Two fast algorithms for enclosing all eigenvalues and invariant subspaces in generalized eigenvalue problems are proposed. In these algorithms, individual eigenvectors and invariant subspaces are enclosed when eigenvalues are well separated and closely clustered, respectively. The first algorithm involves only cubic complexity and automatically determines eigenvalue clusters. The second algorithm is applicable even for defective eigenvalues. Numerical results show the properties of the proposed algorithms. v w := max i (|v i |/(1 − |w i |)). Let I n , e (i) and ½ be the n × n identity matrix, the ith column of I n , and the column vector with proper dimension all of whose elements are 1, respectively. For M c ∈ C m×n and M r ∈ R m×n , where M r has nonnegative components, M c , M r denotes the interval matrix whose center and radius are M c and M r , respectively. When m = n = 1, especially, M c , M r geometrically means a disk in the complex plane. Let • and ./ be the pointwise multiplication and division, respectively. The notation fl(·) and fl (·) denote results of floating point computations, where all operations inside parentheses are executed by ordinary floating point arithmetic in rounding to nearest and rounding toward +∞ modes, respectively. The notation fl(·) denotes a rigorous upper bound of inside parentheses obtained by rounding mode controlled floating point computations. Let eps, realmin, ⊗ and F be machine epsilon, the smallest positive normalized floating point number (especially eps = 2 −52 and realmin = 2 −1022 in IEEE 754 double precision), the Kronecker product (see [13], e.g.) and the set of all floating point real numbers, respectively. For a Fréchet differentiable matrix function F (X) where X ∈ C m×n , denote the Fréchet derivative of F at X applied to the matrix H by F X (H). We introduce Lemmas 2.1, 2.2, 2.3, and 2.5 and Corollary 2.4.Lemma 2.1 (e.g., Golub and Van Loan [1]). For S ∈ C n×n and 1 ≤ p ≤ ∞, if S p < 1, I n − S is nonsingular. Lemma 2.2 (e.g., Horn and Johnson [13]). For any complex matrices K, L, M , and N with compatible sizes, it holds that (K ⊗ L)(M ⊗ N ) = (KM ⊗ LN ) and vec(LM N ) = (N T ⊗ L)vec(M ). Lemma 2.3 (Minamihata [14]). Let S ∈ C n×n , f ∈ C n and t := |S|½. If t ∞ < 1, then I n − S is nonsingular and |(I n − S) −1 ||f | ≤ |f | + f t t holds.
1207Proof. Lemma 2.1 and S ∞ = t ∞ < 1 give the nonsingularity of I n − S. The Neumann series givesCorollary 2.4. Let S and t be as in Lemma 2.3 and F ∈ C n×k . Assume t ∞ < 1 and define w := ( F :1 t , . . . , F :k t ) T . Then I n − S is nonsingular andProof. The identity |F | = (|F :1 |, . . . , |F :k |) and the applications of Lemma 2.3 to |(I n − S) −1 ||F :j |, j = 1, . . . , k, show the result.
Enclosure utilizing GED.In this section, we establish theories for enclosing all the eigenvalues and invariant subspaces based on the GED AX = BXD, X, D ∈ C n×n , where D is diagonal. If B is nonsingular (this can be verified by the algorithm in this section), the GED exists, although X is not always nonsingular. We also p...
