Skip to main content

An ODE-based method for computing the approximate greatest common divisor of polynomials

Abstract

Computing the greatest common divisor of a set of polynomials is a problem which plays an important role in different fields, such as linear system, control, and network theory. In practice, the polynomials are obtained through measurements and computations, so that their coefficients are inexact. This poses the problem of computing an approximate common factor. We propose an improvement and a generalization of the method recently proposed in Guglielmi et al. (SIAM J. Numer. Anal. 55, 1456–1482, 2017), which restates the problem as a (structured) distance to singularity of the Sylvester matrix. We generalize the algorithm in order to work with more than 2 polynomials and to compute an Approximate GCD (Greatest Common Divisor) of degree k ≥ 1; moreover, we show that the algorithm becomes faster by replacing the eigenvalues by the singular values.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Beckermann, B., Labahn, G.: When are two numerical polynomials relatively prime?. J. Symb. Comput. 26, 677–689 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bini, D., Boito, P.: A fast algorithm for approximate polynomial gcd based on structured matrix computations. In: Numerical methods for structured matrices and applications, pp. 155–173. Basel, Birkhäuser (2010)

  3. 3.

    Bini, D., Boito, P.: Structured matrix-based methods for polynomial 𝜖-GCD: analysis and comparisons. In: ISSAC 2007, pp. 9–16. ACM, New York (2007)

  4. 4.

    Boito, P.: Structured matrix based methods for approximate polynomial GCD. Edizioni della Normale Pisa (2011)

  5. 5.

    Chèze, G., Galligo, A., Mourrain, B., Yacoubsohn, J.-C.: A subdivision method for computing nearest gcd with certification. Theor. Comput. Sci. 412, 4493–4503 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Chin, P., Corless, R.M., Corliss, G.F.: Optimization strategies for the approximate gcd problem. In: Proceedings of ISSAC’98, pp. 228–235. ACM, New York (1998)

  7. 7.

    Corless, R.M., Gianni, P.M., Trager, B.M., Watt, S.M.: The singular value decomposition for polynomial systems. In: Proceedings of ISSAC’95, pp. 195–207. ACM, New York (1995)

  8. 8.

    Dieci, L., Eirola, T.: On smooth decompositions of matrices. SIAM J. Matrix. Anal. and Appl. 20, 800–819 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Duan, X., Zhang, X., Wang, Q.: Computing approximation GCD of several polynomials by structured total least norm. ALAMT 3, 39–46 (2013)

    Article  Google Scholar 

  10. 10.

    Golub, G., Pereyra, V.: Separable nonlinear least squares: the variable projection method and its applications. Inverse Prob. 19, 1–26 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Golub, G., Pereyra, V.: The differentiation of pseudo-inverses and nonlinear least squares problems whose variables separate. SIAM J. Numer. Anal. 10, 413–432 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Guglielmi, N., Markovsky, I.: An ODE based method for computing the distance of coprime polynomials. SIAM J. Numer. Anal. 55, 1456–1482 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Kaltofen, E., Yang, Z., Zhi, L.: Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials. In: Proceedings of ISSAC’06, pp. 169–176. ACM, New York (2006)

  14. 14.

    Karcanias, N., Fatouros, S., Mitrouli, M., Halikias, G.: Approximate greatest common divisor of many polynomials and generalised resultants. Comput. Math. Appl. 51, 1817–1830 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Karmarkar, N.K., Lakshman, Y.N.: On approximate GCDs of univariate polynomials. J. Symbolic Comp. 26, 653–666 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Li, B., Liu, Z., Zhi, L: A fast algorithm for solving the sylvester structured total least squares problem. Signal Process. 87, 2313–2319 (2007)

    Article  MATH  Google Scholar 

  17. 17.

    Li, B., Nie, J., Zhi, L.: Approximate gcds of polynomials and sparse sos relaxations. Theor. Comput. Sci. 409, 200–210 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Magnus, J.R.: On differentiating eigenvalues and eigenvectors. Economet. Theor. 1, 179–191 (1985)

    Article  Google Scholar 

  19. 19.

    Markovsky, I.: Structured low-rank approximation and its applications. Automatica 44, 891–909 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Markovsky, I., Usevich, K.: Software for weighted structured low-rank approximation. Comput. Appl. Math. 256, 278–292 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Markovsky, I., Van Huffel, S: An algorithm for approximate common divisor computation. In: Proceedings of MTNS’06, pp. 274–279. Kyoto, Japan (2006)

  22. 22.

    Maroulas, J., Dascalopoulos, D.: Applications of the generalized Sylvester matrix. Appl. Math. Comput. 8, 121–135 (1981)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Pan, V.Y.: Computation of approximate polynomial GCDs and an extension. Inf. Comput. 167, 71–85 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Qiu, W., Hua, Y., Abed-Meraim, K.: A subspace method for the computation of the GCD of polynomials. Automatica 33, 741–743 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Schost, E., Spaenlehauer, P.-J.: A quadratically convergent algorithm for structured low-rank approximation. Found. Comput. Math. 16, 457–492 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Terui, A.: GPGCD: An iterative method for calculating approximate GCD of univariate polynomials. Theor. Comput. Sci. 479, 127–149 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Usevich, K., Markovsky, I.: Variable projection for affinely structured low-rank approximation in weighted 2-norms. J. Comput. Appl. Math. 272, 430–448 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Usevich, K., Markovsky, I.: Variable projection methods for approximate (greatest) common divisor computations. Theor. Comput. Sci. 681, 176–198 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Vardulakis, A.I.G., Stoyle, P.N.R.: Generalized Resultant Theorem. J. Inst. Maths. Applics. 22, 331–335 (1978)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Zarowsky, C.J., Ma, X., Fairman, F.W.: QR-factorization method for computing the greatest common divisor of polynomials with inexact coefficients. IEEE Trans. Signal Process. 48, 3042–3051 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Zeng, Z.: The approximate GCD of inexact polynomials, Part 1: A univariate algorithm http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.83.645&rep=rep1&type=pdf (2004)

  32. 32.

    Zeng, Z.: The numerical greatest common divisor of univariate polynomials. Contemp. Math. 556, 187–217 (2011)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

Nicola Guglielmi thanks INdAM GNCS for financial support. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement number 258581 “Structured low-rank approximation: Theory, algorithms, and applications” and fund for Scientific Research (FWO-Vlaanderen), FWO projects G028015N ”Decoupling multivariate polynomials in nonlinear system identification” and G090117N ”Block-oriented nonlinear identification using Volterra series”; and the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme initiated by the Belgian Science Policy Office.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Antonio Fazzi.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Fazzi, A., Guglielmi, N. & Markovsky, I. An ODE-based method for computing the approximate greatest common divisor of polynomials. Numer Algor 81, 719–740 (2019). https://doi.org/10.1007/s11075-018-0569-0

Download citation

Keywords

  • Sylvester matrix
  • Iterative methods
  • Approximate GCD
  • Polynomials

Mathematics Subject Classification (2010)

  • 15A24
  • 65K10