Numerical matrix methods in the computation of the greatest common divisor (GCD) of polynomials

2016 ◽  
Author(s):  
Siti Nor Asiah binti Isa ◽  
Nor’aini Aris ◽  
Shazirawati Mohd Puzi
2017 ◽  
Vol 5 (1) ◽  
pp. 202-224 ◽  
Author(s):  
Dimitrios Christou ◽  
Marilena Mitrouli ◽  
Dimitrios Triantafyllou

Abstract This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coeffcients of the GCD through the range of the Bézout matrix. A comparison in terms of computational complexity and numerical effciency of the Bézout-QR, Sylvester-QR, and subspace-SVD methods for the computation of theGCDof sets of several polynomials with real coeffcients is provided.Useful remarks about the performance of the methods based on computational simulations of sets of several polynomials are also presented.


2018 ◽  
Vol 35 (3) ◽  
pp. 22-27
Author(s):  
E. S. Kirenkina ◽  
I. G. Pavlenko

The article is devoted to the issues of ensuring the competitiveness of a trading enterprise through its quantitative assessment. Analytic and graphical methods of evaluation are conventionally identified. The methodical approaches to the evaluation of the competitiveness of the trading enterprise, including the Rosenberg Model, the assessment of the quality level of production, matrix methods, the evaluation of comparative advantages, the multifactor score, the model of competitiveness are described, their strengths and weaknesses are identified.


Author(s):  
A. V. Katernyuk

In all spheres business experts try to raise competitiveness of the company by different ways, for instance at the expense of more efficient redistribution of available resources (costs). Objectives connected with modeling and optimizing resources used in advertising are becoming the most topical. Deeper knowledge in planning and conducting any marketing and advertising campaigns are in demand today among many specialists. The process of searching for and finding optimum costs of advertising in the Internet as a factor of the rise in the company sustainability can be successfully shaped through universal matrix methods of solution (e.g. simplex-method). Objectives which cannot be resolved by this method can be supplemented by such economic indicators, as profitability of investment and return on one ruble. The article summarizes the instrumental base dealing with estimating the efficiency of events connected with customer attraction to such a fast growing industry as internet-services. The author proposes besides traditional ways of expense optimization to take into account economic indicators connected with profitability of each sale channel. The following tools were used in the research: modeling, induction method, investment analysis, methods of statistics and formal logics, multi-criteria optimization, specific software meant for solving similar tasks, in particular special macros for excel table.  


Author(s):  
B. G.-Tóth ◽  
E. M. T. Hendrix ◽  
L. G. Casado

AbstractOver the last decades, algorithms have been developed for checking copositivity of a matrix. Methods are based on several principles, such as spatial branch and bound, transformation to Mixed Integer Programming, implicit enumeration of KKT points or face-based search. Our research question focuses on exploiting the mathematical properties of the relative interior minima of the standard quadratic program (StQP) and monotonicity. We derive several theoretical properties related to convexity and monotonicity of the standard quadratic function over faces of the standard simplex. We illustrate with numerical instances up to 28 dimensions the use of monotonicity in face-based algorithms. The question is what traversal through the face graph of the standard simplex is more appropriate for which matrix instance; top down or bottom up approaches. This depends on the level of the face graph where the minimum of StQP can be found, which is related to the density of the so-called convexity graph.


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