A Taxonomy for Conjugate Gradient Methods
Steven F. Ashby, Thomas A. Manteuffel, Paul E. Saylor · SIAM Journal on Numerical Analysis · 1990
The conjugate gradient method of Hestenes and Stiefel is an effective method for solving large, sparse Hermitian positive definite (hpd) systems of linear equations, $Ax = b$. Generalizations to non-hpd matrices have long been sought. The recent theory of Faber and Manteuffel gives necessary and sufficient conditions for the existence of a CG method. This paper uses these conditions to develop and organize such methods. It is shown that any CG method for $Ax = b$ is characterized by an hpd inner product matrix B and a left preconditioning matrix C. At each step the method minimizes the B-norm of the error over a Krylov subspace. This characterization is then used to classify known and new methods. Finally, it is shown how eigenvalue estimates may be obtained from the iteration parameters, generalizing the well-known connection between CG and Lanczos. Such estimates allow implementation of a stopping criterion based more nearly on the true error.