CS 205A class_4 notes

Scientific Computing

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CS205 - Class 4 QR Continued (Readings Heath pp120-137) 1. Transition to Householder. So far, we’ve seen two different ways of finding a least squares solution, namely the normal equations and the (modified) Gram-Schmidt method. Using the normal equations squares the condition number of the A matrix, which could be bad to begin with. The Gram-Schmidt method does not suffer from the same numerical problems, but can still be unstable. If the columns of A are “nearly linearly-dependent,” then this method can suffer from numerical instabilities. The QR decomposition is exactly what the Gram-Schmidt procedure does couched in the language of matrix factorization. There is an algorithm to performs the QR factorization that does not suffer from the numerical instabilities mentioned above and this is the one that is quite often used by practitioners. 2. A Householder transform is defined by T T vv v v I H 2 = for some vector 0 v . Note that 1 T H H H = = , and thus H is orthogonal. a. For a vector a , we define k H using 2 ˆ ˆ ( ) k k k v a S a a e = + where ˆ (0, ,0, , , ) T k m a a a = " " and ( ) 1 k S a = ± is the sign function. Then k H a zeroes out the entries of a below k a . b. Let 2 1 2 a ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ . Then ( ) ( ) = + = + = 2 1 5 0 0 1 9 2 2 1 2 ˆ ˆ 1 2 1 1 S e a a S a v . c. Note that v v v a v a Ha T T 2 = so we never need to form H explicitly, but instead only need to find the vector v . d. ( )( ) ( )( ) ⎡− = × = = 0 0 3 2 1 5 30 15 2 2 1 2 2 1 5 2 , 1 , 5 2 , 1 , 5 2 , 1 , 2 2 , 1 , 5 2 2 1 2 2 1 2 1 T T H .
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