CS 205A class_10 notes

Scientific Computing

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CS205 – Class 10 Covered in class: All Reading: Shewchuk Paper on course web page 1. Conjugate Gradient Method – this covers more than just optimization, e.g. we’ll use it later as an iterative solver to aid in solving pde’s 2. Let’s 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 TT f xx A x b x c =− + d. If A is symmetric, positive definite then f(x) is minimized by the solution x to Ax=b! i. 11 22 T f xA x A x b A x b ∇= + = since A is symmetric ii. () 0 fx is equivalent to Ax=b 1. this makes sense considering the scalar equivalent 2 1 2 f xa x b x c + where the line of symmetry is / x ba = 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 , or Ax=b is a minimum 1.
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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_10 notes - CS205 Class 10 Covered in class:...

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