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HouseQR

# HouseQR - Householder QR Algorithmically this method is...

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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 - βu 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 - βuu 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 ± x 2 e 1 , and that Hx = ( I - βuu t ) x = ± x 2 e 1 , with β = 2 / ( u t u ). As with G.E. we can view the k th step as A ( k ) = R ( k ) X ( k ) 0 ˜ A ( k ) = I 0 0 ˜ H k R ( k - 1) X ( k - 1) 0 ˜ A ( k - 1) = H k A ( k - 1) , where R ( k ) is k × k upper triangular, and ˜ A ( k ) is ( m - k ) × ( n - k ). Here u t k = (0 t , ˜ u t k ), where ˜ u
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