By Barrett R.
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Example text
Householder transformations, which are relatively costly but stable, have also been proposed. The Householder approach results in a three-fold increase in work associated with inner products and vector updates (not with matrix vector products); however, convergence may be better, especially for ill-conditioned systems (see Walker [214]). FVom the point of view of parallelism, Gram-Schmidt orthogonalization may be preferred, giving up some stability for better parallelization properties (see Demmel, Heath and Van der Vorst [67]).
Computational costs per iteration are similar to BiCG, but the method doesn't require the transpose matrix. Unlike BiCG, the two matrix-vector products are not independent, so the number of synchronization points in a parallel environment is larger. 9. Biconjugate Gradient Stabilized (Bi-CGSTAB) Applicable to nonsymmetric matrices. Computational costs per iteration are similar to BiCG and CGS, but the method doesn't require the transpose matrix. An alternative for CGS that avoids the irregular convergence patterns of CGS while maintaining about the same speed of convergence; as a result we often observe less loss of accuracy in the updated residual.
This requires a few extra vectors of storage and vector update operations per iteration, but it avoids expending a matrix-vector product on the residual calculation. Also, the algorithm has been modified so that only two full preconditioning steps are required instead of three. Computation of the residual is done for the convergence test. If one uses right (or post) preconditioning, that is MI = /, then a cheap upper bound for \\r^\\ can be computed in each iteration, avoiding the recursions for r^.