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So the aim of preconditioning gmres might be to

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Unformatted text preview: . . , en } , and let vi (x) be the minimum polynomial of ei with respect to C. Observe that v1 (x) = p(x) because Cej = ej +1 for j = 1, . . . , n − 1, so {e1 , Ce1 , C2 e1 , . . . , Cn−1 e1 } = {e1 , e2 , e3 , . . . , en } and n−1 Cn e1 = Cen = C∗n = − n−1 αj ej +1 = − j =0 αj Cj e1 =⇒ v1 (x) = p(x). j =0 Since v1 (x) divides the LCM of all vi (x) ’s (which we know from (7.11.5) to be the minimum polynomial m(x) for C ), we conclude that p(x) divides m(x). But m(x) always divides p(x) —recall (7.11.4)—so m(x) = p(x). Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.11 Minimum Polynomials and Krylov Methods http://www.amazon.com/exec/obidos/ASIN/0898714540 649 Example 7.11.2 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Poor Man’s Root Finder. The companion matrix is the source of what is often called the poor man’s root finder because any general purpose algorithm designed to compute eigenvalues (e.g., the QR iteration on p. 535) can be applied to the companion matrix for a polynomial p(x) to compute the roots of p(x). When used in conjunction with (7.1.12) on p. 497, the companion matrix is also a poor man’s root bounder . For example, it follows that if λ is a root of p(x), then |λ| ≤ C ∞ = max{|α0 |, 1 + |α1 |, . . . , 1 + |αn−1 |} ≤ 1 + max |αi |. D E The results on p. 647 insure that the minimum polynomial v (x) for every nonzero vector b relative to A ∈ C n×n divides the minimum polynomial m(x) for A, which in turn divides the characteristic polynomial c(x) for A, so it follows that every v (x) divides c(x). This suggests that it might be possible to construct c(x) as a product of vi (x) ’s. In fact, this is what Krylov did in 1931, and the following example shows how he did it. T H IG R Example 7.11.3 Krylov’s method for constructing the characteristic polynomial for A ∈ C n×n as a product of minimum polynomials for vectors is as follows. k−1 Sta...
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