Advertisement

Acceleration of the Inversion of Triangular Toeplitz Matrices and Polynomial Division

  • Brian J. Murphy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6885)

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.

Keywords

Reciprocal of a polynomial modulo a power Polynomial division Triangular Toeplitz matrix inversion Banded triangular Toeplitz matrices 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bini, D., Pan, V.Y.: Polynomial and Matrix Computations, Volume 1. Fundamental Algorithms. Birkhäuser, Boston (1994)Google Scholar
  3. 3.
    Pan, V.Y.: Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser/Springer, Boston/New York (2001)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chan, R.: Toeplitz preconditioners for Toeplitz systems with nonnegative generating functions. IMA J. Numer. Anal. 11, 333–345 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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. 6.
    Bini, D., Pan, V.Y.: Polynomial Division and Its Computational Complexity. Journal of Complexity 2, 179–203 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Pan, V.Y.: Complexity of Computations with Matrices and Polynomials. SIAM Review 34(2), 225–262 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sieveking, M.: An Algorithm for Division of Power Series. Computing 10, 153–156 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bini, D.: Parallel Solution of Certain Toeplitz Linear Systemns. SIAM J. on Computing 13(2), 268–276 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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. 11.
    Reif, J.H., Tate, S.R.: Optimum Size Division Circuits. SIAM J. on Computing 19(5), 912–925 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Aho, A.V., Hopcroft, J.E., Ullman, J.D.: The Design and Analysis of Algorithms. Addison-Wesley, Reading (1974)zbMATHGoogle Scholar
  14. 14.
    Borodin, A.B., Munro, I.: Computational Complexity of Algebraic and Numeric Problems. American Elsevier, New York (1975)zbMATHGoogle Scholar
  15. 15.
    von zur Gathen, J., Gerhard, J.: Modern Computer Algebra, 2nd edn. Cambridge University Press, Cambridge (2003) (first edition, 1999)Google Scholar
  16. 16.
    Commenges, D., Monsion, M.: Fast inversion of triangular Toeplitz matrices. IEEE Trans. Automat. Control AC-29, 250–251 (1984)Google Scholar
  17. 17.
    Bernstein, D.J.: Removing redundancy in high-precision Newton iteration (2004), http://cr.yp.to/fastnewton.html#fastnewton-paper
  18. 18.
    van der Hoeven, J.: Newton’s method and FFT trading. Journal of Symbolic Computing 45(8), 857–878 (2010)Google Scholar
  19. 19.
    Hanrot, G., Zimmermann, P.: Newton iteration revisited, http://www.loria.fr/~zimmerma/papers/fastnewton.ps.gz
  20. 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. 21.
    Schonhage, A.: Variations on computing reciprocals of power series. Inf. Process. Lett. 74(1-2), 41–46 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Harvey, D.: aster algorithms for the square root and reciprocal of power series. Math. Comp. 80, 387–394 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Murphy, B.: Short-circuited FFTs for computing paritally known convolutions (2010), http://comet.lehman.cuny.edu/bmurphy/research/ScFFT.pdf
  24. 24.
    Dongarra, J., Hammarling, S., Sorensen, D.: Block reduction of matrices to condensed form for eigenvalue computations. J. Comp. Appl. Math. 27, 215 (1989)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Brian J. Murphy
    • 1
  1. 1.Department of Mathematics and Computer ScienceLehman College of the City University of New YorkBronxUSA

Personalised recommendations

Cite paper

Buy options