A Numerical Method for Computing Border Curves of Bi-parametric Real Polynomial Systems and Applications

For a bi-parametric real polynomial system with parameter values restricted to a finite rectangular region, under certain assumptions, we introduce the notion of border curve. We propose a numerical method to compute the border curve, and provide a numerical error estimation.

[1]  Fabrice Rouillier,et al.  Solving parametric polynomial systems , 2004, J. Symb. Comput..

[2]  Tien-Yien Li,et al.  Solving real polynomial systems with real homotopies , 1993 .

[3]  Volker Weispfenning,et al.  Comprehensive Gröbner Bases , 1992, J. Symb. Comput..

[4]  Yong Feng,et al.  Computing real witness points of positive dimensional polynomial systems , 2017, Theor. Comput. Sci..

[5]  G. Sacks.  A DECISION METHOD FOR ELEMENTARY ALGEBRA AND GEOMETRY , 2003 .

[6]  T. Y. Li.  Numerical solution of multivariate polynomial systems by homotopy continuation methods , 2008 .

[7]  Solen Corvez,et al.  Using Computer Algebra Tools to Classify Serial Manipulators , 2002, Automated Deduction in Geometry.

[8]  Wenjun Wu,et al.  Basic principles of mechanical theorem proving in elementary geometries , 1986, Journal of Automated Reasoning.

[9]  G. Stewart.  Perturbation theory for the singular value decomposition , 1990 .

[10]  W. Böge,et al.  Quantifier Elimination for Real Closed Fields , 1985, AAECC.

[11]  Dongming Wang,et al.  Stability analysis of biological systems with real solution classification , 2005, ISSAC.

[12]  M. Morari,et al.  Parametric optimization and optimal control using algebraic geometry methods , 2006 .

[13]  A. Tarski,et al.  Sur les ensembles définissables de nombres réels , 1931 .

[14]  Fabrice Rouillier,et al.  Finding at Least One Point in Each Connected Component of a Real Algebraic Set Defined by a Single Equation , 2000, J. Complex..

[15]  Wenyuan Wu,et al.  Finding points on real solution components and applications to differential polynomial systems , 2013, ISSAC '13.

[16]  Xiao-Shan Gao,et al.  Computations with parametric equations , 1991, ISSAC '91.

[17]  Andrew J. Sommese,et al.  The numerical solution of systems of polynomials - arising in engineering and science , 2005 .

[18]  Charles W. Wampler,et al.  Finding All Real Points of a Complex Curve , 2006 .

[19]  J. Yorke,et al.  The cheater's homotopy: an efficient procedure for solving systems of polynomial equations , 1989 .

[20]  Jan Verschelde,et al.  Sweeping algebraic curves for singular solutions , 2010, J. Comput. Appl. Math..

[21]  Bruno Buchberger,et al.  Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal , 2006, J. Symb. Comput..

[22]  Tomás Recio,et al.  Sturm-Habicht sequence , 1989, ISSAC '89.

[23]  Éric Schost,et al.  A Baby Step–Giant Step Roadmap Algorithm for General Algebraic Sets , 2012, Found. Comput. Math..

[24]  J. Hauenstein.  Numerically Computing Real Points on Algebraic Sets , 2011, Acta Applicandae Mathematicae.

[25]  Guillaume Moroz,et al.  Complexity of the resolution of parametric systems of polynomial equations and inequations , 2006, ISSAC '06.

[26]  Changbo Chen,et al.  Comprehensive Triangular Decomposition , 2007, CASC.

[27]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[28]  Jonathan D. Hauenstein,et al.  Cell decomposition of almost smooth real algebraic surfaces , 2013, Numerical Algorithms.

[29]  Bican Xia,et al.  Real Solution Classification for Parametric Semi-Algebraic Systems , 2005, Algorithmic Algebra and Logic.

[30]  Éric Schost,et al.  Computing Parametric Geometric Resolutions , 2003, Applicable Algebra in Engineering, Communication and Computing.