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Papers by Laureano Gonzalez-vega

Continuous extensions are now routinely provided by many IVP solvers, for graphical output, error... more Continuous extensions are now routinely provided by many IVP solvers, for graphical output, error control, or event location. Recent developments suggest that a uniform, stable and convenient interpolant may be provided directly by value and derivative data (Hermite data), because a new companion matrix for such data allows stable, robust and convenient root-finding by means of (usually built-in) generalized eigenvalue solvers such as eig in Matlab or Eigenvalues in Maple. Even though these solvers are not as efficient as a special-purpose Hermite interpolant root-finder might be, being O(d 3 ) in cost instead of O(d 2 ), for low or moderate degrees d they are efficient enough. More, because all roots are found, the first root (and hence the event) is guaranteed to be found. Further, the excellent conditioning properties (compared to the monomial basis or to divided differences) suggest that the results will be as accurate as possible. The techniques of this paper apply to polynomial or rational interpolants such as the shape-preserving interpolants of Brankin and Gladwell. We give a sketch of barycentric Hermite interpolation and a sketch of the theory of conditioning of such interpolants. Moreover, we present the construction of the Hermite interpolation polynomial companion matrix pencil and a discussion of scaling and precomputation. We remark that the Bézout matrix can be used used to solve more complicated event location problems involving more than one polynomial, via polynomial eigenvalue problems.

Using a new formulation of the Bézout matrix, we construct bivariate matrix polynomials expressed... more Using a new formulation of the Bézout matrix, we construct bivariate matrix polynomials expressed in a tensor-product Lagrange basis. We use these matrix polynomials to solve common tasks in computer-aided geometric design. For example, we show that these bivariate polynomials can serve as stable and efficient implicit representations of plane curves for a variety of curve intersection problems.

Computer Aided Geometric Design, 2013

This paper is devoted to introducing a new approach for computing the topology of a real algebrai... more This paper is devoted to introducing a new approach for computing the topology of a real algebraic plane curve presented either parametrically or defined by its implicit equation when the corresponding polynomials which describe the curve are known only "by values". This approach is based on the replacement of the usual algebraic manipulation of the polynomials (and their roots) appearing in the topology determination of the given curve with the computation of numerical matrices (and their eigenvalues). Such numerical matrices arise from a typical construction in Elimination Theory known as the Bézout matrix which in our case is specified by the values of the defining polynomial equations on several sample points.

... If the reader is not satisfied with what he/she finds, it is, perhaps, the fault of The Edito... more ... If the reader is not satisfied with what he/she finds, it is, perhaps, the fault of The Editors, Laureano Gonzalez-Vega and Tomas Recio Page 12. 2 M.-E. Alonso, E. Becker, M.-F. Roy, T. Wormann According to the multiplicities of the points, the variety V (I) is split into subsets V^(J ...

RAIRO - Theoretical Informatics and Applications

Comptes Rendus de l Académie des Sciences - Series I - Mathematics

It is shown that Maple is an appropriate problem solving environment for experimenting with algor... more It is shown that Maple is an appropriate problem solving environment for experimenting with algorithms for computer aided geometric design and for performing computations which can benefit from a mixture of numeric and symbolic computation.

ACM SIGSAM Bulletin, 2004

Continuous extensions are now routinely provided by many IVP solvers, for graphical output, error... more Continuous extensions are now routinely provided by many IVP solvers, for graphical output, error control, or event location. Recent developments suggest that a uniform, stable and convenient interpolant may be provided directly by value and derivative data (Hermite data), because a new companion matrix for such data allows stable, robust and convenient root-finding by means of (usually built-in) generalized eigenvalue solvers such as eig in Matlab or Eigenvalues in Maple. Even though these solvers are not as efficient as a special-purpose Hermite interpolant root-finder might be, being O(d 3 ) in cost instead of O(d 2 ), for low or moderate degrees d they are efficient enough. More, because all roots are found, the first root (and hence the event) is guaranteed to be found. Further, the excellent conditioning properties (compared to the monomial basis or to divided differences) suggest that the results will be as accurate as possible. The techniques of this paper apply to polynomial or rational interpolants such as the shape-preserving interpolants of Brankin and Gladwell. We give a sketch of barycentric Hermite interpolation and a sketch of the theory of conditioning of such interpolants. Moreover, we present the construction of the Hermite interpolation polynomial companion matrix pencil and a discussion of scaling and precomputation. We remark that the Bézout matrix can be used used to solve more complicated event location problems involving more than one polynomial, via polynomial eigenvalue problems.

Using a new formulation of the Bézout matrix, we construct bivariate matrix polynomials expressed... more Using a new formulation of the Bézout matrix, we construct bivariate matrix polynomials expressed in a tensor-product Lagrange basis. We use these matrix polynomials to solve common tasks in computer-aided geometric design. For example, we show that these bivariate polynomials can serve as stable and efficient implicit representations of plane curves for a variety of curve intersection problems.

Computer Aided Geometric Design, 2013

This paper is devoted to introducing a new approach for computing the topology of a real algebrai... more This paper is devoted to introducing a new approach for computing the topology of a real algebraic plane curve presented either parametrically or defined by its implicit equation when the corresponding polynomials which describe the curve are known only "by values". This approach is based on the replacement of the usual algebraic manipulation of the polynomials (and their roots) appearing in the topology determination of the given curve with the computation of numerical matrices (and their eigenvalues). Such numerical matrices arise from a typical construction in Elimination Theory known as the Bézout matrix which in our case is specified by the values of the defining polynomial equations on several sample points.

... If the reader is not satisfied with what he/she finds, it is, perhaps, the fault of The Edito... more ... If the reader is not satisfied with what he/she finds, it is, perhaps, the fault of The Editors, Laureano Gonzalez-Vega and Tomas Recio Page 12. 2 M.-E. Alonso, E. Becker, M.-F. Roy, T. Wormann According to the multiplicities of the points, the variety V (I) is split into subsets V^(J ...

RAIRO - Theoretical Informatics and Applications

Comptes Rendus de l Académie des Sciences - Series I - Mathematics

It is shown that Maple is an appropriate problem solving environment for experimenting with algor... more It is shown that Maple is an appropriate problem solving environment for experimenting with algorithms for computer aided geometric design and for performing computations which can benefit from a mixture of numeric and symbolic computation.

ACM SIGSAM Bulletin, 2004

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