class_12

class_12 - CS 205 Class 12 Readings: Same as last Covered...

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CS 205 – Class 12 Readings : Same as last Covered in class : All 1. finding the A-orthogonal directions with Gram-Schmidt a. given a vector V k , construct k s by subtracting out the “A-overlap” of V k with 1 s to 1 k s so that 0 ki s As  for i=1,k-1 b. we define j k j j j j k k k s As s As V V s 1 1 i. note that i j k j j j j k i k i k As s As s As V As V As s 1 1 and then all the terms in the sum vanish except for one leaving 0 i i i i i k i k i k As s As s As V As V As s as desired c. for ik , 1 1 k kj k i k i j i k i j jj V s r V r s r V r s , where the summation vanishes because the residual at step i is orthogonal to all the previous search directions i. when k=i this leads to k k k k s r V r and k k k k k kkkk s r V r s s   (we’ll use this below) ii. when k < i, 0 Vr  , i.e. the residual is orthogonal to all the previous k V as well (we’ll use this below) 2. Each new direction V is chosen in the steepest decent fashion, i.e. () k k k V f x r   . a. 1 1 1 1 1 1 1 1 1 k k k k k k k k k As s r r As s r s because k k k k k k r r r V r s b.
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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.

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class_12 - CS 205 Class 12 Readings: Same as last Covered...

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