QR - Two QR Factorizations We compare two techniques for QR...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 12/18/2010 for the course PHYS 5073 taught by Professor Mark during the Fall '10 term at Arkansas.

Ask a homework question - tutors are online