Error norm estimation and stopping criteria in preconditioned conjugate gradient iterations

Some techniques suitable for the control of the solution error in the preconditioned conjugate gradient method are considered and compared. The estimation can be performed both in the course of the iterations and after their termination.The importance of such techniques follows from the non‐existence of some reasonable a priori error estimate for very ill‐conditioned linear systems when sufficient information about the right‐hand side vector is lacking. Hence, some a posteriori estimates are required, which make it possible to verify the quality of the solution obtained for a prescribed right‐hand side. The performance of the considered error control procedures is demonstrated using real‐world large‐scale linear systems arising in computational mechanics. Copyright © 2001 John Wiley & Sons, Ltd.