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Early Ending in Homotopy Path-Tracking for Real Roots

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Artificial Intelligence and Symbolic Computation (AISC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 11110))

Abstract

For computing only the isolated real solutions to a given polynomial system, a heuristic test is proposed to decide whether one homotopy path will converge to a real root, which is based on the asymptotic behavior of an angle defined by two points on the homotopy path. The data that the test requires is easily obtained from the points along the curve-following procedure in homotopy methods. The homotopy path-tracking may be sped up if we start the test before the endgames, since most divergent paths and paths heading to complex roots can be stopped tracking earlier and unnecessary endgames are avoided. Experiments show that the test works pretty well on tested examples.

The work is partly supported by the projects NSFC Grants 11471307, 61732001, 61532019 and CAS Grant QYZDB-SSW-SYS026.

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References

  1. Garcia, C.B., Zangwill, W.I.: Finding all solutions to polynomial systems and other systems of equations. Math. Program. 16(1), 159–176 (1979)

    Article  MathSciNet  Google Scholar 

  2. Drexler, F.J.: Eine methode zur berechnung sämtlicher lösungen von polynomgleichungssystemen. Numerische Mathematik 29(1), 45–58 (1977)

    Article  MathSciNet  Google Scholar 

  3. Sommese, A.J., Verschelde, J., Wampler, C.W.: Numerical algebraic geometry. In: The Mathematics of Numerical Analysis. Lectures in Applied Mathematics, vol. 32, pp. 749–763 AMS (1996)

    Google Scholar 

  4. Allgower, E.L., Georg, K.: Introduction to numerical continuation methods. Reprint of the 1979 Original. Society for Industrial and Applied Mathematics (2003)

    Google Scholar 

  5. Li, T.Y.: Numerical solution of polynomial systems by homotopy continuation methods. In: Handbook of Numerical Analysis, vol. 11, pp. 209–304. Elsevier (2003)

    Google Scholar 

  6. Sommese, A.J., Wampler, C.W.: The Numerical Solution of Systems of Polynomials Arising in Engineering and Science. World Scientific, Singapore (2005)

    Book  Google Scholar 

  7. Morgan, A.: Solving Polynominal Systems Using Continuation for Engineering and Scientific Problems. Society for Industrial and Applied Mathematics, Philadelphia (2009)

    Book  Google Scholar 

  8. Hauenstein, J.D., Sommese, A.J.: What is numerical algebraic geometry? J. Symb. Comput. 79, 499–507 (2017). SI: Numerical Algebraic Geometry

    Article  Google Scholar 

  9. Huber, B., Sturmfels, B.: A polyhedral method for solving sparse polynomial systems. Math. Comput. 64(212), 1541–1555 (1995)

    Article  MathSciNet  Google Scholar 

  10. Huber, B., Sturmfels, B.: Bernstein’s theorem in affine space. Discret. Comput. Geom. 17(2), 137–141 (1997)

    Article  MathSciNet  Google Scholar 

  11. Verschelde, J., Verlinden, P., Cools, R.: Homotopies exploiting newton polytopes for solving sparse polynomial systems. SIAM J. Numer. Anal. 31(3), 915–930 (1994)

    Article  MathSciNet  Google Scholar 

  12. Bates, D.J., Haunstein, J.D., Sommese, A.J., Wampler, C.W.: Numerically Solving Polynomial Systems with Bertini. Society for Industrial and Applied Mathematics, Philadelphia (2013)

    MATH  Google Scholar 

  13. Lee, T.L., Li, T.Y., Tsai, C.H.: HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method. Computing 83(2), 109 (2008)

    Article  MathSciNet  Google Scholar 

  14. Leykin, A.: Numerical algebraic geometry for macaulay2. https://msp.org/jsag/2011/3-1/p02.xhtml

  15. Verschelde, J.: Algorithm 795: PHCpack: a general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Softw. 25(2), 251–276 (1999)

    Article  Google Scholar 

  16. Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975). https://doi.org/10.1007/3-540-07407-4_17

    Chapter  Google Scholar 

  17. Seidenberg, A.: A new decision method for elementary algebra. Ann. Math. 60(2), 365–374 (1954)

    Article  MathSciNet  Google Scholar 

  18. Safey El Din, M., Schost, E.: Polar varieties and computation of one point in each connected component of a smooth real algebraic set. In: Proceedings of the 2003 International Symposium on Symbolic and Algebraic Computation, ISSAC 2003, pp. 224–231. ACM, New York (2003)

    Google Scholar 

  19. Safey El Din, M., Spaenlehauer, P.J.: Critical point computations on smooth varieties: degree and complexity bounds. In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2016, pp. 183–190. ACM, New York (2016)

    Google Scholar 

  20. Bank, B., Giusti, M., Heintz, J., Pardo, L.M.: Generalized polar varieties and an efficient real elimination. Kybernetika 40(5), 519–550 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Bank, B., Giusti, M., Heintz, J., Pardo, L.: Generalized polar varieties: geometry and algorithms. J. Complex. 21(4), 377–412 (2005)

    Article  MathSciNet  Google Scholar 

  22. Rouillier, F., Roy, M.F., Safey El Din, M.: Finding at least one point in each connected component of a real algebraic set defined by a single equation. J. Complex. 16(4), 716–750 (2000)

    Article  Google Scholar 

  23. El Din, M.S., Schost, É.: Properness defects of projections and computation of at leastone point in each connected component of a real algebraic set. Discret. Comput. Geom. 32(3), 417–430 (2004)

    Article  Google Scholar 

  24. Li, T.Y., Wang, X.: Solving real polynomial systems with real homotopies. Math. Comput. 60(202), 669–680 (1993)

    Article  MathSciNet  Google Scholar 

  25. Brake, D.A., Hauenstein, J.D., Liddell, A.C.: Numerically validating the completeness of the real solution set of a system of polynomial equations, February 2016

    Google Scholar 

  26. Cifuentes, D., Parrilo, P.A.: Sampling algebraic varieties for sum of squares programs. 27, November 2015

    Google Scholar 

  27. Lu, Y., Bates, D.J., Sommese, A.J., Wampler, C.W.: Finding all real points of a complex curve. Technical report, In Algebra, Geometry and Their Interactions (2006)

    Google Scholar 

  28. Bates, D.J., Sottile, F.: Khovanskii-Rolle continuation for real solutions. Found. Comput. Math. 11(5), 563–587 (2011)

    Article  MathSciNet  Google Scholar 

  29. Besana, G.M., Rocco, S., Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Cell decomposition of almost smooth real algebraic surfaces. Numer. Algorithms 63(4), 645–678 (2013)

    Article  MathSciNet  Google Scholar 

  30. Hauenstein, J.D.: Numerically computing real points on algebraic sets. Acta Applicandae Mathematicae 125(1), 105–119 (2013)

    Article  MathSciNet  Google Scholar 

  31. Shen, F., Wu, W., Xia, B.: Real root isolation of polynomial equations based on hybrid computation. In: Feng, R., Lee, W., Sato, Y. (eds.) ASCM 2009, pp. 375–396. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-43799-5_26

    Chapter  Google Scholar 

  32. Wang, Y., Wu, W., Xia, B.: A special homotopy continuation method for a class of polynomial systems. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2017. LNCS, vol. 10490, pp. 362–376. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66320-3_26

    Chapter  Google Scholar 

  33. Wu, W., Reid, G.: Finding points on real solution components and applications to differential polynomial systems. In: Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ISSAC 2013, pp. 339–346. ACM, New York (2013)

    Google Scholar 

  34. Hauenstein, J.D., Regan, M.H.: Adaptive strategies for solving parameterized systems using homotopy continuation. Appl. Math. Comput. 332, 19–34 (2018)

    MathSciNet  Google Scholar 

  35. Morgan, A.P., Sommese, A.J.: Coefficient-parameter polynomial continuation. Appl. Math. Comput. 29(2), 123–160 (1989)

    MathSciNet  MATH  Google Scholar 

  36. Li, T.Y.: Numerical solution of multivariate polynomial systems by homotopy continuation methods. 6, 399–436, January 1997

    Google Scholar 

  37. Chow, S.N., Mallet-Paret, J., Yorke, J.A.: A homotopy method for locating all zeros of a system of polynomials. 730, January 1979

    Google Scholar 

  38. Morgan, A.P.: A method for computing all solutions to systems of polynomials equations. ACM Trans. Math. Softw. 9(1), 1–17 (1983)

    Article  MathSciNet  Google Scholar 

  39. Wright, A.H.: Finding all solutions to a system of polynomial equations. Math. Comput. 44(169), 125–133 (1985)

    Article  MathSciNet  Google Scholar 

  40. Zulehner, W.: A simple homotopy method for determining all isolated solutions to polynomial systems. Math. Comput. 50(181), 167–177 (1988)

    Article  MathSciNet  Google Scholar 

  41. Morgan, A., Sommese, A.: A homotopy for solving general polynomial systems that respects m-homogeneous structures. Appl. Math. Comput. 24(2), 101–113 (1987)

    MathSciNet  MATH  Google Scholar 

  42. Wampler, C., P. Morgan, A., Sommese, A.: Complete solution of the nine-point path synthesis problem for four-bar linkages. 114, March 1992

    Google Scholar 

  43. Verschelde, J., Haegemans, A.: The GBQ-algorithm for constructing start systems of homotopies for polynomial systems. SIAM J. Numer. Anal. 30(2), 583–594 (1993)

    Article  MathSciNet  Google Scholar 

  44. Verschelde, J., Cools, R.: Symbolic homotopy construction. 4, 169–183, September 1993

    Google Scholar 

  45. Morgan, A.P., Sommese, A., Wampler, C.: A product-decomposition theorem for bounding Bezout numbers, March 2018

    Google Scholar 

  46. Bernshtein, D.N.: The number of roots of a system of equations. Funct. Anal. Appl. 9(3), 183–185 (1975)

    Article  MathSciNet  Google Scholar 

  47. Hauenstein, J.D., Sommese, A.J., Wampler, C.W.: Regeneration homotopies for solving systems of polynomials. Math. Comp. 80(273), 345–377 (2011)

    Article  MathSciNet  Google Scholar 

  48. Huber, B., Verschelde, J.: Polyhedral end games for polynomial continuation. Numer. Algorithms 18(1), 91–108 (1998)

    Article  MathSciNet  Google Scholar 

  49. Sosonkina, M., Watson, L.T., Stewart, D.: Note on the end game in homotopy zero curve tracking. 22, 281–287, September 1996

    Google Scholar 

  50. Gerdt, V., Blinkov, Y., Yanovich, D.: GINV Project. http://invo.jinr.ru/ginv/

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Author information

Authors and Affiliations

  1. LMAM and School of Mathematical Sciences, Peking University, Beijing, China

    Yu Wang & Bican Xia

  2. Chongqing Institute of Green and Intelligent Technology, Chinese Academy of Sciences, Chongqing, China

    Wenyuan Wu

Authors
  1. Yu Wang
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  2. Wenyuan Wu
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  3. Bican Xia
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Corresponding authors

Correspondence to Yu Wang , Wenyuan Wu or Bican Xia .

Editor information

Editors and Affiliations

  1. University of Edinburgh, Edinburgh, United Kingdom

    Jacques Fleuriot

  2. Beihang University, Beijing, China

    Dongming Wang

  3. Karlsruhe Institute of Technology, Karlsruhe, Germany

    Jacques Calmet

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Wang, Y., Wu, W., Xia, B. (2018). Early Ending in Homotopy Path-Tracking for Real Roots. In: Fleuriot, J., Wang, D., Calmet, J. (eds) Artificial Intelligence and Symbolic Computation. AISC 2018. Lecture Notes in Computer Science(), vol 11110. Springer, Cham. https://doi.org/10.1007/978-3-319-99957-9_12

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  • DOI: https://doi.org/10.1007/978-3-319-99957-9_12

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