# Second term is 0 if and only if x 0 ie xn x thus e a

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Unformatted text preview: ve deﬁnite, the ﬁrst term is larger or equal to 0 The second term is 0 if and only if ∆x = 0, i.e., xn = x Thus e A is minimal if and only if xn = x as claimed The monotonicity property is a consequence of the inclusion Kn ⊆ Kn+1 , and since Kn is a subset of IRm of dimension n as long as convergence has not yet been achieved, convergence must be achieved in at most m steps That is, each step of conjugate direction cuts down the error term component by component 5 / 25 Optimality of conjugate gradients (cont’d) The guarantee that the CG iteration converges in at most m steps is void in ﬂoating point arithmetic For arbitrary matrices A on a real computer, no decisive reduction in en A will necessarily be observed at all when n = m In practice, however, CG is used not for arbitrary matrices but for matrices whose spectra are well behaved (partially due to preconditioning) that convergence to a desired accuracy is achieved for n m The theoretical exact convergence at n = m has no relevance to this use of the CG iteration in scientiﬁc computing 6 / 25 Conjugate gradients as an optimization algorithm The CG iteration has a certain optimality property: it minimizes en A at step n over all vectors x ∈ Kn A standard form for minimizing a nonlinear function of x ∈ IRm At the heart of the iteration is the formula xn = xn −1 + αn p n −1 A familiar equation in optimization, in which a current approximation xn−1 is updated to a new approximation xn by movi...
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