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the numerical radius
μ(A) = and the the numerical abscissa
FIELD OF VALUES OF A MATRIX INVERSE 63
(a) Kk 1(eiαA+e−iαA∗),e1 for each α; and 2
(b) Kk(A,e1),
are all equal to span{e1,...,ek} and, consequently, the approximations to the
field of values will be exactly the same in both approaches.
In the remainder of this paper we concentrate on the field of values for the inverse of a large sparse invertible matrix A. Hereby, we do not want to compute A−1 explicitly, since this is often prohibitively expensive, but we also want to avoid the costs of accurately solving a linear system of the form Ax = b. Instead, we will use a single Krylov space generated by A to approximate W (A−1 ).
3. The field of values of the inverse of a matrix: applications
We review a few applications where estimates for the field of values of A−1 may be useful. First, we recall the definitions (see, e.g., [14]) of the inner nu- merical radius
ν(A) =
min |z|, z∈W (A)
max |z|, z∈W (A)
ω(A) =
From (2.1) we know that ω(A) = 1 λmax(A + A∗); this quantity is also called
2
the logarithmic norm of A. We will now review some areas where it may be of interest to approximate the quantities ν(A−1), μ(A−1), or ω(A−1).
The first area we consider is the study of the convergence of iterative methods for linear systems. Consider the linear system Ax = b with x0 as initial guess, r0 = b − Ax0 as corresponding initial residual, and the Krylov space Uk = K(A,r0) as search space. In minimum residual methods, such as MINRES for symmetric A or GMRES for nonsymmetric A, one approximates x by
+ ∗∗ −1∗∗ x≈xk =(AUk) b:=(UkAAUk) UkA b,
where we assume that AUk is of full rank k. This choice minimizes the residual over the search space: xk = argminyk∈Uk ∥b − Ayk∥. Eiermann and Ernst [2, Section 6.1.3], [1, Cor. 6.2] show that for this minimum residual
∥rMR∥:=∥b−Ax ∥ kk
max Re(z). z∈W (A)