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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 [181], and Dongarra, * et
al.* [68]).
Specifically, as soon as some segment of is computed, we may begin
computing, in sequence, corresponding segments of , , and .

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.