Nonlinear systems solver in floating-point arithmetic using LP reduction

Proceedings - SPM 2009: SIAM/ACM Joint Conference on Geometric and Physical Modeling

Volumn , Issue , 2009, Pages 123-134

  Fünfzig, Christoph ^a   Michelucci, Dominique ^a   Foufou, Sebti ^a  

^a UNIVERSITÉ DE BOURGOGNE   (France)

Author keywords

Distance computation; Geometric constraint solving; Intersection computation; Interval arithmetic; Linear programming; Subdivision solver

Indexed keywords

ALGEBRAIC EQUATIONS; BERNSTEIN BASIS; BERNSTEIN POLYNOMIAL; CONSTRAINED SYSTEMS; CONSTRAINT SOLVING; DISTANCE COMPUTATION; EXPONENTIAL NUMBERS; FLOATING POINTS; FLOATING-POINT ARITHMETIC; GEOMETRIC CONSTRAINT SOLVING; INTERVAL ARITHMETIC; NONLINEAR INEQUALITY; NUMERICAL RESOLUTION; POLYTOPES; PRIMAL-DUAL; SUB-SYSTEMS; SUBDIVISION SOLVER; SYSTEM EQUATIONS; SYSTEM VARIABLES;

AMBER; COMPUTER SCIENCE; DIGITAL ARITHMETIC; DYNAMIC PROGRAMMING; LINEAR PROGRAMMING; LINEARIZATION; MODELS; NONLINEAR SYSTEMS; OPTIMIZATION;

NONLINEAR EQUATIONS;

EID: 70350623069     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/1629255.1629271     Document Type: Conference Paper

References (22)

1. B. Bruderlin and D. Roller, editors. Geometric Constraint Solving and Applications. Springer, 1998.
2. V. Chvatal. Linear Programming (Series of Books in the Mathematical Sciences). W. H. Freeman, September 1983.
3. I. Duff, A. Erisman, and J. Reid. Direct Methods for Sparse Matrices. Clarendon Press, Oxford, 1986.
4. C. B. Durand. Symbolic and Numerical Techniques for Constraint Solving. PhD thesis, Purdue University, 1998.
5. G. Elber and M.-S. Kim. Geometric constraint solver using multivariate rational spline functions. In SMA'01: Proc. of the 6th ACM Symp. on Solid Modeling and Applications, pages 1-10, New York, NY, USA, 2001. ACM Press.
6. G. Farin. Curves and Surfaces for CAGD: A Practical Guide. Academic Press Professional, San Diego, California, 1988.
7. J. J. H. Forrest and J. A. Tomlin. Updating triangular factors of the basis to maintain sparsity in the product form simplex method. Mathematical Programming, (2):263-278, 1972.
8. T. Grandine. Geometry processing (chapter 24). In Handbook of CAGD (Farin, Hoschek, Kim eds.), pages 603-623. Elsevier, 2002.
9. C. Hoffmann and B. Yuan. There are 12 common tangents to four spheres. 2000. http://www.cs. purdue.edu/homes/cmh/distribution/ SphereTangents.htm.
10. R. Kearfott. Rigorous Global Search: Continuous Problems. Kluwer, Dordrecht, Netherlands, 1996.
11. H. Lamure and D. Michelucci. Solving constraints by homotopy. In Proc. of the Symp. on Solid Modeling Foundations and CAD/CAM Applications, pages 263-269, May 1995.
12. D. Michelucci. Linear Programming for Interval Newton Solvers - Extended abstract on a work in progress. In Automatic Deduction in Geometry, ADG 2008, Shanghai, 2008.
13. D. Michelucci and S. Foufou. Using Cayley-Menger determinants for geometric constraint solving. In SM '04: Proceedings of the ninth ACM Symposium on Solid Modeling and Applications, pages 285-290, Aire-la-Ville, Switzerland, Switzerland, 2004. Eurographics Association.
14. D. Michelucci and S. Foufou. Bernstein basis for interval analysis: application to geometric constraints systems solving. 8th Conference on Real Numbers and Computers (Bruguera and Daumas, eds.), pages 37-46, July 2008.
15. B. Mourrain and J.-P. Pavone. Subdivision methods for solving polynomial equations. Technical Report RR-5658, INRIA, August 2005.
16. M. Reuter, T. S. Mikkelsen, E. C. Sherbrooke, T. Maekawa, and N. M. Patrikalakis. Solving nonlinear polynomial systems in the barycentric Bernstein basis. The Visual Computer, 24(3):187-200, 2008.
17. E. C. Sherbrooke and N. M. Patrikalakis. Computation of the solutions of nonlinear polynomial systems. Comput. Aided Geom. Des., 10(5):379-405, 1993.
18. D. Stewart. A Platform with Six Degrees of Freedom. UK Institution of Mechanical Engineers Proceedings, 180 Part 1(15):371-386, 1965-66.
19. G. Varadhan, S. Krishnan, T. Sriram, and D. Manocha. Topology preserving isosurface extraction for geometry processing. In Second Eurographics Symposium on Geometry Processing, pages 235-244, 2004.
20. C. Wampler, A. Morgan, and A. Sommese. Numerical continuation methods for solving polynomial systems arising in kinematics. ASME J. on Design, (112):59-68, 1990.
21. J. Wilkinson. The algebraic eigenvalue problem. Oxford University Press, 1965.
22. R. Wunderling. Paralleler und objektorientierter Simplex-Algorithmus. PhD thesis, TU Berlin, 1996.