Iteratively constructing preconditioners via the conjugate gradient method

We consider the problem of solving a symmetric, positive def-inite system of linear equations. The most well-known and widely-used method for solving such systems is the precondi-tioned Conjugate Gradient method. The performance of this method depends crucially on knowing a good preconditioner matrix. We show that the Conjugate Gradient method itself can produce good preconditioners as a by-product. These preconditioners allow us to derive new asymptotic bounds on the time to solve multiple related linear systems