Lecture 21 Summary

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Brief summary of some preconditioning ideas: multigrid, incomplete LU/Cholesky, Jacobi/block-Jacobi. (Since Jacobi preconditioners only have short-range interactions, they tend not to work well for matrices that come from finite-difference/finite-element discretizations on grids, but they can work well for diagonally dominant matrices that arise in spectral and integral-equation methods.)

BiCG (bi-conjugate gradient), derived (as in the van der Vorst notes below) via preconditioned "CG" on a symmetric-indefinite system of twice the size. Hence derived algorithm 39.1 in Trefethen, motivated why it works (why residual is still orthogonal to an expanding Krylov space), but also explained the two sources of breakdown. Briefly discussed refinements: QMR and BiCGSTAB(ell).

Began talking about sparse-direct solvers.