Abstract
A new algorithm for real root isolation of zero-dimensional nonsingular square polynomial systems based on hybrid computation is presented in this paper. First, approximate the (complex) roots of the given polynomial equations via homotopy continuation method. Then, for each approximate root, an initial box relying on the Kantorovich theorem is constructed, which contains the corresponding accurate root. Finally, the Krawczyk interval iteration with interval arithmetic is applied to the initial boxes so as to check whether or not the corresponding approximate roots are real and to obtain the real root isolation boxes. Moreover, an empirical construction of initial box is provided for speeding-up the computation in practice. Our experiments on many benchmarks show that the new hybrid method is very efficient. The method can find all real roots of any given zero-dimensional nonsingular square polynomial systems provided that the homotopy continuation method can find all complex roots of the equations.
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Notes
- 1.
See reference [26], Sect. 11, Automatic differentiation.
- 2.
In Matlab2008b that we do the experiments, the zero threshold is 2.2204e\(-\)016.
- 3.
As mentioned before, the zero threshold in Matlab2008b is 2.2204e\(-\)016, which is almost the same order of magnitude of those radiuses.
References
Beltran, C.: A continuation method to solve polynomial systems, and its complexity. Numerische Mathematik. Online rst. doi:10.1007/s00211- 010-0334-3
Beltran, C., Leykin, A.: Certified numerical homotopy tracking. Exp. Math. 21(1), 69–83 (2012)
Beltran, C., Leykin, A.: Robust certified numerical homotopy tracking. Found. Comput. Math. 13(2), 253–295 (2013)
Benchmarks: http://hom4ps.math.msu.edu/HOM4PS_soft_files/equations.zip
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1997)
Boulier, F., Chen, C., Lemaire, F., Moreno Maza, M.: Real root isolation of regular chains. In: Proceedings of ASCM’2009, pp. 15–29 (2009)
Cheng, J.-S., Gao, X.-S., Guo, L.-L.: Root isolation of zero-dimensional polynomial systems with linear univariate representation. J. Symbolic Comput. 47(7), 843–858 (2012)
Cheng, J.-S., Gao, X.-S., Li, J.: Root isolation for bivariate polynomial systems with local generic position method. In: Procedings of ISSAC’2009, pp. 103–110
Cheng, J.-S., Gao, X.-S., Yap, C.-K.: Complete numerical isolation of real zeros in zero-dimensional triangular systems. In: Proceedings of ISSAC’2007, pp. 92–99 (2007)
Collins, G.E., Akritas, A.G.: Polynomial real root isolation using Descartes’ rule of signs. In: Proceedings of SYMSAC, pp. 272–275 (1976)
Collins, G.E., Loos, R.: Real zeros of polynomials. In: Buchberger, B., Collins, G.E., Loos, R. (eds.) Computer Algebra: Symbolic and Algebraic Computation, pp. 83–94. Springer, New York (1982)
Dayton, B., Li, T.Y., Zeng, Z.G.: Multiple zeros of nonlinear systems. Math. Comp. 80, 2143–2168 (2011)
Emiris, I.Z., Mourrain, B., Tsigaridas, E.P.: The DMM bound: multivariate (aggregate) separation bounds. In: Proceedings of ISSAC’2010, pp. 243–250
Gragg, G.W., Tapia, R.A.: Optimal error bounds for the Newton-Kantorovich theorem. SIAM J. Numer. Anal. 11(1), 10–13 (1974)
Hauenstein, J.D., Sottile, F.: Algorithm 921: alphaCertified: certifying solutions to polynomial systems. ACM Trans. Math. Softw.(TOMS) 38(4), 28 (2012)
Leykin, A., Verschelde, J., Zhao, A.L.: Newton’s method with deflation for isolated singularities of polynomial systems. Theoret. Comput. Sci. 359, 111–122 (2006)
Leykin, A., Verschelde, J., Zhao, A.L.: Higher-order deflation for polynomial systems with isolated singular solutions. In: Dickenstein, A., Schreyer, F., Sommese, A. (eds.) Algorithms in Algeraic Geometry, pp. 79–97. Springer, New York (2008)
Li, T.Y.: Numerical solution of multivariate polynomial systems by homotopy continuation methods. Acta Numerica 6, 399–436 (1997)
Li, T. Y.: HOM4PS-2.0. http://hom4ps.math.msu.edu/HOM4PS_soft.htm (2008)
Li, T.Y., Wang, X.S.: Solving real polynomial systems with real homotopies. Math. Comput. 60(202), 669–680 (1993)
Mantzaflaris, A., Mourrain, B., Tsigaridas, E.P.: On continued fraction expansion of real roots of polynomial systems, complexity and condition numbers. Theor. Comput. Sci. (TCS) 412(22), 2312–2330 (2011)
Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. Society for Industrial and Applied Mathematics, Philadelphia (2009)
Mujica, J.: Complex Analysis in Banach Spaces. North-Holland Mathematics Studies. Elsevier, Amsterdam (1986)
Rouillier, F.: Solving zero-dimensional systems through the rational univariate representation. Appl. Algebra Eng. Commun. Comput. 9, 433–461 (1999)
Rump, S.M.: INTLAB—INTerval LABoratory. In: Csendes, T. (ed.) Developments in Reliable Computing, pp. 77–104. Kluwer Academic Publishers. http://www.ti3.tu-harburg.de/rump/ (1999)
Rump, S.M.: Verification methods: rigorous results using floating-point arithmetic. Acta Numerica 19, 287–449 (2010)
Shen, F.: The real roots isolation of polynomial system based on hybrid computation. Master degree thesis, Peking University (April 2012)
Sommese, A., Wampler, C.: The Numerical Solution of Systems of Polynomials: Arising in Engineering and Science. World Scientific, Singapore (2005)
Strzebonski, A.W., Tsigaridas, E.P.: Univariate real root isolation in multiple extension fields. In: Proceedings of ISSAC’2012, pp. 343–350
Verschelde, J.: PHCpack. http://homepages.math.uic.edu/jan/PHCpack/phcpack.html (1999)
Wampler, C.: HomLab. http://nd.edu/cwample1/HomLab/main.html
Wu, X., Zhi, L.: Computing the multiplicity structure from geometric involutive form. In: Proceedings of ISSAC’2008, pp. 325–332 (2008)
Xia, B.: DISCOVERER: a tool for solving semi-algebraic systems. ACM Commun. Comput. Algebra 41(3), 102–103 (2007)
Xia, B., Zhang, T.: Real solution isolation using interval arithmetic. Comput. Math. Appl. 52, 853–860 (2006)
Yang, L., Xia, B.: An algorithm for isolating the real solutions of semi-algebraic systems. J. Symbolic Comput. 34, 461–477 (2002)
Yang, Z., Zhi, L., Zhu, Y.: Verified error bounds for real solutions of positive-dimensional polynomial systems. In: Proceedings of ISSAC’2013 (2013)
Zhang, T.: Isolating real roots of nonlinear polynomial. Master degree thesis, Peking University (2004)
Zhang, T., Xiao, R., Xia, B.: Real solution isolation based on interval Krawczyk operator. In: Sung-il, P., Park, H. (eds.) Proceedings of ASCM’2005, pp. 235–237 (2005)
Acknowledgments
The work is partly supported by the ANR-NSFC project EXACTA (ANR-09-BLAN-0371-01/60911130369), NSFC-11001040, NSFC-11271034 and the project SYSKF1207 from ISCAS. The authors especially thank professor Dongming Wang for the early discussion on this topic in 2010 and also thank professor T.Y. Li for his helpful suggestions and his team’s work on Hom4ps2-Matlab interface. Thanks also go to Ting Gan who computed the 15 systems in Table 3 and provided us suggestion on possible improvements on our program. Thank Zhenyi Ji who shared with us his insights on our hybrid method. Thank the referees for their valuable constructive comments which help improve the presentation greatly.
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Shen, F., Wu, W., Xia, B. (2014). Real Root Isolation of Polynomial Equations Based on Hybrid Computation. In: Feng, R., Lee, Ws., Sato, Y. (eds) Computer Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43799-5_26
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