An iterative algorithm for approximate orthogonalisation of symmetric matrices

Abstract

In a previous article, one of the authors presented an extension of an iterative approximate orthogonalisation algorithm, due to Z. Kovarik, for arbitrary rectangular matrices. In the present article, we propose a modified version of this extension for the class of arbitrary symmetric matrices. For this new algorithm, the computational effort per iteration is much smaller than for the initial one. We prove its convergence and also derive an error reduction factor per iteration. In the second part of the article, we show that we can eliminate the matrix inversion required by the previous algorithm in each iteration, by replacing it with a polynomial matrix expression. Some numerical experiments are also presented for a collocation discretisation of a first kind integral equation.

^*All the computations were made with the Numerical Linear Algebra software package OCTAVE, freely available under the terms of the GNU General Public License, see www.octave.org

Keywords:

• Orthogonalisation algorithm
• Symmetric matrix
• Approximate inverse

Acknowledgments

The article was supported by NATO through collaborative linkage grant PST.CLG.977924 and by the DAAD via a grant that one of the authors had as a visiting professor at the Friedrich-Alexander-Universität Erlangen–Nürnberg, Germany, in the period October 2002–August 2003.

Notes

^*All the computations were made with the Numerical Linear Algebra software package OCTAVE, freely available under the terms of the GNU General Public License, see www.octave.org