Equivalent to c 1 ac c x c 1 b the matrix in brackets

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Unformatted text preview: ¡ £¡¢£ Preconditioning The convergence of a matrix iteration depends on the properties of the matrix - the eigenvalues, the singular values, or sometimes other information In many cases, the problem of interest can be transformed so that the properties of the matrix are improved drastically The process of preconditioning is essential to most successful applications of iterative methods 18 / 25 Preconditioning for Ax = b Suppose we want to solve m × m nonsingular system Ax = b (3) For any nonsingular m × m matrix M , the system M −1 Ax = M −1 b (4) has the same solution If we solve the (4) iteratively, however, the convergence will depend on the properties of M −1 A instead of A If this preconditioner M is well chosen, (4) may be solved much more rapidly than (3) For this idea to be useful, it must be possible to compute M −1 A efficiently As usual in numerical linear algebra, this does not mean an explicit construction of the inverse M −1 , but the solution of system of equations in this form My = c (5) 19 / 25 Preconditioning for Ax = b (cont’d) Two extreme cases: If M = A, then (5) is the same as (3), a...
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