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Unformatted text preview: Homework 4
1 A different kind of norms
1. Derive a version of the normal equations to minimize the M norm Let r(x) = Ax - b, then we can define the directional derivative of ||r||2 as ||r||2 d (x) = ||r(x + tu)||2 M u dt where ||r(x + tu)||2 = ||r(x) + tAu||2 M M = r + tAu, r + tAu = ||r||2 + 2t Au, r M + t2 ||Au||2 M M so for t = 0, the original equation becomes 2 < Au, r >M = 2uT AT M (Ax - b) = 0 AT M (Ax - b) = 0 2. Show the 2-norm form equivalence of the M form ||z||2 = z T M z M = (z T L)(LT z) = (LT z)T (LT z) =0 t=0 1 ...
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- Spring '09