ARGONNE NATIONAL LABORATORY 9700 South Cass

Incomplete factorization has been shown to be a good preconditioner for the conjugate gradient method on a wide variety of problems. It is well known that allowing some ll-in during the incomplete factorization can signiicantly reduce the number of iterations needed for convergence. Allowing ll-in, however, increases the time for the factorization and for the triangular system solves. In addition, it is diicult to predict a priori how much ll-in to allow and how to allow it. The unpredictability of the required storage/work and the unknown beneets of the additional ll-in make such strategies impractical to use in many situations. In this paper we motivate, and then present, two \black-box" strategies that signiicantly increase the eeectiveness of incomplete Cholesky factorization as a preconditioner. These strategies require no parameters from the user and do not increase the cost of the triangular system solves. EEcient implementations for these algorithms are described. These algorithms are shown to be successful for a variety of problems from the Harwell-Boeing sparse matrix collection.