Ana Marco

Implicitization of rational surfaces by means of polynomial interpolation

A method for finding the implicit equation of a surface given by rational parametric equations is presented. The method is based on an efficient computation of the resultant by means of classical multivariate polynomial interpolation. The used approach considerably reduces the problem of intermediate expression swell and it can easily be implemented in parallel.

Parallel computation of determinants of matrices with polynomial entries

Abstract An algorithm for computing the determinant of a matrix whose entries are multivariate polynomials is presented. It is based on classical multivariate Lagrange polynomial interpolation, and it exploits the Kronecker product structure of the coefficient matrix of the linear system associated with the interpolation problem. From this approach, the parallelization of the algorithm arises naturally. The reduction of the intermediate expression swell is also a remarkable feature of the algorithm.

Computing curve intersection by means of simultaneous iterations

A new algorithm is proposed for computing the intersection of two plane curves given in rational parametric form. It relies on the Ehrlich–Aberth iteration complemented with some computational tools like the properties of Sylvester and Bézout matrices, a stopping criterion based on the concept of pseudo-zero, an inclusion result and the choice of initial approximations based on the Newton polygon. The algorithm is implemented as a Fortran 95 module. From the numerical experiments performed with a wide set of test problems it shows a better robustness and stability with respect to the Manocha–Demmel approach based on eigenvalue computation. In fact, the algorithm provides better approximations in terms of the relative error and performs successfully in many critical cases where the eigenvalue computation fails.

ACCURATE COMPUTATIONS WITH TOTALLY POSITIVE BERNSTEIN-VANDERMONDE MATRICES ∗

The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive Bernstein-Vandermonde matrix is considered. Bernstein-Vandermonde ma- trices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials of degree less than or equal to n the Bernstein basis instead of the monomial basis. The approach in this paper is based on the computation of the bidiagonal factorization of a totally positive Bernstein-Vandermonde matrix or of its inverse. The explicit expressions obtained for the determinants involved in the process make the algorithm both fast and accurate. The error analysis of this algorithm for computing this bidiagonal factorization and the perturbation theory for the bidiagonal factorization of totally positive Bernstein-Vandermonde matrices are also carried out. Several applications of the computation with this type of matrices are also pointed out.

Structured matrices in the application of bivariate interpolation to curve implicitization

A non-standard application of bivariate polynomial interpolation is discussed: the implicitization of a rational algebraic curve given by its parametric equations. Three different approaches using the same interpolation space are considered, and their respective computational complexities are analyzed. Although the techniques employed are usually associated to numerical analysis, in this case all the computations are carried out using exact rational arithmetic. The power of the Kronecker product of matrices in this application is stressed.

Accurate polynomial interpolation by using the Bernstein basis

The problem of polynomial interpolation with the Lagrange-type data when using the Bernstein basis instead of the monomial basis is addressed. The extension to the bivariate case, which leads to the use of a generalized Kronecker product, is also developed. In addition to the matricial description of the solution and the proof of unisolvence, algorithms for the computation of the coefficients of the interpolating polynomial are presented. Numerical experiments illustrating the advantage of computing with Bernstein-Vandermonde matrices instead of with Vandermonde matrices are included.

Least squares problems involving generalized Kronecker products and application to bivariate polynomial regression

A method for solving least squares problems (A ⊗ Bi)x = b whose coefficient matrices have generalized Kronecker product structure is presented. It is based on the exploitation of the block structure of the Moore-Penrose inverse and the reflexive minimum norm g-inverse of the coefficient matrix, and on the QR method for solving least squares problems. Firstly, the general case where A is a rectangular matrix is considered, and then the special case where A is square is analyzed. This special case is applied to the problem of bivariate polynomial regression, in which the involved matrices are structured matrices (Vandermonde or Bernstein-Vandermonde matrices). In this context, the advantage of using the Bernstein basis instead of the monomial basis is shown. Numerical experiments illustrating the good behavior of the proposed algorithm are included.

A total positivity property of the Marchenko-Pastur Law

A property of the Marchenko-Pastur measure related to total positivity is presented. The theoretical results are applied to the accurate computation of the roots of the corresponding orthogonal polynomials, an important issue in the construction of Gaussian quadrature formulas.