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Implementation of a Near-Optimal Complex Root Clustering Algorithm

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Mathematical Software – ICMS 2018 (ICMS 2018)

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Abstract

We describe Ccluster, a software for computing natural \(\varepsilon \)-clusters of complex roots in a given box of the complex plane. This algorithm from Becker et al. (2016) is near-optimal when applied to the benchmark problem of isolating all complex roots of an integer polynomial. It is one of the first implementations of a near-optimal algorithm for complex roots. We describe some low level techniques for speeding up the algorithm. Its performance is compared with the well-known MPSolve library and Maple.

Rémi’s work has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No. 676541.

Victor’s work is supported by NSF Grants # CCF-1116736 and # CCF-1563942 and by PSC CUNY Award 698130048.

Chee’s work is supported by NSF Grants # CCF-1423228 and # CCF-1564132.

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Notes

  1. 1.

    Irina Voiculescu informed us that her student Dan-Andrei Gheorghe has independently implemented the same algorithm in a Masters Thesis Project (May 18, 2017) at Oxford University. Sewon Park and Martin Ziegler at KAIST, Korea, have implemented a modified version of Becker et al. (2016) for polynomials having only real roots being the eigenvalues of symmetric square matrices with real coefficients. See the technical report CS-TR-2018-415 at https://cs.kaist.ac.kr/research/techReport.

  2. 2.

    https://julialang.org/. Download our code in https://github.com/rimbach/Ccluster.

  3. 3.

    http://arblib.org/. Download our code in https://github.com/rimbach/Ccluster.jl.

  4. 4.

    We treat two-valued predicates for simplicity; the discussion could be extended to predicates (like \(\widetilde{T}^{G}_{*}\)) which returns a finite set of values.

References

  1. Becker, R., Sagraloff, M., Sharma, V., Xu, J., Yap, C.: Complexity analysis of root clustering for a complex polynomial. In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, pp. 71–78. ACM (2016)

    Google Scholar 

  2. Becker, R., Sagraloff, M., Sharma, V., Yap, C.: A near-optimal subdivision algorithm for complex root isolation based on the pellet test and newton iteration. J. Symb. Comput. 86, 51–96 (2018)

    Article  MathSciNet  Google Scholar 

  3. Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algorithms 23(2–3), 127–173 (2000)

    Article  MathSciNet  Google Scholar 

  4. Bini, D.A., Robol, L.: Solving secular and polynomial equations: a multiprecision algorithm. J. Comput. Appl. Math. 272, 276–292 (2014)

    Article  MathSciNet  Google Scholar 

  5. Brönnimann, H., Burnikel, C., Pion, S.: Interval arithmetic yields efficient dynamic filters for computational geometry. Discrete Appl. Math. 109(1–2), 25–47 (2001)

    Article  MathSciNet  Google Scholar 

  6. Emiris, I.Z., Pan, V.Y., Tsigaridas, E.P.: Algebraic algorithms. In: Computing Handbook, Third Edition: Computer Science and Software Engineering, pp. 10:1–10:30. Chapman and Hall/CRC (2014)

    Google Scholar 

  7. Fortune, S.: An iterated eigenvalue algorithm for approximating roots of univariate polynomials. J. Symb. Comput. 33(5), 627–646 (2002)

    Article  MathSciNet  Google Scholar 

  8. Giusti, M., Lecerf, G., Salvy, B., Yakoubsohn, J.-C.: On location and approximation of clusters of zeros of analytic functions. Found. Comput. Math. 5(3), 257–311 (2005)

    Article  MathSciNet  Google Scholar 

  9. Gourdon, X.: Combinatoire, Algorithmique et Géométrie des Polynomes. Ph.D. thesis, École Polytechnique (1996)

    Google Scholar 

  10. Hribernig, V., Stetter, H.J.: Detection and validation of clusters of polynomial zeros. J. Symb. Comput. 24(6), 667–681 (1997)

    Article  MathSciNet  Google Scholar 

  11. Kobel, A., Rouillier, F., Sagraloff, M.: Computing real roots of real polynomials... and now for real! In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, pp. 303–310. ACM (2016)

    Google Scholar 

  12. Niu, X.-M., Sakurai, T., Sugiura, H.: A verified method for bounding clusters of zeros of analytic functions. J. Comput. Appl. Math. 199(2), 263–270 (2007)

    Article  MathSciNet  Google Scholar 

  13. Pan, V.Y.: Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding. J. Symb. Comput. 33(5), 701–733 (2002)

    Article  MathSciNet  Google Scholar 

  14. Rouillier, F., Zimmermann, P.: Efficient isolation of polynomial’s real roots. J. Comput. Appl. Math. 162(1), 33–50 (2004)

    Article  MathSciNet  Google Scholar 

  15. Sagraloff, M., Mehlhorn, K.: Computing real roots of real polynomials. J. Symb. Comput. 73, 46–86 (2016)

    Article  MathSciNet  Google Scholar 

  16. Yap, C., Sagraloff, M., Sharma, V.: Analytic root clustering: a complete algorithm using soft zero tests. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds.) CiE 2013. LNCS, vol. 7921, pp. 434–444. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39053-1_51

    Chapter  Google Scholar 

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Imbach, R., Pan, V.Y., Yap, C. (2018). Implementation of a Near-Optimal Complex Root Clustering Algorithm. In: Davenport, J., Kauers, M., Labahn, G., Urban, J. (eds) Mathematical Software – ICMS 2018. ICMS 2018. Lecture Notes in Computer Science(), vol 10931. Springer, Cham. https://doi.org/10.1007/978-3-319-96418-8_28

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  • DOI: https://doi.org/10.1007/978-3-319-96418-8_28

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