Reorthogonalization Methods in ABS Classes

Abstract

In this paper we analyze two subclasses of ABS class of methods which produce orthogonal projection vectors. We theoretically prove that the twice is enough selective reorthogonalization criterion of Parlett-Kahan (14) and of Hegedus (8) can be used in the various ABS classes. Here we also provide a detailed numerical analysis of these ABS- based algorithms. We revealed that the ABS-based algorithm combined with the modified Parlett-Kahan criterion by Hegedus provided more accurate results in the three considered cases (the rank of the coefficient matrix, the determination of the orthogonal bases, and the QR factorization) than the built-in rank and qr MATLAB functions.