W-GCV for the Bidiagonal Sample Clauses

W-GCV for the Bidiagonal. System In the previous section we discussed W-GCV in the context of Tikhonov regularization on the original (full) system of equations involving A and b. This allowed us to provide a general description, but our aim is to apply W-GCV to choosing regularization parameters for the projected problem, (2.12). In this case, the W-GCV function has the form: G (ω, λ) = kǁ(I − BkB†k,λ )βe1ǁ2 .Σ . trace(I − ωBkBk,λ) Bk, βe1 . † Σ2 kβ2 = k x x λ2 i δ2 + λ2 ΣPT e1 Σ(i)Σ2 + .ΣPT e1 Σ(k+1)Σ2Σ , k Σ(1 − ω)δ + λ .1 + Σi=1 i i δ2 + λ2 k where, using the notation introduced in (2.16), Pk is an orthogonal matrix containing the left singular vectors of Bk, δi is the ith largest singular value of Bk, and W T b = βe1 with β = ǁbǁ. Note that this reduces to the expression in (2.16) when ω = 1.
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