SLICOT Working Note

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1 SLICOT Working Note 3-3 MB4BV A FORTRAN 77 Subroutine to Compute the Eigenvectors Associated to the Purely Imaginary Eigenvalues of Skew-Hamiltonian/Hamiltonian Matrix Pencils Peihong Jiang Matthias Voigt 3 September 3 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-CT97-54), 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 University of Rochester, New York, USA; pjiang3@u.rochester.edu 3 Max Planck Institute for Dynamics of Complex Technical Systems, 396 Magdeburg, Germany; voigtm@mpi-magdeburg.mpg.de

2 Abstract We implement a structure-preserving numerical algorithm for extracting the eigenvectors associated to the purely imaginary eigenvalues of skew-hamiltonian/hamiltonian matrix pencils. We compare the new algorithm with the QZ algorithm using random examples with dierent diculty. The results show that the new algorithm is signicantly faster, more robust, and more accurate, especially for hard examples. Keywords: Eigenvalues, eigenvectors, skew-hamiltonian/hamiltonian matrix pencil, reliability, structure-preserving algorithm.

3 Introduction Skew-Hamiltonian/Hamiltonian matrix pencils have a wide range of applications in systems and control theory, for instance in linear-quadratic optimal control 7, H optimal control 9, for computing system norms, or analyzing system properties, such as passivity or contractivity 9, 8,, or a counterclockwise input/output dynamics 6, 6. In these applications, dierent spectral information is needed, such as eigenvalues, eigenvectors, and deating subspaces. The results presented in this paper are motivated by the following problem. In electrical engineering it is common to model electrical circuits by dierential-algebraic equations or descriptor systems. Electrical circuits have many important properties, for instance they cannot internally generate energy which is called passivity. This property should also be reected in the equations that model the circuit. However, due to errors introduced in the modeling process, e.g., by linearization, model approximation, or model order reduction, the resulting model is not always passive. In this case we need to perform a post-processing of the equations in order to restore the passivity property which is important to obtain meaningful results when simulating this model. This process is called passivity enforcement and is typically done by perturbing the system or parts of it, 4, 5. Passivity can be characterized by the purely imaginary eigenvalues of an associated skew-hamiltonian/hamiltonian matrix pencil. Passivity is then enforced by perturbing this pencil which results in a movement of the purely imaginary eigenvalues. This process is iteratively repeated until no purely imaginary eigenvalues exist. To compute optimal perturbations not only the eigenvalues have to be computed but also the corresponding eigenvectors, see, e.g.,, 4, 5. This report focuses on the actual computation of these eigenvectors. Standard numerical methods to compute eigenvalues, eigenvectors and deating subspaces rely on the generalized Schur decomposition 7. However, this factorization does not respect the structure of the pencil. Therefore, one is interested in a skew-hamiltonian/hamiltonian Schur-like form. Unfortunately, when purely imaginary eigenvalues exist, diculties arise because of the possible non-existence of such a structured Schur form. To avoid this problem, the pencil is embedded into a skew-hamiltonian/hamiltonian pencil of double size. A structure-preserving algorithm for computing the purely imaginary eigenvalues in a very accurate and reliable manner is presented in. An algorithm for the computation of the corresponding eigenvectors based on the structured canonical forms given in, is stated in 5. This paper describes the FORTRAN 77 routine MB4BV that has been implemented to extract the eigenvectors corresponding to the simple, nite, purely imaginary eigenvalues. We give a brief overview of the theoretical foundations, the algorithm outline, details of the implementation, and present numerical results obtained by MB4BV. Theory In this report we deal with the following matrix structures. Denition. Let J := I n, where In is the n n identity matrix. I n i) A matrix H R n n is Hamiltonian if HJ ) T = HJ. ii) A matrix N R n n is skew-hamiltonian if N J ) T = N J. iii) A real matrix pencil λn H is skew-hamiltonian/hamiltonian if N is skew-hamiltonian and H is Hamiltonian. Skew-Hamiltonian/Hamiltonian pencils satisfy the Hamiltonian spectral symmetry, i.e., eigenvalues occur in pairs {λ, λ} if they are purely real or imaginary, or otherwise in quadruples { λ, λ, λ, λ },

4 Figure : Hamiltonian eigensymmetry see Figure. Numerical algorithms should preserve this structure in order to get meaningful results. To compute the eigenvalues of a skew-hamiltonian/hamiltonian matrix pencil, we use the fact that J -congruence transformations of the form λñ H := J Q T J T λn H)Q with a nonsingular matrix Q preserve the skew-hamiltonian/hamiltonian structure. Therefore, we hope that we can compute an orthogonal matrix Q such that J Q T J T N N λn H)Q = λ H H N T H T is in skew-hamiltonian/hamiltonian Schur form, i.e., the subpencil λn H is in generalized Schur form. Unfortunately, not every skew-hamiltonian/hamiltonian pencil has this form, since pairs of simple purely imaginary eigenvalues cannot be represented in this structure. In this case, we embed the matrix pencil into another double-sized matrix pencil to solve the problem as follows. We introduce the orthogonal matrices Y = In I n, P = I n I n Then we dene the double-sized matrix pencils and λb N B H := λ I n I n I n, I n N H, N H λ B N B H := X T λb N B H ) X. X = YP. The 4n 4n matrix pencil λ B N B H is again real skew-hamiltonian/hamiltonian with the same eigenvalues with double algebraic, geometric, and partial multiplicities) as the original pencil. To compute the eigenvalues one uses the generalized symplectic URV decomposition which is summarized in the following theorem see the real-case versions of the results in ).

5 Theorem. Let λn H be a regular real n n skew-hamiltonian/hamiltonian pencil. Then there exist real orthogonal n n matrices Q, Q such that Q T N J Q J T N N = N T, J Q T J T M M N Q = M T := M, ) Q T H H HQ = H T, where N, M, and H are upper triangular, H is upper quasi-triangular, N and M are skewsymmetric and the generalized matrix product N H M H is in real periodic Schur form 7. Then, by using the matrix decomposition from ) we can compute an orthogonal matrix Q such that λ ˆB N ˆB H : = J Q T J T λ B N B ) H Q N N = λ M M N T M T H H H H H T H T ), with Q = P T J Q J T Q P. Note, that we never explicitly construct the embedded pencils. It is sucient to compute the necessary parts of the matrices in ). The spectrum of λn H is given by ΛN, H) = ±i Λ N H M H ), which can be determined by evaluating the entries on the and diagonal blocks of the matrices only. In particular, the simple, nite, purely imaginary eigenvalues correspond to the diagonal blocks of this matrix product. Provided that the pairwise distance of the simple, nite, purely imaginary eigenvalues is suciently large, they can be computed in a robust way without any error in the real part. However, if a purely imaginary eigenvalue has an algebraic multiplicity larger than one or if two purely imaginary eigenvalues are close, they might still be perturbed o the imaginary axis. This depends on the sign-characteristic of the involved eigenvalues, similarly as in 8. Therefore, for the rest of the paper we assume for simplicity that all nite, purely imaginary eigenvalues are simple. To compute the eigenvectors corresponding to the nite, purely imaginary eigenvalues we will make use of the structure of λ ˆB N ˆB H. As in passivity enforcement we only need the positive imaginary eigenvalues, i.e., those with positive imaginary parts, we restrict ourselves to the computation of the eigenvectors corresponding to these eigenvalues. To derive an algorithm for computing the desired eigenvectors we make use of the following two lemmas. In the following we assume that the matrix pencil λ N M H H is regular which is also equivalent to the regularity of the pencils λ N M H H and λ ˆBN ˆB H in ). Lemma 3. The vector v v is a right eigenvector of the matrix pencil λ N M H H corresponding to the eigenvalue ω if and only if iv v is a right eigenvector of the matrix pencil λ N M H H corresponding to the eigenvalue iω. Proof. Let v v be a right eigenvector of λ N M H H we have ω N v = H v, ω M v = H v. corresponding to the eigenvalue ω. Then 3

6 This is equivalent to iω N iv ) = H v, iω M v = H iv ). In other words, iv v is a right eigenvector of the matrix pencil λ N M H H corresponding to the eigenvalue iω. The converse statement can be proven in a completely analogous manner. Lemma 4. The vector v is a right eigenvector of the matrix pencil λ N M H H to the eigenvalue λ if and only if the vector v is a right eigenvector of the skew-hamiltonian/hamiltonian matrix pencil λ ˆB N ˆB H in ) corresponding to the eigenvalue λ. Proof. Trivial. As an intermediate step, we compute a matrix X whose columns contain the eigenvectors to the positive real eigenvalues of the pencil λ N M H H. This is done by the following basic steps already summarized in 5. Step : Reorder the positive real eigenvalues of the generalized matrix product P := N the top, i.e., compute orthogonal matrices U i = U ) i U ), i =,..., 4, such that i N ) N ) H ) H ) U T N U = U T 4 M U 3 = N ) M ) M ) M ), U T H U 3 =, U T 4 H U = H ) H ) H ) H ), H M H to are still in upper quasi-)triangular form, but the eigenvalues of the generalized matrix product ) ) P ) := N ) ) H M ) ) H are the positive real ones of P 3. N Step : Reorder the eigenvalues λ ) H ) by computing orthogonal matrices V M ) H ) = V ) V ), V = V ) V ) such that ) V T N ) λ H ) M ) R R H ) V = λ S S, R S where ΛR, S ) R + and ΛR, S ) R. Step 3: Compute the eigenvectors of λr S, i.e., compute a matrix W such that S W = R W D, where D is an appropriate diagonal matrix composed of the eigenvalues of λr S. Step 4: Collect the information contained in the relevant columns of the transformation matrices to obtain X ) U ) X := X ) := U ) V ) W. 3 Now, using Lemma 3 it turns out that ix ) X := X ) 4

7 contains the eigenvectors corresponding to the positive imaginary eigenvalues of the pencil λ N M. Then, by employing Lemma 4, the columns of the matrix X contain eigenvectors to the H H positive imaginary eigenvalues of the pencil λ ˆB N ˆB H. Note that all eigenvalues of this pencil have double algebraic, geometric, and partial multiplicities. So the matrix X contains only half of the eigenvectors to each positive imaginary eigenvalue of λ ˆB N ˆB H. However, this is no problem, since by later turning over to the original pencil λn H, we do not need the other half of the eigenvectors. Now, the corresponding eigenvectors to the positive imaginary eigenvalues of the double-sized matrix pencil λb N B H are given by iq ) X ) + Q ) X ) Y X Y := = X Q = iq ) X ) + Q ) X ) Y iq ) X ) + Q ) X ), iq ) X ) + Q ) X ) where Q := Q ) Q ) Q ) Q ) with Q ij) R n n and Q := Q ) Q ) Q ) Q ) with Q ij) R n n. In the equation above, only Y contains the desired eigenvectors of the matrix pencil λn H and Y has not to be computed explicitly. More specically, we can express Y as Y := iq ) X ) + Q ) X ) iq ) X ) + Q ) X ) Within the next sections we will focus on the implementation of the SLICOT routine MB4BV that enables us to compute Y.. 3 Specication SUBROUTINE MB4BV N, N, LDN, M, LDM, H, LDH, H, LDH, $ Q, LDQ, Q, LDQ, NEIG, ALPHAI, BETA, EVEC, $ LDEVEC, DWORK, LDWORK, BWORK, INFO ) C C.. Scalar Arguments.. INTEGER INFO, LDEVEC, LDH, LDH, LDM, LDN, $ LDQ, LDQ, LDWORK, N, NEIG C C.. Array Arguments.. LOGICAL BWORK * ) DOUBLE PRECISION ALPHAI * ), BETA * ), DWORK * ), $ H LDH, * ), H LDH, * ), $ M LDM, * ), N LDN, * ), Q LDQ, * ), $ Q LDQ, * ) COMPLEX*6 EVEC LDEVEC, * ) 4 Argument List 4. Input/Output Parameters N - input) INTEGER 5

8 The order of the matrix pencil. N has to be greater than or equal to. N also must be even. N - input) DOUBLE PRECISION array, dimension LDN, N/) On entry, the leading N/-by-N/ part of this array must contain the upper triangular matrix N in ). LDN - INTEGER The leading dimension of the array N. max {, N/}. The parameter LDN has to be greater or equal than M - input) DOUBLE PRECISION array, dimension LDM, N/) On entry, the leading N/-by-N/ part of this array must contain the upper triangular matrix M in ). LDM - INTEGER The leading dimension of the array M. max {, N/}. The parameter LDM has to be greater or equal than H - input) DOUBLE PRECISION array, dimension LDH, N/) On entry, the leading N/-by-N/ part of this array must contain the upper triangular matrix H in ). LDH - INTEGER The leading dimension of the array H. The parameter LDH has to be greater or equal than max {, N/}. H - input) DOUBLE PRECISION array, dimension LDH, N/) On entry, the leading N/-by-N/ part of this array must contain the upper quasi-triangular matrix H in ). LDH - INTEGER The leading dimension of the array H. The parameter LDH has to be greater or equal than max {, N/}. Q - input) DOUBLE PRECISION array, dimension LDQ, N) On entry, the leading N-by-N part of this array must contain the orthogonal transformation matrix Q in ). LDQ - INTEGER The leading dimension of the array Q. max {, N}. The parameter LDQ has to be greater or equal than Q - input) DOUBLE PRECISION array, dimension LDQ, N) On entry, the leading N-by-N part of this array must contain the orthogonal transformation matrix Q in ). LDQ - INTEGER The leading dimension of the array Q. The parameter LDQ has to be greater or equal than 6

9 max {, N}. NEIG - output) INTEGER The number of simple, nite, positive imaginary eigenvalues. ALPHAI - output) DOUBLE PRECISION array, dimension N/), BETA - output) DOUBLE PRECISION array, dimension N/) On exit, the ratio i ALPHAIj)/BETAj), j =,..., NEIG, represents the simple, nite, positive imaginary eigenvalues of the matrix pencil. EVEC - output) COMPLEX*6 array, dimension LDEVEC, N/) On exit, the leading N-by-NEIG part of this array contains the eigenvectors corresponding to the eigenvalues represented by the arrays ALPHAI and BETA in the same order. LDEVEC - INTEGER The leading dimension of the array EVEC. The parameter LDEVEC has to be greater or equal than max {, N}. 4. Workspace DWORK - DOUBLE PRECISION array, dimension LDWORK) On exit, if INFO =, DWORK) returns the optimal LDWORK. On exit, if INFO =, DWORK) returns the minimum value of LDWORK. LDWORK - INTEGER The dimension of the array DWORK. LDWORK 6N/) + max { 3N + 9N + 6, 7 }. For good performance LDWORK should be generally larger. If LDWORK =, then a workspace query is assumed; the routine only calculates the optimal size of the DWORK array, returns this value as the rst entry of the DWORK array, and no error message related to LDWORK is issued by XERBLA. BWORK - LOGICAL array, dimensionn/) 4.3 Error Indicator INFO - INTEGER INFO = : Successful exit; INFO < : If INFO = i, the i-th argument had an illegal value; INFO = : The eigenvalue reordering in MB3KD failed; INFO = : The triangularization or reordering in DGGES failed; INFO = 3: The eigenvector computation in DGGEV failed. 7

10 5 Numerical Experiments 5. Setup of Test Examples In order to test the performance of MB4BV, we feed it with random examples which have purely imaginary eigenvalues. We randomly generate matrices E, A R n n, B R n m, C R m n, D R m m, and dene the transfer function Gs) = CsE A) B + D. Let λn H = λ E E T A B C D γi m A T C T γi m B T D T. 3) Under some assumptions, the matrix pencil λn H is guaranteed to have purely imaginary eigenvalues if min ω R σ max Giω)) < γ < G L, where σ max ) denotes the largest singular value and L is the L -norm 9. Theoretically, when the distance between γ and G L decreases, the diculty of the example will increase, in the sense that the eigenvalues will be increasingly sensitive to perturbations, as the numerical results will later demonstrate. This is due to the fact, that there are two purely imaginary eigenvalues which almost form a non-trivial Jordan block in the Weierstraÿ canonical form. Then, the transformation matrices will be ill-conditioned, which leads to a higher sensitivity of these eigenvalues. 5. Environment and Conguration The tests have been performed on a generic-pae Ubuntu machine with Intel R Core TM Duo CPU with 3.GHz and 4GB RAM. The algorithms have been implemented and tested in MATLAB Ra). A FORTRAN MEX-le as an interface between MATLAB and MB4BV has also been written for testing purposes and improving the user-friendliness. Using MATLAB, one can easily generate random examples, compute the L -norm of the descriptor system, determine the eigenvalues by the structure-preserving solver MB4BD see 3, 4 for implementation details) and the corresponding MEX-le skewhamileig, and prole the performance. The MATLAB interface for MB4BV is given by alpha, beta, evecs = shh_evecs N, M, H, H, Q, Q ). Here, N, M, H, H, Q, and Q are the matrices N, M, H, H, Q, and Q in ), respectively. The purely imaginary eigenvalues are given by i αi β i, and the corresponding eigenvectors are stored in the columns of evecs in the same order as the eigenvalues. In this section we also compare our structure-preserving approach with the standard one for general eigenvalue problems, namely the QZ algorithm with eigenvalue reordering). In order to have a fair comparison of the performance of both methods, we also implemented a FORTRAN 77 subroutine which combines the QZ algorithm with reordering the purely imaginary eigenvalues to the top by using the LAPACK subroutines DGGES and DGGEV. For testing, a corresponding MEX-le has been written. 5.3 Numerical Results 5.3. Performance Comparison In this paragraph we will discuss the behavior of the new approach compared to the QZ algorithm. We do this by constructing random pencils of the form 3) with n = and m = 5. Table shows the performance results of both algorithms when computing both desired eigenvalues and eigenvectors for 8

11 Table : Performance comparison new algorithm QZ algorithm γ runtime avg. rel. residual runtime avg. rel. residual failure rate G ) L e e-3.% G ) L e e-3.% G ) L e e-.8% G ) L e e- 4.6% G ) L e e- 8.9% G ) L e e- 78.7% dierent values of γ, from the easier examples to the harder ones. Each row contains the results for a thousand test runs. The accuracy of the results is measured by computing the average of the relative residuals given by λ i N H) v i / v i. The runtime is given in seconds, using the tic and toc commands in MATLAB. Furthermore, for the QZ algorithm we have an additional column that indicates the percentage of examples that could not be solved. This is due to the fact that eigenvalues might be perturbed o the imaginary axis and will not be considered as purely imaginary when the distance to the imaginary axis exceeds a certain threshold. For our tests this value is set to e-. First of all, Table shows that the QZ algorithm needs about 5% more time to execute than the new algorithm. However, the most important aspect of the new algorithm is the improved reliability. We can see that the failure rate of the QZ algorithm is dramatically increasing when the examples become more ill-conditioned. By failure we mean that the algorithm extracts a dierent number of eigenvectors, compared to the actual number of purely imaginary eigenvalues. Moreover, even in the cases where the QZ algorithm successfully extracts the eigenvectors, the average relative residual becomes signicantly larger when the condition gets worse. On the other hand, the performance of the new algorithm is far more reliable and accurate. Its runtime and accuracy remain at the same level from the easy examples to the harder ones On the Failure of the QZ Algorithm We now briey describe the nature of QZ algorithm's failure by examining a small example. Consider the randomly generated 6 6 skew-hamiltonian/hamiltonian matrix pencil λn H = λ , with γ = G L 6 ). The spectrum is given by Λ = {97.5i, 97.5i,.6,.6,, }. In the spectrum, only the purely imaginary eigenvalue 97.5i is interesting to us. The new algorithm successfully extracts one eigenvector with relative residual.8594e-5. 9

12 However, the QZ algorithm fails to extract the eigenvector. Since it does not respect the structure of the pencil, all purely imaginary eigenvalues will be perturbed o the imaginary axis. When we are selecting the purely imaginary eigenvalues, we cannot expect that the real part of an eigenvalue is exactly zero, as it is for the structure-preserving algorithm. Alternatively, what we do is to ask if the real part is smaller than some tolerance. This tolerance is empirically set to e-. This is a rather tight bound, but it illustrates the behavior of the QZ algorithm quite well. However the real part of this eigenvalue after perturbation is.479e-. Because the real part is slightly larger than the tolerance, this eigenvalue is not selected and thus no eigenvector is extracted. 6 Conclusions In this report we have presented a FORTRAN 77 implementation of a new algorithm for computing the eigenvectors of a skew-hamiltonian/hamiltonian matrix pencil associated to the purely imaginary eigenvalues. The performed numerical tests clearly indicate that compared to the QZ algorithm the new method. is more robust, especially for ill-conditioned examples;. is comparably accurate for well-conditioned examples, and signicantly more accurate for ill-conditioned examples; 3. needs only about /3 of time to execute. Acknowledgment The main part of this work was done while the rst author visited the second author as an intern in summer. The nancial support from the RISE program of the Deutscher Akademischer Austauschdienst is gratefully acknowledged. References P. Benner, R. Byers, V. Mehrmann, and H. Xu. Numerical computation of deating subspaces of skew-hamiltonian/hamiltonian pencils. SIAM J. Matrix Anal. Appl., 4):659,. P. Benner, V. Sima, and M. Voigt. L -norm computation for continuous-time descriptor systems using structured matrix pencils. IEEE Trans. Automat. Control, 57):3338, January. 3 P. Benner, V. Sima, and M. Voigt. FORTRAN 77 subroutines for the solution of skew- Hamiltonian/Hamiltonian eigenproblems Part I: Algorithms and applications. SLICOT Working Note 3-, August 3. 4 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 3-, August 3. 5 P. Benner and M. Voigt. On the computation of particular eigenvectors of Hamiltonian matrix pencils. Proc. Appl. Math. Mech., ):753754,. 6 P. Benner and M. Voigt. Spectral characterization and enforcement of negative imaginariness for descriptor systems. Linear Algebra Appl., 4394):49, August 3. 7th Conference of the International Linear Algebra Society, Braunschweig, Germany, August.

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