Preconditioning

• One can change A and b by preconditioning

• M1-1 A (M2-1 M2)x = M1-1 b

• is same equation as before for any choice of matrices M1 and M2

• All these choices are designed to accelerate convergence of iterative methods

• Anew = M1-1 A M2-1

• xnew = M2 x

• bnew = M1-1 b

• We choose M1 and M2to make our standard methods perform better

• There are specialized preconditioning ideas and perhaps better general approaches such as multigrid and Incomplete LU (ILU) decomposition

Anew xnew = bnewhas same form as aboveand we can apply any ofthe methods that we used onA x = b

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