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Unformatted text preview: Householder QR Algorithmically, this method is very close to G.E. with no pivoting. There, a Gauss transform M k = I + m k e t k was used to introduce zeros below the ( k,k ) element of A ( k- 1) , giving A ( k ) G = M k A ( k- 1) G . Here, a Householder reflector H k = I- u k u t k replaces the Gauss transform as the operator that introduces zeros below the ( k,k ) element: A ( k ) H = H k A ( k- 1) H . Let A R m n , with m n , and let p = min ( n,m- 1). The Householder QR factorization of A can be coarsely described as A (0) = A For k = 1:p Compute u so that ( I- uu t ) A ( k- 1) has zeros below its ( k,k ) entry Compute A ( k ) = H k A ( k- 1) End There are some important details to consider yet, but it is essentially this simple. We know that if u is a Householder vector for x , then it is a multiple of x k x k 2 e 1 , and that Hx = ( I- uu t ) x = k x k 2 e 1 , with = 2 / ( u t u ). As with G.E. we can view the k th step as A ( k ) = R ( k ) X ( k ) A (...
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This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.
- Fall '10