FORTRAN 77 Subroutines for the Solution of Skew-Hamiltonian/Hamiltonian Eigenproblems Part I: Algorithms and Applications 1

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1 SLICOT Working Note FORTRAN 77 Subroutines for the Solution of Skew-Hamiltonian/Hamiltonian Eigenproblems Part I: Algorithms and Applications 1 Peter Benner 2 Vasile Sima 3 Matthias Voigt 4 August This document presents research results obtained during the development of the SLICOT (Subroutine Library in Systems and Control Theory) software. The SLICOT Library and the related CACSD tools based on SLICOT were partially developed within the Numerics in Control Network (NICONET) funded by the European Community BRITE-EURAM III RTD Thematic Networks Programme (contract number BRRT-CT ) see SLICOT can be used free of charge by academic users see for solving analysis and synthesis problems of modern and robust control. This report is available from 2 Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany; benner@mpi-magdeburg.mpg.de 3 National Institute for Research & Development in Informatics Bucharest 1 Romania; vsima@ici.ro 4 Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany; voigtm@mpi-magdeburg.mpg.de

2 Abstract Skew-Hamiltonian/Hamiltonian matrix pencils λs H appear in many applications including linear quadratic optimal control problems H -optimization certain multi-body systems and many other areas in applied mathematics physics and chemistry. In these applications it is necessary to compute certain eigenvalues and/or corresponding deating subspaces of these matrix pencils. Recently developed methods exploit and preserve the skew-hamiltonian/hamiltonian structure and hence increase reliability accuracy and performance of the computations. In this paper we describe the corresponding algorithms whose implementations have been included in the Subroutine Library in Control Theory (SLICOT). Furthermore we address some of their applications. We describe variants for real and complex problems with versions for factored and unfactored matrices S. Keywords: Deating subspaces eigenvalue reordering generalized eigenvalues generalized Schur form skew-hamiltonian/hamiltonian matrix pencil software structure-preservation.

3 1 Introduction In this paper we discuss algorithms for the solution of generalized eigenvalue problems with skew- Hamiltonian/Hamiltonian structure. We are interested in the computation of certain eigenvalues and corresponding deating subspaces. We have to deal with the following algebraic structures [4]. 0 In Denition 1.1. Let J := where I I n 0 n is the n n identity matrix. For brevity of notation we do not indicate the dimension with the matrix J and use it for all possible values of n. (i) A matrix H C 2n 2n is Hamiltonian if (HJ ) H = HJ. The Lie algebra of Hamiltonian matrices in C 2n 2n is denoted by H 2n. (ii) A matrix S C 2n 2n is skew-hamiltonian if (SJ ) H = SJ. The Jordan algebra of skew- Hamiltonian matrices in C 2n 2n is denoted by SH 2n. (iii) A matrix pencil λs H C 2n 2n is skew-hamiltonian/hamiltonian if S SH 2n and H H 2n. (iv) A matrix S C 2n 2n is symplectic if SJ S H = J. The Lie group of symplectic matrices in C 2n 2n is denoted by S 2n. (v) A matrix U C 2n 2n is unitary symplectic if UJ U H = J and UU H = I 2n. The compact Lie group of unitary symplectic matrices in C 2n 2n is denoted by US 2n. Note that a similar denition can be given for real matrices. As a convention all following considerations also hold for real skew-hamiltonian/hamiltonian matrix pencils. Then all matrices H must be replaced by T all (skew-)hermitian matrices become (skew-)symmetric and unitary matrices become orthogonal. More signicant differences to the complex case are explicitly mentioned. Skew-Hamiltonian/Hamiltonian matrix pencils satisfy certain properties which [ we will ] briey [ state. ] A D B F Every skew-hamiltonian/hamiltonian matrix pencil can be written as λs H = λ E A H G B H with skew-hermitian matrices D E and Hermitian matrices F G. If λ is a (generalized) eigenvalue of λs H so is also λ. In other words eigenvalues which are not purely imaginary occur in pairs. For real skew-hamiltonian/hamiltonian matrix pencils we also have a pairing of complex conjugate eigenvalues i.e. if λ is an eigenvalue of λs H so are also λ λ λ. This leads to eigenvalue pairs (λ λ) if λ is purely real or purely imaginary or otherwise to eigenvalue quadruples ( λ λ λ λ ). The structure of skew-hamiltonian/hamiltonian matrix pencils is preserved under J -congruence transformations that is λ S H := J P H J T (λs H)P with nonsingular P is again skew-hamiltonian/hamiltonian. If we choose P unitary we additionally preserve the condition of the problem. In this way there is hope that we can choose a unitary J -congruence transformation to transform λs H into a condensed form which reveals its eigenvalues and deating subspaces. A suitable candidate for this condensed form is the structured Schur form i.e. we compute a unitary matrix Q such that J Q H J T S11 S (λs H)Q = λ H11 H 0 S11 H 0 H11 H with the subpencil λs 11 H 11 in generalized Schur form where S 11 is upper triangular H 11 is upper triangular (upper quasi-triangular in the real case) S is skew-hermitian and H is Hermitian. However a structured Schur form does not necessarily exist. Conditions for the existence are proven in [17 18] for the complex case or in [26] for the real case. This problem can be circumvented by embedding λs H into a skew-hamiltonian/hamiltonian matrix pencil of double dimension in an appropriate way as explained in Section 3. Throughout this paper we denote by Λ (S H) Λ 0 (S H) Λ + (S H) the set of nite eigenvalues of λs H with negative zero and positive real parts respectively. The set of innite eigenvalues is denoted by Λ (S H). Multiple eigenvalues are repeated in Λ (S H) Λ 0 (S H) Λ + (S H) and 1

4 Λ (S H) according to their algebraic multiplicity. The set of all eigenvalues counted according to multiplicity is Λ(S H). Similarly we denote by Def (S H) Def 0 (S H) Def + (S H) and Def (S H) the right deating subspaces corresponding to Λ (S H) Λ 0 (S H) Λ + (S H) and Λ (S H) respectively. 2 Applications 2.1 Linear-Quadratic Optimal Control First we consider the continuous-time innite horizon linear-quadratic optimal control problem: choose a control function u(t) to minimize the cost functional H x(t) Q S x(t) S c := u(t) S H dt (1) R u(t) subject to the linear time-invariant descriptor system t 0 Eẋ(t) = Ax(t) + Bu(t) x(t 0 ) = x 0. (2) Here u(t) C m is control input vector x(t) C n is the descriptor vector and E A C n n B C n m Q = Q H C n n R = R H C m m S C n m. For well-posedness the (m + n) (m + n) weighting matrix Q S R = S H R must be Hermitian and positive semidenite. Typically in addition to minimizing (1) the control u(t) must make x(t) asymptotically stable. Under some conditions the application of the maximum principle [19 22] yields as a necessary condition that the control u satises the two-point boundary value problem of Euler-Lagrange equations ẋ(t) x(t) E c µ(t) = A c µ(t) x(t 0 ) = x 0 lim E H µ(t) = 0 (3) t u(t) u(t) with the matrix pencil E 0 0 A 0 B λe c A c = λ 0 E H 0 Q A H S S H B H R Assuming that the matrix R is nonsingular we can substitute u(t) = R ( 1 S H x(t) + B H µ(t) ) and system (3) simplies to [ẋ(t) ] x(t) S = H x(t µ(t) µ(t) 0 ) = x 0 lim E H µ(t) = 0 t with the skew-hamiltonian/hamiltonian matrix pencil [ E 0 A BR 1 S H BR 1 B H ] λs H = λ 0 E H SR 1 S H Q ( A BR 1 S H) H. (4) The generalized algebraic Riccati equation associated to the skew-hamiltonian/hamiltonian matrix pencil is given by [14] 0 = Q SR 1 S H + X H ( A BR 1 S H) + ( A BR 1 S H) H X X H ( BR 1 B H) X E H X = X H E. (5) 2

5 Under certain conditions the optimal control u (t) that stabilizes the descriptor system (2) can be constructed by using a stabilizing solution X of (5). The matrix X can be obtained by computing the deating subspace of (4) associated to the nite eigenvalues with negative real parts and to some purely imaginary and innite eigenvalues. Note that when the matrix R is singular the problem becomes much more involved. Then one has to consider so-called (generalized) Lur'e equations instead of Riccati equations. However there is also a connection between Lur'e equations and skew-hamiltonian/hamiltonian and related even matrix pencils [24 25]. 2.2 H -Optimization Similar structures as in Subsection 2.1 occur in H -optimization [16]. Consider a descriptor system of the form Eẋ(t) = Ax(t) + B 1 w(t) + B 2 u(t) P : z(t) = C 1 x(t) + D 11 w(t) + D u(t) x(t 0 ) = x 0 (6) y(t) = C 2 x(t) + D 21 w(t) + D 22 u(t) where E A R n n B i R n mi C i R pi n and D ij R pi mj for i j = 1 2. In this system x(t) R n is the (generalized) state vector u(t) R m2 is the control input vector and w(t) R m1 is an exogenous input that may include noise linearization errors and unmodeled dynamics. The vector y(t) R p2 contains measured outputs while z(t) R p1 is a regulated output or an estimation error. The H control problem is usually formulated in the frequency domain. For this we need the space H p m which consists of all C p m -valued functions that are analytic and bounded in the open right half-plane C +. For F H p m the H -norm is dened by F H := sup s C + σ max (F (s)) where σ max (F (s)) denotes the maximal singular value of the matrix F (s). In robust control F H is used as a measure of the worst-case inuence of the disturbances w on the output z where in this case F is the transfer function mapping noise or disturbance inputs to error signals [27]. Solving the optimal H control problem is the task of designing a dynamic controller K : {Ê ˆx(t) = ˆx(t) + ˆBy(t) u(t) = Ĉ ˆx(t) + ˆDy(t) (7) with Ê Â RN N ˆB R N p 2 Ĉ R m2 N ˆD R m 2 p 2 such that the closed-loop system resulting from inserting (7) into (6) that is ) ) Eẋ(t) = (A + B 2 ˆDZ1 C 2 x(t) + B 2 Z 2 Ĉ ˆx(t) + (B 1 + B 2 ˆDZ1 D 21 w(t) Ê ˆx(t) = ˆBZ ) 1 C 2 x(t) + ( + ˆBZ1 D 22 Ĉ ˆx(t) + ˆBZ 1 D 21 w(t) (8) ) ) z(t) = (C 1 + D Z 2 ˆDC2 x(t) + D Z 2 Ĉ ˆx(t) + (D 11 + D ˆDZ1 D 21 w(t) with Z 1 = ( 1 ( I p2 D 22 ˆD) and Z2 = I m2 ˆDD ) 1 22 has the following properties: x(t) (i) System (8) is internally stable that is the solution of the system with w 0 is asymptotically ˆx(t) x(t) stable i.e. lim = 0. t ˆx(t) 3

6 (ii) The closed-loop transfer function T zw from w to z satises T zw H p1 m1 and is minimized in the H -norm. Closely related to the optimal H control problem is the modied optimal H control problem. For a given descriptor system of the form (6) we search the inmum value γ for which there exists an internally stabilizing dynamic controller of the form (7) such that the corresponding closed-loop system (8) satises T zw H p1 m1 with T zw H < γ. For the construction of optimal controllers one can make use of the following even matrix pencils (see [23] for a denition and related software) and λn H M H (γ) = λn J M J (γ) = 0 λe T A T 0 0 C T 1 λe A 0 B 1 B B T 1 γ 2 I m1 0 D T 11 0 B T D T C 1 0 D 11 D I p1 0 λe A 0 0 B 1 λe T A T 0 C1 T C2 T 0 0 C 1 γ 2 I p1 0 D 11 0 C D B1 T 0 D11 T D T I m1 (9) (10) which can be transformed to skew-hamiltonian/hamiltonian structure by using the method used in [3 26]. Using appropriate deating subspaces of the matrix pencils (9) and (10) it is possible to state conditions for the existence of an optimal H controller. Then we can check if these conditions are fullled for a given value of γ. Using a bisection scheme we can iteratively rene γ until a wanted accuracy is achieved (see [16 5] for details). Note that the transformation to skew-hamiltonian/hamiltonian structure is done in order to compute the deating subspaces in a structure-preserving manner which is still an open problem for even matrix pencils. Finally when a suboptimal value γ has been found one can compute the actual controller. The controller formulas are rather cumbersome and are therefore omitted. For details see [15]. 2.3 L -Norm Computation Finally we briey describe a method to compute the L -norm of an LTI system using skew-hamiltonian/hamiltonian matrix pencils [26 6 7]. This norm plays an important role in robust control or model order reduction (see [ ] and references therein). Consider a descriptor system Eẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) (11) with E A R n n B R n m C R p n D R p m and descriptor vector x(t) R n control vector u(t) R m and output vector y(t) R p. For such a system its transfer function is given by G(s) := C (se A) 1 B + D which directly maps inputs to outputs in the frequency domain [10]. We dene the space RL p m of all proper rational p m-matrix-valued transfer functions which are bounded on the imaginary axis. The natural norm of this space is the L -norm dened by G L := sup σ max (G(iω)). ω R 4

7 Consider the skew-hamiltonian/hamiltonian matrix pencils E 0 A BR λn M(γ) = λ 0 E T 1 D T C γbr 1 B T γc T S 1 C A T + C T DR 1 B T with the matrices R = D T D γ 2 I m and S = DD T γ 2 I p. It can be shown that if λe A has no purely imaginary eigenvalue and γ > min σ max (G(iω)) is not a singular value of D then G ω R L γ if and only if λn M(γ) has purely imaginary eigenvalues. In this way we can again use an iterative scheme to improve the value of γ until a wanted accuracy for the L -norm is achieved. () 3 Theory and Algorithm Description In this section we briey describe the theory behind the algorithms that we will use. We refer to [4 2] for a very detailed analysis of the algorithms. We consider complex and real problems separately since there are signicant dierences in the theory. We also distinguish the cases of unfactored and factored skew-hamiltonian matrices S. Note that the skew-hamiltonian matrices in (4) () and the skew- Hamiltonian matrices resulting from appropriate transformations of the skew-symmetric matrices in (9) (10) are block-diagonal and hence admit a factorization S = J Z H J T Z. (13) E 0 I 0 For example if S = 0 E H then Z = 0 E H. The factorization (13) can be understood as a Cholesky-like decomposition of S with respect to the indenite inner product x y := x H J y since J Z H J T is the adjoint of Z with respect to. We also say that a skew-hamiltonian matrix S is J -semidenite if it admits a factorization of the form (13). Hence in our implementation we distinguish the cases that the full matrix S or just its Cholesky factor Z is given. In all cases we apply an embedding strategy to the matrix pencil λs H to avoid the problem of non-existence of a structured Schur form. 3.1 The Complex Case Let λs H be a given complex skew-hamiltonian/hamiltonian matrix pencil with J -semidenite skew- Hamiltonian part S = J Z H J T Z. We split the skew-hamiltonian matrix ih =: N = N 1 + in 2 where N 1 is real skew-hamiltonian and N 2 is real Hamiltonian i.e. F1 G N 1 = 1 H 1 F1 T G 1 = G T 1 H 1 = H1 T F2 G N 2 = 2 H 2 F2 T G 2 = G T 2 H 2 = H2 T and F j G j H j R n n for j = 1 2. We dene the matrices I n I2n ii Y c = 2n P = 0 0 I n 0 2 I 2n ii 2n 0 I n 0 0 X c = Y c P. (14) I n By using the embedding B N := diag ( N N ) we obtain that B c N := X H c B N X c = F 1 F 2 G 1 G 2 F 2 F 1 G 2 G 1 H 1 H 2 H 2 H 1 F1 T F2 T F2 T F1 T 5 (15)

8 is a real 4n 4n skew-hamiltonian matrix. Similarly we dene [ Z 0 J Z H J T 0 B Z := B 0 Z T := 0 J Z H J T It can be shown that ] B S := S 0 = B 0 S T B Z. B c Z := X H c B Z X c B c T := X H c B T X c B c S := X H c B S X c (16) are all real. Hence λb c S B c N = X H c (λb S B N ) X c = X H c () λs N 0 0 λ S N X c is a real 4n 4n skew-hamiltonian/skew-hamiltonian matrix pencil. To compute the eigenvalues of this matrix pencil we can compute the structured decomposition of the following theorem [4]. Theorem 3.1. Let λs N be a real regular skew-hamiltonian/skew-hamiltonian matrix pencil with S = J Z T J T Z. Then there exist a real orthogonal matrix Q R 2n 2n and a real orthogonal symplectic matrix U R 2n 2n such that U T Z11 Z ZQ = 0 Z 22 (17) J Q T J T N11 N N Q = 0 N11 T where Z 11 and Z22 T are upper triangular N 11 is upper quasi triangular and N is skew-symmetric. Moreover J Q T J T Z T (λs N ) Q = λ 22 Z 11 Z22Z T ZZ T 22 N11 N 0 Z11Z T 22 0 N11 T S11 S =: λ N11 N 0 S11 T 0 N11 T (18) is a J -congruent skew-hamiltonian/skew-hamiltonian matrix pencil. Proof. See [4]. By dening B H = and using Theorem 3.1 we can compute factorizations B Z c : = U T BZQ c Z11 Z = 0 Z 22 H 0 0 H BH c = Xc H B H X c B c H : = J Q T J T B c HQ = J Q T J T ( ib c N ) Q = i B c N = in11 in 0 ( in 11 ) H where λ B S c B H c = J QT J T (λbs c Bc H ) Q are J -congruent complex skew-hamiltonian/hamiltonian matrix pencils and λ B S c B H c is in a structured quasi-triangular form. Then the structured Schur form can be obtained by further triangularizing the diagonal 2 2 blocks of λ B S c B H c via a J -congruence transformation. From the symmetry of the eigenvalues if follows that Λ (S H) = Λ ( ) Z22Z H 11 in 11. Now we can reorder the eigenvalues of λ B S c B H c to the top in order to compute the desired deating subspaces (corresponding to the eigenvalues with negative real parts). The following theorem makes statements about the deating subspaces [4]. 6

9 Theorem 3.2. Let λs H C 2n 2n be a skew-hamiltonian/hamiltonian matrix pencil with J -semidenite skew-hamiltonian matrix S = J Z H J T Z. Consider the extended matrices B Z = diag ( Z Z ) ) B T = diag (J Z H J T J Z H J T B S = B T B Z = diag ( S S ) B H = diag ( H H ). Let U V W be unitary matrices such that U H Z11 Z B Z V = =: R 0 Z Z 22 W H T11 T B T U = =: R 0 T T 22 W H H11 H B H V = =: R 0 H H 22 where Λ (B S B H ) Λ (T 11 Z 11 H 11 ) and Λ (T 11 Z 11 H 11 [ ) ] Λ + (B S B H ) =. Here Z 11 T 11 H 11 C m m V1. Suppose Λ (S H) contains p eigenvalues. If C V 4n m are the rst m columns of V 2 2p m 2n 2p then there are subspaces L 1 and L 2 such that range V 1 = Def (S H) + L 1 range V 2 = Def + (S H) + L 2 [ U1 L 1 Def 0 (S H) + Def (S H) L 2 Def 0 (S H) + Def (S H). ] W1 If Λ (T 11 Z 11 H 11 ) = Λ (B S B H ) and are the rst m columns of U W respectively then U 2 W 2 there exist unitary matrices Q U Q V Q W such that U 1 = [ P U 0 ] Q U U 2 = [ 0 P + ] U QU V 1 = [ P V 0 ] Q V V 2 = [ 0 P + ] V QV W 1 = [ P W 0] Q W W 2 = [ 0 P + W ] QW and the columns of P V and P + V form orthogonal bases of Def (S H) and Def + (S H) respectively. Moreover the matrices P U P + U P W and P + W have orthonormal columns and the following relations are satised ZP V = P U Z 11 J Z H J T P U = P W T 11 HP V = P W H 11 ZP + V = P + U Z 22 J Z H J T P + U = P + W T 22 HP + V = P + W H 22. Here Z kk T kk and H ( kk k = 1 2 satisfy Λ T11 Z11 H ) ( 11 = Λ T22 Z22 H ) 2 = Λ (S H). Proof. See [4]. So the algorithm for computing the stable deating subspaces of a complex skew-hamiltonian/hamiltonian matrix pencil λs H with S = J Z H J T Z is as follows [4]. ALGORITHM 1. Computation of stable deating subspaces of complex skew-hamiltonian/hamiltonian matrix pencils in factored form Input: Hamiltonian matrix H and the factor Z of S SH n. Output: Structured Schur form of the extended skew-hamiltonian/hamiltonian matrix pencil λbs c BH c eigenvalues of λs H orthonormal bases P V P U of the deating subspace Def (S H) and the companion subspace respectively as in Theorem

10 1: Set N = ih and determine the matrices B c Z B c N as in (16) and (15) respectively. Perform Algorithm 2 to compute the factorization ˆB Z c = U T BZQ c Z11 Z = 0 Z 22 ˆB N c = J Q T J T BN c N11 N Q = 0 N11 T where Q is real orthogonal U is real orthogonal symplectic Z 11 Z T 22 are upper triangular and N 11 is upper quasi triangular. 2: Apply the periodic QZ algorithm [9 13] to the 2 2 diagonal blocks of the matrix pencil λz22z H 11 N 11 to determine unitary matrices Q 1 Q 2 U such that U H Z 11Q 1 Q H 2 Z22U H Q H 2 N 11Q 1 are all upper triangular. Dene Û := diag (U U) ˆQ := diag (Q1 Q 2) and set B c Z = Û H ˆBc Z ˆQ Bc N = J ˆQ H J T ˆBc N ˆQ. 3: Use Algorithm 3 to determine a unitary matrix Q and a unitary symplectic matrix Ũ such that Z11 Ũ H Z Bc Z Q = 0 Z22 J Q ( H J T i B ) N c H11 H Q = 0 H11 H where Z 11 Z 22 H H 11 are upper triangular such that Λ (J ( B Z) c H J T Bc Z i B ) N c is contained in the spectrum of the 2p 2p leading principal subpencil of λ Z 22 H Z 11 H 11. 4: Set V = [ I 2n 0 ] I2p X cq ˆQ Q U = [ I 0 2n 0 ] I2p X cuûũ and compute P 0 V P + U orthogonal bases of range V and range U respectively using any numerically stable orthogonalization scheme. Next we briey discuss the algorithms which are used in Algorithm 1. ALGORITHM 2. Computation of a structured matrix factorization for real skew-hamiltonian/skew-hamiltonian matrix pencils in factored form Input: A real skew-hamiltonian matrix N R 2n 2n and the factor Z R 2n 2n of S. Output: A real orthogonal matrix Q a real orthogonal symplectic matrix U and the structured factorization (17). 1: Set Q = U = I 2n. By changing the elimination order in the classical RQ decomposition determine an orthogonal matrix Q 1 such that Z11 Z Z := ZQ 1 =: 0 Z 22 where Z 11 Z T 22 are upper triangular. Update N = J Q T 1 J T N Q 1 Q := QQ 1. 2: Compute an orthogonal matrix Q 1 and an orthogonal symplectic matrix U 1 such that Z : = U1 T Z11 Z ZQ 1 =: 0 Z 22 N : = J Q T 1 J T N11 N N Q 1 =: 0 N11 T where Z 11 Z T 22 are upper triangular and N 11 is upper Hessenberg. Update Q := QQ 1 and U := UU 1. This step is performed by using a sequence of orthogonal and orthogonal symplectic matrices to annihilate the elements in N in a specic order without destroying the structure of Z (see [4] for details). 8

11 3: Apply the periodic QZ algorithm [9 13] to the matrix pencil λz T 22Z 11 N 11 to determine orthogonal matrices Q 1 Q 2 U such that U T Z 11Q 1 Q T 2 Z T 22U are both upper triangular and Q T 2 N 11Q 1 is upper quasi triangular. Set U 1 := diag (U U) Q 1 := diag (Q 1 Q 2). Update Z := U T 1 ZQ 1 N := J Q T 1 J T N Q 1 Q := QQ 1 U := UU 1. After performing Algorithm 2 the eigenvalues of the complex skew-hamiltonian/hamiltonian matrix pencil λs H can be determined by the diagonal 1 1 and 2 2 blocks of the matrices Z 11 Z 22 and N 11. Next we describe the eigenvalue reordering technique to reorder the nite stable eigenvalues to the top of the matrix pencil which enables us to compute the corresponding deating subspaces. ALGORITHM 3. Eigenvalue reordering for complex skew-hamiltonian/hamiltonian matrix pencils in factored form Input: Regular 2n 2n complex skew-hamiltonian/hamiltonian matrix pencil λs H with S = J Z H J T Z Z = Z W H D H = 0 T 0 H H with upper triangular Z T H and H. Output: A unitary matrix Q a unitary symplectic matrix U and the transformed matrices U H ZQ J Q H J T HQ which have still the same triangular form as Z and H respectively but the eigenvalues in Λ (S H) are reordered such that they occur in the leading principal subpencil of J Q H J T (λs H) Q. 1: Set Q = U = I 2n. Reorder the eigenvalues in the subpencil λt H Z H. a) Determine unitary matrices Q 1 Q 2 Q 3 such that T H := Q H 3 T H Q 2 Z := Q H 2 ZQ 1 H := Q H 3 HQ 1 are still upper triangular but the m eigenvalues with negative real part are reordered to the top of λt H Z H. Set Q 1 := diag (Q 1 Q 3) U 1 := diag (Q 2 Q 2) and update Q := QQ 1 U := UU 1. b) Determine unitary matrices Q 1 Q 2 Q 3 such that T H := Q H 3 T H Q 2 Z := Q H 2 ZQ 1 H := Q H 3 HQ 1 are still upper triangular but the m + eigenvalues with positive real part are reordered to the bottom of λt H Z H. Set Q 1 := diag (Q 1 Q 3) U 1 := diag (Q 2 Q 2) and update Q := QQ 1 U := UU 1. 2: Reorder the remaining n m eigenvalues with negative real parts which are now in the bottom right subpencil of λs H. Determine a unitary matrix Q 1 and a unitary symplectic matrix U 1 such that the eigenvalues of top left subpencil of λs H with positive real parts and those of the bottom right subpencil of λs H with negative real parts are interchanged. Update Q := QQ 1 U := UU 1. If the matrix S is not given in factored form we can use the following algorithm for the computation of the deating subspaces [4]. ALGORITHM 4. Computation of stable deating subspaces of complex skew-hamiltonian/hamiltonian matrix pencils in unfactored form Input: Complex skew-hamiltonian/hamiltonian matrix pencil λs H. Output: Structured Schur form of the extended skew-hamiltonian/hamiltonian matrix pencil λbs c BH c eigenvalues of λs H orthonormal basis P V of the deating subspace Def (S H) as in Theorem : Set N = ih and determine the matrices BS c BN c as in (16) and (15) respectively. Perform Algorithm 5 to compute the factorization ˆB S c = J Q T J T BSQ c S11 S = 0 S11 T ˆB N c = J Q T J T BN c N11 N Q = 0 N11 T where Q is real orthogonal S 11 is upper triangular and N 11 is upper quasi triangular. 9

12 2: Apply the QZ algorithm [11] to the 2 2 diagonal blocks of the matrix pencil λs 11 N 11 to determine unitary matrices Q 1 Q 2 such that Q H 2 S 11Q 1 Q H 2 N 11Q 1 are both upper triangular. Dene ˆQ := diag (Q 1 Q 2) and set B c S = J ˆQ H J T ˆBc S ˆQ Bc N = J ˆQ H J T ˆBc N ˆQ. 3: Use Algorithm 6 to determine a unitary matrix Q such that J Q S11 H J T S Bc S Q = 0 SH 11 J Q ( H J T i B ) N c H11 H Q = 0 H11 H where S ( 11 H 11 are upper triangular such that Λ Bc S i B ) N c is contained in the spectrum of the 2p 2p leading principal subpencil of λ S 11 H 11. 4: Set V = [ I 2n 0 ] I2p X cq ˆQ Q and compute P 0 V an orthogonal basis of range V using any numerically stable orthogonalization scheme. Now we present the algorithm for the computation of the structured matrix factorization for complex matrix pencils in unfactored form. ALGORITHM 5. Computation of a structured matrix factorization for real skew-hamiltonian/skew-hamiltonian matrix pencils in unfactored form Input: A real skew-hamiltonian/skew-hamiltonian matrix pencil λs N. Output: A real orthogonal matrix Q and the structured factorization (18). 1: Set Q = I 2n. Reduce S to skew-hamiltonian triangular form i.e. determine an orthogonal matrix Q 1 such that S := J Q T 1 J T S11 S SQ 1 = 0 S11 T with an upper triangular matrix S 11. Update N := J Q T 1 J T N Q 1 Q := QQ 1. This step is performed by applying a sequence of Householder reections and Givens rotations in a specic order see [4] for details. 2: Reduce N to skew-hamiltonian Hessenberg form. Determine an orthogonal matrix Q 1 such that S : = J Q T 1 J T S11 S SQ 1 = 0 S11 T N : = J Q T 1 J T N11 N N Q 1 = 0 N11 T where S 11 is upper triangular and N 11 is upper Hessenberg. Update Q := QQ 1. This step is performed by applying an appropriate sequence of Givens rotations to annihilate the elements in N in a specic order without destroying the structure of S for details see [4]. 3: Apply the QZ algorithm to the matrix pencil λs 11 N 11 to determine orthogonal matrices Q 1 and Q 2 such that Q T 2 S 11Q 1 is upper triangular and Q T 2 N 11Q 1 is upper quasi triangular. Set Q 1 := diag (Q 1 Q 2) and update S := J Q T 1 J T SQ 1 N := J Q T 1 J T N Q 1 Q := QQ 1. Again similar to the factored case the eigenvalues are determined by the diagonal 1 1 and 2 2 blocks of S 11 and N 11. Also the following eigenvalue reordering routine is similar to the one of the factored case. 10

13 ALGORITHM 6. Eigenvalue reordering for complex skew-hamiltonian/hamiltonian matrix pencils in unfactored form Input: Regular 2n 2n complex skew-hamiltonian/hamiltonian matrix pencil λs H of the form S = H D H = 0 H H with upper triangular S H. S W 0 S H Output: A unitary matrix Q and the transformed matrices J Q H J T SQ J Q H J T HQ which have still the same triangular form as S and H respectively but the eigenvalues in Λ (S H) are reordered such that they occur in the leading principal subpencil of J Q H J T (λs H) Q. 1: Set Q = I 2n. Reorder the eigenvalues in the subpencil λs H. a) Determine unitary matrices Q 1 Q 2 such that S := Q H 2 SQ 1 H := Q H 2 HQ 1 are still upper triangular but the m eigenvalues with negative real part are reordered to the top of λs H. Set Q 1 := diag (Q 1 Q 2) and update Q := QQ 1. b) Determine unitary matrices Q 1 Q 2 such that S := Q H 2 SQ 1 H := Q H 2 HQ 1 are still upper triangular but the m + eigenvalues with positive real part are reordered to the bottom of λs H. Set Q 1 := diag (Q 1 Q 2) and update Q := QQ 1. 2: Reorder the remaining n m eigenvalues with negative real parts which are now in the bottom right subpencil of λs H. Determine a unitary matrix Q 1 such that the eigenvalues of top left subpencil of λs H with positive real parts and those of the bottom right subpencil of λs H with negative real parts are interchanged. Update Q := QQ The Real Case We also briey recall the theory for the real case which has some signicant dierences compared to the complex case. For a very detailed description we refer to [2]. Let λs H be a real skew- Hamiltonian/Hamiltonian matrix pencil with J -semidenite skew-hamiltonian part S = J Z T J T Z where Z = Z11 Z H = Z 21 Z 22 [ F G H F T ]. We introduce the orthogonal matrices 2 I2n I Y r = 2n X 2 I 2n I r = Y r P 2n with P as in (14). Now we dene the double-sized matrices Z 0 B Z : = 0 Z J Z B T : = T J T 0 0 J Z T J T = S 0 B S : = = B 0 S T B Z H 0 B H : =. 0 H J 0 B T 0 J Z T J 0 0 J 11

14 Furthermore we dene Z 11 0 Z 0 BZ r : = Xr T B Z X r = 0 Z 11 0 Z Z 21 0 Z Z 21 0 Z 22 B r T : = X T r B T X r = J (B r Z) T J T BS r : = Xr T B S X r = J (BZ) r T J T BZ r B r H : = X T r B H X r = 0 F 0 G F 0 G 0 0 H 0 F T H 0 F T 0. It can be easily observed that the 4n 4n matrix pencil λbs r Br H is again real skew-hamiltonian/hamiltonian. For the computation of the eigenvalues of λs H we apply the following structured matrix factorization which is also often referred to as generalized symplectic URV decomposition [2]. Theorem 3.3. Let λs H be a real skew-hamiltonian/hamiltonian matrix pencil with S = J Z T J T Z. Then there exist orthogonal matrices Q 1 Q 2 and orthogonal symplectic matrices U 1 U 2 such that Q T ( 1 J Z T J T ) T11 T U 1 = 0 T 22 U2 T Z11 Z ZQ 2 = (19) 0 Z 22 Q T H11 H 1 HQ 2 = 0 H 22 with the formal matrix product T 1 11 H 11Z 1 11 Z T 22 HT 22T T 22 in real periodic Schur form [9 13] where T 11 Z 11 H 11 T T 22 Z T 22 are upper triangular and H T 22 is upper quasi triangular. Proof. The proof is constructive see [2]. By using Theorem 3.3 (with the same notation) we get the following factorization of the embedded matrix pencil λbs r Br H with factored matrix Br S. We can compute an orthogonal matrix Q 1 and an orthogonal symplectic matrix Ũ such that T22 T 0 T T 0 Ũ T BZ r Q = 0 Z 11 0 Z Z T11 T 0 =: Z 0 Z Z 22 (20) 0 H 11 0 H J Q T J T BH r Q = H22 T 0 H 0 H H 22 =: H 0 H T 0 0 H11 T 11 0 where Q J = P T Q1 J T 0 P 0 Q Ũ = U1 0 PT P. From the condensed form (20) we can immediately get the eigenvalues of λs H as 2 0 U 2 ( Λ(S H) = Λ ZT 22 Z11 H ) 11 = ±i Λ ( T11 1 H 11Z11 1 Z T 22 HT 22 T 22 T ). (21)

15 Note that all matrices of the product are upper triangular except H T 22 which is upper quasi triangular. Hence the eigenvalue information can be extracted directly from the diagonal 1 1 or 2 2 blocks of the main diagonals. Note that the nite simple purely imaginary eigenvalues of the initial matrix pencil correspond to the positive eigenvalues of the generalized matrix product. Hence these eigenvalues can be computed without any error in their real parts. This leads to a high robustness in algorithms which require these eigenvalues e.g. in the L -norm computation [6]. However if two purely imaginary eigenvalues are very close they might still be slightly perturbed from imaginary axis. This essentially depends on the Kronecker structure of a close-by skew-hamiltonian/hamiltonian matrix pencil with double purely imaginary eigenvalues. This problem is similar to the Hamiltonian matrix case see [21]. To compute the deating subspaces we are interested in it is necessary to compute the structured Schur form of the embedded matrix pencils λbs r Br H. This can be done by computing a nite number of similarity transformations to the subpencil λ Z 22 T Z 11 H 11 to put H 11 into upper quasi triangular form. That is we compute orthogonal matrices Q 3 Q 4 U 3 such that H 11 = Q T H 3 11 Q 4 Z 11 = U3 T Z 11 Q 4 Z 22 = U3 T Z 22 Q 3 where Z 11 Z22 T are upper triangular and H 11 is upper quasi triangular. By setting Q = Q Q4 0 0 Q 3 U = Ũ U3 0 Z 0 U = U T Z 3 Q 3 and H = Q T H 3 Q 3 we obtain the structured Schur form of 3 λb r S Br H as λ B r S B r H with B r S = J ( Br Z ) T J T Br Z and B Z r : = U T BZQ r Z11 Z = 0 Z 22 B H r : = J Q T J T BHQ r H11 H = 0 H11 T. Now we can reorder the eigenvalues of λ B S r B H r to the top in order to compute the desired deating subspaces which is similar to the complex case. Then for the deating subspaces we nd a similar result as Theorem 3.2 which we do not state here for brevity. If the matrix S is not given in factored form we need the following slightly modied version of Theorem 3.3 from [2]. Theorem 3.4. Let λs H be a real skew-hamiltonian/hamiltonian matrix pencil. Then there exist orthogonal matrices Q 1 Q 2 such that Q T 1 SJ Q 1 J T S11 S = 0 S11 T SH 2n J Q T 2 J T T11 T SQ 2 = 0 T11 T =: T SH 2n (22) Q T H11 H 1 HQ 2 = 0 H 22 with the formal matrix product S 1 11 H 11T 1 11 HT 22 in real periodic Schur form where S 11 T 11 H 11 are upper triangular and H T 22 is upper quasi triangular. Proof. The proof is done by construction see [2]. 13

16 Then we can compute an orthogonal matrix Q such that S 11 0 S 0 J Q T J T BS r Q = 0 T 11 0 T S S11 T 0 =: S 0 ST T11 T 0 H 11 0 H [ J Q T J T BH r Q = H22 T 0 H 0 H H 22 =: 0 0 H11 T 0 with Q J = P T Q1 J T 0 P. The spectrum of λs H is given by 0 Q 2 Λ(S H) = ±i Λ ( S11 1 H 11T11 1 ) HT 22 ] H 0 H 11 T which can be determined by evaluating the entries on the 1 1 and 2 2 diagonal blocks of the matrices only. To put the matrix pencil formed of the matrices in (23) into structured Schur form we have to triangularize λ S 11 H 11 i.e. we determine orthogonal matrices Q 3 and Q 4 such that S 11 = Q T 4 S 11 Q 3 H 11 = Q T 4 H 11 Q 3 are upper triangular and upper quasi triangular respectively. By setting Q = Q Q T S 4 Q 4 and H = Q T H 4 Q 4 we obtain the structured Schur form as B S r : = J Q T J T S11 S B S Q = 0 S11 T B H r : = J Q T J T H11 H B H Q = 0 H11 T. (23) Q3 0 S 0 Q = 4 By properly reordering the eigenvalues we can compute the desired deating subspaces as explained above. As for the complex case we give a brief description of the used algorithms for the real case from [2]. ALGORITHM 7. Computation of stable deating subspaces of real skew-hamiltonian/hamiltonian matrix pencil in factored form Input: Real Hamiltonian matrix H and the factor Z of S. Output: Structured Schur form of the extended skew-hamiltonian/hamiltonian matrix pencil λbs r BH r eigenvalues of λs H orthonormal bases P V P U of the deating subspace Def (S H) and the companion subspace respectively as in Theorem : Apply Algorithm 8 to the matrices Z J Z T J T and H and determine orthogonal matrices Q 1 Q 2 and orthogonal symplectic matrices U 1 U 2 such that ( ) Q T 1 J Z T J T T11 T U 1 = 0 T 22 U2 T Z11 Z ZQ 2 = 0 Z 22 Q T H11 H 1 HQ 2 = 0 H 22 with the formal matrix product T 1 11 H11Z 1 Z T 22 HT 22T T 22 in real periodic Schur form where T 11 Z 11 H 11 T22 T Z22 T are upper triangular and H22 T is upper quasi triangular. 14

17 T 2: Apply Algorithm 9 to determine orthogonal matrices Q 3 Q 4 U 3 such that Z 11 = U3 T T 22 0 Q 4 and 0 Z 11 T Z 22 = U3 T T Q 3 are upper triangular and H 11 = Q T H Z 22 H22 T Q 4 is upper quasi triangular. 0 3: Update Z := U3 T T T 0 Q 3 H := Q T 0 H 3 0 Z H T Q 3 0 and set B Z r Z11 Z H11 H = Br 0 Z H = 22 0 H11 T. ( Apply the real eigenvalue reordering method in Algorithm 10 to the pair Br Z B ) H r to determine an orthogonal matrix ˆQ and an orthogonal symplectic matrix Û such that Û T Br Z ˆQ J ˆQT J T Br H ˆQ are in structured triangular ( ) T form and Λ (J Br Z J T Br Z H) B r is contained in the leading 2p 2p principal subpencil of λz22z T 11 H 11. 4: Set V = [ I 2n 0 ] ( J Q1J T 0 Q3 0 Y r P ˆQ) I2p 0 Q 2 0 Q 4 0 U = [ I 2n 0 ] ( ) U1 0 U3 0 I2p Y r P Û 0 U 2 0 U 3 0 and compute P V P U orthogonal bases of range V and range U respectively using any numerically stable orthogonalization scheme. The next algorithm describes the computation of the generalized symplectic URV decomposition which can e.g. be used to compute the eigenvalues of a real skew-hamiltonian/hamiltonian matrix pencil in factored form. ALGORITHM 8. Generalized symplectic URV decomposition Input: A real 2n 2n matrix pencil λt Z H. Output: Orthogonal matrices Q 1 Q 2 orthogonal symplectic matrices U 1 U 2 (19). and the structured factorization 1: Set Q 1 = Q 2 = U 1 = U 2 = I 2n. By using dierent elimination orders in QR and RQ like decompositions determine orthogonal matrices Q 1 and Q 2 such that T := Q T T11 T 1 T =: Z := Z 0 T Q Z11 Z 2 =: 22 0 Z 22 where T 11 T22 T Z 11 Z22 T are n n and upper triangular. Update H = Q T 1 H Q 2 Q 1 := Q 1 Q1 Q 2 := Q 2 Q2. 2: Compute orthogonal matrices Q 1 Q2 and orthogonal symplectic matrices Ũ1 Ũ2 such that T : = Q T 1 T Ũ1 =: T11 T 0 T 22 Z : = Ũ 2 T Z Q Z11 Z 2 =: 0 Z 22 H : = Q T 1 H Q H11 H 2 =: 0 H 22 where T 11 T T 22 Z 11 Z T 22 H 11 are upper triangular and H T 22 is upper Hessenberg. Update Q 1 := Q 1 Q1 Q 2 := Q 2 Q2 U 1 := U 1 Ũ 1 and U 2 := U 2 Ũ 2. This step is performed by using a sequence of orthogonal and orthogonal symplectic matrices to annihilate the elements in H in a specic order without destroying the structure of T and Z (see [2] for details). 15

18 3: Apply the periodic QZ algorithm [9 13] to the formal matrix product T 1 11 H 11Z 1 11 Z T 22 H22T T T 22 to determine orthogonal matrices V 1 V 2 V 3 V 4 V 5 V 6 such that V2 T T 11V 1 V2 T H 11V 3 V4 T Z ( ) 11V 3 V4 T T Z 22V 5 ( ) V T T ( ) 6 T 22V 1 are all upper triangular and V T T 6 H 22V 5 is upper quasi triangular. Set Q 1 := diag (V 2 V 6) Q2 := diag (V 3 V 5) Ũ 1 := diag (V 1 V 1) Ũ 2 := diag (V 4 V 4) and update T := Q T 1 T Ũ1 Z := Ũ T 2 Z Q 2 H := Q T 1 H Q 2 Q 1 := Q 1 Q1 Q 2 := Q 2 Q2 U 1 := U 1 Ũ 1 U 2 := U 2 Ũ 2. Note that the algorithm above applies to any (unstructured) matrix pencil of the form λt Z H but the application of the eigenvalue formula (21) requires the structural assumption that the pencil is skew- Hamiltonian/Hamiltonian. Next we present the triangularization procedure needed for Step 2 of Algorithm 7. ALGORITHM 9. Triangularization procedure for special matrix pencils in factored form A11 0 B D Input: A real matrix pencil λab D = λ where the formal matrix 0 A 22 0 B 22 D 21 0 product A 1 11 DB 1 22 A 1 22 D21B 1 11 is in real periodic Schur form with upper triangular A11 A22 B11 B22 D and upper quasi triangular D 21. Output: Orthogonal matrices Q 1 Q 2 Q 3 such that Q T 3 AQ 2 Q T 2 BQ 1 are upper triangular and Q T 3 DQ 1 is upper quasi triangular. 1: Apply the periodic eigenvalue reordering method introduced in [] to the formal matrix product A 1 11 D B 1 22 A 1 22 D 21B 1 11 to determine orthogonal matrices V 1 V 2 V 3 V 4 V 5 V 6 such that V2 T A 11V 1 V2 T D V 3 V4 T B 22V 3 V5 T A 22V 4 V5 T D 21V 6 V1 T B 11V 6 keep their upper (quasi) triangular structure but they can be partitioned into 2 2 blocks with the last diagonal blocks corresponding to all nonpositive eigenvalues of the formal product and the rst diagonal blocks corresponding to the other eigenvalues. 2: Set Q 1 := diag(v 6 V 3) Q 2 := diag(v 1 V 4) Q 3 := diag(v 2 V 5) and update A : = Q T 3 AQ 2 =: B : = Q T 2 BQ 1 =: D : = Q T 3 DQ 1 =: A 11 A A A 33 A A 44 B 11 B B B 33 B B D 13 D D 24 D 31 D D where A 1 22 D24B 1 44 A 1 44 D42B 1 22 has only nonpositive real eigenvalues. 16

19 3: Let P be an appropriate permutation matrix such that A : = P T AP = B : = P T BP = D : = P T DP = A 11 0 A 0 0 A 33 0 A A A 44 B 11 0 B 0 0 B 33 0 B B B 44 0 D 13 0 D 14 D 31 0 D D D 42 0 [à =: [ B =: [ D =: ] 0  ] 0 ˆB ] 0 ˆD and update Q 1 := Q 1P Q 2 := Q 2P Q 3 := Q 3P. [à ] 4: Triangularize λã B D i.e. compute orthogonal matrices Q 1 Q2 Q3 such that A := Q T 3 A Q 2 =: 0  B B := Q T 2 B Q D 1 =: D := 0 ˆB Q T 3 D Q 1 =: with upper triangular 0 ˆD à B upper quasi triangular D and unchanged  ˆB ˆD. Update Q1 = Q 1 Q1 Q 2 = Q 2 Q2 Q 3 = Q 3 Q3. 5: Triangularize λâ ˆB ˆD with an appropriate permutation matrix ˆP i.e. A := ˆP T A ˆP =: B ˆP T B ˆP =: D := 0 ˆB ˆP T D ˆP =: unchanged à B D. Update Q1 = Q 1 ˆP Q2 = Q 2 ˆP Q3 = Q 3 ˆP. [à ] B := 0  D with upper triangular 0 ˆD  ˆB upper quasi triangular ˆD and Note that the separation of the nonpositive from the other eigenvalues of the formal matrix product A 1 11 D B22 1 A 1 22 D 21B11 1 is performed in order to avoid perturbations of the purely imaginary eigenvalues of skew-hamiltonian/hamiltonian matrix pencils. This follows from the connection of the nonpositive eigenvalues of the matrix product and the matrix pencil λab D similar to (21). When the nonpositive eigenvalues are separated the triangularization of the corresponding part of λab D can be done by only applying permutation matrices. When the matrix pencil is triangularized we apply the following eigenvalue reordering algorithm. ALGORITHM 10. Eigenvalue reordering for real skew-hamiltonian/hamiltonian matrix pencils in factored form Input: Regular 2n 2n real skew-hamiltonian/hamiltonian matrix pencil λs H with S = J Z T J T Z Z = Z W H D H = 0 T 0 H T with upper triangular Z and T T and upper quasi triangular H. Output: An orthogonal matrix Q an orthogonal symplectic matrix U and the transformed matrices U T ZQ J Q T J T HQ which have still the same triangular form as Z and H respectively but the eigenvalues in Λ (S H) are reordered such that they occur in the leading principal subpencil of J Q T J T (λs H) Q. 1: Set Q = U = I 2n. Reorder the eigenvalues in the subpencil λt T Z H. a) Determine orthogonal matrices Q 1 Q 2 Q 3 such that T T := Q T 3 T T Q 2 Z := Q T 2 ZQ 1 H := Q T 3 HQ 1 are still upper (quasi) triangular but the m eigenvalues with negative real part are reordered to the top of λt T Z H. Set Q 1 := diag (Q 1 Q 3) U 1 := diag (Q 2 Q 2) and update Q := QQ 1 U := UU 1. b) Determine orthogonal matrices Q 1 Q 2 Q 3 such that T T := Q T 3 T T Q 2 Z := Q T 2 ZQ 1 H := Q T 3 HQ 1 are still upper (quasi) triangular but the m + eigenvalues with positive real part are reordered to the bottom of λt T Z H. Set Q 1 := diag (Q 1 Q 3) U 1 := diag (Q 2 Q 2) and update Q := QQ 1 U := UU 1. 17

20 2: Reorder the remaining n m eigenvalues with negative real parts which are now in the bottom right subpencil of λs H. Determine an orthogonal matrix Q 1 and an orthogonal symplectic matrix U 1 such that the eigenvalues of top left subpencil of λs H with positive real parts and those of the bottom right subpencil of λs H with negative real parts are interchanged. Update Q := QQ 1 U := UU 1. In case that we have to deal with skew-hamiltonian/hamiltonian matrix pencils λs H with unfactored matrix S we use the following algorithms. ALGORITHM 11. Computation of stable deating subspaces of real skew-hamiltonian/hamiltonian matrix pencil in unfactored form Input: Real skew-hamiltonian/hamiltonian matrix pencil λs H. Output: Structured Schur form of the extended skew-hamiltonian/hamiltonian matrix pencil λbs r BH r eigenvalues of λs H orthonormal basis P V of the deating subspace Def (S H) as in Theorem : Apply Algorithm to the matrices S and H and determine orthogonal matrices Q 1 Q 2 such that Q T 1 SJ Q 1J T S11 S = 0 S11 T SH 2n J Q T 2 J T T11 T SQ 2 = 0 T11 T SH 2n Q T H11 H 1 HQ 2 = 0 H 22 with the formal matrix product S H11T11 HT 22 in real periodic Schur form where S 11 T 11 H 11 are upper triangular and H22 T is upper quasi triangular. 2: Apply Algorithm 13 to determine orthogonal matrices Q 3 Q 4 such that S 11 = Q T S Q 3 is upper triangular and H 11 = Q T 4 3: Update [ 0 H11 H22 T 0 ] Q 3 is upper quasi triangular. S := Q T S 0 4 Q 0 T 4 H := Q T 4 0 H H T Q T 11 and set B S r S11 S H11 H = 0 S11 T Br H = 0 H11 T. ( Apply the real eigenvalue reordering method in Algorithm 14 to the pair Br S B ) H r to determine an orthogonal matrix ˆQ such that J ˆQ T J (λ T B S r B ( H) r ˆQ is in structured Schur form and Λ Br S B ) H r is contained in the leading 2p 2p principal subpencil of λs 11 H 11. 4: Set V = [ I 2n 0 ] ( J Q1J T 0 Q3 0 Y r P 0 Q 2 0 Q 4 ˆQ) I2p 0 and compute P V orthogonal basis of range V using any numerically stable orthogonalization scheme. The following algorithm is used to compute a structured matrix pencil decomposition which is similar to the generalized symplectic URV decomposition. ALGORITHM. Variant of the generalized symplectic URV decomposition for unfactored real skew-hamiltonian/hamiltonian matrix pencils 18

21 Input: A real 2n 2n skew-hamiltonian/hamiltonian matrix pencil λs H. Output: Orthogonal matrices Q 1 Q 2 and the structured factorization (22). 1: Set Q 1 = Q 2 = I 2n. Reduce S to skew-hamiltonian triangular form i.e. determine an orthogonal matrix Q 1 such that S := Q T 1 SJ Q 1J T S11 S = 0 S11 T with an upper triangular matrix S 11. Update H := Q T 1 HJ Q 1J T Q 1 := Q 1 Q1. This step is performed by applying a sequence of Householder reections and Givens rotations in a specic order see [2] for details. 2: Set T := S Q 2 := J Q 1J T. Perform eliminations in H i.e. compute orthogonal matrices Q 1 Q2 such that S : = Q T 1 SJ Q 1J T S11 S = 0 S11 T SH 2n T : = J Q T 2 J T T Q T11 T 2 = 0 T11 T SH 2n H : = Q T 1 H Q H11 H 2 = 0 H 22 where S 11 T 11 H 11 are upper triangular and H T 22 is upper Hessenberg. Update Q 1 := Q 1 Q1 Q 2 := Q 2 Q2. This step is performed by applying an appropriate sequence of Givens rotations to annihilate the elements in H in a specic order without destroying the structure of S and T for details see [2]. 3: Apply the periodic QZ algorithm [9 13] to the formal matrix product S 1 11 H 11T 1 11 H T 22 to determine orthogonal matrices V 1 V 2 V 3 V 4 such that V T 1 S 11V 3 V T 1 H 11V 4 V T 2 T 11V 4 are all upper triangular and ( V T 3 H 22V 2 ) T is upper quasi triangular. Set Q 1 := diag (V 1 V 3) Q2 := diag (V 4 V 2) and update S := Q T 1 SJ Q 1J T T := J Q T 2 J T T Q 2 H := Q T 1 H Q 2 Q 1 := Q 1 Q1 Q 2 := Q 2 Q2. Now we present the triangularization algorithm. All remarks which have been made for the factored case analogously hold for the unfactored case. ALGORITHM 13. Triangularization procedure for special matrix pencils in unfactored form A B Input: A real matrix pencil λa B = λ where the formal matrix product A A 22 B 21 0 BA 1 22 B21 is in real periodic Schur form with upper triangular A 11 A 22 B and upper quasi triangular B 21. Output: Orthogonal matrices Q 1 Q 2 such that Q T 2 AQ 1 is upper triangular and Q T 2 BQ 1 is upper quasi triangular. 1: Apply the periodic eigenvalue reordering method introduced in [] to the formal matrix product A 1 11 B A 1 22 B 21 to determine orthogonal matrices V 1 V 2 V 3 V 4 such that V2 T A 11V 1 V2 T B V 3 V4 T A 22V 3 V4 T B 21V 1 keep their upper (quasi) triangular structure but they can be partitioned into 2 2 blocks with the last diagonal blocks corresponding to all nonpositive real eigenvalues of the formal product and the rst diagonal blocks corresponding to the other eigenvalues. 19

22 2: Set Q 1 := diag(v 1 V 3) Q 2 := diag(v 2 V 4) and update A : = Q T 2 AQ 1 =: B : = Q T 2 BQ 1 =: where A 1 22 B24A 1 44 B42 has only nonpositive real eigenvalues. 3: Let P be an appropriate permutation matrix such that A : = P T AP = B : = P T BP = A 11 A A A 33 A A B 13 B B 24 B 31 B B A 11 0 A 0 0 A 33 0 A A A 44 0 B 13 0 B 14 B 31 0 B B B 42 0 [à =: [ B =: ] 0  ] 0 ˆB and update Q 1 := Q 1P Q 2 := Q 2P. [à ] 4: Triangularize λã B i.e. compute orthogonal matrices Q 1 Q2 such that A := Q T 2 A Q 1 =: B := 0  B Q T 2 B Q 1 =: with upper triangular 0 ˆB à upper quasi triangular B and unchanged  ˆB. Update Q1 = Q 1 Q1 Q 2 = Q 2 Q2. [à ] 5: Triangularize λâ ˆB with an appropriate permutation matrix ˆP i.e. A := ˆP T A ˆP =: B := 0  B ˆP T B ˆP =: with upper triangular 0 ˆB  upper quasi triangular ˆB and unchanged à B. Update Q1 = Q 1 ˆP Q2 = Q 2 ˆP. Finally we describe the reordering of the eigenvalues. ALGORITHM 14. Eigenvalue reordering for real skew-hamiltonian/hamiltonian matrix pencils in unfactored form Input: Regular 2n 2n real skew-hamiltonian/hamiltonian matrix pencil λs H of the form S = H D H = 0 H T with upper triangular S an upper quasi triangular H. S W 0 S T Output: An orthogonal matrix Q and the transformed matrices J Q T J T SQ J Q T J T HQ which have still the same (quasi) triangular form as S and H respectively but the eigenvalues in Λ (S H) are reordered such that they occur in the leading principal subpencil of J Q T J T (λs H) Q. 1: Set Q = I 2n. Reorder the eigenvalues in the subpencil λs H. a) Determine orthogonal matrices Q 1 Q 2 such that S := Q T 2 SQ 1 H := Q T 2 HQ 1 are still upper (quasi) triangular but the m eigenvalues with negative real part are reordered to the top of λs H. Set Q 1 := diag (Q 1 Q 2) and update Q := QQ 1. b) Determine orthogonal matrices Q 1 Q 2 such that S := Q T 2 SQ 1 H := Q T 2 HQ 1 are still upper (quasi) triangular but the m + eigenvalues with positive real part are reordered to the bottom of λs H. Set Q 1 := diag (Q 1 Q 2) and update Q := QQ 1. 20

23 2: Reorder the remaining n m eigenvalues with negative real parts which are now in the bottom right subpencil of λs H. Determine an orthogonal matrix Q 1 such that the eigenvalues of top left subpencil of λs H with positive real parts and those of the bottom right subpencil of λs H with negative real parts are interchanged. Update Q := QQ 1. 4 Conclusion We have presented algorithms which can be used to compute the eigenvalues and deating subspaces of skew-hamiltonian/hamiltonian matrix pencils in a structure-preserving way which may lead to higher accuracy reliability and computational performance. Applications which are based on matrix pencils of this structure have been introduced to show the importance of our considerations. In Part II of this paper [8] we describe details of the implementation for the Subroutine Library in Control Theory (SLICOT). We furthermore present results of some numerical experiments in order to show the superiority of our method compared to standard approaches. References [1] A. C. Antoulas. Approximation of Large-Scale Dynamical Systems. Adv. Des. Control. SIAM Philadelphia PA [2] P. Benner R. Byers P. Losse V. Mehrmann and H. Xu. Numerical solution of real skewhamiltonian/hamiltonian eigenproblems. Unpublished report November [3] P. Benner R. Byers V. Mehrmann and H. Xu. Numerical computation of deating subspaces of embedded Hamiltonian pencils. Technical Report SFB393/99-15 Fakultät für Mathematik TU Chemnitz Germany Available online from html. [4] P. Benner R. Byers V. Mehrmann and H. Xu. Numerical computation of deating subspaces of skew-hamiltonian/hamiltonian pencils. SIAM J. Matrix Anal. Appl. 24(1): [5] P. Benner P. Losse V. Mehrmann L. Poppe and T. Reis. γ-iteration for descriptor systems using structured matrix pencils. In Proc. International Symposium on Mathematical Theory of Networks and Systems Blacksburg VA USA [6] P. Benner V. Sima and M. Voigt. L -norm computation for continuous-time descriptor systems using structured matrix pencils. IEEE Trans. Automat. Control 57(1): January 20. [7] P. Benner V. Sima and M. Voigt. Robust and ecient algorithms for L -norm computation for descriptor systems. In Proc. 7th IFAC Symposium on Robust Control Design pages Aalborg Denmark June 20. [8] P. Benner V. Sima and M. Voigt. FORTRAN 77 subroutines for the solution of skew- Hamiltonian/Hamiltonian eigenproblems Part II: Implementation and numerical results. SLICOT Working Note August [9] A. Bojanczyk G.H. Golub and P. Van Dooren. The periodic Schur decomposition. Algorithms and applications. In F.T. Luk editor Advanced Signal Processing Algorithms Architectures and Implementations III volume 1770 of Proc. SPIE pages