A Class of Modifications for Kovarik's Method

Abstract

The approximate orthogonalization method for a finite set of linearly independent vectors was introduced by Z. Kovarik in which it is necessity to compute explicitly the inverse of a matrix in every iteration. It is proved that Kovarik's method converges quadratically. Several modifications have been proposed for Kovarik's method, all of which try to eliminate the necessity of explicit computation of the inverse. Most of these methods are linear convergent. The best modification, with a good convergent behaviour, is Petcu and Popa's, although they did not express any satisfactory reason for the origin of this modification. In this paper, we present a class of modifications for Kovarik's method which consists of Petcu and Popa's method. We prove that the methods from this class are, generally, linear convergent, while, only for the special case of the Petcu and Popa's method, it is quadratic convergent. Therefore, we show that Petcu and Popa's method, in contrast with their claim, is not linear but quadratic convergent, turning it into an optimal method in this class.