The Mathematica book (4th edition)
The Mathematica book (4th edition)
Polynomial Algorithms in Computer Algebra
Polynomial Algorithms in Computer Algebra
A method computing multiple roots of inexact polynomials
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Numerical Polynomial Algebra
C++ gui programming with qt 4, second edition
C++ gui programming with qt 4, second edition
Subdivision methods for solving polynomial equations
Journal of Symbolic Computation
Rational Algebraic Curves: A Computer Algebra Approach
Rational Algebraic Curves: A Computer Algebra Approach
Approximate parametrization of plane algebraic curves by linear systems of curves
Computer Aided Geometric Design
A Symbolic-Numeric Algorithm for Computing the Alexander Polynomial of a Plane Curve Singularity
SYNASC '10 Proceedings of the 2010 12th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing
An adapted version of the Bentley-Ottmann algorithm for invariants of plane curves singularities
ICCSA'11 Proceedings of the 2011 international conference on Computational science and its applications - Volume Part III
A regularization method for computing approximate invariants of plane curves singularities
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Rational Hausdorff divisors: A new approach to the approximate parametrization of curves
Journal of Computational and Applied Mathematics
| Hi-index | 5.23 |
We address the algebraic problem of analyzing the local topology of each singularity of a plane complex algebraic curve defined by a squarefree polynomial with both exact (i.e. integers or rationals) and inexact data (i.e. numerical values). For the inexact data, we associate a positive real number that measures the noise in the coefficients. This problem is ill-posed in the sense that tiny changes in the input produce huge changes in the output. We design a regularization method for estimating the local topological type of each singularity of a plane complex algebraic curve. Our regularization method consists of the following: (i) a symbolic-numeric algorithm that computes the approximate local topological type of each singularity; (ii) and a parameter choice rule, i.e. a function in the noise level. We prove that the symbolic-numeric algorithm together with the parameter choice rule computes an approximate solution, which satisfies the convergence for noisy data property. We implement our algorithm in a new software package called GENOM3CK written in the Axel free algebraic geometric modeler and in the Mathemagix free computer algebra system. For our purpose, both of these systems provide modern graphical capabilities, and algebraic and geometric tools for exact and inexact input data.