Numerical solution of symmetric least-squares problems with an inversion-free Kovarik-type algorithm

Abstract

In a previous paper we presented two variants of Kovarik's approximate orthogonalization algorithm for arbitrary symmetric matrices, one with and one without explicit matrix inversion. Here we propose another inverse-free version that has the advantage of a smaller bound on the convergence factor, while the computational costs per iteration are even less than in the initial inverse-free variant.

We then investigate the application of the new algorithm for the numerical solution of linear least-squares problems with a symmetric matrix. The basic idea is to modify the right-hand side of the equation during the transformation of the matrix. We prove that the sequence of vectors generated in this way converges to the minimal norm solution of the problem.

Numerical tests with the collocation discretization of a first-kind integral equation demonstrate a mesh-independent behaviour and stability with respect to numerical errors introduced by the use of numerical quadrature.

Keywords:

• Collocation discretization
• First-kind integral equation
• Linear least-squares problem
• Orthogonalization algorithm
• Symmetric matrix

Acknowledgements

The work of C. Popa is supported by the Grant CEEX 05-D11-25/2005 and the DAAD via a grant as a visiting professor at FAU Erlangen-Nuremberg, Germany (December 2005–January 2006). Part of this research was conducted when M. Mohr was a member of the DFG Junior Research Group: Inverse Problems in Piezoelectricity and its Applications (grant Ka 1778/1). All computations were performed with the Numerical Linear Algebra software package OCTAVE, freely available under the terms of the GNU General Public License, see www.octave.org.