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One can change A and b by preconditioning
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M1-1 A (M2-1 M2)x = M1-1 b
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is same equation as before for any choice of matrices M1 and M2
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All these choices are designed to accelerate convergence of iterative methods
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Anew = M1-1 A M2-1
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xnew = M2 x
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bnew = M1-1 b
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We choose M1 and M2 to make our standard methods perform better
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There are specialized preconditioning ideas and perhaps better general approaches such as multigrid and Incomplete LU (ILU) decomposition
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Anew xnew = bnew has same form as above and we can apply any of the methods that we used on A x = b
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