A NEW TAXONOMY OF CONJUGATE-GRADIENT METHODS

In this paper, a framework is proposed for conjugate gradient and conj ugate direction algorithms based on an indefinite quadratic form. By t he choice of the Hessian G of this form and of another arbitrary matri x K, no fewer than nineteen distinct cg algorithms may be succinctly d efined. Four of these are believed to be new. The choice of certain ot her vectors leads either to a two-term or a three-term recurrence form ula 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 t he two-term methods.