Future Generation Computer Systems 19 (2003) 1231–1242 New algorithms for the iterative refinement of estimates of invariant subspaces K. Hüper a,∗, P. Van Dooren b a Department of Mathematics, Würzburg University, Am Hubland, D-97074 Würzburg, Germany b Department of Mathematical Engineering, Université Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium Abstract New methods for refining estimates of invariant subspaces of a non-symmetric matrix are presented. We use global analysis to show local quadratic convergence of our method under mild conditions on the spectrum of the matrix. © 2003 Elsevier B.V. All rights reserved. Keywords: Non-symmetric eigenvalue problem; Invariant subspace; Riccati equations; Sylvester equations; Global analysis; Local quadratic convergence 1. Refining estimates of invariant subspaces mations, which display local quadratic convergence to an upper block triangular form. The formulation The computation of invariant subspaces has re- of the algorithms and their convergence analysis are ceived a lot of attention in the numerical linear valid for different diagonal block sizes, as long as algebra literature. In this paper we present a new these blocks have disjoint spectrum, i.e., as long as algorithm that borrows ideas of two well established the corresponding invariant subspaces are well de- methods: the Riccati-/Sylvester-like iteration, and fined. An important special case is the computation the Jacobi-like iteration. Even though these tech- of the real Schur form, which groups complex con- niques are traditionally associated with very different jugate eigenvalues in 2 × 2 diagonal blocks. It can be eigenvalue algorithms, the algorithm that we de- obtained by our method, provided these eigenvalue rive in this paper makes a nice link between them. pairs are disjoint. We always work over R, but the Our method refines estimates of invariant subspaces generalization to C is immediate and we state with- of real non-symmetric matrices which are already out proof that all the results from this paper directly “nearly” upper block triangular, but not in condensed apply to the complex case. The outline of this paper form. It uses the Lie group of real unipotent lower is as follows. After introducing some notation we block triangular (n × n)-matrices as similarities on a will focus on an algorithm consisting of similarity nearly upper block triangular matrix. We develop a transformations by unipotent lower block triangular class of algorithms based on such similarity transfor- matrices. In order to improve numerical accuracy, we then use orthogonal transformations instead. The ∗ Corresponding author. convergence properties of the orthogonal algorithm is E-mail addresses: [email protected] shown to be an immediate consequence of the former (K. Hüper), [email protected] (P. Van Dooren). one. 0167-739X/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0167-739X(03)00048-7 1232 K. Hüper, P. Van Dooren / Future Generation Computer Systems 19 (2003) 1231–1242 2. Lower unipotent block triangular l ∈ ln and for arbitrary a ∈ V transformations Dσ(I, A) · (l, a) = lA − Al + a. (6) n×n Let V ⊂ Rn×n denote the vector space of real upper We show that for any h ∈ R the linear system (n × n) block triangular -matrices lA − Al + a = h (7) n×n V :={X ∈ R |Xij = 0ni×nj ∀1 ≤ j<i≤ r} (1) has a solution in terms of l ∈ ln and a ∈ V .By ni×ni with square diagonal sub-blocks Xii ∈ R , i = decomposing into upper block triangular and strictly r ,...,r n = n L h = h . + h . 1 and i=1 i . Let n denote the Lie lower block triangular parts upp bl str low bl group of real unipotent lower block triangular (n × and because a ∈ V is already upper block triangular it n)-matrices partitioned conformably with V : remains to show that the strictly lower block triangular part of (7) L ={X ∈ Rn×n|X = I ∀ ≤ k ≤ r, X n : kk nk 1 ij (lA − Al)str.low.bl. = hstr.low.bl. (8) = 0ni×nj ∀1 ≤ i<j≤ r}. (2) can be solved for l ∈ ln. We partition into “blocks of Given a real upper block triangular matrix A ∈ V ,we sub-blocks” consider the orbit MLn of A under similarity action σ L l 0 A A of n: l = 11 ,A= 11 12 , l l A n×n 21 22 0 22 σ : Ln × V → R , (h ) (L, X) → σ(L, X) = −1, 11 str.low.bl. 0 LXL (3) hstr.low.bl. = , h (h )str.low.bl. n×n −1 21 22 MLn :={X ∈ R |X = LAL ,L∈ Ln}. (4) n ×n A ∈ R 1 1 l = accordingly, i.e., 11 and 11 0n1 as be- The following (generic) assumption will be crucial in l l fore. Thus one has to solve for 21 and 22 . Considering our analysis. the (21)-block of (8) gives l A − A l = h . Assumption 2.1. Let A ∈ V , then its diagonal 21 11 22 21 21 (9) A i = ,... ,r sub-blocks ii, 1 have mutually disjoint By Assumption 2.1, the Sylvester equation (9) has spectra. l a unique solution 21 . In order to prove the result, we proceed analogously with the (22 )-block of (8), Our first result shows that any matrix lying in a i.e. l A − A l =−l A + (h )str.low.bl., and sufficiently small neighborhood of A which fulfils 22 22 22 22 21 12 22 continue inductively (l := l , A := A , etc.) by L 22 22 Assumption 2.1 is an element of an n-orbit of some partitioning the remaining diagonal blocks A , i = B ii other matrix, say , which also fulfils Assumption 2.1. 2,... ,r into smaller blocks of sub-blocks. ᮀ We first show that under Assumption 2.1 the smooth σ mapping satisfies the following lemma. Let A ∈ Rn×n fulfil Assumption 2.1, then the next lemma characterizes the Ln-orbit of the matrix A. Lemma 2.1. The mapping σ defined by (3) is locally surjective around (I, A). n×n −1 Lemma 2.2. MLn :={X ∈ R |X = LAL ,L ∈ Ln} is diffeomorphic to Ln. Proof. Let ln denote the Lie algebra of real lower block triangular (n × n)-matrices Proof. The set MLn is a smooth manifold because n×n it is the orbit of a semi-algebraic group action, see ln :={X ∈ R |X = 0n ∀1 ≤ k ≤ r, X kk k ij [10, p. 353]. We will show that the stabilizer subgroup = ∀ ≤ i<j≤ r}. 0ni×nj 1 (5) stab(A) ⊂ Ln equals the identity {I} in Ln, i.e. that the only solution in terms of L ∈ Ln for It is sufficient to show that the derivative Dσ(I, A) : n×n −1 ln × V → R is locally surjective. For arbitrary LAL = A ⇔ [L, A] = 0 (10) K. Hüper, P. Van Dooren / Future Generation Computer Systems 19 (2003) 1231–1242 1233 . is L = I. Partition L and A conformably as such that (for nα=n1 +···+nα−1): In 0 A A I L = 1 ,A= 11 12 , nα 00 L L 0 A (α) (α)−1 21 22 22 L XL = 0 Inα 0 X P(α) I where L ∈ Ln−n . The (21)-block of [L, A] = 0 0 n¯ α 22 1 L A − A L = L = yields 21 11 22 21 0, implying 21 0by In 00 Assumption 2.1 on the spectrum of A. By recursive α × 0 In 0 = Z, (15) application of this argument to the (22 )-block of (10) α −P(α) I the result follows. Therefore, L = I implies stab(A) = 0 n¯ α {I} M =∼ L / (A) = L ᮀ and hence Ln n stab n. where Z is of the form Z1,1 ··· ··· ··· ··· ··· Z1,r . 3. Algorithms, main ideas . .. . .Z. α−1,α−1 The algorithms presented in this section for the . Z = . , iterative refinement of invariant subspaces of non- . .Zα,α . symmetric real matrices are driven by the following . . 0 Zα+ ,α+ . ideas. Let the matrix A ∈ V satisfy Assumption 2.1 1 1 . .. and consider an X ∈ MLn sufficiently close to A, i.e., . Zr, ··· Zr,α− 0 Zr,α+ ··· Zr,r X − A <∆λ, (11) 1 1 1 (16) where Z := tr(ZZ ) and ∆λ denotes the absolute Z value of the smallest difference of any eigenvalues of i.e., the blocks below the diagonal block α,α are zero. two different diagonal sub-blocks of A. Obviously, For convenience we first assume without loss of gen- erality that r = 2. In this case we want to solve the In span 1 (12) (21)-block of 0(n2+···+nr)×n1 I X X I 0 · 11 12 · 0 is then a good approximation of an n1-dimensional (1) (1) P I X21 X22 −P I right invariant subspace of X, because by assumption Z Z (11) on X, the blocks Xji are small for all j>i. = 11 12 (α) Z (17) Consider an L ∈ Ln of the following partitioned 0 22 form: ( ) in terms of P 1 , i.e., we want to solve the matrix I n1 valued algebraic Riccati equation . .. (1) (1) (1) (1) P X11 + X21 − P X12P − X22P = 0. (18) I nα L(α) = . , Since (18) is in general not solvable in closed form, : (α+ ,α) . (13) p 1 . several authors have suggested different approaches to . . .. solve (18) iteratively; see [4] for Newton-type itera- p(r,α) I tions on the non-compact Stiefel manifold and [8,17] nr for iterations like where empty blocks are considered to be zero. We P X − X P = P X P − X ,P= .