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The Numerical Factorization of Polynomials

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Abstract

Polynomial factorization in conventional sense is an ill-posed problem due to its discontinuity with respect to coefficient perturbations, making it a challenge for numerical computation using empirical data. As a regularization, this paper formulates the notion of numerical factorization based on the geometry of polynomial spaces and the stratification of factorization manifolds. Furthermore, this paper establishes the existence, uniqueness, Lipschitz continuity, condition number, and convergence of the numerical factorization to the underlying exact factorization, leading to a robust and efficient algorithm with a MATLAB implementation capable of accurate polynomial factorizations using floating point arithmetic even if the coefficients are perturbed.

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Notes

  1. http://homepages.neiu.edu/~naclab.html.

  2. http://homepages.neiu.edu/~zzeng/NumFactorTests.zip.

References

  1. E. Kaltofen, Polynomial factorization: a success story, Proc. of ISSAC’03, ACM Press (2003), 3–4.

  2. E. Kaltofen, Challenges of symbolic computation: My favorite open problems, J. Symb. Comput., 29 (2000), 161–168.

  3. L. Blum, F. Cucker, M. Shub and S. Smale, Complexity and Real Computation, Springer-Verlag, New York, 1998.

    Book  MATH  Google Scholar 

  4. J.-P. Dedieu and M. Shub, Newton’s method for overdetermined system of equations, Mathematics of Computation 69 (2002), 1099–1115.

  5. S. Smale, Newton’s method estimates from data at one point, in The Merging of Disciplines: New Directions in Pure, Applied, and Computational Mathematics (R. Ewing, K. Gross, and C. Martin, eds.), Springer-Verlag, New York, 1986, pp. 185–196.

  6. Z. Zeng, Computing multiple roots of inexact polynomials, Math. Comp., 74 (2005), 869–903.

  7. P. Bürgisser and F. Cucker, Condition: The Geometry of Numerical Algorithms, Series: Grundlehren der mathematischen Wissenschaften 349, Springer, 2013.

  8. W. Kahan, Conserving confluence curbs ill-condition, Technical Report, Computer Science Department, University of California, Berkeley, 1972.

  9. D.J. Bates, J.D. Hauenstein, A.J. Sommese and C.W. Wampler, Numerically Solving Polynomial Systems with Bertini, SIAM Publications, 2013.

  10. T.Y. Li, Solving polynomial systems by the homotopy continuation method, in Handbook of Numerical Analysis Vol. XI (P.G. Ciarlet, J.L. Lions eds.), Elsevier B.V. 2003, pp. 209–304.

  11. R. Corless, M. Giesbrecht, M. Van Hoeij, I. Kotsireas and S. Watt,  Towards Factoring bivariate Approximate Polynomials, Proc. of ISSAC’01, ACM Press (2001), 85–92.

  12. R. Corless, A. Galligo, I. Kotsireas and S. Watt, A geometric-numeric algorithm for absolute factorization of multivariate polynomials, Proc. of ISSAC’02, ACM Press (2002), 37–45.

  13. A. Galligo and M. van Hoeij, Approximate bivariate factorization, a geometric viewpoint,  Proceedings of SNC’07, ACM Press (2007), 1–10.

  14. Z. Zeng, Regularization and matrix computation in numerical polynomial algebra, in Approximate Commutative Algebra, Texts and Monographs in Symbolic Computation (L. Robbiano, and J. Abbott eds.), Springer Vienna, 2009, pp. 125–162.

  15. T. Sasaki, M. Suzuki, M. Kolar, and M. Sasaki, Approximate factorization of multivariate polynomials and absolute irreducibility testing, Japan Journal of Industrial and Applied Mathematics, 8–3 (1991), 357–375.

  16. M. van Hoeij, Factoring polynomials and the knapsack problem, Journal of Number Theory, 95–2 (2002), 167–189.

  17. A. Lenstra, H. Lenstra, and L. Lovasz, Factoring polynomials with rational coefficients, Mathematische Annalen, 261–4 (1982), 515–534.

  18. J. von zur Gathen and E. Kaltofen, Factoring sparse multivariate polynomials, Journal of Computer and System Sciences, 31–2 (1985), 265–287.

  19. E. Kaltofen, Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization, SIAM Journal on Computing, 14–2 (1985), 469–489.

  20. G. Lecerf, New recombination algorithms for bivariate polynomial factorization based on Hensel lifting, Applicable Algebra in Engineering, Communication and Computing, 21–2 (2010), 151–176.

  21. Y. Huang, H.J. Stetter, W. Wu and L. Zhi, Pseudofactors of multivariate polynomials, Proc. of ISSAC’00, ACM Press (2000), 161–168.

  22. A. Galligo and S. Watt, A numerical absolute primality test for bivariate polynomials,  Proceedings of ISSAC’97, ACM Press (1997), 217–224.

  23. E. Kaltofen and J. May, On approximate irreducibility of polynomials in several variables,  Proc. of ISSAC’03, ACM Press (2003), 161–168.

  24. T. Sasaki, Approximate multivariate polynomial factorization based on zero-sum relations, Proceedings of ISSAC 2001 ACM Press (2001), 284–291.

  25. S. Gao, E. Kaltofen, J. May, Z. Yang and L. Zhi, Approximate factorization of multivariate polynomials via differential equations, Proc. of ISSAC’04, ACM Press (2004), 167–174.

  26. A. Sommese, J. Verschelde and C. Wampler, Numerical factorization of multivariate complex polynomials, Theoret. Comput. Sci. 315 (2004), 651–669.

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

  28. W. Ruppert, Reducibility of polynomials  \(f(x,y)\),  J. Number Theory, 77 (1999), 62–70.

  29. S. Gao, Factoring multivariate polynomials via partial differential equations, Mathematics of Computation, 72–242 (2003), 801–822.

  30. E. Kaltofen, J. May, Z. Yang, and L. Zhi, Approximate factorization of multivariate polynomials using singular value decomposition, J. Symb. Comput., 43–5 (2008), 359–376.

  31. G. W. Stewart, Matrix Algorithms. Volume I: Basic Decompositions, SIAM publications, 1998.

  32. Z. Zeng, The Gauss–Newton iteration and Tubular Neighborhood Theorem, Preprint, 2012,  http://homepages.neiu.edu/~zzeng/Papers/tnt

  33. Z. Zeng, ApaTools: A Maple and Matlab toolbox for approximate polynomial algebra,  Software for Algebraic Geometry, IMA Volume 148, (M. Stillman, N. Takayama and J. Verschelde, eds.) Springer, 2008, pp. 149–167.

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Authors and Affiliations

  1. Chongqing Key Laboratory of Automated Reasoning and Cognition, Chongqing Institute of Green and Intelligent Technology, CAS, Chongqing, China

    Wenyuan Wu

  2. Department of Mathematics, Northeastern Illinois University, Chicago, IL, 60625, USA

    Zhonggang Zeng

Authors
  1. Wenyuan Wu
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  2. Zhonggang Zeng
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Corresponding author

Correspondence to Zhonggang Zeng.

Additional information

Communicated by Felipe Cucker.

Wenyuan Wu: This work is partially supported by grant NSFC 11471307, 11171053.

Zhonggang Zeng: Research supported in part by NSF under Grant DMS-0715127.

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Wu, W., Zeng, Z. The Numerical Factorization of Polynomials. Found Comput Math 17, 259–286 (2017). https://doi.org/10.1007/s10208-015-9289-1

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