ABSTRACT
Both the efficiency and robustness of iterative techniques can be very much improved by using preconditioning. Preconditioning is simply a means of transforming the original linear system into one equivalent. It is natural to consider preconditioners based on iterations of other iterative methods, possibly of another Krylov subspace method. The latter case provides an inner-outer Krylov method, that can be viewed as having a polynomial preconditioner with a polynomial that changes from one step to the next and is defined implicitly by the polynomial generated by the (inner) Krylov subspace method. The so-called Approximate inverses or Approximate inverse preconditioners are preconditioners approximating directly A-1 and not A, as usual. They do not require solving a linear system to be applied and often have very interesting parallel potentialities. Factored approximate inverse preconditioners for general sparse matrices can be efficiently constructed by means of a generalization of the Gram-Schmidt process known as biconjugation.
