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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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