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 δA ∈ R m × n with A + δA = ˜ Q ˜ R , where k δA k = k A k O( μ ), and k ˜ Q t ˜ Q- I k = κ ( A )O( μ ). 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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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