Page 77 - Textos de Matemática Vol. 44

FIELD OF VALUES OF A MATRIX INVERSE 67
5. Two numerical approximation methods for W(A−1) To approximate the field of values of A−1, note that we have
W(A−1) ⊇ W(U k∗A−1U k)
for any n×k matrix U k with orthonormal columns that form a basis for a space
Uk. To avoid matrix inversion of a large matrix, one idea is to take Uk = AUk.

In what follows we will assume that this space is of full dimension k. We can
compute the resulting approximation efficiently as follows. If Hk = QkRk is
the reduced QR-decomposition of H , then AU R−1 = U H R−1 = U Q k k k k+1 k k k+1 k
has orthonormal columns, and
(5.1) W(A−1) ⊇ W(R−∗U∗A∗A−1AU R−1) = W(R−∗H∗R−1).
kk kk kkk
This approximation is a true subset of W(A−1) and behaves monotonically as function of k; cf. also Proposition 2.1.
Proposition 5.1. Write Hk = R−∗H∗R−1, and suppose that Uk ⊂ Uk+1. kkk
Then
W(H k) ⊆ W(H k+1) ⊆ W(A−1).
Proof. This follows from the fact that W(H k) is the restriction of W(A−1)
to the subspace Uk.
We note that Manteuffel and Starke [8] also suggest this approximation via a
slightly different derivation. An advantage of expression (5.1) compared with [8]
is that from (5.1) it is clear that this approximation to W(A−1) is itself a field
of values of a low-dimensional matrix R−∗H∗R−1. (In [8], this approximation kkk
is described as the intersection of strips in the complex plane determined by the minimal and maximal eigenvalues of generalized eigenvalue problems.)
In the following we will assume that Hk is invertible. The eigenvalues of H k :=(Uk∗A∗Uk)−1Uk∗A∗AUk =H−∗Rk∗Rk
k
are the harmonic Ritz values of A with respect to search space Uk and target
τ = 0 (see [10] and Section 3). It is not difficult to prove that the eigenvalues
of this matrix are the inverses of the eigenvalues of R−∗H∗R−1. Since the kkk
eigenvalues of H k are the harmonic Ritz values, our conclusion is that after k steps we know that W(A−1) contains the convex hull of the inverses of the harmonic Ritz values.