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The why and how

As we have seen, the convergence rate of iterative methods depends on spectral properties of the coefficient matrix. Hence one may attempt to transform the linear system into one that is equivalent in the sense that it has the same solution, but that has more favorable spectral properties. A preconditioner is a matrix that effects such a transformation.

For instance, if a matrix approximates the coefficient matrix in some way, the transformed system

has the same solution as the original system , but the spectral properties of its coefficient matrix may be more favorable.

In devising a preconditioner, we are faced with a choice between finding a matrix that approximates , and for which solving a system is easier than solving one with , or finding a matrix that approximates , so that only multiplication by is needed. The majority of preconditioners falls in the first category; a notable example of the second category will be discussed in §.