Page 75 - Textos de Matemática Vol. 44

FIELD OF VALUES OF A MATRIX INVERSE 65
all harmonic Ritz values that may be obtained after one step of the Arnoldi iteration (see [14, p. 166] for a similar statement about W (A) and Ritz values). We will come back to the relation of harmonic Ritz values and the field of values in Section 5.
For another area of application, we consider the linear differential-algebraic equation with constant coefficients
Au′(t) = u(t), u(0) = u0.
If A is expensive to invert, this equation may be considered “effectively implicit” in the sense that it is computationally not easy to explicitly express u′(t) in terms of u(t). The numerical abscissa of A−1 determines the behavior of ∥etA−1 ∥ as t → 0 (see, e.g., [14, (14.2)]):
d ∥etA−1 ∥t=0 = ω(A−1). dt
In fact, we also have ∥etA−1 ∥ ≤ etω(A−1 ) for all t ≥ 0; see, e.g., [14, Thm. 17.1]. We can also use W(A−1) for estimates for the norm of the matrix inverse
and the condition number κ(A) = ∥A∥ ∥A−1∥. We have μ(A−1) ≤ ∥A−1∥ ≤ 2 μ(A−1)
(and a similar result for powers of A−1, cf. [14, (17.6)]). Combined with an estimate for ∥A∥, for instance using μ(A) ≤ ∥A∥ ≤ 2 μ(A), this gives an ap- proximation for the condition number; see also the following section.
The field of values of A and of A−1 may also be used to determine inclusion regions for matrix eigenvalues. Particularly, Manteuffel and Starke [8] mention the inclusion region
Λ(A)⊆W(A) ∩ 1 . W(A−1)
Here 1/S for a subset S of the complex numbers is defined elementwise: 1/S = {z−1 : z ∈ S}. This approach was extended to inverses of shifted matrices in [6] to give very useful and efficiently generated spectral inclusion regions.
Finally, we note that W (A−1) also arises in the location of certain parameters in the recently developed IDR method [11].
4. Relations involving μ(A), ν(A) μ(A−1), and ν(A−1)
Before considering the practical approximation of the field of values of A−1 in the next section, we look at some bounds for ν(A−1) and μ(A−1), and their relations with the quantities μ(A), ν(A), ∥A∥, and ∥A−1∥. Eiermann and