Reorthogonalization Methods Revisited
Acta Polytechnica Hungarica · 2015
New theoretical background of Parlett-Kahan's "twice is enough" algorithm for computing accurate vectors in Gram-Schmidt orthogonalization is given.An unorthodox type of error analysis is applied by considering lost digits in cancellation.The resulting proof is simple and that makes it possible to calculate the number of accurate digits after all reorthogonalization steps.Self improving nature of projection matrices is found giving a possible explanation for the stability of some ABS methods.The numerical tests demonstrate the validity and applicability of the theoretical results for the CGS, MGS and rank revealing QR algorithms.