  This paper develops the preconditioning technique as a method to address the
accuracy issue caused by ill-conditioning. Given a preconditioner $M$ for an
ill-conditioned linear system $Ax=b$, we show that, if the inverse of the
preconditioner $M^{-1}$ can be applied to vectors accurately, then the linear
system can be solved accurately. A stability concept called inverse-equivalent
accuracy is introduced to describe higher accuracy that is achieved and an
error analysis will be presented. As an application, we use the preconditioning
approach to accurately compute a few smallest eigenvalues of certain
ill-conditioned matrices. Numerical examples are presented to illustrate the
error analysis and the performance of the methods.
