, Volume 49, Issue 4, pp 339–347 | Cite as

Iterative Toeplitz solvers with local quadratic convergence

  • E. Linzer
  • M. Vetterli


We study an iterative, locally quadratically convergent algorithm for solving Toeplitz systems of equations from [R. P. Brent, F. G. Gustavson and D. Y. Y. Yun. “Fast solution of Toeplitz systems of equations and computation of Padé approximations”,J. Algorithms, 1:259–295, 1980]. We introduce a new iterative algorithm that is locally quadratically convergent when used to solve symmetric positive definite Toeplitz systems. We present a set of numerical experiments on randomly generated symmetric positive definite Toeplitz matrices. In these experiments, our algorithm performed significantly better than the previously proposed algorithm.

AMS (MOS) Subject Classifications

65F10 65B99 

Key words

Toeplitz iterative methods steepest descent quadratic convergence 

Iterative Toeplitz Solver mit lokal quadratischer Konvergenz


Wir studieren einen iterativen, lokal quadratisch konvergenten Algorithmus für die Lösung von Toeplitz-Systemen von Gleichungen von [R. P. Brent, F. G. Gustavson und D. Y. Y. Yun, “Fast solution of Toeplitz systems of equations and computation of Padé approximations”,J. Algorithms, 1:259–295, 1980]. Wir führen einen neuen iterativen Algorithmus ein, der lokal quadratisch konvergent ist, wenn er für positiv definite Toeplitz-Systeme gebraucht wird. Wir präsentieren eine Anzahl von numerischen Experimenten mit zufallsgenerierten, symmetrischen, positiv definiten Toeplitz-Matrizen. In diesen Experimenten ist unser Algorithmus entscheidend besser als der früher vorgeschlagene Algorithmus.


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  1. [1]
    Bunch, J. R.: Stability of methods for solving Toeplitz systems of equations. SIAM J. Sci. Stat. Comput.6, 349–364 (1985).Google Scholar
  2. [2]
    Blahut, R. E.: Fast algorithms for digital signal processing. Reading, MA: Addison-Wesley 1986.Google Scholar
  3. [3]
    Golub, G. H., Van Loan, C. F.: Matrix computations. Baltimore, MD: Johns Hopkins 1983.Google Scholar
  4. [4]
    Levinson, N.: The Wiener rms error criterion in filter design and prediction. J. Math. Phys.25, 261–278 (1947).Google Scholar
  5. [5]
    Trench, W. F.: An algorithm for the inversion of finite Toeplitz matrices. J. SIAM12, 512–522 (1964).Google Scholar
  6. [6]
    Cybenko, G., Berry, M.: Hyperbolic Housholder algorithm for factoring structured matrices. SIAM J. Matrix Anal. Appl.11, 499–520 (1990).Google Scholar
  7. [7]
    Ammar, G. S., Gragg, W. B.: Superfast solution of real positive definite Toeplitz systems.9, 61–76 (1988).Google Scholar
  8. [8]
    Ammar, G. S., Gragg, W. B.: The generalized Schur algorithm for the superfast solution of Toeplitz systems. In: Pindor, M., Gilewicz, J., Siemaszko, W. (eds.) Rational approximation and its application in mathematics and physics. Springer 1986.Google Scholar
  9. [9]
    Bitmead, R. R., Anderson, B. D. O.: Asymptotically fast solution of Toeplitz and related systems of linear equations. Linear Algebera Appl.34, 103–116 (1980).Google Scholar
  10. [10]
    Brent, R. P., Gustavson, F. G., Yun, D. Y. Y.: Fast solution of Toeplitz systems equations and computation of Padé approximations. J. Algorithms1, 259–295 (1980).Google Scholar
  11. [11]
    Strang, G.: A proposal for Toeplitz matrix calculations. Stud. Appl. Math.74, 171–176 (1986).Google Scholar
  12. [12]
    Rino, C.: The inversion of covariance matrices by finite Fourier transformations. IEEE Trans. Inform. Theory16, 230–232 (1970).Google Scholar
  13. [13]
    Chan, R. H.: The spectrum of a family of circulant preconditioned Toeplitz systems. SIAM J. Numer. Anal.26, 503–506 (1989).Google Scholar
  14. [14]
    Ku, T., Kuo, J.: Design and analysis of Toeplitz preconditioners. Proc. IEEE Int. Conf. Acoust. Speech Sig. Proc., pp. 1811–1814 (1990).Google Scholar
  15. [15]
    Pan, V.: Fast and efficient parallel inversion of Toeplitz and block Toeplitz matrices. Operator Theory: Adv. Appl.40, 359–389 (1989).Google Scholar
  16. [16]
    Pan, V., Schrieber, R.: A fast, preconditioned conjugate gradient Toeplitz solver. Technical report 89.14, RIACS, NASA Ames Research Center, March 1989.Google Scholar
  17. [17]
    Linzer, E.: Arithmetic complexity and numerical properties of algorithms involving Toeplitz matrices. PhD thesis, Columbia University, New York, NY, October 1990.Google Scholar
  18. [18]
    Strang, G.: Introduction to applied mathematics. Wellesley, MA: Wellesley-Cambridge 1986.Google Scholar
  19. [19]
    Gohberg, I. C., Fel'dman, I. A.: Convolution equations and projection methods for their solution. Providence, RI: American Mathematical Society 1974.Google Scholar
  20. [20]
    Iohvidov, I. S.: Hankel and Toeplitz matrices and forms. Boston, MA: Birkhauser 1982.Google Scholar
  21. [21]
    Cybenko, G.: Error analysis of some signal processing algorithms. Princeton, NJ: PhD thesis, Princeton University 1978.Google Scholar
  22. [22]
    Chan, R. H., Strang, G.: Toeplitz equations by conjugate gradients with circulant preconditioner. SIAM J. Sci. Stat. Comput.10, 104–119 (1989).Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • E. Linzer
    • 1
  • M. Vetterli
    • 2
  1. 1.IBM ResearchYorktown Heights
  2. 2.Department of Electrical Engineering and Center for Telecommunications ResearchColumbia UniversityNew York

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