In many scientific applications, one needs to solve linear systems
Ax=b
with A square and non-singular, often symmetric, or positive definite, or an M-matrix. For a variety of reasons, one can choose to solve this system by an iterative method, rather than by some form of Gaussian elimination.Iterative methods feature a preconditioning step, that is, the exact solution of a linear system
Cx=b
where C in some sense approximates A. In general, a more accurate choice of C will yield a more speedy solution of the iterative process, but such a choice will also be more costly to construct and to apply.
For general background information about iterative methods
and preconditioners, see the
`Templates
' book [2], and books by
Axelsson [1], Hackbusch [7],
and Saad [10].
More specifically for domain decomposition methods and Schwarz methods,
see, e.g., [4,11].