Algorithm 960: Polynomial: An object-oriented matlab library of fast and efficient algorithms for polynomials

The design and implementation of a Matlab object-oriented software library for working with polynomials is presented. The construction and evaluation of polynomials in Bernstein form are motivated and justified. Efficient constructions for the coefficients of a polynomial in Bernstein form when the polynomial is not given with this representation are provided. The presented adaptive evaluation algorithm uses the VS (Volk and Schumaker) algorithm, the de Casteljau algorithm, and a compensated VS algorithm. In addition, we have completed the library with other algorithms to perform other usual operations with polynomials in Bernstein form

Desingularization in Computational Applications and Experiments

After briefly recalling some computational aspects of blowing up and of representation of resolution data common to a wide range of desingularization algorithms (in the general case as well as in special cases like surfaces or binomial varieties), we shall proceed to computational applications of resolution of singularities in singularity theory and algebraic geometry, also touching on relations to algebraic statistics and machine learning. Namely, we explain how to compute the intersection form and dual graph of resolution for surfaces, how to determine discrepancies, the log-canoncial threshold and the topological Zeta-function on the basis of desingularization data. We shall also briefly see how resolution data comes into play for Bernstein-Sato polynomials, and we mention some settings in which desingularization algorithms can be used for computational experiments. The latter is simply an invitation to the readers to think themselves about experiments using existing software, whenever it seems suitable for their own work.Comment: notes of a summer school talk; 16 pages; 1 figur

Structure-Preserving Matrix Methods for Computations on Univariate and Bivariate Bernstein Polynomials

Curve and surface intersection finding is a fundamental problem in computer-aided geometric design (CAGD). This practical problem motivates the undertaken study into methods for computing the square-free factorisation of univariate and bivariate polynomials in Bernstein form. It will be shown how these two problems are intrinsically linked and how finding univariate polynomial roots and bivariate polynomial factors is equivalent to finding curve and surface intersection points. The multiplicities of a polynomial’s factors are maintained through the use of a square free factorisation algorithm and this is analogous to the maintenance of smooth intersections between curves and surfaces, an important property in curve and surface design. Several aspects of the univariate and bivariate polynomial factorisation problem will be considered. This thesis examines the structure of the greatest common divisor (GCD) problem within the context of the square-free factorisation problem. It is shown that an accurate approximation of the GCD can be computed from inexact polynomials even in the presence of significant levels of noise. Polynomial GCD computations are ill-posed, in that noise in the coefficients of two polynomials which have a common factor typically causes the polynomials to become coprime. Therefore, a method for determining the approximate greatest common divisor (AGCD) is developed, where the AGCD is defined to have the same degree as the GCD and its coefficients are sufficiently close to those of the exact GCD. The algorithms proposed assume no prior knowledge of the level of noise added to the exact polynomials, differentiating this method from others which require derived threshold values in the GCD computation. The methods of polynomial factorisation devised in this thesis utilise the Sylvester matrix and a sequence of subresultant matrices for the GCD finding component. The classical definition of the Sylvester matrix is extended to compute the GCD of two and three bivariate polynomials defined in Bernstein form, and a new method of GCD computation is devised specifically for bivariate polynomials in Bernstein form which have been defined over a rectangular domain. These extensions are necessary for the computation of the factorisation of bivariate polynomials defined in the Bernstein form