Connections between Sylvester resultant matrix and Bezout matrix for Bernstein polynomials basis

Concepts from algebraic geometry such as cones and fans are related to toric varieties and can be applied to determine the convex polytopes and homogeneous coordinate rings of multivariate polynomial systems. The homogeneous coordinates of a system in its projective vector space can be associated with the entries of the resultant matrix of the system under consideration. This paper presents some conditions for the homogeneous coordinates of a certain system of bivariate polynomials through the construction and implementation of the Sylvester-Bèzout hybrid resultant matrix formulation. This basis of the implementation of the Bèzout block applies a combinatorial approach on a set of linear inequalities, named 5-rule. The inequalities involved the set of exponent vectors of the monomials of the system and the entries of the matrix are determined from the coefficients of facets variable known as brackets. The approach can determine the homogeneous coordinates of the given system and the entries of the Bèzout block. Conditions for determining the homogeneous coordinates are also given and proven.

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The Powers and their Bernstein Polynomials

Real Analysis Exchange ◽

10.2307/44153570 ◽

1983 ◽

Vol 9 (2) ◽

pp. 578

Author(s):

He

Keyword(s):

Bernstein Polynomials

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Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

Fractal and Fractional ◽

10.3390/fractalfract5010008 ◽

2021 ◽

Vol 5 (1) ◽

pp. 8

Author(s):

Cundi Han ◽

Yiming Chen ◽

Da-Yan Liu ◽

Driss Boutat

Keyword(s):

Numerical Analysis ◽

Fractional Order ◽

Governing Equation ◽

Numerical Solutions ◽

Bernstein Polynomials ◽

Matrix Product ◽

Algebraic Equations ◽

Rotating Beam ◽

The Matrix ◽

The Time Domain

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

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