Some Applications of Fourier Analysis and Calculus of Probability to the Study of Real Roots of Algebraic Equations

1944 ◽  
Vol 56 (3) ◽  
pp. 470
Author(s):  
Alfred M. Peiser
Keyword(s):  
Fourier Analysis ◽  
Algebraic Equations ◽  
Real Roots

On Graeffe's Method for Complex Roots of Algebraic Equations

1924 ◽  
Vol 22 (2) ◽  
pp. 83-87 ◽  
Author(s):  
S. Brodetsky ◽  
G. Smeal
Keyword(s):  
Nineteenth Century ◽  
Real Axis ◽  
Practical Method ◽  
Algebraic Equations ◽  
Argand Diagram ◽  
Considerable Difficulty ◽  
The Real ◽  
Real Roots ◽  
Bipolar Coordinates ◽  
Complex Roots

The only really useful practical method for solving numerical algebraic equations of higher orders, possessing complex roots, is that devised by C. H. Graeffe early in the nineteenth century. When an equation with real coefficients has only one or two pairs of complex roots, the Graeffe process leads to the evaluation of these roots without great labour. If, however, the equation has a number of pairs of complex roots there is considerable difficulty in completing the solution: the moduli of the roots are found easily, but the evaluation of the arguments often leads to long and wearisome calculations. The best method that has yet been suggested for overcoming this difficulty is that by C. Runge (Praxis der Gleichungen, Sammlung Schubert). It consists in making a change in the origin of the Argand diagram by shifting it to some other point on the real axis of the original Argand plane. The new moduli and the old moduli of the complex roots can then be used as bipolar coordinates for deducing the complex roots completely: this also checks the real roots.

scholarly journals Equivalence among orbital equations of polynomial maps

2018 ◽  
Vol 29 (09) ◽  
pp. 1850082 ◽  
Author(s):  
Jason A. C. Gallas
Keyword(s):  
Equations Of Motion ◽  
Algebraic Equations ◽  
Nonlinear Transformations ◽  
Unique Representation ◽  
Real Roots ◽  
Polynomial Maps ◽  
Infinite Sequences

This paper shows that orbital equations generated by iteration of polynomial maps do not necessarily have a unique representation. Remarkably, they may be represented in an infinity of ways, all interconnected by certain nonlinear transformations. Five direct and five inverse transformations are established explicitly between a pair of orbits defined by cyclic quintic polynomials with real roots and minimum discriminant. In addition, infinite sequences of transformations generated recursively are introduced and shown to produce unlimited supplies of equivalent orbital equations. Such transformations are generic and valid for arbitrary dynamics governed by algebraic equations of motion.

scholarly journals Symbolic-Numerical Modeling of the Influence of Damping Moments on Satellite Dynamics

2018 ◽  
Vol 173 ◽  
pp. 05008
Author(s):  
Sergey A. Gutnik ◽  
Vasily A. Sarychev
Keyword(s):  
Numerical Modeling ◽  
Asymptotic Stability ◽  
Gröbner Basis ◽  
Algebraic Equations ◽  
Grobner Basis ◽  
Real Roots ◽  
Spatial Oscillations ◽  
Damping Torque ◽  
Orbital Coordinate System

The dynamics of a satellite on a circular orbit under the influence of gravitational and active damping torques, which are proportional to the projections of the angular velocity of the satellite, is investigated. Computer algebra Gröbner basis methods for the determination of all equilibrium orientations of the satellite in the orbital coordinate system with given damping torque and given principal central moments of inertia were used. The conditions of the equilibria existence depending on three damping parameters were obtained from the analysis of the real roots of the algebraic equations spanned by the constructed Gröbner basis. Conditions of asymptotic stability of the satellite equilibria and the transition decay processes of the spatial oscillations of the satellite at different damping parameters have also been obtained.

scholarly journals Dynamics of a System of Two Connected Bodies Moving along a Circular Orbit around the Earth

2020 ◽  
Vol 226 ◽  
pp. 02010
Author(s):  
Sergey A. Gutnik ◽  
Vasily A. Sarychev
Keyword(s):  
Circular Orbit ◽  
Orbital Plane ◽  
Radius Vector ◽  
Algebraic Equations ◽  
Real Roots ◽  
Principal Axes ◽  
Special Cases ◽  
Spherical Hinge ◽  
Algebra Method ◽  
Two Bodies

Symbolic–numeric methods are used to investigate the dynamics of a system of two bodies connected by a spherical hinge. The system is assumed to move along a circular orbit under the action of gravitational torque. The equilibrium orientations of the two-body system are determined by the real roots of a system of 12 algebraic equations of the stationary motions. Attention is paid to the study of the conditions of existence of the equilibrium orientations of the system of two bodies refers to special cases when one of the principal axes of inertia of each of the two bodies coincides with either the normal of the orbital plane, the radius vector or the tangent to the orbit. Nine distinct solutions are found within an approach which uses the computer algebra method based on the algorithm for the construction of a Gröbner basis.

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