E rn1 0 then pn ae0 a has a unique solution pn pn

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Unformatted text preview: and Pn is again defined as GMRES (i.e., the set of polynomials p of degree ≤ n with p (0) = 1) 11 / 25 CG and polynomial approximation Theorem If the CG iteration has not already converged before step n (i.e., rn−1 = 0), then pn (A)e0 A has a unique solution pn ∈ Pn , and the iterate xn has error en = pn (A)e0 for this same polynomial pn . Consequently, we have en e0 A A = inf p ∈Pn p (A)e0 e0 A A ≤ inf max |p (λ)| p ∈Pn λ∈Λ(A) (2) where Λ(A) denotes the spectrum of A From theorem in the last lecture, it follows that en = p (A)e0 for some p ∈ Pn The equality is a consequence of (2) and monotonic convergence (1) 12 / 25 CG and polynomial approximation (cont’d) As for the inequality, e0 = m aj vj is an expansion of e0 in j =1 orthonormal eigenvectors of A, then we have p (A)e0 = m aj p (λj )vj and thus j =1 m e0 2 A m aj2 λj , = j =1 p (A)e0 2 A aj2 λj (p (λj ))2 = j =1 These identities imply p (A)e0 2 / e0 2 ≤ maxλ∈Λ(A) |p (λ)|2 , which A A implies the inequality The rate of convergence of the CG iteration is determined by the location of the spectrum of A A good spectrum is one on which polynomials pn ∈ Pn can be very small, with size decreasing rapidly with n Roughly speaking, this may happen for either or both of two reasons: the eigenvalues may be grouped in small clusters, or they may lie well separated in a relative sense from the origin The two best known corollaries address these two ideas in their extreme forms 13 / 25 Rate...
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This document was uploaded on 02/10/2014.

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