This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CS205 – Class 11 Covered in class : Everything Readings : Shewchuk Paper on course web page and Heath 473478 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...
View
Full
Document
This note was uploaded on 01/29/2008 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.
 Fall '07
 Fedkiw

Click to edit the document details