Numerische Mathematik

, Volume 109, Issue 3, pp 365–383

Approximate iterations for structured matrices

Authors

    • Max-Planck-Institut für Mathematik in den Naturwissenschaften
  • Boris N. Khoromskij
    • Max-Planck-Institut für Mathematik in den Naturwissenschaften
  • Eugene E. Tyrtyshnikov
    • Institute of Numerical MathematicsRussian Academy of Sciences
Open AccessArticle

DOI: 10.1007/s00211-008-0143-0

Cite this article as:
Hackbusch, W., Khoromskij, B.N. & Tyrtyshnikov, E.E. Numer. Math. (2008) 109: 365. doi:10.1007/s00211-008-0143-0

Abstract

Important matrix-valued functions f (A) are, e.g., the inverse A −1, the square root \({\sqrt{A}}\) and the sign function. Their evaluation for large matrices arising from pdes is not an easy task and needs techniques exploiting appropriate structures of the matrices A and f (A) (often f (A) possesses this structure only approximately). However, intermediate matrices arising during the evaluation may lose the structure of the initial matrix. This would make the computations inefficient and even infeasible. However, the main result of this paper is that an iterative fixed-point like process for the evaluation of f (A) can be transformed, under certain general assumptions, into another process which preserves the convergence rate and benefits from the underlying structure. It is shown how this result applies to matrices in a tensor format with a bounded tensor rank and to the structure of the hierarchical matrix technique. We demonstrate our results by verifying all requirements in the case of the iterative computation of A −1 and \({\sqrt{A}}\).

Mathematics Subject Classification (2000)

65F30 65F50 65N35 65F10

Copyright information

© The Author(s) 2008