Open Access
Access granted
Subscription Access
Vol 43, No 2 (2017)
- Year: 2017
- Articles: 9
- URL: https://journal-vniispk.ru/0361-7688/issue/view/10836
Article
Revealing matrices of linear differential systems of arbitrary order
Abstract
If the leading matrix of a linear differential system is nonsingular, then its determinant is known to bear useful information about solutions of the system. Of interest is also the frontal matrix. However, each of these matrices (we call them revealing matrices) may occur singular. In the paper, a brief survey of algorithms for transforming a system of full rank to a system with a nonsingular revealing matrix of a desired type is given. The same transformations can be used to check whether the rank of the original system is full. A Maple implementation of these algorithms (package EGRR) is discussed, and results of comparison of estimates of their complexity with actual operation times on a number of examples are presented.
67-74
A real variety with boundary and its global parameterization
Abstract
An algebraic variety in R3 is studied that plays an important role in the investigation of the normalized Ricci flow on generalized Wallach spaces related to invariant Einstein metrics. A procedure for obtaining a global parametric representation of this variety is described, which is based on the use of the intersection of this variety with the discriminant set of an auxiliary cubic polynomial as the axis of parameterization. For this purpose, elimination theory and computer algebra are used. Three different parameterization of the variety are obtained; each of them is valid for certain noncritical values of one of the parameters.
75-83
Discretization of quasilinear evolution equations by computer algebra methods
Abstract
An algorithmic approach to construction of finite difference schemes on regular grids developed by the first two authors is applied to quasilinear evolution equations in one spatial variable. The approach combines the finite volumes method, numerical integration, and difference elimination, which is done by means of computer algebra. As a concrete example, a difference scheme for the Korteweg-de Vries equation is constructed. This scheme is strongly consistent and absolutely stable. The numerical behavior of the scheme obtained is illustrated by solving a Cauchy problem.
84-89
Application of computer algebra methods for investigation of stationary motions of a gyrostat satellite
Abstract
With the help of computer algebra methods, properties of a non-linear algebraic system that determines the equilibrium orientations of a gyrostat satellite moving along a circular orbit were investigated. The main attention is paid to the study of equilibrium orientations of the satellite with given principal central moments of inertia and given gyrostatic torque in special cases, when the projection of the vector of gyrostatic torque is located in one of the principal central planes of inertia of the satellite. Computer algebra method based on the algorithm for the construction of a Gröbner basis were applied to reduce the satellite stationary motion system of nine algebraic equations with nine variables to a single algebraic equation in one variable that determines all the equilibrium orientations of the satellite. Bifurcation curves in the space of system parameters that determine boundaries of domains with a fixed number of equilibria of the satellite were obtained symbolically. A comparative analysis of the effectiveness of different algorithms of Gröbner bases computation in solving the problem was carried out.
90-97
Application of computer algebra for the reconstruction of surfaces from their photometric stereo images
Abstract
The problem of reconstructing a Lambertian surface from its two photometric stereo images is discussed. Previously, the solution to this problem was only obtained for a special choice of two light source directions. In this paper, using the computer algebra system Mathematica, the necessary and sufficient conditions for the unique reconstruction of the surface from its two images is analyzed in a more general setting. Photometric images of various surfaces are simulated, and the validity of the theoretical results is demonstrated.
98-104
Evaluating the rational generating function for the solution of the Cauchy problem for a two-dimensional difference equation with constant coefficients
Abstract
We propose an algorithm for evaluation of rational generating functions for solutions of the Cauchy problems for two-dimensional difference equations with constant coefficients. The coefficients of onedimensional difference equations and the initial data are used to solve the corresponding Cauchy problems. The algorithm is implemented in the Maple computer algebra system.
105-111
MathPartner computer algebra
Abstract
In this paper, we describe general characteristics of the MathPartner computer algebra system (CAS) and Mathpar programming language thereof. MathPartner can be used for scientific and engineering calculations, as well as in high schools and universities. It allows one to carry out both simple calculations (acting as a scientific calculator) and complex calculations with large-scale mathematical objects. Mathpar is a procedural language; it supports a large number of elementary and special functions, as well as matrix and polynomial operators. This service allows one to build function images and animate them. MathPartner also makes it possible to solve some symbolic computation problems on supercomputers with distributed memory. We highlight main differences of MathPartner from other CASs and describe the Mathpar language along with the user service provided.
112-118
Partial algorithms for satellite unknowns determination
Abstract
The concept of satellite unknowns with respect to a set of selected unknowns in linear homogeneous differential systems was introduced earlier by the author of this paper, and an algorithm for satellite unknowns testing was proposed. On one of the stages of this algorithm, it is required to compare Picard–Vessiot extensions of two differential systems constructed in the course of the algorithm operation. The commonly accepted method of solving this problem, which is based on Hrushovski’s algorithm, has rather high algorithmic complexity, which hampers its use in practice. In the paper, some partial algorithms for satellite unknowns testing are described. These algorithms are not always applicable but have relatively low computational complexity and can be implemented in computer algebra systems.
119-125
On testing the existence of universal denominators for partial differential and difference equations
Abstract
We consider the problem of testing the existence of a universal denominator for partial differential or difference equations with polynomial coefficients and prove its algorithmic undecidability. This problem is closely related to finding rational function solutions in that the construction of a universal denominator is a part of the algorithms for finding solutions of such form for ordinary differential and difference equations.
126-129