Skip to main content
SpringerLink
Log in

From implicit to recursive equations

  • Original Paper
  • Published:
Applicable Algebra in Engineering, Communication and Computing Aims and scope

Abstract

The technique of relaxed power series expansion provides an efficient way to solve so called recursive equations of the form \(F = \varPhi (F)\), where the unknown F is a vector of power series, and where the solution can be obtained as the limit of the sequence \(0, \varPhi (0), \varPhi (\varPhi (0)), \ldots \). With respect to other techniques, such as Newton’s method, two major advantages are its generality and the fact that it takes advantage of possible sparseness of \(\varPhi \). In this paper, we consider more general implicit equations of the form \(\varPhi (F) = 0\). Under mild assumptions on such an equation, we will show that it can be rewritten as a recursive equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Bostan, A., Chyzak, F., Ollivier, F., Salvy, B., Schost, É., Sedoglavic, A.: Fast computation of power series solutions of systems of differential equation. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms, pp. 1012–1021. New Orleans, Louisiana, USA, January (2007)

  2. Brent, R.P., Kung, H.T.: Fast algorithms for manipulating formal power series. J. ACM 25, 581–595 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berthomieu, J., Lebreton, R.: Relaxed \(p\)-adic hensel lifting for algebraic systems. In: van der Hoeven, J., van Hoeij, M. (eds.) Proceedings of the ISSAC ’12, pp. 59–66. Grenoble, France, July (2012)

  4. Cantor, D.G., Kaltofen, E.: On fast multiplication of polynomials over arbitrary algebras. Acta Inf. 28, 693–701 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fischer, M.J., Stockmeyer, L.J.: Fast on-line integer multiplication. In: Proceedings 5th ACM Symposium on Theory of Computing, vol. 9, pp. 67–72 (1974)

  7. van der Hoeven, J.: Lazy multiplication of formal power series. In: Küchlin, W.W. (ed.) Proceedings of the ISSAC ’97, pp. 17–20. Maui, Hawaii, July (1997)

  8. van der Hoeven, J.: Relax, but don’t be too lazy. JSC 34, 479–542 (2002)

    MathSciNet  MATH  Google Scholar 

  9. van der Hoeven, J.: Relaxed multiplication using the middle product. In: Bronstein, M. (ed.) Proceedings of the ISSAC ’03, pp. 143–147. Philadelphia, USA, August (2003)

  10. van der Hoeven, J.: New algorithms for relaxed multiplication. JSC 42(8), 792–802 (2007)

    MathSciNet  MATH  Google Scholar 

  11. van der Hoeven, J.: Relaxed resolution of implicit equations. Technical Report, HAL (2009). http://hal.archives-ouvertes.fr/hal-00441977. Accessed Aug 2018

  12. van der Hoeven, J.: Newton’s method and FFT trading. JSC 45(8), 857–878 (2010)

    MathSciNet  MATH  Google Scholar 

  13. van der Hoeven, J.: From implicit to recursive equations. Technical Report, HAL (2011). http://hal.archives-ouvertes.fr/hal-00583125. Accessed Aug 2018

  14. van der Hoeven, J.: Faster relaxed multiplication. In: Proceedings of the ISSAC ’14, pp. 405–412. Kobe, Japan, July (2014)

  15. Lebreton, R.: Relaxed hensel lifting of triangular sets. JSC 68(2), 230–258 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Lebreton, R., Schost, É.: A simple and fast online power series multiplication and its analysis. JSC 72, 231–251 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Sedoglavic, A.: Méthodes seminumériques en algèbre différentielle; applications à l’étude des propriétés structurelles de systèmes différentiels algébriques en automatique. Ph.D. thesis, École polytechnique (2001)

  18. Schönhage, A., Strassen, V.: Schnelle Multiplikation großer Zahlen. Computing 7, 281–292 (1971)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We wish to thank the referees for their careful reading and their comments and suggestions.

Author information

Authors and Affiliations

  1. LIX, CNRS, École polytechnique, 91128, Palaiseau Cedex, France

    Joris van der Hoeven

Authors
  1. Joris van der Hoeven
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joris van der Hoeven.

Additional information

This work has been supported by the ANR-09-JCJC-0098-01 MaGiX and ANR-10-BLAN-0109 projects, as well as a Digiteo 2009-36HD Grant and Région Ile-de-France.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

van der Hoeven, J. From implicit to recursive equations. AAECC 30, 243–262 (2019). https://doi.org/10.1007/s00200-018-0370-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00200-018-0370-2

Keywords

Mathematics Subject Classification

Search

Navigation