Skip to main content
SpringerLink
Log in

Multigrid methods for toeplitz matrices

  • Published:
CALCOLO Aims and scope Submit manuscript

Abstract

We introduce a class of Multigrid methods for solving banded, symmetric Toeplitz systems Ax=b. We use a, special choice of the projection operator whose coefficients simply depend on some spectral properties of A. This choice leads to an iterative Multigrid method with convergence rate smaller than 1 independent of the condition number K2(A) and of the dimension of the matrix. In the second part the B0 class is introduced: this class, of Toeplitz matrices contains the linear space generated by the matrices arising from the finite differences discretization of the differential operators

, m∈N +. To sum up we present an adaptive algorithm which has a input the coefficients of A and return an iterative Multigrid method with convergence speed independent of the mesh spacing h and with an asymptotical cost of O(n).

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.

Similar content being viewed by others

References

  1. W. Hackbursch,Multigrid Methods and Applications, 1985, Springer Verlag, Berlin.

    Google Scholar 

  2. W. Hackbusch,Multigrid methods II, Lecture notes in Math. 1228 (1987).

  3. K. Stuben, U. Trottenberg,Multigrid methods, Lecture notes in Math. 960 (1981).

  4. D. Bini, M. Capovani,Spectral and Computational Properties of Band Simmetric Matrices, Linear Algebra Appl. 52/53, 99–125 (1983).

    MathSciNet  Google Scholar 

  5. R. S. Varga,Matrix Iterative Analysis, 1962, Prentice/Hall.

  6. A. Greenbaum,Analysis of a multigrid method as an iterative technique for solving linear systems, SIAM J. Numer. Anal. 21 (3), 473–458 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Stoer, R. Bulirsch,Introduction to Numerical Analysis, 1980, Springer Verlag, Berlin.

    Google Scholar 

  8. U. Grenander, G. Szego,Toeplitz Forms and Their Applications, 1984, Second edition, Chelsea, New York.

    MATH  Google Scholar 

  9. I. S. Iokhvidov,Hankel and Toeplitz forms: algebraic theory, 1982, Birkhauser, Boston.

    MATH  Google Scholar 

  10. S. Serra,On multi-iterative methods, To appear.

  11. S. Serra,Proprietà algebriche e computazionali delle matrici di Toeplitz e metodi Multigrid, Thesis in computer science, University of Pisa 1990.

  12. G. Fiorentino,Formulazione algebrica dei metodi multigrid. Applicazioni alla risoluzione di sistemi di Toeplitz a bonda, Thesis in computer science, University of Pisa 1990.

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienze dell'Informazione, Università di Pisa, corso Italia 90, 56100, Pisa, Italy

    G. Fiorentino & S. Serra

Authors
  1. G. Fiorentino
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Serra
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fiorentino, G., Serra, S. Multigrid methods for toeplitz matrices. Calcolo 28, 283–305 (1991). https://doi.org/10.1007/BF02575816

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02575816

Keywords

Search

Navigation