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Corollary of 2 since a polynomial p x n1 1 xj

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Unformatted text preview: of CG convergence First, we suppose that the eigenvalues are perfectly clustered but assume nothing about the locations of these clusters Theorem If A has only n distinct eigenvalues, then the CG iteration converges in at most n steps This is a corollary of (2), since a polynomial p (x) = n=1 (1 − x/λj ) ∈ Pn exists that is zero at any specified set of j n points {λj } At the other extreme, suppose we know nothing about any clustering of the eigenvalues but only that their distances from the origin vary by at most a factor κ ≥ 1 In other words, suppose we know only the 2-norm condition number κ = λmax /λmin , where λmax and λmin are the extreme eigenvalues of A 14 / 25 Rate of CG convergence (cont’d) Theorem Let the CG iteration be applied to a symmetric positive definite matrix problem Ax = b, where A has 2-norm condition number κ. Then the A-norm of the errors satisfy √ √ √ κ+1 n κ + 1 −n κ−1 n en A √ +√ ≤2 ≤2 √ e0 A κ−1 κ−1 κ+1 See text for proof using Chebyshev polynomials Since √ κ−1 2 √ ∼1− κ+1 κ as κ → ∞, it implies that if κ is large but not too large, convergence √ to a specified tolerance can be expected in O ( κ) iterations An upper bound, and convergence may be faster for special right hand sides or if the spectrum is clustered 15 / 25 Example: CG convergence Consider a 500 × 500 sparse matrix A where we have 1’s on the diagonal and a random number from the uniform distribution on [−1,...
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