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Factored-Q

# Factored-Q - The Action of Householder Reectors Let A Rmn m...

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The Action of Householder Reflectors Let A R m × n , m n . You may recall that the Householder QR factorization can be written as H p · · · H 2 H 1 A = R, where p = min ( n, m - 1) and H k = I - ( 2 u t k u k ) u k u t k is a Householder reflector which introduces zeros into positions k + 1 to m of the k th column of the matrix A ( k - 1) = H k - 1 · · · H 2 H 1 A . This gives A = QR , where we define Q = H 1 H 2 · · · H p . The H j are not explicitly formed, since that is very inefficient, and since they are completely determined by the u vectors. The matrix Q is never formed, either. What we do is save the u j ’s. If we let Q u be the array whose j th column is u j , it would be called the ”factored form” of Q (I think of it as an array, not a matrix). In Matlab, if u is our variable name for u j , then we might write Qu (: , j ) = u . In a memory efficient code, we would use the lower triangle of A (plus an n-vector) to store all of the u j s . Later we will need to find the vector
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• Fall '10
• MARK
• Vector Motors, Singular value decomposition, Householder transformation, householder reﬂectors, Householder QR factorization

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