TABLE OF CONTENTS We have mentioned that the purpose of preconditioning (1.1) by (1.2) or (1.5) is to ensure that the preconditioned system has better spectral properties, that are required by the iterative methods in later chapters.
Here we give a short list of common theories on preconditioned systems that a reader can find in the literature. These theories, while useful and mathematically interesting, are not always relevant to faster iterative solution.
For a symmetric matrix A ? 
n n, any of the following proves that the symmetric M is an effective preconditioner.
Conditioned number estimates for a SPD case ?( MA) ? O(1) for any n.
A common form of conditioned number estimates is presented by
where c 1, c 2 > 0, and M = M 1/2 M 1/2 and M 1/2 are both SPD. This implies that
To explain this result, first note that (setting y = M 1/2 x and z = M ?1/2 x) M ?1/2 AM ?1/2 x = ?x implies M ?1/2 Az = ?M 1/2 z and also M ?1 Az = ?z so ?( M ?1 A) = ?( M ?1/2 AM ?1/2). Second, the above inequalities may be written as
Using the min max theorem (on Rayleigh quotients) for symmetric matrices,
