Numerical Algorithms

, Volume 7, Issue 2, pp 183–199

Toeplitz approximate inverse preconditioner for banded Toeplitz matrices

  • Martin Hanke
  • James G. Nagy

DOI: 10.1007/BF02140682

Cite this article as:
Hanke, M. & Nagy, J.G. Numer Algor (1994) 7: 183. doi:10.1007/BF02140682


In this paper we introduce a new preconditioner for banded Toeplitz matrices, whose inverse is itself a Toeplitz matrix. Given a banded Hermitian positive definite Toeplitz matrixT, we construct a Toepliz matrixM such that the spectrum ofMT is clustered around one; specifically, if the bandwidth ofT is β, all but β eigenvalues ofMT are exactly one. Thus the preconditioned conjugate gradient method converges in β+1 steps which is about half the iterations as required by other preconditioners for Toepliz systems that have been suggested in the literature. This idea has a natural extension to non-banded and non-Hermitian Toeplitz matrices, and to block Toeplitz matrices with Toeplitz blocks which arise in many two dimensional applications in signal processing. Convergence results are given for each scheme, as well as numerical experiments illustrating the good convergence properties of the new preconditioner.

Key words

Circulant matrices conjugate gradient preconditioner Toeplitz matrices 

Subject classification

AMS(MOS) 65F10 65F15 

Copyright information

© J.C. Baltzer AG, Science Publishers 1994

Authors and Affiliations

  • Martin Hanke
    • 1
  • James G. Nagy
    • 2
  1. 1.Institut für Praktische MathematikUniversität KarlsruheKarlsruheGermany
  2. 2.Department of MathematicsSouthern Methodist UniversityDallasUSA

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