In Chebyshev Iteration the iteration parameters
are known as soon
as one knows the ellipse containing the eigenvalues (or rather, the
field of values) of the operator. Therefore the computation of
inner products, as is necessary in methods like GMRES or CG,
is avoided .
This avoids the synchronization points required of CG-type methods, so
machines with hierarchical or distributed memory may achieve higher
performance (it also suggests strong parallelization properties ; for a
discussion of this see Saad [185], and Dongarra, et
al. [71]).
Specifically, as soon as some segment of 
is computed, we may begin
computing, in sequence, corresponding segments of 
, 
, and 
.
Figure: The Preconditioned Chebyshev Method
The pseudocode for the Preconditioned Chebyshev
Method with preconditioner 
is given in Figure 
.
It handles the case of a symmetric
positive definite coefficient matrix 
. The
eigenvalues of 
are assumed to be all real and in the
interval 
, which does not include
zero.