An accurate parallel block Gram-Schmidt algorithm without reorthogonalization

The modified Gram-Schmidt (MGS) orthogonalization process-used for example in the Arnoldi algorithm-often constitutes the bottleneck that Limits parallel efficiencies. Indeed, a number of communications, proportional to the square of the problem size, are required to compute the dot-products. A block formulation is attractive but it suffers from potential numerical instability. In this paper, we address this issue and propose a simple procedure that allows the use of a block Gram-Schmidt algorithm while guaranteeing a numerical accuracy close to that of MGS. The main idea is to determine the size of the blocks dynamically. The main advantages of this dynamic procedure are two-fold: first, high performance matrix-vector multiplications can be used to decrease the execution time. Next, in a parallel environment, the number of communications is reduced. Performance comparisons with the alternative Iterated CGS also show an improvement for a moderate number of processors. Copyright (C) 2000 John Wiley & Sons, Ltd.