9 research outputs found
B\'ezier representation of the constrained dual Bernstein polynomials
Explicit formulae for the B\'ezier coefficients of the constrained dual
Bernstein basis polynomials are derived in terms of the Hahn orthogonal
polynomials. Using difference properties of the latter polynomials, efficient
recursive scheme is obtained to compute these coefficients. Applications of
this result to some problems of CAGD is discussed.Comment: 10 page
Continued Fraction Expansion of Real Roots of Polynomial Systems
We present a new algorithm for isolating the real roots of a system of
multivariate polynomials, given in the monomial basis. It is inspired by
existing subdivision methods in the Bernstein basis; it can be seen as
generalization of the univariate continued fraction algorithm or alternatively
as a fully analog of Bernstein subdivision in the monomial basis. The
representation of the subdivided domains is done through homographies, which
allows us to use only integer arithmetic and to treat efficiently unbounded
regions. We use univariate bounding functions, projection and preconditionning
techniques to reduce the domain of search. The resulting boxes have optimized
rational coordinates, corresponding to the first terms of the continued
fraction expansion of the real roots. An extension of Vincent's theorem to
multivariate polynomials is proved and used for the termination of the
algorithm. New complexity bounds are provided for a simplified version of the
algorithm. Examples computed with a preliminary C++ implementation illustrate
the approach.Comment: 10 page
Generating Surfaces of Variable Eccentricity Within a Ray Tracer
Polynomial surfaces used in ray tracing have recently been improved upon allowing for three dimensional applications. Among these are surfaces that have a varying eccentricity. This paper will discuss a method for finding real roots of polynomials [allowing us to create these surfaces]. First, we will give the reader a basic comprehension of the workings of a ray tracer, a general understanding of three dimensional polynomial surfaces, how this newly implemented root finder functions, and how these concepts enable us to create surfaces of variable eccentricity. Then, examples will be provided to demonstrate the capabilities of the program
The computation of multiple roots of a Bernstein basis polynomial
This paper describes the algorithms of Musser and Gauss for the computation of multiple roots of a theoretically exact Bernstein basis polynomial ˆ 5 f(y) when the coefficients of its
given form f(y) are corrupted by noise. The exact roots of f(y) can therefore be assumed to be simple, and thus the problem reduces to the calculation of multiple roots of a polynomial f˜(y) that is near f(y), such that the backward error is small. The algorithms require many greatest common divisor (GCD) computations and polynomial deconvolutions, both of which are implemented by a structure-preserving matrix method. The motivation of these algorithms arises from the unstructured and structured condition numbers of a multiple root of a polynomial. These condition numbers have an elegant interpretation in terms of the pejorative manifold of ˆ 12 f(y), which allows the geometric significance of the GCD computations and polynomial deconvolutions to be considered. A variant of the Sylvester resultant matrix is used for the GCD computations because it yields better results than the standard form of this matrix, and the polynomial deconvolutions can be computed in several different ways, sequentially or simultaneously, and with the inclusion or omission of the preservation of the structure of the coefficient matrix. It is shown that Gauss’ algorithm yields better results than Musser’s algorithm, and the reason for these superior results is explained
On Continued Fraction Expansion of Real Roots of Polynomial Systems, Complexity and Condition Numbers
International audienceWe elaborate on a correspondence between the coeffcients of a multivariate polynomial represented in the Bernstein basis and in a tensor-monomial basis, which leads to homography representations of polynomial functions, that use only integer arithmetic (in contrast to Bernstein basis) and are feasible over unbounded regions. Then, we study an algorithm to split this representation and we obtain a subdivision scheme for the domain of multivariate polynomial functions. This implies a new algorithm for real root isolation, MCF, that generalizes the Continued Fraction (CF) algorithm of univariate polynomials. A partial extension of Vincent's Theorem for multivariate polynomials is presented, which allows us to prove the termination of the algorithm. Bounding functions, projection and preconditioning are employed to speed up the scheme. The resulting isolation boxes have optimized rational coordinates, corresponding to the first terms of the continued fraction expansion of the real roots. Finally, we present new complexity bounds for a simplified version of the algorithm in the bit complexity model, and also bounds in the real RAM model for a family of subdivision algorithms in terms of the real condition number of the system. Examples computed with our C++ implementation illustrate the practical aspects of our method