BIT Numerical Mathematics

, Volume 39, Issue 3, pp 417–438

Bounds for the Entries of Matrix Functions with Applications to Preconditioning

  • Michele Benzi
  • Gene H. Golub

DOI: 10.1023/A:1022362401426

Cite this article as:
Benzi, M. & Golub, G.H. BIT Numerical Mathematics (1999) 39: 417. doi:10.1023/A:1022362401426


Let A be a symmetric matrix and let f be a smooth function defined on an interval containing the spectrum of A. Generalizing a well-known result of Demko, Moss and Smith on the decay of the inverse we show that when A is banded, the entries of f(A)are bounded in an exponentially decaying manner away from the main diagonal. Bounds obtained by representing the entries of f(A)in terms of Riemann-Stieltjes integrals and by approximating such integrals by Gaussian quadrature rules are also considered. Applications of these bounds to preconditioning are suggested and illustrated by a few numerical examples.

Matrix functionsquadrature rulesLanczos processband matricesexponential decaypreconditioned conjugate gradients

Copyright information

© Swets & Zeitlinger 1999

Authors and Affiliations

  • Michele Benzi
    • 1
  • Gene H. Golub
    • 2
  1. 1.Scientific Computing Group, CIC-19Los Alamos National LaboratoryLos AlamosUSA. email:
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA. email: