CS 205A class_11 notes

Scientific Computing

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Unformatted text preview: CS205 – Class 11 Covered in class : Everything Readings : Shewchuk Paper on course web page and Heath 473-478 1. Steepest Decent for Ax=b ( continued) a. Suppose that our initial guess is such that the error term, exact e x x = − , is an eigenvector of the matrix A i. Then r A e e λ = − = − ii. 1 ( ) ( ) k k k k k k k k k k k k k k k r r r r r r x x r x e x e r Ar r A e r Ae λ λ + ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⋅ ⋅ ⋅ = + = + − = + ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⋅ ⋅ − ⋅ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ and then 1 ( ) k k k k k k k k e x a c t k k k r r r r x x e x e x e x r e r r λ + ⎛ ⎞ ⎛ ⎞ ⋅ ⋅ = + = + = − = ⎜ ⎟ ⎜ ⎟ ⋅ ⋅ − ⎝ ⎠ ⎝ ⎠ and we’re done! iii. In this case, we lie exactly on one of the coordinate axis of the ellipsoid and f ∇ and e point in the same direction: f r A e e λ −∇ = = − = − b. When all the eigenvalues are equal, we have circles instead of ellipses. Then f ∇ and e always point in the same direction, and the steepest decent method converges in one...
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This note was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.

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CS 205A class_11 - CS205 – Class 11 Covered in class Everything Readings Shewchuk Paper on course web page and Heath 473-478 1 Steepest Decent

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