1031 to write y p lim ak n lim p n o c k k p1 p2 ipp

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Unformatted text preview: ue (go back and look at (7.10.7)). But what really is needed is an indication of how many digits of accuracy can be expected to be gained per iteration. So as not to obscure the simple underlying idea, assume that Hn×n is diagonalizable with σ (H) = {λ1 , λ2 , . . . , λs } , D E T H where 1 &gt; |λ1 | &gt; |λ2 | ≥ |λ3 | ≥ · · · ≥ |λs | (which is frequently the case in applications), and let (k ) = x(k ) − x denote the error after the k th iteration. Subtracting x = Hx + d (a consequence of (7.10.18)) from x(k ) = Hx(k − 1) + d produces (for large k ) IG R (k ) = H (k − 1) = Hk (0) = (λk G1 + λk G2 + · · · + λk Gs ) (0) ≈ λk G1 (0), 1 2 s 1 where the Gi ’s are the spectral projectors occurring in the spectral decomposition (pp. 517 and 520) of Hk . Similarly, (k − 1) ≈ λk−1 G1 (0), so comparing 1 the ith components of (k − 1) and (k ) reveals that after several iterations, Y P − 1) 1 1 ≈ = |λ1 | ρ (H) i (k ) i (k for each i = 1, 2, . . . , n. To understand the signiﬁcance of this, suppose for example that O C | i (k − 1)| = 10−q and | i (k )| = 10−p with p ≥ q &gt; 0, so that the error in each entry is reduced by p − q digits per iteration. Since p − q = log10 − 1) ≈ − log10 ρ (H) , i (k ) i (k we see that − log10 ρ (H) provides us with an indication of the number of digits of accuracy that can be expected to be eventually gained on each iteration. For this reason, the number R = − log10 ρ (H) (or, alternately, R = − ln ρ (H)) is called the asymptotic rate of convergence, and this is the primary tool for comparing diﬀerent linear stationary iterative algorithms. The trick is to ﬁnd splittings that guarantee rapid convergence while insuring that H = M−1 N and d = M−1 b can be computed easily. The following three examples present the classical splittings. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 622 Chapter 7 Eigenvalues and Eigenvectors http://...
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