# 5 i on the diagonal aij 1 on the sub and super

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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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