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Unformatted text preview: Two QR Factorizations We compare two techniques for QR factorizations of a full-rank matrix A R m n , with m n . While there are a few other methods available for use, we will talk here about the modified Gram-Schmidt process (MGS), and the Householder QR factorization (HQR). MGS thin QR factorization: A = QR, where Q R m n satisfies Q t Q = I and R R n n is upper triangular. The cost is 2 mn 2 + O( mn ) flops. If A is overwritten by Q , then only 1 2 n 2 + O( n ) words of memory are required. If Q and R are the computed versions of Q and R , then there exists A R m n with A + A = Q R , where k A k = k A k O( ), and k Q t Q- I k = ( A )O( ). HQR factored- Q full QR factorization: A = QR, where Q R m m satisfies Q t Q = QQ t = I and R R m n is upper triangular. We say factored here because HQR does not produce Q , but instead produces u 1 ,u 2 ,...,u n , where H k = H ( u k ) and Q = H 1 H 2 H s . The cost is....
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- Fall '10