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.