4 research outputs found
A formal study of Bernstein coefficients and polynomials
International audienceBernstein coefficients provide a discrete approximation of the behavior of a polynomial inside an interval. This can be used for example to isolate real roots of polynomials. We prove a criterion for the existence of a single root in an interval and the correctness of the de Casteljau algorithm to compute efficiently Bernstein coefficients
New Operational Matrices of Seventh Degree Orthonormal Bernstein Polynomials
Based on analyzing the properties of Bernstein polynomials, the extended orthonormal Bernstein polynomials, defined on the interval [0, 1] for n=7 is achieved. Another method for computing operational matrices of derivative and integration D_b and R_(n+1)^B respectively is presented. Also the result of the proposed method is compared with true answers to show the convergence and advantages of the new method
Theorem of three circles in Coq
The theorem of three circles in real algebraic geometry guarantees the
termination and correctness of an algorithm of isolating real roots of a
univariate polynomial. The main idea of its proof is to consider polynomials
whose roots belong to a certain area of the complex plane delimited by straight
lines. After applying a transformation involving inversion this area is mapped
to an area delimited by circles. We provide a formalisation of this rather
geometric proof in Ssreflect, an extension of the proof assistant Coq,
providing versatile algebraic tools. They allow us to formalise the proof from
an algebraic point of view.Comment: 27 pages, 5 figure
New Operational Matrices of Seventh Degree Orthonormal Bernstein Polynomials
Based on analyzing the properties of Bernstein polynomials, the extended orthonormal Bernstein polynomials, defined on the interval [0, 1] for n=7 is achieved. Another method for computing operational matrices of derivative and integration D_b and R_(n+1)^B respectively is presented. Also the result of the proposed method is compared with true answers to show the convergence and advantages of the new method