Page 78 - Textos de Matemática Vol. 44

68 M. E. HOCHSTENBACH, D. A. SINGER AND P. F. ZACHLIN
However, in spite of Proposition 5.1, W(R−∗H∗R−1) may not always be an kkk
approximation to W(A−1) of good quality. We now present a new, alternative, approach to approximate W(A−1):
(5.2) W(A−1) ⊇ W(U∗A−1U ) ≈ W(H−1). kkk
The key idea is now to approximate W(A−1) using a Krylov subspace generated
by A. In what is to follow we assume the Arnoldi decomposition (2.2). We prove
some results on the quality of the approximation W(A−1) ≈ W(H−1) in the k
following Proposition. We will see that this approximation may be superior to W(H k) in Example 5.3. Let dist(z,S) denote the distance from z ∈ C to a closedandboundedsetS⊂C:dist(z,S)=min{|z−s| : s∈S}.
Proposition 5.2. (a) W(H−1)=W(U∗(A−1+Ek)Uk), with kk
(5.3) Ek = hk+1,k (A−1uk+1) (e∗kH−1Uk∗). k
(b) W(H−1)=W(U∗A−1(Ik+Fk)Uk),where kk
Fk = hk+1,k uk+1 (e∗kH−1Uk∗). k
(c) For all z ∈ W(H−1) we have k
dist(z, W(U∗A−1U )) ≤ |h |∥U∗A−1u ∥∥H−∗e ∥. k k k+1,k k k+1 k k
Proof. To start with, to derive (5.2) we rewrite (2.2): A−1Uk = UkH−1 − hk+1,kA−1uk+1e∗kH−1,
so that
(5.4) U∗A−1U = H−1 − h U∗A−1u e∗H−1.
kk
k k k k+1,k k k+1 k k
Discarding the last term on the right-hand side gives the approximation
U∗A−1U ≈ H−1. kkk
In fact, we see that
H−1 =U∗(A−1+E )U =U∗A−1(I +F )U ,
kkkkkkkk
which implies (a) and (b). Finally, from (5.4) we have y∗H−1y=y∗(U∗A−1U )y+h (y∗U∗A−1u )(e∗H−1y)
k kkk+1,kkk+1kk
from which (c) follows.