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Unformatted text preview: CS205 Class 9 Covered in class: All Reading: Shewchuk Paper on course web page 1. Conjugate Gradient Method this covers more than just optimization, e.g. well use it later as an iterative solver to aid in solving pdes 2. Lets go back to linear systems of equations Ax=b. a. Assume that A is square, symmetric, positive definite b. If A is dense we might use a direct solver, but for a sparse A, iterative solvers are better as they only deal with nonzero entries c. Quadratic Form 1 ( ) 2 T T f x x Ax b x c =- + d. If A is symmetric, positive definite then f(x) is minimized by the solution x to Ax=b! i. 1 1 ( ) 2 2 T f x Ax A x b Ax b = +- =- since A is symmetric ii. ( ) f x = is equivalent to Ax=b 1. this makes sense considering the scalar equivalent 2 1 2 ( ) f x ax bx c =- + where the line of symmetry is / x b a = which is the solution of ax=b and the location of the maximum or minimum iii. The Hessian is H=A, and since A is symmetric, positive definite so is H, and a solution to...
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This note was uploaded on 01/05/2012 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.
- Fall '07