Preconditioning for accurate solutions of ill-conditioned linear systems

This article develops the preconditioning technique as a method to address the accuracy issue caused by ill-conditioning. Given a preconditionerMfor an ill-conditioned linear systemAx=b, we show that, if the inverse of the preconditionerM(-1)can be applied to vectorsaccurately, then the linear system can be solvedaccurately. A stability concept calledinverse-equivalentaccuracy is introduced to describe the high accuracy that is achieved and an error analysis will be presented. Numerical examples are presented to illustrate the error analysis and the performance of the methods.