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Preconditioning

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Administrivia: November 16, 2009 Administrivia: November 16, 2009 Reading in A Multigrid Tutorial : Chapters 1-2 (introduction and smoothers) Saad (2 nd edition only) also has a chapter on multigrid.

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Conjugate gradient iteration Conjugate gradient iteration One matrix-vector multiplication per iteration Two vector dot products per iteration Four n-vectors of working storage x 0 = 0, r 0 = b, d 0 = r 0 for k = 1, 2, 3, . . . α k = (r T k-1 r k-1 ) / (d T k-1 Ad k-1 ) step length x k = x k-1 + α k d k-1 approx solution r k = r k-1 α k Ad k-1 residual β k = (r T k r k ) / (r T k-1 r k-1 ) improvement d k = r k + β k d k-1 search direction
Conjugate gradient: Convergence Conjugate gradient: Convergence In exact arithmetic, CG converges in n steps (completely unrealistic!!) Accuracy after k steps of CG is related to: consider polynomials of degree k that are equal to 1 at 0. how small can such a polynomial be at all the eigenvalues of A? Thus, eigenvalues close together are good. Condition number: κ (A) = ||A|| 2 ||A -1 || 2 = λ max (A) / λ min (A) Residual is reduced by a constant factor by O( κ 1/2 (A)) iterations of CG.

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The Landscape of Sparse Ax=b Solvers The Landscape of Sparse Ax=b Solvers Pivoting LU GMRES , BiCGSTAB , Cholesky Conjugate gradient Direct A = LU Iterative y’ = Ay Non- symmetric Symmetric positive definite More Robust Less Storage More Robust More General D
Other Krylov subspace methods Other Krylov subspace methods Nonsymmetric linear systems: GMRES: for i = 1, 2, 3, . . .

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