A new taxonomy of conjugate gradient methods

In this paper, a framework is proposed for conjugate gradient and conjugate direction algorithms based on an indefinite quadratic form. By the choice of the Hessian G of this form and of another arbitrary matrix K, no fewer than nineteen distinct cg algorithms may be succinctly defined. Four of these are believed to be new. The choice of certain other vectors leads either to a two-term or a three-term recurrence formula and it is shown that to every two-term formula there corresponds a three-term formula, although the converse is not generally true. The two matrices G and K also determine the stability characteristics of the two-term methods.