Several authors have found ways to eliminate some of the synchronization points induced by the inner products in methods such as CG. One strategy has been to replace one of the two inner products typically present in conjugate gradient-like methods by one or two others in such a way that all inner products can be performed simultaneously. The global communication can then be packaged. A first such method was proposed by Saad [182] with a modification to improve its stability suggested by Meurant [156]. Recently, related methods have been proposed by Chronopoulos and Gear [55], D'Azevedo and Romine [62], and Eijkhout [88]. These schemes can also be applied to nonsymmetric methods such as BiCG. The stability of such methods is discussed by D'Azevedo, Eijkhout and Romine [61].
Another approach is to generate a number of
successive Krylov vectors (see §
) and
orthogonalize these as a block (see
Van Rosendale [210], and Chronopoulos and
Gear [55]).