◽  
Limited Power ◽  
Limited Power Supply ◽  
Basins Of Attractions

We discuss dynamics of a vibro-impact system consisting of a cart with an piecewise-linear restoring force, which vibrates under driving by a source with limited power supply. From the point of view of dynamical systems, vibro-impact systems exhibit a rich variety of phenomena, particularly chaotic motion. In our analyzes, we use bifurcation diagrams, basins of attractions, identifying several non-linear phenomena, such as chaotic regimes, crises, intermittent mechanisms, and coexistence of attractors with complex basins of attraction.

scholarly journals Accurate Computation of Periodic Regions' Centers in the General M-Set with Integer Index Number

2010 ◽  
Vol 2010 ◽  
pp. 1-12 ◽  
Author(s):  
Wang Xingyuan ◽  
He Yijie ◽  
Sun Yuanyuan
Keyword(s):  
Polynomial Equation ◽  
Negative Integer ◽  
The Other ◽  
Index Number ◽  
Polynomial Equations ◽  
Decimal Point ◽  
Numerical Accuracy ◽  
Index Numbers ◽  
Accurate Computation ◽  
The Relationship

This paper presents two methods for accurately computing the periodic regions' centers. One method fits for the general M-sets with integer index number, the other fits for the general M-sets with negative integer index number. Both methods improve the precision of computation by transforming the polynomial equations which determine the periodic regions' centers. We primarily discuss the general M-sets with negative integer index, and analyze the relationship between the number of periodic regions' centers on the principal symmetric axis and in the principal symmetric interior. We can get the centers' coordinates with at least 48 significant digits after the decimal point in both real and imaginary parts by applying the Newton's method to the transformed polynomial equation which determine the periodic regions' centers. In this paper, we list some centers' coordinates of general M-sets'k-periodic regions(k=3,4,5,6)for the index numbersα=−25,−24,…,−1, all of which have highly numerical accuracy.

Forward displacement analysis of a new 1CCC–5SPS parallel mechanism using Gröbner theory

2009 ◽  
Vol 223 (5) ◽  
pp. 1233-1241 ◽  
Author(s):  
D Gan ◽  
Q Liao ◽  
J S Dai ◽  
S Wei ◽  
L D Seneviratne
Keyword(s):  
Parallel Mechanism ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
Geometrical Constraint ◽  
Degree Polynomial ◽  
Forward Displacement ◽  
Constraint Equations ◽  
Moving Platform ◽  
Forward Kinematic ◽  
Forward Displacement Analysis

A new parallel mechanism 1CCC–5SPS which has distance and angle constraints is introduced in this article. Degree of freedom and forward kinematic analysis of this new parallel mechanism are presented, in which four equivalent polynomial equations are obtained from the original six geometrical constraint equations. The Gröbner basis theory is used with the four equations and the problem of forward displacement is reduced to a 40th degree polynomial equation in a single unknown from a constructed 10 × 10 Sylvester's matrix which is small in size, from which 40 different locations of the moving platform can be derived. A numerical example confirms the efficiency of the procedure.

Direct Kinematic Singularities of 3-3 Stewart-Gough Platforms

2005 ◽  
Vol 219 (4) ◽  
pp. 311-324 ◽  
Author(s):  
C H Liu ◽  
J Chiu
Keyword(s):  
Parallel Manipulator ◽  
Degrees Of Freedom ◽  
Polynomial Equation ◽  
Numerical Results ◽  
Polynomial Equations ◽  
Kinematic Singularity ◽  
Cubic Polynomial ◽  
Singular Configuration ◽  
Kinematic Singularities ◽  
Stewart Gough Platform

In this article, a method to locate direct kinematic singularities of a 3-3 Stewart-Gough parallel manipulator (called a Stewart manipulator henceforth) is proposed. The Stewart manipulator is first replaced by an analogous manipulator, the 3PRPS parallel manipulator, and as the first three active joints of this manipulator remain fixed, this manipulator reduces to an asymmetric 3RPS parallel manipulator. With all moving platform's degrees of freedom, except its height, properly specified, there exists at least one height that gives rise to direct kinematic singularity of the asymmetric 3RPS manipulator and this height is a root of a cubic polynomial equation. The procedure to locate direct kinematic singularities thus reduces to solving cubic polynomial equations. Numerical results show that every singular configuration of the asymmetric 3RPS manipulator thus-determined is also a singular configuration of the 3-3 Stewart-Gough platform.

scholarly journals A combined methodology for the approximate estimation of the roots of the general sextic polynomial equation

2021 ◽  
Author(s):  
Giorgos P. Kouropoulos
Keyword(s):  
Mathematical Model ◽  
Regression Analysis ◽  
Polynomial Equation ◽  
Approximate Solutions ◽  
The Other ◽  
Polynomial Equations ◽  
Polynomial Coefficients ◽  
Approximate Estimation ◽  
Predetermined Interval

Abstract In this article we present the methodology, according to which it is possible to derive approximate solutions for the roots of the general sextic polynomial equation as well as some other forms of sextic polynomial equations that normally cannot be solved by radicals; the approximate roots can be expressed in terms of polynomial coefficients. This methodology is a combination of two methods. The first part of the procedure pertains to the reduction of a general sextic equation H(x) to a depressed equation G(y), followed by the determination of solutions by radicals of G(y) which does not include a quintic term, provided that the fixed term of the equation depends on its other coefficients. The second method is a continuation of the first and pertains to the numerical correlation of the roots and the fixed term of a given sextic polynomial P(x) with the radicals and the fixed term of the sextic polynomial Q(x), where the two polynomials P(x) and Q(x) have the same coefficients except for the fixed term which might be different. From the application of the methodology presented above, the following formulation is derived; For any given general sextic polynomial equation P with coefficients within the interval [a, b], a defined polynomial equation Q corresponds which has equal coefficients to P except for its fixed term which might be different and dependent on the other coefficients so that Q has radical solutions. If we assume a pair of equations P, Q with coefficients within a predetermined interval [a, b], the numerical correlation through regression analysis of the radicals of Q, the roots of P and the fixed terms of P, Q, leads to the derivation of a mathematical model for the approximate estimation of the roots of sextic equations whose coefficients belong to the interval [a, b].

scholarly journals Exact solutions of sixth and fifth degree equations

2021 ◽  
Author(s):  
Mohammed El Amine MONIR
Keyword(s):  
Exact Solutions ◽  
Algebraic Polynomial ◽  
Polynomial Equation ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
The Real ◽  
Absolute Method ◽  
Quintic Equation ◽  
Degree Equation

Abstract The real problematic with algebraic polynomial equations is how to exactly solve any sixth and fifth degree polynomial equations. In this study, we give a new absolute method that presents a new decomposition to exactly solve a sixth degree polynomial equation, while the corresponding fifth degree equation can be easily transformed into a sixth degree equation of this kind (sixth degree equation solvable by this method), then the sextic equation (sixth degree equation) obtained will be solved by applying the principles of this method; moreover, the solutions of the quintic equation (fifth degree equation) will be easily deduced.

scholarly journals Praxéologies du professeur : analyse comparative de manuels scolaires dans l'enseignement des équations polynomiales du premier degréTeacher's Praxeologies: Comparative Analysis of the Textbook in Teaching First-Degree Polynomial Equations

2020 ◽  
Vol 22 (4) ◽  
pp. 224-231
Author(s):  
Edelweis Jose Tavares Barbosa ◽  
Anna Paula de Avelar Brito Lima
Keyword(s):  
Polynomial Equation ◽  
Classroom Teachers ◽  
Polynomial Equations ◽  
Degree Polynomial ◽  
Mots Clés ◽  
Anthropological Theory ◽  
The Common ◽  
Analyse Comparative ◽  
Théorie Anthropologique Du Didactique ◽  
Anthropological Theory Of Didactics

RésuméLe but de cet article est d'analyser, de manière comparative, les livres didactiques et les praxéologies mises en place par les enseignants dans leur pratique pédagogique, concernant l'enseignement des équations polynomiales du premier degré. Cette étude est faite dans le cadre de la théorie anthropologique du didactique (TAD) proposée par Yves Chevallard et ses collaborateurs (1999, 2002, 2009, 2010). La méthodologie est basée sur une approche ethnographique qualitative, dans laquelle les organisations mathématiques et didactiques de trois enseignants sont analysées en les comparant à celles des livres de référence. Les résultats indiquent qu'il existe une certaine conformité entre les praxéologies à enseigner, proposées par les auteurs des manuels scolaires et les praxéologies effectivement enseignées par les professeurs en classe. Les enseignants sont les organisateurs des tâches, des techniques et de la technologie de complexité croissante (FONSECA, 2004) qui sont rendus routinières ou problématiques en classe. La résolution d’une équation polynomiale du premier degré du type ax+b=c a été le point commun des trois professeurs, bien que deux des trois enseignants aient aussi travaillé des équations du type a1x+b1=a2x+b2.Mots-clés : Livres didactiques, Équations polynomiales du premier degré, Théorie Anthropologique du didactique.AbstractThe aim of this article was to analyze, comparatively, praxeologies in didactic books and praxeologies carried out by the teacher, concerning the teaching of polynomial equations of the first degree. This study is done within the framework of the Anthropological Theory of Didactics (ATD), proposed by Yves Chevallard and his collaborators (1999, 2002, 2009, 2010). The methodology consists of a qualitative ethnographic approach, in which the mathematical and didactic organizations of three teachers were compared with those of their reference books. The results indicate that there is some conformity between the praxeologies to be taught, proposed by the authors of the textbooks, and the praxeologies effectively taught by the teachers in the classroom. Teachers are the organizers of tasks, techniques, and technology of increasing complexity (FONSECA, 2004) that are made routine or problematic in the classroom. The resolution of a first-degree polynomial equation of the type ax+b=c was the common point among the three teachers, although two of the three teachers also worked on equations of the type a1x+b1=a2x+b2.Keywords: Didactic books, Polynomial equations of the first degree, Anthropological theory of didactics.

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