Since using a preconditioner in an iterative method incurs some extra cost, both initially for the setup, and per iteration for applying it, there is a trade-off between the cost of constructing and applying the preconditioner, and the gain in convergence speed. Certain preconditioners need little or no construction phase at all (for instance the SSOR preconditioner), but for others, such as incomplete factorizations, there can be substantial work involved. Although the work in scalar terms may be comparable to a single iteration, the construction of the preconditioner may not be vectorizable/parallelizable even if application of the preconditioner is. In that case, the initial cost has to be amortized over the iterations, or over repeated use of the same preconditioner in multiple linear systems.

Most preconditioners take in their application an amount of work proportional to the number of variables. This implies that they multiply the work per iteration by a constant factor. On the other hand, the number of iterations as a function of the matrix size is usually only improved by a constant. Certain preconditioners are able to improve on this situation, most notably the modified incomplete factorizations and preconditioners based on multigrid techniques.

On parallel machines there is a further trade-off between the efficacy of a preconditioner in the classical sense, and its parallel efficiency. Many of the traditional preconditioners have a large sequential component.

Mon Nov 20 08:52:54 EST 1995