A Generalized Gram{schmidt Procedure for Parallel Applications

The Gram-Schmidt procedure is used to orthogonalize one vector against a set of vectors or to construct a QR factorization of a matrix. In the Classical Gram-Schmidt algorithm (CGS), orthogonal vectors are produced via matrix{vector updates, which is desirable for parallel computers. Unfortunately, this algorithm exhibits a poor numerical stability behavior and the loss of orthogonality cannot be bounded in the presence of roundoo errors. In the Modiied Gram-Schmidt algorithm (MGS), the numerical stability is controlled by sequentially orthogonalizing the current vector to the others, but this destroys the parallel properties of the algorithm. Here, we propose an extremely simple Generalized Gram-Schmidt algorithm (GGS) that combines a good parallel behavior with an acceptable numerical accuracy. This algorithm can be seen as a block Gram-Schmidt. It possess the same parallel properties of CGS and, for small block sizes, the accuracy is comparable to this of MGS. However, the stability of our algorithm is not known and we believe that the loss of orthogonality can be arbitrarily bad.