Jin-San Cheng
ABSTRACT
In this paper, we present a new method for isolating real roots of a bivariate polynomial system. Our method is a subdivision method which is based on real root isolation of univariate polynomials and analyzing the local geometrical properties of the given system. We propose the concept of the orthogonal monotone system in a box and use it to determine the uniqueness and the existence of a simple real zero of the system in the box. We implement our method to isolate the real zeros of a given bivariate polynomial system. The experiments show the effectivity and efficiency of our method, especially for systems with high degrees and sparse terms. Our method also works for non-polynomial systems.
- O. Aberth. 2007. Introduction to precise numerical methods .Elsevier. Google Scholar
Digital Library
- E. Berberich, P. Emeliyanenko, and M. Sagraloff. 2011. An Elimination Method for Solving Bivariate Polynomial Systems: Eliminating the Usual Drawbacks. In Meeting on Algorithm Engineering and Expermiments . Google ScholarDigital Library
- Y. Bouzidi, S. Lazard, G. Moroz, M. Pouget, F. Rouillier, and M. Sagraloff. 2016. Solving bivariate systems using rational univariate representations. Journal of Complexity, Vol. 37 (2016), 34--75. Google ScholarDigital Library
- M. Burr, S. W. Choi, B. Galehouse, and C. Yap. 2012. Complete subdivision algorithms, II: Isotopic meshing of singular algebraic curves. Journal of Symbolic Computation, Vol. 47, 2 (2012), 131--152. Google ScholarDigital Library
- M. Burr, S. Gao, and E. Tsigaridas. 2017. The complexity of an adaptive subdivision method for approximating real curves. In Proceedings of ISSAC'2017. ACM, 61--68. Google ScholarDigital Library
- L. Busé, H. Khalil, and B. Mourrain. 2005. Resultant-based methods for plane curves intersection problems. In International Workshop on Computer Algebra in Scientific Computing. Springer, 75--92. Google ScholarDigital Library
- J. S. Cheng, X. S. Gao, and J. Li. 2009a. Root isolation for bivariate polynomial systems with local generic position method. In Proceedings of ISSAC'2009. ACM, 103--110. Google ScholarDigital Library
- J. S. Cheng, X. S. Gao, and C. K. Yap. 2009b. Complete numerical isolation of real roots in zero-dimensional triangular systems. Journal of Symbolic Computation, Vol. 44, 7 (2009), 768--785. Google ScholarDigital Library
- J. S. Cheng and K. Jin. 2015. A generic position based method for real root isolation of zero-dimensional polynomial systems. Journal of Symbolic Computation, Vol. 68 (2015), 204--224. Google ScholarDigital Library
- J. S. Cheng, J. Wen, and W. Zhang. 2018. Plotting Planar Implicit Curves and Its Applications. In International Congress on Mathematical Software. Springer, 113--122.Google Scholar
- R. M. Corless, P. M. Gianni, and B. M. Trager. 1997. A reordered Schur factorization method for zero-dimensional polynomial systems with multiple roots. In ISSAC, Vol. 97. 133--140. Google ScholarDigital Library
- D. I. Diochnos, I. Z. Emiris, and E. P. Tsigaridas. 2009. On the asymptotic and practical complexity of solving bivariate systems over the reals. Journal of Symbolic Computation, Vol. 44, 7 (2009), 818--835. Google ScholarDigital Library
- I. Z. Emiris and E. P. Tsigaridas. 2005. Real solving of bivariate polynomial systems. In International Workshop on Computer Algebra in Scientific Computing. Springer, 150--161. Google ScholarDigital Library
- J. Garloff and A.P. Smith. 2000. Investigation of a subdivision based algorithm for solving systems of polynomial equations. (2000).Google Scholar
- J. Garloff and A.P. Smith. 2001. Solution of systems of polynomial equations by using Bernstein expansion. In Symbolic Algebraic Methods and Verification Methods. Springer, 87--97.Google Scholar
- H. Hong, M. Shan, and Z. Zeng. 2008. Hybrid method for solving bivariate polynomial system. SRATC'2008 (2008).Google Scholar
- R. Imbach. 2016. A Subdivision Solver for Systems of Large Dense Polynomials. arXiv:1603.07916 (2016).Google Scholar
- J. B. Kioustelidis. 1978. Algorithmic error estimation for approximate solutions of nonlinear systems of equations. Computing, Vol. 19, 4 (1978), 313--320.Google ScholarCross Ref
- A. Kobel and M. Sagraloff. 2015. On the complexity of computing with planar algebraic curves. Journal of Complexity, Vol. 31, 2 (2015), 206--236.Google ScholarCross Ref
- R. Krawczyk. 1969. Newton-algorithmen zur bestimmung von nullstellen mit fehlerschranken. Computing, Vol. 4, 3 (1969), 187--201.Google ScholarCross Ref
- J. M. Lien, V. Sharma, G. Vegter, and C. Yap. 2014. Isotopic Arrangement of Simple Curves: An Exact Numerical Approach Based on Subdivision. In ICMS 2014. Springer, 277--282. LNCS No. 8592. Download from http://cs.nyu.edu/exact/papers/ for version with Appendices and details on MK Test.Google Scholar
- Z. Lu, B. He, Y. Luo, and L. Pan. 2005. An algorithm of real root isolation for polynomial systems. Proceedings of Symbolic Numeric Computation (2005), 94--107.Google Scholar
- A. Mantzaflaris, B. Mourrain, and E. Tsigaridas. 2011. On continued fraction expansion of real roots of polynomial systems, complexity and condition numbers. Theoretical Computer Science, Vol. 412, 22 (2011), 2312--2330. Google ScholarDigital Library
- C. Miranda. 1940. Un'osservazione su un teorema di Brouwer .Consiglio Nazionale delle Ricerche.Google Scholar
- R. E. Moore. 1966. Interval analysis . Vol. 4. Prentice-Hall Englewood Cliffs, NJ.Google Scholar
- R. E. Moore. 1977. A test for existence of solutions to nonlinear systems. SIAM J. Numer. Anal., Vol. 14, 4 (1977), 611--615.Google ScholarDigital Library
- R. E. Moore and J. B. Kioustelidis. 1980. A simple test for accuracy of approximate solutions to nonlinear (or linear) systems. SIAM J. Numer. Anal., Vol. 17, 4 (1980), 521--529.Google ScholarDigital Library
- B. Mourrain and J. P. Pavone. 2009. Subdivision methods for solving polynomial equations. Journal of Symbolic Computation, Vol. 44, 3 (2009), 292--306. Google ScholarDigital Library
- S. Plantinga and G. Vegter. 2004. Isotopic approximation of implicit curves and surfaces. In Proceedings of the 2004 Eurographics/ACM SIGGRAPH symposium on Geometry processing. ACM, 245--254. Google ScholarDigital Library
- X. Qin, Y. Feng, J. Chen, and J. Zhang. 2013. Parallel computation of real solving bivariate polynomial systems by zero-matching method. Appl. Math. Comput., Vol. 219, 14 (2013), 7533--7541. Google ScholarDigital Library
- W. Rudin. 1976. Principles of mathematical analysis. (1976).Google Scholar
- S. M. Rump. 1983. Solving algebraic problems with high accuracy. In A new approach to scientific computation. Elsevier, 51--120. Google ScholarDigital Library
- E. C. Sherbrooke and N. M. Patrikalakis. 1993. Computation of the solutions of nonlinear polynomial systems. Computer Aided Geometric Design, Vol. 10, 5 (1993), 379--405. Google ScholarDigital Library
- S. Smale. 1986. Newton's method estimates from data at one point. In The merging of disciplines: new directions in pure, applied, and computational mathematics . Springer, 185--196.Google Scholar
- A. Strzebonski. 2008. Real root isolation for exp-log functions. In Proceedings of ISSAC'2008. ACM, 303--314. Google ScholarDigital Library
- A. Strzebonski. 2009. Real root isolation for tame elementary functions. In Proceedings of ISSAC'2009. ACM, 341--350. Google ScholarDigital Library
- A. Strzebonski. 2012. Real root isolation for exp--log--arctan functions. Journal of Symbolic Computation, Vol. 47, 3 (2012), 282--314. Google ScholarDigital Library
- S. Telen, B. Mourrain, and M. V. Barel. 2019. Truncated normal forms for solving polynomial systems. ACM Communications in Computer Algebra, Vol. 52, 3 (2019), 78--81. Google ScholarDigital Library
- B. Xia and T. Zhang. 2006. Real solution isolation using interval arithmetic. Computers and Mathematics with Applications, Vol. 52, 6--7 (2006), 853--860. Google ScholarDigital Library
Index Terms
- Certified Numerical Real Root Isolation for Bivariate Polynomial Systems
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