Skip to main content
SpringerLink
Log in

Acceleration of the Inversion of Triangular Toeplitz Matrices and Polynomial Division

  • Conference paper
Computer Algebra in Scientific Computing (CASC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6885))

Included in the following conference series:

Abstract

Computing the reciprocal of a polynomial in z modulo a power z n is well known to be closely linked to polynomial division and equivalent to the inversion of an n×n triangular Toeplitz matrix. The degree k of the polynomial is precisely the bandwidth of the matrix, and so the matrix is banded iff k ≪ n. We employ the above equivalence and some elementary but novel and nontrivial techniques to obtain minor yet noticeable acceleration of the solution of the cited fundamental computational problems.

Supported by PSC CUNY Awards 609 62400–0040 and 393 6327000–41.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Kailath, T., Kung, S.Y., Morf, M.: Displacement Ranks of Matrices and Linear Equations. Journal Math. Analysis and Appls. 68(2), 395–407 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bini, D., Pan, V.Y.: Polynomial and Matrix Computations, Volume 1. Fundamental Algorithms. Birkhäuser, Boston (1994)

    Google Scholar 

  3. Pan, V.Y.: Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser/Springer, Boston/New York (2001)

    Book  MATH  Google Scholar 

  4. Chan, R.: Toeplitz preconditioners for Toeplitz systems with nonnegative generating functions. IMA J. Numer. Anal. 11, 333–345 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lin, F.R., Ching, W.K.: Inverse Toeplitz preconditioners for Hermitian Toeplitz systems. Numer. Linear Algebra Appl. 12, 221–229 (2005)

    Google Scholar 

  6. Bini, D., Pan, V.Y.: Polynomial Division and Its Computational Complexity. Journal of Complexity 2, 179–203 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Pan, V.Y.: Complexity of Computations with Matrices and Polynomials. SIAM Review 34(2), 225–262 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sieveking, M.: An Algorithm for Division of Power Series. Computing 10, 153–156 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bini, D.: Parallel Solution of Certain Toeplitz Linear Systemns. SIAM J. on Computing 13(2), 268–276 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bini, D., Pan, V.Y.: Improved Parallel Polynomial Division. SIAM J. on Computing 22(3), 617–627 (1993); Proc. version in FOCS 1992, pp. 131–136. IEEE Computer Society Press, Los Alamitos (1992)

    Google Scholar 

  11. Reif, J.H., Tate, S.R.: Optimum Size Division Circuits. SIAM J. on Computing 19(5), 912–925 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pan, V.Y., Landowne, E., Sadikou, A.: Polynomial Division with a Remainder by Means of Evaluation and Interpolation. Information Processing Letters 44, 149–153 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Algorithms. Addison-Wesley, Reading (1974)

    MATH  Google Scholar 

  14. Borodin, A.B., Munro, I.: Computational Complexity of Algebraic and Numeric Problems. American Elsevier, New York (1975)

    MATH  Google Scholar 

  15. von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003) (first edition, 1999)

    Google Scholar 

  16. Commenges, D., Monsion, M.: Fast inversion of triangular Toeplitz matrices. IEEE Trans. Automat. Control AC-29, 250–251 (1984)

    Google Scholar 

  17. Bernstein, D.J.: Removing redundancy in high-precision Newton iteration (2004), http://cr.yp.to/fastnewton.html#fastnewton-paper

  18. van der Hoeven, J.: Newton’s method and FFT trading. Journal of Symbolic Computing 45(8), 857–878 (2010)

    Google Scholar 

  19. Hanrot, G., Zimmermann, P.: Newton iteration revisited, http://www.loria.fr/~zimmerma/papers/fastnewton.ps.gz

  20. Schoenhage, A., Vetter, E.: A New Approach to Resultant Computations and Other Algorithms with Exact Division. In: European Symposium on Algorithms, pp. 448–459 (1994)

    Google Scholar 

  21. Schonhage, A.: Variations on computing reciprocals of power series. Inf. Process. Lett. 74(1-2), 41–46 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  22. Harvey, D.: aster algorithms for the square root and reciprocal of power series. Math. Comp. 80, 387–394 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Murphy, B.: Short-circuited FFTs for computing paritally known convolutions (2010), http://comet.lehman.cuny.edu/bmurphy/research/ScFFT.pdf

  24. Dongarra, J., Hammarling, S., Sorensen, D.: Block reduction of matrices to condensed form for eigenvalue computations. J. Comp. Appl. Math. 27, 215 (1989)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Lehman College of the City University of New York, Bronx, NY, 10468, USA

    Brian J. Murphy

Authors
  1. Brian J. Murphy
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies (LIT), Joint Institute for Nuclear Research (JINR), 141980, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Wolfram Koepf

  3. Institut für Informatik, Lehrstuhl für Effiziente Algorithmen, Technische Universität München, Boltzmannstraße 3, 85748, Garching, Germany

    Ernst W. Mayr

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Murphy, B.J. (2011). Acceleration of the Inversion of Triangular Toeplitz Matrices and Polynomial Division. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2011. Lecture Notes in Computer Science, vol 6885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23568-9_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-23568-9_25

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23567-2

  • Online ISBN: 978-3-642-23568-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation