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**Unformatted text preview: **nd nothing has been gained
If M = I , then (4) is the same as (3), and then it is a trivial solution Between these two extremes lie the useful preconditioners,
structured enough (5) can be solved quickly
but close enough to A in some sense that an iteration for (4) converges
more quickly than an iteration for (3) What does it mean for M to be “close enough” to A?
If the eigenvalues of M −1 A are close to 1 and M −1 A − I 2 is small,
then any of the iterations we have discussed can be expected to
converge quickly
However, preconditioners that do not satisfy such a strong condition
may also perform well
A simple rule of thumb: preconditioner M is good if M −1 A is not too
far from normal and its eigenvalues are clustered
20 / 25 Left, right and Hermitian preconditioners
What we have described may be more precisely terms as left
preconditioner
Another idea is to transform Ax = b into AM −1 y = b with x = M −1 y
in which case M is called a right preconditioner
If A is Hermitian positive deﬁnite, then it is usual to preserve this
property in preconditioning
Suppose M is als...

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