Hyperbolic Pairs in the Method of Conjugate Gradients

Abstract

(starting from an arbitrary x1 and r1 = b Ax1, Pt = r1) converges to the solution of(1) in n steps or less. Theoretically, the process (2) can be carried out in such a manner that only one multiplication of A times a vector is required at each step, although in practice two such multiplications are often employed. In many applications we are led to equations of the form (1) where A is nonsingular and symmetric but not positive definite. For example the problem of minimizing the quadratic form