Projections and preconditioning for inconsistent least-squares problems

Abstract

In this paper we describe two iterative algorithms for the numerical solution of linear least-squares problems. They are based on a combination between an extension of the classical Kaczmarzs projections method (Popa [5]) and an approximate orthogonalization technique due to Kovarik. We prove that both new algorithms converge to any solution of an inconsistent and rank-defficient least-squares problem (with respect to the choice of the initial approxi-mation), the convergence being much faster than for the classical Kaczmarz - like methods. Some numerical experiments on a first kind integral equation are described.

Keywords:

• Inconsistent Least-Squares Problems
• Kaczmarzs Iteration
• Approx- Imate Orthogonalization
• Preconditioning

AMS Subject Classifications:

• 65F10
• 65F20

C.R. Category:

• G.1.3