STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES IN STRUCTURED PENCILS

Transcription

1 Electronic Transactions on Numerical Analysis Volume 44 pp Copyright c 215 ISSN ETNA STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES IN STRUCTURED PENCILS VOLKER MEHRMANN AND HONGGUO XU Abstract The long standing problem is discussed of how to deflate the part associated with the eigenvalue infinity in a structured matrix pencil using structure preserving unitary transformations We derive such a deflation procedure and apply this new technique to symmetric Hermitian or alternating pencils and in a modified form to (anti)-palindromic pencils We present a detailed error and perturbation analysis of this and other deflation procedures and demonstrate the properties of the new algorithm with several numerical examples Key words structured staircase form structured Kronecker canonical form symmetric pencil Hermitian pencil alternating pencil palindromic pencil linear quadratic control H control AMS subject classifications 65F15 15A21 93B4 1 Introduction In this paper we develop numerical methods to calculate a structure preserving deflation under unitary (or in the real case real orthogonal) equivalence transformations for the infinite eigenvalue part in eigenvalue problems for structured matrix pencils (11) (λn M)x = where N = σ N N M = σ M M K nn σ N σ M {±1} and is either T the transpose or the conjugate transpose Here K stands for either R or C Eigenvalue problems of this form arise in many applications in particular in the context of linear quadratic optimal control problems (see eg ) H control problems (see eg ) and other applications; see eg 1 13 In most applications one needs information about the Kronecker structure (eigenvalues eigenvectors multiplicities Jordan chains sign characteristic) of these pencils; see 18 Numerically in order to obtain such information structure preserving methods (using congruence rather than equivalence) are preferred because only these preserve all possible characteristics and they reveal the symmetries in the Kronecker structure of a structured pencil of the form (11) A structure preserving method may also save computational time and work since it can make use of the symmetries in the matrix structures to simplify the computation In order to compute the Kronecker structure of (11) in a numerically backward stable way a structured staircase form under unitary structure preserving transformations has been derived in 5 and implemented as production software in 4 In the real case the unitary transformations can be chosen to be real orthogonal but to simplify in the following when speaking of unitary transformations we implicitly mean real orthogonal transformations in the case of real matrices The following result summarizes the staircase form Received May Accepted October Published online on January Recommended by D Szyld Supported by Deutsche Forschungsgemeinschaft through the DFG Research Center MATHEON Mathematics for Key Technologies in Berlin and by the University of Kansas Dept of Mathematics during a research visit in 213 The second author is partially supported by the Alexander von Humboldt Foundation and the Deutsche Forschungsgemeinschaft through the DFG Research Center MATHEON Mathematics for Key Technologies in Berlin Institut für Mathematik MA 4-5 TU Berlin Str des 17 Juni 136 D-1623 Berlin Germany (mehrmann@mathtu-berlinde) Department of Mathematics University of Kansas Lawrence KS 6645 USA (xu@mathkuedu) 1

2 2 V MEHRMANN AND H XU THEOREM 11 (Structured staircase form) For a pencil λn M with N = σ N N M = σ M M K nn there exists a real orthogonal or unitary matrix U K nn respectively such that U NU = N 11 N 1m N 1m+1 N 1m+2 N 12m Nm 1m+2 σ N N 1m N mm N mm+1 σ N N 1m+1 σ N N mm+1 N m+1m+1 σ N N 1m+2 σ N N m 1m+2 σ N N 12m n 1 n m l q ṃ q 2 q 1 (12) U MU = M 11 M 1m M 1m+1 M 1m+2 M 12m+1 σ M M 1m M mm M mm+1 M mm+2 σ M M 1m+1 σ M M mm+1 M m+1m+1 σ M M 1m+2 σ M M mm+2 σ M M 12m+1 n 1 n m l q ṃ q 1 where q 1 n 1 q 2 n 2 q m n m N j2m+1 j K njqj+1 1 j m 1 N m+1m+1 = = σ N K 2p2p M j2m+2 j = Γ j K njqj Γ j R njnj 1 j m Σ M m+1m+1 = 11 Σ 12 σ M Σ Σ 11 = σ M Σ 11 K 2p2p 12 Σ 22 Σ 22 = σ M Σ 22 K l 2pl 2p and the blocks Σ 22 and Γ j j = 1 m are nonsingular Proof The proof for the real case σ N = 1 and σ M = 1 has been given in 5; the other cases follow analogously It is clear that the transformation introduced in Theorem 11 preserves the structure of the pencil but note that in the real case or in the complex case with being the complex conjugate this transformation is a congruence transformation while in the complex case with being the transpose this is just a structure preserving equivalence transformation but not a congruence transformation This staircase form allows to deflate the singular part and some of the infinite eigenvalue parts of the pencil in a structure preserving way so that for eigenvalue eigenvector and

3 STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 3 invariant subspace computations only the central block λn m+1m+1 M m+1m+1 of the form (12) has to be considered which is regular and of index at most one ie it has only finite eigenvalues plus infinite eigenvalues with Kronecker blocks of size one due to the fact that and Σ 22 are nonsingular The staircase form has recently been extended to parameter dependent matrix pencils arising in the control of differential-algebraic systems with variable coefficients 9 where however the transformation is more complex From the staircase form it is clear how to deflate all the parts associated with higher (than one) indices and singular parts using unitary transformations (despite the usual difficulties with rank decisions) However for a long time it remained an open problem how to deflate the remaining index one part associated with the eigenvalue infinity in a structure preserving way using unitary structure preserving transformations ie to reduce the central subpencil to a subpencil λn f M f of the same structure (with a nonsingular N f ) that contains all the information about the finite eigenvalues In this paper we solve this problem and develop a new technique for structure preserving deflation of infinite eigenvalues via unitary transformations We present an error and perturbation analysis and also several numerical examples that demonstrate the properties of the new method and also compare it to non-unitary structure preserving transformations 2 Deflation of the index one part A simple way to achieve a structure preserving but non-unitary deflation is to use the Schur complement Compute a factorization (21) U N1 N = of N with N 1 of full row rank This can be done via the rank-revealing QR decomposition or the singular value decomposition; see 8 The critical part in this factorization is the decision about the rank of N which is done in the usual way by setting those part to zero that are associated to the singular values which are smaller than the machine precision times the norm of N; see 6 for a detailed discussion of this topic Using the structure of the pencil we have (22) Ñ = U Ñ11 NU = M = U M11 M12 MU = σ M M 12 M22 where Ñ11 = σ N Ñ11 M ii = σ M M ii i = 1 2 By construction Ñ 11 is invertible and since λn M is regular and of index at most one M 22 is also invertible Then with I (23) L = 1 σ M M M I one has (24) with the Schur complement (U L) N(U L) = L Ñ11 Ñ L = (U L) M(U L) = L S M L = M22 (25) S = M 11 σ M M12 M 1 22 M 12

4 4 V MEHRMANN AND H XU Hence λn M is equivalent to the decoupled block diagonal pencil (λñ11 S) (λ M 22 ) This deflation procedure via a non-unitary transformation preserves the structure but if M 22 is ill-conditioned with respect to inversion then the eigenvalues of the pencil λñ11 S may be corrupted due to the ill-conditioning in the computation of the Schur complement S We analyze the properties of the Schur complement approach in Section 4 and present some numerical examples in Section 7 Another possibility to deflate the part associated with the infinite eigenvalues is to use equivalence transformations that are unitary but not structure preserving Starting again with the factorization (21) we form U M1 M = partitioned analogously and let V be a unitary M 2 matrix such that (26) M 2 V = M22 with M 22 square nonsingular Setting (27) N = U NV = N11 N12 M = U MV = M11 M12 M22 then since N 11 and M 22 in (27) are nonsingular we can eliminate N 12 and M 12 simultaneously by block Gaussian eliminations It follows that λn M is equivalent to the decoupled block diagonal pencil (λ N 11 M 11 ) (λ M 22 ) The computation of the subpencil λ N 11 M 11 can be performed in a backward stable way but it has lost its symmetry structures and as a consequence it is hard to obtain the resulting symmetric structure in the finite spectrum of λn M in further computations Thus this method is inadequate for many of the tasks where the exact eigenvalue symmetry is needed We present some numerical examples to demonstrate this deficiency in Section 7 Furthermore in the resulting pencil the sign characteristics which are invariants of the pencil under structure preserving transformations 18 are not preserved 3 Deflation under unitary structure preserving transformations To derive a structure preserving deflation under unitary structure preserving transformations consider the unitary matrices U V obtained in (21) and (26) Form (31) V NV =: N11 N 12 σ N N V MV =: 12 N 22 M11 M 12 σ M M 12 M 22 Then we have the following surprising result THEOREM 31 Consider a structured pencil of the form (11) which is regular and of index at most one Let Q = U Q11 Q V = 12 be partitioned conformably to the block Q 21 Q 22 structure in (31) where U V are obtained in (21) and (26) Then Q 11 is invertible and λn 11 M 11 = Q 11(λÑ11 S)Q 11 = Q 11(λ N 11 M 11 )

5 STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 5 where the subpencils λn 11 M 11 λñ11 S and λ N 11 M 11 are given in (31) (24) and (27) respectively Consequently the finite eigenvalues of λn M are exactly the finite eigenvalues of the structured subpencil λn 11 M 11 Proof By definition of Q one has (32) λv NV V MV = Q (λ N M) and (33) λv NV V MV = Q (λñ M)Q Based on (22) one has N1 = U N = Ñ11 U M1 = U M = M 2 M11 M12 σ M M 12 M22 U and then (26) becomes σ M M 12 M22 Q = M 22 From this relation we obtain that (34) σ M M 12 Q 11 + M 22 Q 21 = and (35) σ M M 12 = M 22 Q 12 M22 = M 22 Q 22 Suppose that Q 11 is singular then there exists a nonzero vector x such that Q 11 x = By post-multiplying x to (34) and using the fact that M 22 is invertible one has Q 21 x = Then Q11 x = contradicting the fact that Q is unitary Therefore Q Q 11 must be invertible 21 Comparing the (11) block on both sides of (32) leads to the first equivalence relation λn 11 M 11 = Q 11(λ N 11 M 11 ) Since M 22 is invertible from the second relation in (35) so are M 22 and Q 22 From (34) it follows that (36) Q 21 Q 1 11 = σ 1 M M M Then the matrix L defined in (23) becomes I L = Q 21 Q 1 11 I and defining R = Q11 Q 12 Q 22

6 6 V MEHRMANN AND H XU and using the fact that Q is unitary it follows that Hence (33) can be written as Q = L R (37) λv NV V MV = R (λ L Ñ L L M L) R and comparing the (1 1) block on both sides of (37) leads to the second equivalence relation λn 11 M 11 = Q 11(λÑ11 S)Q 11 One can also obtain another perception of how the subpencils λñ11 S and λn 11 M 11 are related to the pencils λv NV V MV and λñ M The relation (37) implies that Setting R (λv NV V MV ) R 1 = L (λñ M) L = (λñ11 S) (λ M 22 ) Q11 L = = I L Q 21 we have Q = LR and also Q11 I Q 1 R = 11 Q 12 I Q = 22 1 Q11 R I (38) R (λv NV V MV )R 1 = L (λñ M)L = (λn 11 M 11 ) (λ M 22 ) which has the pencil (λn 11 M 11 ) in the (1 1) position The pencil λv NV V MV however is unitarily congruent or equivalent in the complex transpose case to the original pencil and the finite eigenvalue part can be simply extracted from the (1 1) block It follows from (38) that the columns of the matrices Q11 I Q 21 form orthonormal bases for the right deflating subspaces corresponding to the finite and infinite eigenvalues of λñ M respectively In contrast to this (24) shows that the Schur complement method simply uses the non-orthonormal basis with the columns of I Q 21 Q 1 11 ie the first block column of L for a structure preserving transformation to block diagonalize λñ M For the original pencil λn M with V = V 1 V 2 and U = U 1 U 2 the columns of V11 Q11 U12 V 1 = = U and U 2 = V 21 Q 21 span the right deflating subspaces corresponding to the finite and infinite eigenvalues respectively The minimal angle between the two right deflating subspaces is U 22 Θ min = min x range V 1 y range U 2 cos 1 x y x y = cos 1 V 1 U 2 = cos 1 Q 21

7 STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 7 where denotes the Euclidean norm for vectors and the spectral norm for matrices The minimal angle measures the closeness of the two deflating subspaces For Q there exists a CS decomposition; see 8 If the size n 1 of Q 11 is larger than or equal to the size n 2 of Q 22 then this has the form W 1 W 2 Q11 Q 12 Q 21 Q 22 Z1 = Z 2 Φ Ψ I n1 n 2 Ψ Φ where W 1 W 2 Z 1 Z 2 are unitary matrices and the diagonal matrices Φ = diag(φ 1 φ n2 ) and Ψ = diag(ψ 1 ψ n2 ) have nonnegative diagonal entries satisfying Φ 2 + Ψ 2 = I n2 If n 1 n 2 then one has W 1 W 2 Q11 Q 12 Q 21 Q 22 Z1 = Z 2 Φ Ψ Ψ Φ I n2 n 1 Suppose that the singular values of Q 21 are cos θ 1 cos θ min{n1n 2} then except for possibly some extra singular values at 1 the singular values of Q 11 are sin θ 1 sin θ min{n1n 2} where θ 1 θ min{n1n 2} π/2 Thus we have Introducing Θ min = min θ j (39) ρ := M 1 22 M 12 = M 12 M 1 22 and using (36) one has and hence cot Θ min = max cot θ j = ρ j Θ min = cot 1 ρ Thus the smallest singular values of Q 11 and Q 22 are given by (31) σ min (Q 11 ) = σ min (Q 22 ) = sin Θ min = ρ 2 and Q 12 = Q 21 = ρ If ρ is large which happens commonly when M 1+ρ 2 22 is nearly singular then the numerical computation of the Schur complement S in (25) may be subject to large errors On the other hand in this case the block N 11 = Q 11Ñ11Q 11 may be close to a singular matrix as well which may also lead to big numerical problems in particular when deciding about the rank of N Let us summarize the methods for computing the subpencils λn 11 M 11 defined in (31) in the following algorithms The first algorithm Algorithm 1 uses the factorizations (21) and (26) to compute U and V and eventually the subpencil λn 11 M 11 The second algorithm Algorithm 2 is just a different version which computes the factorization (22) explicitly and the factorization (31) is computed using Q = U V instead of V

8 8 V MEHRMANN AND H XU ALGORITHM 1 Consider a regular structured pencil λn M of index at most one with N = σ N N and M = σ M M 1 Use a rank revealing QR or singular value decomposition to compute a unitary matrix U such that U N1 N = where N 1 is of full row rank Partition U = U 1 U 2 accordingly and set M 2 = U 2 M 2 Use a rank revealing QR or singular value decomposition to determine a unitary matrix V such that M 2 V = M 22 det M 22 3 Compute V N11 N NV = 12 σ N N V M11 M MV = N 22 σ M M 12 M 22 and return λn 11 M 11 as the pencil associated with the finite spectrum of λn M Algorithm 1 requires two singular value decompositions or rank revealing QR factorizations as well as the associated matrix-matrix multiplications The most difficult part is the rank decision for the matrix N in step 1 which will affect the number of finite eigenvalues and the size and therefore also the conditioning of M 2 An alternative version of Algorithm 1 is given in the following algorithm ALGORITHM 2 Consider a regular structured pencil λn M of index at most one with N = σ N N and M = σ M M 1 Use a Schur-like decomposition of N to compute a unitary matrix U such that U NU = Ñ11 =: Ñ det Ñ11 and compute M = U M11 M12 MU = σ M M 12 M22 2 Use a rank revealing QR or singular value decomposition to compute a unitary matrix Q11 Q Q = 12 partitioned accordingly such that Q 21 Q 22 σ M M 12 M22 Q = M22 3 Compute Q N11 N ÑQ = 12 σ N N Q M11 M MQ = N 22 σ M M 12 M 22 or if only λn 11 M 11 is needed then N 11 = Q 11Ñ11Q 11 M 11 = Q 11( M 11 Q 11 + M 12 Q 21 )

9 STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 9 Algorithm 2 uses in contrast to Algorithm 1 a structured Schur-like decomposition in the first step and performs two unitary transformations on the pencil The other costs are comparable to those of Algorithm 1 We will perform the error and perturbation analysis for these algorithms as well as that of the Schur complement approach in the next section 4 Error and perturbation analysis In this section we present a detailed error and perturbation analysis for the different deflation procedures presented in the last sections Our error analysis for the structure preserving method under unitary transformations will be based on Algorithm 2 The error analysis for Algorithm 1 is essentially the same Let u be the unit roundoff and denote for each of the matrices defined in the previous section the computed counterpart by the corresponding calligraphic letter eg let U Ñ 11 be the computed U and Ñ11 in step 1 of Algorithm 2 Following standard backward error analysis (8) there exists a unitary matrix Ũ such that Ũ Ñ11 (N + δn u )Ũ = = Ñ where Ũ U = O(u) δn u = σ N δn u and δn u = O( N u) For the computed M we have M11 M = Ũ (M + δm u )Ũ = M12 σ M M 12 M22 with δm u = σ M δm u δm u = O( M u) Setting X = U Ũ we obtain from the exact factorization (22) that (41) Ñ = Ñ11 Ñ11 = X (Ñ + U δn u U)X = X + δñ11 δñ12 σ N δñ 12 δñ22 Assume that δñ11 < σ min (Ñ11) Then Ñ11 + δñ11 is nonsingular One can show that for I G K = G = (Ñ11 + δñ11) 1 δñ12 I X one has Ñ11 K + δñ11 σ N δñ 12 δñ12 δñ22 K = Ñ11 + δñ11 Ñ11 + δñ11 δñ22 σ N δñ = 12G Introduce the unitary matrix I G (I + GG X = ) 1 2 G I (I + G G) 1 2 where (I + GG ) 1 2 and (I + G G) 1 2 are the positive definite square roots of I + GG and I + G G respectively Then one has K = X (I + GG ) 1 2 (I + G G) 1 2 G (I + G G) 1 2

10 1 V MEHRMANN AND H XU and then Ñ11 X + δñ11 σ N δñ 12 δñ12 δñ22 ( ) X = (I + GG ) 1 2 (N11 + δn 11 )(I + GG ) 1 2 Comparing with (41) one can show that X = X diag(x 1 X 2 ) where X 1 X 2 are unitary Without loss of generality we set X = X I + δx11 δx =: I + δx =: 12 δx 21 I + δx 22 and introduce the condition number Since G = O(κ N u) and one has κ N = N Ñ = Ñ11 Ñ11 δx 12 = G(I + G G) 1 2 δx21 = G (I + GG ) 1 2 X 11 = (I + GG ) 1 2 I X22 = (I + G G) 1 2 I δx 12 δx 21 = O( G ) = O(κ N u) δx 11 δx 22 = O( G 2 ) = O(κ 2 Nu 2 ) Clearly here κ N measures the sensitivity of the null space of N Hence Ñ 11 = (I + δx 11 ) Ñ 11 (I + δx 11 ) + O( N u) = Ñ11 + δñ11 δñ11 = O ( (κ 2 Nu + 1) N u ) and it follows from M = X ( M + U δm u U)X that (42) M = M + δ M δ M = O(κN M u) Partition M conformable to M assume that δ M 22 < σ min ( M 22 ) so that M 22 is invertible and let S be the computed Schur complement based on M Then under the assumption that a backward stable linear system solver is used for computing M 1 22 M 12 columnwise one has S = M 11 σ M M12 M 1 22 M 12 + E s with (( E s = O M 11 + ρ M 12 + ρ 2 M ) ) ( ) 22 u = O (1 + ρ) 2 M u where ρ is defined in (39) Some elementary calculations and (42) lead to S = S + E s + F s =: S + δs

11 STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 11 with ( ) F s = O (1 + ρ) 2 κ N M u where F s results from the errors introduced in M by performing the transformation with Ũ Note that due to the structure of the error we could absorb E s into F s Altogether we have λñ11 S = λ(ñ11 + δñ11) + (S + δs) with δñ11 = O ( ( ) (κ 2 N u + 1) N u) and δs = O (1 + ρ) 2 κ N M u It then follows that the columns of I σ M 1 M M and I span the right deflating subspaces corresponding to the finite and infinite eigenvalues respectively of the pencil ( ) λñ Es M + and λñ11 S λ M 22 are the corresponding computed subpencils associated to these bases Let Q be unitary such that Then σ M M 12 M22 Q = M 22 σ M M 1 22 M 12 = Q 21 Q 1 11 and therefore the right deflating subspace associated with the finite eigenvalues is spanned by the orthonormal columns of the matrix it follows from (42) that (43) M 22 = Q11 Q 21 Y = Q Q Introducing ) σ M M 12 M22 QY = ( M22 + E Y with E = E 1 E 2 = σ M δ M 12 δ M 22 Q Similar to the matrix X one may express I + δy11 δy Y = 12 I G = Y (I + G Y G Y ) 1 2 δy 21 I + δy 22 G Y I (I + G Y G Y ) 1 2 where G Y = ( M 22 + E 2 ) 1 E 1 One then has δy 12 δy 21 = O (κ N κ M u) δy 11 δy 22 = O ( κ 2 N κ 2 M u 2) where κ M = M 22 1 M 1

12 12 V MEHRMANN AND H XU Since Ũ ( λ (N + δn u ) (M + δm u ) ) Ũ Q is block upper triangular and because of the structure of the pencil the columns of Q11 Ũ Ũ I Q 21 are the right deflating subspaces of λ(n + δn u ) (M + δm u ) corresponding to the finite and infinite eigenvalues respectively Hence the minimal angle between these two subspaces is given by Θ min = cos 1 Q 21 and since Q 21 = Q 21 (I + δy 11 ) + Q 22 δy 21 it follows that and hence Q 21 = Q 21 + O(κ N κ M u) Θ min = Θ min + O(csc Θ min κ N κ M u) The perturbation in the deflating subspace associated to the finite eigenvalues can then be measured by δ f := sin 1 f with f = ( ) Q12 U Ũ Q 22 Q11 Q 21 = Q12 Q 22 X Q11 and the perturbation in the deflating subspace of the eigenvalue infinity can be measured by δ := sin 1 with = Inserting the obtained norm bounds we have ( ) I U Ũ I Q 21 δ f = O( δy 21 + δx ) = O(κ N κ M u) δ = O( δx 12 ) = O(κ N u) It should be noted that in the case that N is already in block diagonal form as Ñ which is for example the case when N arises from the middle block of the staircase form (12) then U = Ũ = I and the computed subpencil corresponding to the finite eigenvalues is given by ( ) λñ11 S S = S + E s E s = O (1 + ρ) 2 M u where E s is arising just from the computation of S In step 2 of Algorithm 2 for the computed version Q of Q there exists a unitary matrix Q such that Q Q = O(u) and (44) ( with E = O M ) ( ) 22 u = O M 22 u ) (σ M M 12 M22 + E Q = M 22

13 Introducing then similar to (43) we obtain σ M M 12 M22 Q = STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 13 M22 + Y = Q Q σ M δ M 12 δ M 22 Q =: M22 + E with E = O(κ N M u) Then one has ) ( M22 + EQ + E Y = M22 Partitioning EQ + E = E 11 to Y we may express I + δy11 δy Y = 12 = δy 21 I + δy 22 (( ) ) E 12 then E11 = O M 22 + κ N M u Similar I G Y (I + G Y G Y ) 1 2 G Y I (I + G Y G Y ) 1 2 where G Y = ( M 22 + E 12 ) 1 E 11 We obtain the estimates ( ) ( ) δy 21 δy 12 = O E 11 M 22 1 = O + κ (κ M22 N κ M )u and ( δy 11 δy 22 = O + κ (κ M22 N κ M ) 2 u 2) := M 22 M 22 1 Using these estimates the computed N 11 and M 11 can be expressed as where κ M22 with δn 11 = O ( Q 11 2 N u ) and N 11 = Q 11Ñ11 Q 11 + δn 11 M 11 = Q 11( M 11 Q11 + M 12 Q21 ) + δm 11 (( ) ) with δm 11 = O M 11 Q 11 + M 12 Q 21 Q 11 u Using the relation between λñ M and λñ M one has N 11 = Q 11Ñ11 Q 11 + N with N = O (( κ 2 N u + 1) N Q 11 2 u ) and M 11 = Q 11( M 11 Q11 + M 12 Q21 ) + M1 (( ) ) with M1 = O κ N M + M 11 Q 11 + M 12 Q 21 Q 11 u Since Q 21 Q 1 11 = (Q 21 (I + δy 11 ) + Q 22 δy 21 ) (Q 11 (I + δy 11 ) + Q 12 δy 21 ) 1 ( ) ( = Q 21 Q Q 22δY 21 (I + δy 11 ) 1 Q 1 11 = Q 21 Q Q 22 δy 21(I + δy 11 ) 1 Q 1 11 I + Q 12 δy 21 (I + δy 11 ) 1 Q 1 11 ) 1 ( I + Q 12 δy 21 (I + δy 11 ) 1 Q 1 11 ) 1

14 14 V MEHRMANN AND H XU using (36) one obtains M 11 = Q 11S Q 11 + M1 + M2 where by using (35) M2 = Q 11(Q 21 Q 1 11 ) M22 δy 21 (I + δy 11 ) 1 Q 1 11 ( I + Q 12 δy 21 (I + δy 11 ) 1 Q 1 11 ) 1 Q11 = Q 21 M 22 δy 21 (I + δy 11 ) 1 (I + Q 1 11 Q 12δY 21 (I + δy 11 ) 1 ) 1 Q 1 11 Q 11 Using δy 21 (I + δy 11 ) 1 = G Y = ( M 22 + E 12 ) 1 E 11 it follows that M2 = Q 21E 11 + o(u) and hence ( ( ) ) M2 = O Q 21 M 22 + κ N M u and M 11 = Q 11S Q 11 + M with M = O ((κ N M + M 22 Q 21 + M ) ) 11 Q 11 2 u Here we have used the fact that M 12 M 12 M 1 22 M 22 = Q 21 M 22 In the situation that only the finite eigenvalues are considered one can reduce the effort to compute λn 11 M 11 only So if we set σ N N 12 = Q Q 12Ñ11 11 σ M M 12 = Q ( 12 M11 Q11 + M ) 12 Q21 then the columns of Q11 Q 21 form an orthonormal basis for the right deflating subspace of λñ M 1 corresponding to the finite eigenvalues where M 1 is a perturbed M of order O( M 11 u) because of the inexact QR factorization (44) The resulting subpencil is λ Q Q 11Ñ11 11 Q ( 11 M11 Q11 + M ) 12 Q21 and the pencil λn 11 M 11 is obtained by adding the extra error pencil λ N M introduced by the numerical computation of N 11 and M 11 Similarly the columns of span the right deflating subspace of λñ I M 1 corresponding to the eigenvalue infinity and the columns of the matrices Q11 Ũ Ũ I Q 21 span the corresponding deflating subspaces of a pencil slightly perturbed from λn M Setting ( ) Q12 Q11 Q12 Q11 f = U Ũ = X ɛ Q 22 Q f := sin 1 f 22 Q 21 Q 21

15 and STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 15 = ( U ) I Ũ we have ɛ = δ = O(κ N u) and similarly ɛ f = O ( δy 21 + δx ) = O ɛ I := sin 1 ) ) + κ ((κ M22 N κ M u and the minimal angle between the two perturbed subspaces is given by ) ) Θ min = Θ min + O (csc Θ min + κ (κ M22 N κ M u If λn M is already in the form λñ M then U = Ũ = I and in this case N = O ( N Q 11 2 u ) (( M = O M 22 Q 21 + M 11 Q 11 2) ) u and it follows that with ɛ = ɛ f = O(κ M22 u) Θmin = Θ min + O(csc Θ min κ M22 u) We may express N 11 and M 11 as By using (31) we have N 11 = Q 11(Ñ11 + N ) Q 11 M 11 = Q 11(S + M ) Q 11 N = Q 11 1 N Q 11 M = Q 11 1 M Q 11 N = O (( κ 2 Nε + 1 ) κ 2 Q 11 N u ) (( ) ) M = O κ N Q M + κ 2 Q 11 M 11 + ρ Q 1 11 M 22 u where κ Q11 = Q 11 Q 1 11 Recalling from (31) that Q 1 11 = 1 + ρ 2 and comparing the errors in the subpencils λñ11 S and λn 11 M 11 it follows that M and δs have the same order while N can be larger than δñ11 by a factor κ 2 Q 11 The latter will be of equal order only when κ 2 Q 11 is not too large On the other hand if we transform λñ11 S to then we have Q 11(λÑ11 S) Q 11 = Q 11(λÑ11 S) Q 11 + Q 11(λδÑ11 δs) Q 11 Q Q 11δÑ11 11 = O (( κ 2 N u + 1 ) Q 11 2 N u ) Q 11δS Q ( ) 11 = O (1 + ρ) 2 Q 11 2 κ N M u The first quantity is of the same order as N while the second one can be larger than M by a factor κ 2 Q 11 unless κ Q11 is not too large So in terms of numerical stability the error analysis does not show which method has an advantage over the other This is an unusual circumstance in numerical analysis where usually the methods based on unitary transformations in the worst case situation have smaller errors than the ones based on non-unitary transformations We will demonstrate this effect in the numerical examples in Section 7

16 16 V MEHRMANN AND H XU 5 An alternative point of view Another way to understand the new unitary structure preserving method of deflating the infinite eigenvalue part is to consider the extension trick introduced in 9 Partition the matrix U in (21) as U = U 1 U 2 with U1 of the same size as N1 T For any eigenvector x of λn M corresponding to a finite eigenvalue λ it then follows from U (λn M)x = that U 2 Mx = So the original eigenvalue problem is equivalent to the non-square non-symmetric eigenvalue problem ( ) N M (51) λ U x = 2 M and we are only looking for eigenvectors in the nullspace of U 2 M Let z := V z1 x =: be partitioned according to the block structure in (31) Then the eigenvalue problem (51) is equivalent to the extended eigenvalue problem V ( ) N M λ I U V z 2 M N 11 N 12 = λ σ N N 12 N 22 Clearly then it follows that z 2 = and z 2 (λn 11 M 11 )z 1 = M 11 M 12 σ M M 12 M 22 z1 = z 2 M22 Hence an eigenvector of λn M corresponding to a finite eigenvalue has the form z1 x = V where z 1 is an eigenvector of the subpencil λn 11 M 11 More generally In1 the columns of V form an orthonormal basis for the deflating subspace of λn M corresponding to the finite eigenvalues 6 Palindromic and anti-palindromic pencils Structured pencils closely related to those discussed before are the palindromic/anti-palindromic pencils of the form λa σa where A is an arbitrary real or complex square matrix and σ = ±1; see 1 16 for a detailed discussion of the relationship Assume further that the eigenvalue σ (if it occurs) is semisimple which corresponds to the property of being regular and of index at most one for the structured pencils discussed before Applying a Cayley transformation (see 1 11) the pencil λn M with N := A σa = σn M := A + σa = σm is a structured pencil of the form (11) with σ N = σ and σ M = σ and the semi-simple eigenvalue σ becomes the eigenvalue and the pencil has index at most one To this pencil we can apply the discussed transformations V U as defined in (21) and (26) Then (61) U (A σa) = N1 U (A + σa) = M1 M 2

17 Partitioning STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 17 conformably one has N1 A = 1 σb 1 A 2 σb 2 and hence AU = A 1 A 2 U B1 A = B 2 M1 M 2 A = 1 + σb 1 A 2 + σb 2 A 2 = σb 2 M 2 = 2σB 2 Furthermore applying V to M 2 we obtain M 2 V = M 22 and hence B 2 V = B 22 with B 22 = σ M 22 /2 Setting then V (λn M)V = λ and thus V A11 A AV = 12 A 21 A 22 A 11 σa 11 A 21 σa 12 A A 12 σa 21 A 11 + σa 11 A 21 + σa σa 22 A 12 + σa 21 A 22 + σa 22 λn 11 M 11 = λ(a 11 σa 11 ) (A 11 + σa 11 ) is the subpencil of λn M with the finite eigenvalues only By taking the inverse Cayley transformation this is equivalent to the property that the palindromic subpencil λa 11 σa 11 contains all the eigenvalues of λa σa except the eigenvalue σ which has been deflated It should be noted though that the Cayley transformation or its inverse may lead to cancellation errors if there are eigenvalues close to σ but not equal to σ Note when some eigenvalues are close σ but not equal to σ since A σa has to be computed explicitly the rank decision in computing the first factorization (61) is more difficult to make 7 Numerical examples In this section we present several numerical tests to compare the computed finite eigenvalues of the subpencils generated by the three methods: structured unitary equivalence transformation (Algorithms 1 and 2) Schur complement transformation (by using (24)) and the non-structured equivalence transformation (with (27)) All the tests were performed in MATLAB (Version 82) with double precision All the tested pencils were real and we use relative errors to measure the accuracy of the computed finite eigenvalues where the exact" eigenvalues are computed with the Matlab code eig in extended precision vpa The tested pencils are all skew-symmetric/symmetric The finite eigenvalues of the subpencil extracted with the two structure preserving methods are computed with the MEX code skewhamileig (2 3) If the pencil associated with the finite eigenvalues is extracted via non-structured equivalence transformations we use the MATLAB code qz We also display the quantities Θ min δ f ɛ f ɛ as well as ρ defined in (39) Since Θ min and Θ min are always very close to Θ min we do not display them We do not show δ either since it is the same as ɛ The following quantities were obtained from the numerical tests

18 18 V MEHRMANN AND H XU Eu max Es max Eh max : the maximum relative error of the finite eigenvalues for a given pencil with the structured unitary equivalence transformation method the Schur complement method and the non-structured equivalence transformation method respectively Eu min Es min Eh min : the minimum relative error of the finite eigenvalues for a given pencil with the structured unitary equivalence transformation method the Schur complement method and the non-structured equivalence transformation method respectively Eu GM Es GM Eh GM : the geometric mean of the relative errors of the finite eigenvalues for a given pencil with the structured unitary equivalence transformation method the Schur complement method and the non-structured equivalence transformation method respectively Re hat : the maximum real part of the eigenvalues computed by the non-structured equivalence transformation method in the case when all the exact finite eigenvalues are purely imaginary EXAMPLE 1 We tested a set of skew-symmetric/symmetric pencils X T (λn M)X where 1 N = 1 β M = β α α 2 and X is a randomly generated nonsingular matrix The finite eigenvalues of such a pencil are always ±i 6 and ±i 6/β independent of X and α If β is small in modulus then N is close to a singular matrix and when α is small in modulus then it can be expected that the eigenvalue infinity part will affect the finite eigenvalues Note that when β = 1 then the pencil has two pairs of double eigenvalues We have performed the computations 1 times for each pair of (α β) where each time a different random matrix X is generated Tables display the results for each pair of (α β) The three methods determine the finite eigenvalues with about the same order of relative errors The non-structured method produces slightly less accurate eigenvalues Because it forms non-structured subpencils the computed eigenvalues are no longer purely imaginary The real parts of the computed finite eigenvalues grow as β is getting smaller For all three methods in this example the errors seem insensitive to the value of α while they increase when β decreases However α does affect the deflating subspace associated with the finite eigenvalues EXAMPLE 2 We tested a set of skew-symmetric/symmetric pencils X T (λn M)X where 1 2 β N = 1 1 M = 2 β α 1 β α 1 β and X is a randomly generated matrix The finite eigenvalues of such a pencil are always ±β with both algebraic and geometric multiplicities of 2 When β is tiny then all the finite eigenvalues are close to zero and the eigenvalue condition number is expected to increase Tables display the results for 4 different sets of (α β) (with 1 pencils for each set as in Example 1) illustrating that the three methods have essentially the same accuracy The results also show that the relative errors do increase

19 STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 19 TABLE 71 (α β) = (1 3 1): eigenvalue errors errors in deflating subspaces Θ ρ Eu max 4e-15 9e-15 2e-14 4e-14 6e-14 6e-14 7e-14 8e-14 2e-13 4e-13 Es max 7e-15 4e-14 1e-14 6e-14 6e-14 6e-14 7e-14 1e-13 6e-14 5e-13 Eh max 1e-14 8e-14 2e-14 1e-13 6e-14 8e-14 6e-14 2e-13 1e-12 5e-13 Eu min 2e-15 9e-15 3e-15 1e-14 5e-15 6e-15 2e-14 1e-14 9e-16 3e-15 Es min 1e-14 5e-15 9e-15 4e-15 4e-15 2e-14 6e-15 4e-15 4e-15 Eh min 1e-15 4e-14 1e-14 1e-14 6e-15 3e-15 5e-15 9e-14 6e-15 2e-15 Eu GM 2e-15 9e-15 8e-15 2e-14 2e-14 2e-14 3e-14 3e-14 1e-14 3e-14 Es GM 2e-14 8e-15 2e-14 2e-14 2e-14 4e-14 3e-14 2e-14 4e-14 Eh GM 4e-15 6e-14 2e-14 4e-14 2e-14 2e-14 2e-14 1e-13 8e-14 4e-14 Re hat 2e-14 2e-13 2e-14 2e-13 1e-14 4e-14 9e-15 2e-13 4e-13 1e-13 ɛ f 3e-11 3e-11 2e-12 2e-11 4e-11 4e-11 1e-11 2e-11 1e-1 7e-11 δ f 3e-11 2e-11 2e-12 7e-12 2e-11 2e-11 1e-11 2e-11 1e-1 6e-11 ɛ 6e-16 6e-16 5e-16 7e-16 5e-16 5e-16 3e-16 3e-16 2e-16 5e-16 Θ 6e-1 2e-1 2e-1 2e-1 8e-2 3e-1 2e-1 8e-2 8e-2 1e-1 ρ TABLE 72 (α β) = ( ): eigenvalue errors errors of deflating subspaces and Θ ρ Eu max 8e-13 3e-12 6e-12 7e-12 9e-12 2e-11 8e-11 8e-11 3e-1 2e-9 Es max 3e-12 3e-12 7e-12 9e-12 4e-11 3e-11 4e-11 4e-11 6e-1 2e-9 Eh max 8e-12 7e-11 3e-1 8e-12 1e-1 3e-11 1e-1 4e-11 1e-9 3e-9 Eu min 6e-14 4e-16 8e-14 4e-13 2e-14 9e-15 9e-13 3e-15 5e-16 1e-11 Es min 6e-14 7e-16 3e-14 2e-13 2e-15 9e-15 1e-12 6e-15 7e-16 1e-11 Eh min 5e-14 4e-14 2e-12 4e-13 4e-14 8e-15 9e-13 9e-15 2e-15 1e-11 Eu GM 2e-13 3e-14 7e-13 2e-12 4e-13 4e-13 9e-12 5e-13 4e-13 2e-1 Es GM 4e-13 5e-14 4e-13 1e-12 3e-13 5e-13 8e-12 5e-13 7e-13 1e-1 Eh GM 6e-13 2e-12 2e-11 2e-12 2e-12 5e-13 1e-11 6e-13 2e-12 2e-1 Re hat 2e-6 3e-6 7e-5 2e-6 2e-5 7e-6 1e-5 1e-5 3e-4 6e-4 ɛ f 1e-8 8e-8 5e-8 8e-1 2e-7 1e-7 1e-7 1e-7 4e-7 6e-6 δ f 1e-8 8e-8 5e-8 8e-1 2e-7 1e-7 1e-7 1e-7 4e-7 6e-6 ɛ 2e-12 2e-11 6e-12 6e-12 4e-11 1e-11 7e-12 6e-12 1e-1 3e-11 Θ 2e-1 3e-1 2e-1 1e-1 1e-1 1e-1 5e-2 1e-1 8e-2 6e-3 ρ when β is getting tiny and that again the choice of α does not affect the accuracy very much For this example no method yields finite eigenvalues that are exactly real EXAMPLE 3 For skew-symmetric/symmetric pencils of the form X T (λn M)X where 1 2 β N = 1 1 M = 2 β 1 β 1 β α α and X being a randomly generated nonsingular matrix the finite eigenvalues of such a pencil are always ±β with both algebraic and geometric multiplicities of 2 The numerical results are similar to those of Example 2 and are not presented here

20 2 V MEHRMANN AND H XU TABLE 73 (α β) = (1 7 1): eigenvalue errors errors of deflating subspaces and Θ ρ Eu max 3e-15 4e-15 4e-15 6e-15 7e-15 9e-15 9e-15 9e-15 4e-14 6e-14 Es max 1e-15 4e-15 8e-15 5e-15 4e-15 8e-15 4e-15 2e-14 2e-14 4e-14 Eh max 3e-14 8e-15 3e-14 4e-15 5e-15 2e-14 3e-14 3e-14 4e-14 3e-14 Eu min 7e-16 9e-16 2e-16 7e-16 2e-15 5e-15 4e-15 5e-15 2e-15 Es min 2e-16 4e-16 1e-15 9e-16 1e-15 1e-15 2e-15 3e-15 2e-14 3e-15 Eh min 4e-16 4e-15 3e-15 2e-15 4e-15 4e-15 4e-15 1e-15 8e-15 1e-14 Eu GM 2e-15 2e-15 9e-16 2e-15 5e-15 6e-15 6e-15 1e-14 1e-14 Es GM 5e-16 1e-15 3e-15 2e-15 2e-15 3e-15 3e-15 8e-15 2e-14 1e-14 Eh GM 3e-15 5e-15 1e-14 3e-15 4e-15 9e-15 1e-14 7e-15 2e-14 2e-14 Re hat 3e-14 2e-14 3e-14 3e-15 1e-14 1e-14 3e-14 7e-15 3e-14 3e-14 ɛ f 5e-8 4e-8 1e-7 2e-7 2e-8 2e-7 4e-7 1e-7 1e-6 4e-7 δ f 4e-8 3e-8 1e-7 9e-8 2e-8 1e-7 3e-7 1e-7 6e-7 3e-7 ɛ 2e-16 5e-16 3e-16 3e-16 3e-16 7e-16 3e-16 2e-16 2e-15 5e-16 Θ 2e-1 7e-1 3e-1 2e-1 4e-1 3e-1 2e-1 7e-2 2e-1 2e-1 ρ TABLE 74 (α β) = ( ): eigenvalue errors errors of deflating subspaces and Θ ρ Eu max 1e-12 2e-12 9e-12 3e-11 3e-11 3e-11 52e-11 9e-11 1e-1 2e-1 Es max 2e-11 2e-12 7e-12 5e-12 2e-11 3e-11 7e-11 1e-1 8e-11 1e-1 Eh max 2e-11 3e-11 3e-11 7e-11 3e-1 1e-1 6e-11 1e-1 9e-11 1e-1 Eu min 5e-16 1e-15 2e-15 1e-13 4e-15 2e-15 9e-14 1e-14 3e-13 1e-15 Es min 2e-15 1e-14 9e-16 2e-13 5e-15 3e-15 4e-14 4e-15 2e-13 6e-15 Eh min 6e-15 9e-14 7e-15 3e-13 3e-14 1e-14 7e-14 5e-15 7e-13 1e-14 Eu GM 3e-14 4e-14 1e-13 2e-12 4e-13 3e-13 2e-12 1e-12 5e-12 4e-13 Es GM 2e-13 1e-13 8e-14 9e-13 3e-13 3e-13 2e-12 7e-13 4e-12 8e-13 Eh GM 3e-13 2e-12 8e-13 5e-12 2e-12 1e-12 2e-12 8e-13 8e-12 1e-12 Re hat 4e-6 6e-6 2e-5 1e-5 6e-5 2e-5 4e-6 2e-5 2e-5 2e-5 ɛ f 2e-4 4e-4 1e-3 4e-4 2e-3 5e-4 2e-3 2e-4 1e-4 2e-4 δ f 2e-4 4e-4 1e-3 4e-4 2e-3 5e-4 2e-3 2e-4 1e-4 2e-4 ɛ 6e-12 2e-12 2e-11 9e-12 3e-11 2e-11 1e-11 3e-11 4e-11 3e-11 Θ 3e-1 2e-1 5e-1 5e-2 1e-1 4e-1 7e-2 5e-2 2e-1 7e-2 ρ EXAMPLE 4 For skew-symmetric/symmetric pencils of the form X T (λn M)X where 1 β 1 N = 1 1 M = β 1 β α α 1 1 β and X a randomly generated nonsingular matrix there exist two 2 2 Jordan blocks one for the eigenvalue β and another for β In this case it is expected that the computed finite eigenvalues have big errors due to the Jordan blocks and when β becomes smaller then the errors will get larger Our numerical results presented in Tables 79 and 71 confirm these expectations for two sets of (α β) All the numerical tests confirm our expectations obtained from the error analysis and illustrate that surprisingly the Schur complement approach and the approach via orthogonal

21 STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 21 TABLE 75 (α β) = (1 3 1): eigenvalue errors errors of deflating subspaces and Θ ρ Eu max 2e-15 2e-14 2e-14 2e-14 2e-14 2e-14 3e-14 5e-14 6e-14 7e-12 Es max 2e-15 3e-14 3e-14 9e-16 2e-14 2e-14 4e-14 9e-14 7e-14 3e-12 Eh max 2e-14 6e-14 6e-14 5e-14 2e-13 2e-14 9e-14 6e-14 2e-13 6e-12 Eu min 2e-15 2e-14 2e-14 2e-14 2e-14 2e-15 9e-15 2e-14 2e-14 2e-14 Es min 9e-16 3e-14 6e-15 9e-16 2e-14 7e-15 2e-14 2e-14 2e-15 5e-14 Eh min 2e-15 8e-15 2e-14 5e-16 1e-14 5e-16 7e-15 6e-14 1e-14 Eu GM 2e-15 2e-14 2e-14 2e-14 2e-14 6e-15 2e-14 3e-14 3e-14 4e-13 Es GM 2e-15 3e-14 2e-14 9e-16 2e-14 9e-15 3e-14 4e-14 41e-14 4e-13 Eh GM 5e-15 3e-14 3e-14 8e-15 4e-14 5e-15 3e-14 6e-14 4e-14 ɛ f 8e-12 9e-11 9e-12 2e-11 4e-11 4e-12 5e-12 3e-11 9e-12 2e-1 δ f 4e-12 4e-11 9e-12 1e-11 2e-11 4e-12 4e-12 2e-11 6e-12 2e-1 ɛ 4e-16 5e-16 6e-16 3e-16 4e-16 4e-16 2e-16 5e-16 3e-16 5e-16 Θ 3e-1 3e-1 5e-1 2e-1 3e-1 3e-1 3e-1 5e-1 2e-1 3e-2 ρ TABLE 76 (α β) = ( ): eigenvalue errors errors of deflating subspaces and Θ ρ Eu max 9e-5 2e-4 2e-4 3e-4 4e-4 4e-4 5e-4 7e-4 3e-3 2e-2 Es max 5e-5 5e-4 5e-5 2e-4 3e-4 3e-4 6e-4 2e-4 2e-3 6e-3 Eh max 2e-4 3e-4 2e-4 6e-3 4e-4 2e-3 2e-3 2e-3 4e-3 3e-2 Eu min 9e-5 2e-4 5e-6 7e-5 4e-4 5e-5 5e-4 5e-5 2e-5 2e-4 Es min 5e-5 4e-5 5e-5 2e-4 3e-4 5e-5 6e-4 2e-4 5e-5 2e-4 Eh min 2e-4 3e-4 9e-6 2e-4 4e-4 2e-4 3e-5 4e-5 7e-4 9e-4 Eu GM 9e-5 2e-4 3e-5 2e-4 4e-4 2e-4 5e-4 2e-4 2e-4 2e-3 Es GM 5e-5 2e-4 5e-5 2e-4 3e-4 2e-4 6e-4 2e-4 3e-4 8e-4 Eh GM 2e-4 3e-4 5e-5 9e-4 4e-4 4e-4 3e-4 2e-4 2e-3 5e-3 ɛ f 2e-12 3e-11 5e-12 4e-11 2e-11 2e-11 5e-11 6e-12 2e-11 2e-1 δ f 2e-13 2e-11 4e-12 2e-11 2e-11 2e-11 4e-11 5e-12 1e-11 7e-11 ɛ 3e-16 4e-16 3e-16 4e-16 4e-16 3e-16 3e-16 5e-16 4e-16 7e-16 Θ 5e-1 5e-1 3e-1 4e-1 2e-1 3e-1 7e-2 3e-1 2e-1 2e-2 ρ TABLE 77 (α β) = (1 7 1): eigenvalue errors errors of deflating subspaces and Θ ρ Eu max 9e-15 1e-14 1e-14 2e-14 3e-14 3e-14 5e-14 5e-14 2e-13 4e-13 Es max 2e-14 9e-15 3e-14 2e-14 3e-14 5e-14 3e-14 2e-13 2e-14 4e-13 Eh max 2e-14 2e-14 3e-13 3e-13 2e-13 1e-14 6e-14 2e-13 3e-13 4e-13 Eu min 6e-15 1e-14 7e-15 2e-14 3e-14 3e-14 4e-15 5e-14 4e-15 8e-15 Es min 2e-15 3e-15 6e-15 2e-14 3e-14 3e-15 2e-15 4e-15 2e-14 7e-15 Eh min 9e-15 7e-15 3e-14 6e-15 3e-14 7e-15 3e-15 2e-13 2e-15 7e-15 Eu GM 7e-15 1e-14 9e-15 2e-14 3e-14 3e-14 2e-14 5e-14 3e-14 6e-14 Es GM 5e-15 5e-15 2e-14 2e-14 3e-14 2e-14 8e-15 3e-14 2e-14 5e-14 Eh GM 2e-14 2e-14 6e-14 5e-14 7e-14 8e-15 2e-14 2e-13 3e-14 5e-14 ɛ f 9e-8 5e-7 9e-8 2e-7 9e-7 2e-7 5e-7 2e-8 2e-7 7e-7 δ f 7e-8 2e-7 9e-8 9e-8 3e-7 1e-7 5e-7 8e-9 5e-8 7e-7 ɛ 5e-16 3e-16 4e-16 7e-16 4e-16 3e-16 3e-16 4e-16 3e-16 6e-16 Θ 3e-1 5e-1 2e-1 7e-1 3e-1 7e-2 3e-1 7e-2 5e-1 2e-1 ρ 5e

22 22 V MEHRMANN AND H XU TABLE 78 (α β) = ( ): eigenvalue errors errors of deflating subspaces and Θ ρ Eu max 3e-5 3e-5 5e-5 2e-4 3e-4 4e-4 7e-4 3e-3 1e-1 2e-1 Es max 8e-6 3e-5 6e-5 2e-4 6e-4 4e-4 8e-4 2e-3 3e-2 3e-1 Eh max 5e-4 4e-5 4e-3 8e-4 2e-3 4e-4 3e-3 2e-3 8e-2 2e+ Eu min 2e-6 3e-5 5e-5 2e-4 1e-4 4e-4 7e-4 3e-5 5e-5 9e-5 Es min 8e-6 3e-5 6e-5 2e-4 7e-5 4e-4 8e-4 6e-5 4e-5 2e-3 Eh min 2e-5 4e-5 2e-4 2e-4 3e-4 4e-4 7e-4 4e-4 9e-5 2e-4 Eu GM 6e-6 3e-5 5e-5 2e-4 2e-4 4e-4 7e-4 3e-4 3e-3 4e-3 Es GM 8e-6 3e-5 6e-5 2e-4 2e-4 4e-4 8e-4 3e-4 2e-3 2e-2 Eh GM 9e-5 4e-5 8e-4 4e-4 7e-4 4e-4 2e-3 9e-4 3e-3 2e-2 ɛ f 4e-8 2e-8 5e-7 5e-7 2e-7 7e-7 2e-7 6e-7 9e-7 2e-6 δ f 3e-8 3e-8 4e-7 5e-7 3e-7 8e-7 2e-7 3e-7 4e-7 2e-6 ɛ 2e-16 1e-15 3e-16 7e-16 4e-16 7e-16 3e-16 5e-16 6e-16 3e-16 Θ 2e-1 5e-1 6e-1 2e-1 1e-1 8e-2 7e-2 6e-2 2e-2 5e-3 ρ TABLE 79 (α β) = (1 5 1): eigenvalue errors errors of deflating subspaces and Θ ρ Eu max 8e-8 8e-8 9e-8 1e-7 2e-7 2e-7 2e-7 2e-7 2e-7 1e-6 Es max 2e-7 4e-8 7e-8 6e-8 1e-7 2e-7 5e-8 2e-7 2e-7 8e-7 Eh max 2e-7 8e-8 2e-7 2e-7 2e-7 4e-7 3e-7 4e-7 6e-7 2e-6 Eu min 8e-8 8e-8 9e-8 1e-7 2e-7 2e-7 2e-7 2e-7 2e-7 1e-6 Es min 2e-7 4e-8 7e-8 6e-8 1e-7 2e-7 5e-8 2e-7 2e-7 8e-7 Eh min 8e-8 7e-8 2e-7 4e-8 2e-7 3e-7 3e-7 2e-7 5e-7 1e-6 Eu GM 8e-8 8e-8 9e-8 1e-7 2e-7 2e-7 2e-7 2e-7 2e-7 1e-6 Es GM 2e-7 4e-8 7e-8 6e-8 1e-7 2e-7 5e-8 2e-7 2e-7 8e-7 Eh GM 2e-7 7e-8 2e-7 7e-8 2e-7 3e-7 3e-7 3e-7 5e-7 2e-6 ɛ f 2e-9 2e-9 3e-9 8e-1 3e-1 8e-11 7e-9 3e-1 4e-9 3e-9 δ f 1e-9 3e-9 3e-9 9e-1 4e-1 2e-1 2e-9 3e-1 2e-9 3e-9 ɛ 5e-16 4e-16 4e-16 2e-15 1e-15 3e-16 5e-16 6e-16 3e-16 8e-16 Θ 3e-1 4e-1 4e-1 6e-1 2e-1 3e-1 2e-1 5e-1 1e-1 2e-1 ρ TABLE 71 (α β) = ( ): eigenvalue errors errors of deflating subspaces and Θ ρ Eu max 5e-2 6e-2 7e-2 8e-2 1e-1 1e-1 2e-1 3e-1 4e-1 5e-1 Es max 5e-2 5e-2 8e-2 8e-2 8e-2 2e-1 7e-2 3e-1 2e-1 6e-1 Eh max 5e-2 6e-2 3e-1 3e-1 2e-1 9e-2 2e-1 6e-1 3e-1 1e+ Eu min 5e-2 6e-2 7e-2 8e-2 1e-1 1e-1 2e-1 3e-1 4e-1 5e-1 Es min 5e-2 5e-2 8e-2 8e-2 8e-2 2e-1 7e-2 3e-1 2e-1 5e-1 Eh min 2e-2 3e-2 2e-1 9e-2 5e-2 8e-2 2e-1 3e-1 3e-2 2e-1 Eu GM 5e-2 6e-2 7e-2 8e-2 1e-1 1e-1 2e-1 3e-1 4e-1 5e-1 Es GM 5e-2 5e-2 8e-2 8e-2 8e-2 2e-1 7e-2 3e-1 2e-1 6e-1 Eh GM 3e-2 5e-2 3e-1 2e-1 8e-2 9e-2 2e-1 5e-1 9e-2 4e-1 ɛ f 3e-1 9e-1 1e-9 2e-9 3e-9 2e-1 1e-9 3e-9 5e-9 3e-8 δ f 3e-1 8e-1 4e-1 2e-9 2e-9 3e-1 6e-1 3e-9 2e-9 3e-8 ɛ 4e-16 7e-16 5e-16 6e-16 3e-16 9e-16 2e-16 4e-16 7e-16 5e-16 Θ 4e-1 5e-1 2e-1 2e-1 4e-1 3e-1 6e-1 1e-1 6e-2 3e-2 ρ

23 STRUCTURE PRESERVING DEFLATION OF INFINITE EIGENVALUES 23 structure preserving transformations deliver the same accuracy despite possible ill-conditioning The unstructured orthogonal transformation however shows the expected problems with purely imaginary eigenvalues 8 Conclusions We have presented and analyzed several methods to deflate the infinite eigenvalue part in a structured regular pencil of index at most one We have shown via a careful error analysis that it is possible to do this deflation via a structure preserving real orthogonal or unitary transformation as well as with a Schur complement approach Both methods yield similar results in the perturbation and error analysis and this is confirmed in the numerical tests This is surprising and counterintuitive to the general wisdom that unitary transformations typically perform better than transformations with potentially unbounded transformations Nonetheless we suggest to use the unitary structure preserving transformations since they are very much in line with the staircase algorithm Both methods are definitely preferable to the non-structured unitary transformation Acknowledgments This work was started while the first author visited the second and completed while the second author was visiting the first The authors thank both host institutions for their hospitality and generosity The authors also thank the referees for their valuable comments and suggestions REFERENCES 1 P BENNER R BYERS V MEHRMANN AND H XU A robust numerical method for the γ-iteration in H control Linear Algebra Appl 425 (27) pp P BENNER V SIMA AND M VOIGT FORTRAN 77 subroutines for the solutions of skew- Hamiltonian/Hamiltonian eigenproblems part I: algorithms and applications Preprint MPIMD/13-11 Max Planck Institute Magdeburg FORTRAN 77 subroutines for the solutions of skew-hamiltonian/hamiltonian eigenproblems part II: implementation and numerical results Preprint MPIMD/13-12 Max Planck Institute Magdeburg T BRÜLL AND V MEHRMANN STCSSP: A FORTRAN 77 routine to compute a structured staircase form for a (skew-)symmetric/(skew-)symmetric matrix pencil Preprint Institut für Mathematik Technical University Berlin Berlin 27 5 R BYERS V MEHRMANN AND H XU A structured staircase algorithm for skew-symmetric/symmetric pencils Electron Trans Numer Anal 26 (27) pp 1 33 /vol2627/pp1-33dir 6 J DEMMEL AND B KÅGSTRÖM Stably computing the Kronecker structure and reducing subspaces of singular pencils A λb for uncertain data in Large Scale Eigenvalue Problems (Oberlech 1985) J Cullum and R A Willoughby eds vol 127 of North-Holland Math Stud North-Holland Amsterdam 1986 pp P GAHINET AND A J LAUB Numerically reliable computation of optimal performance in singular H control SIAM J Control Optim 35 (1997) pp G H GOLUB AND C F VAN LOAN Matrix Computations 3rd ed Johns Hopkins University Press Baltimore P KUNKEL V MEHRMANN AND L SCHOLZ Self-adjoint differential-algebraic equations Math Control Signals Systems 26 (214) pp D S MACKEY N MACKEY C MEHL AND V MEHRMANN Structured polynomial eigenvalue problems: good vibrations from good linearizations SIAM J Matrix Anal Appl 28 (26) pp Möbius transformations of matrix polynomials Linear Algebra Appl in press doi: 1116/jlaa V MEHRMANN The Autonomous Linear Quadratic Control Problem Theory and Numerical Solution Springer Heidelberg V MEHRMANN AND D WATKINS Structure-preserving methods for computing eigenpairs of large sparse skew-hamiltonian/hamiltonian pencils SIAM J Sci Comput 22 (2) pp V MEHRMANN AND H XU Numerical methods in control J Comput Appl Math 123 (2) pp P H PETKOV N D CHRISTOV AND M M KONSTANTINOV Computational Methods for Linear Control Systems Prentice-Hall Int Hertfordshire C SCHRÖDER Palindromic and Even Eigenvalue Problems Analysis and Numerical Methods PhD Thesis Department of Mathematics Technical University Berlin Berlin 28

24 24 V MEHRMANN AND H XU 17 V SIMA Algorithms for Linear-Quadratic Optimization Marcel Dekker New York R C THOMPSON Pencils of complex and real symmetric and skew matrices Linear Algebra Appl 147 (1991) pp P VAN DOOREN A generalized eigenvalue approach for solving Riccati equations SIAM J Sci Statist Comput 2 (1981) pp K ZHOU J C DOYLE AND K GLOVER Robust and Optimal Control Prentice-Hall Englewood Cliffs 1996