Guglielmi, Nicola; Kressner, Daniel; Lubich, Christian Low rank differential equations for Hamiltonian matrix nearness problems. (English) Zbl 1312.65070 Numer. Math. 129, No. 2, 279-319 (2015). The authors propose and study algorithms for two important optimization problems related to finding nearest Hamiltonian matrices with imaginary eigenvalues, to two specific classes of matrices. Their algorithms consist of a two-lewel procedure, where on the inner level an extremizer for a fixed perturbation size is determined, following the differential equation into a stationary point, and then the outer level to optimize over this size. Reviewer: Constantin Popa (Constanţa) Cited in 16 Documents MSC: 65F30 Other matrix algorithms (MSC2010) 15A18 Eigenvalues, singular values, and eigenvectors 93B36 \(H^\infty\)-control 93B40 Computational methods in systems theory (MSC2010) 49N35 Optimal feedback synthesis 65F15 Numerical computation of eigenvalues and eigenvectors of matrices 93B52 Feedback control 93C05 Linear systems in control theory 65K10 Numerical optimization and variational techniques Keywords:optimization problems; Hamiltonian matrices; two-lewel procedure; differential equations; algorithm; imaginary eigenvalue Software:Eigtool; Seigtool; ARPACK; PSAPSR PDFBibTeX XMLCite \textit{N. Guglielmi} et al., Numer. Math. 129, No. 2, 279--319 (2015; Zbl 1312.65070) Full Text: DOI References: [1] Alam, R., Bora, S., Karow, M., Mehrmann, V., Moro, J.: Perturbation theory for Hamiltonian matrices and the distance to bounded-realness. SIAM J. Matrix Anal. Appl. 32(2), 484-514 (2011) · Zbl 1227.93081 [2] Anderson, B.: Riccati equations, network theory and Brune synthesis: old solutions for contemporary problems, dynamical systems, control, coding, computer vision, series, Progr. Systems Control Theory, pp. 251-25 (1998) · Zbl 1113.74433 [3] Antoulas, A.: Approximation of large-scale dynamical systems. vol. 6, Advances in Design and Control Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2005) · Zbl 1112.93002 [4] Boyd, S., Balakrishnan, V., Kabamba, P.: A bisection method for computing the \[H_\infty H\]∞ norm of a transfer matrix and related problems. Math. Control Signals Syst. 2(3), 207-219 (1989) · Zbl 0674.93020 [5] Byers, R., Kressner, D.: On the condition of a complex eigenvalue under real perturbations. BIT 44(2), 209-214 (2004) · Zbl 1071.15004 [6] Conway, J.B., Halmos, P.R.: Finite dimensional points of continuity of Lat. Linear Algebra Appl. 31, 93-102 (1980) · Zbl 0435.15005 [7] Dieci, L., Pugliese, A.: Two-parameter SVD: coalescing singular values and periodicity. SIAM J. Matrix Anal. Appl. 31(2), 375-403 (2009) · Zbl 1232.65064 [8] Guglielmi, N., Lubich, C.: Differential equations for roaming pseudospectra: paths to extremal points and boundary tracking. SIAM J. Numer. Anal. 49(3), 1194-1209 (2011) · Zbl 1232.15009 [9] Guglielmi, N., Lubich, C.: Erratum/addendum: differential equations for roaming pseudospectra: paths to extremal points and boundary tracking. SIAM J. Numer. Anal. 50(2), 977-981 (2012) · Zbl 1257.15007 [10] Guglielmi, N., Lubich, C.: Low-rank dynamics for computing extremal points of real pseudospectra. SIAM J. Matrix Anal. Appl. 34, 40-66 (2013) · Zbl 1272.65032 [11] Guglielmi, N., Overton, M.: Fast algorithms for the approximation of the pseudospectral abscissa and pseudospectral radius of a matrix. SIAM J. Matrix Anal. Appl. 32(4), 1166-1192 (2011) · Zbl 1248.65034 [12] Hairer, E., Lubich, Ch., Wanner, G.: Geometric Numerical Integration, Springer Series in Computational Mathematics. Structure-preserving algorithms for ordinary differential equations, vol. 31, Springer-Verlag, Berlin (2006) · Zbl 1094.65125 [13] Hespanha, J.: Linear Systems Theory. Princeton University Press, Princeton (2009) · Zbl 1185.93001 [14] Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory. I, Texts in Applied Mathematics. Modelling, state space analysis, stability and robustness, vol. 48, Springer-Verlag, Berlin (2005) · Zbl 1074.93003 [15] Hryniv, R.O., Kliem, W., Lancaster, P., Pommer, C.: A precise bound for gyroscopic stabilization. ZAMM Z. Angew. Math. Mech. 80(8), 507-516 (2000) · Zbl 0981.34036 [16] Karow, M., Kokiopoulou, E., Kressner, D.: On the computation of structured singular values and pseudospectra. Syst. Control Lett. 59(2), 122-129 (2010) · Zbl 1186.93032 [17] Karow, M.: Structured pseudospectra for small perturbations. SIAM J. Matrix Anal. Appl. 32(1), 1383-1398 (2011) · Zbl 1241.15006 [18] Karow, M.: Private communication (2013) [19] Kato, T.: Perturbation Theory for Linear Operators, Classics in Mathematics. Springer-Verlag, Berlin, Reprint of the 1980th edition (1995) · Zbl 0836.47009 [20] Koch, O., Lubich, Ch.: Dynamical low-rank approximation. SIAM J. Matrix Anal. Appl. 29(2), 434-454 (2007) · Zbl 1145.65031 [21] Lehoucq, R.B., Sorensen, D.C., Yang, C.: Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. In: ARPACK users’ guide, Software, Environments, and Tools. vol. 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia (1998) · Zbl 0901.65021 [22] Mehrmann, V., Watkins, D.: Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils. SIAM J. Sci. Comput. 22, 1905-1925 (2000) · Zbl 0986.65033 [23] Mehrmann, V., Xu, H.: Perturbation of purely imaginary eigenvalues of Hamiltonian matrices under structured perturbations. Electron. J. Linear Algebra 17, 234-257 (2008) · Zbl 1161.15005 [24] Meyer, C.D., Stewart, G.W.: Derivatives and perturbations of eigenvectors. SIAM J. Numer. Anal. 25(3), 679-691 (1988) · Zbl 0646.15005 [25] Overton, M.L., Van Dooren, P.: On computing the complex passivity radius. In: Proceedings 4th IEEE Conference on Decision and Control, ECC-CDC-05, Seville, Spain, (2005), 5805, 7960-7964 (2005) · Zbl 1093.15020 [26] Paige, C., Van Loan, C.: A Schur decomposition for Hamiltonian matrices. Linear Algebra Appl. 41, 11-32 (1981) · Zbl 1115.15316 [27] Rump, S.M.: Eigenvalues, pseudospectrum and structured perturbations. Linear Algebra Appl. 413(2-3), 567-593 (2006) · Zbl 1093.15020 [28] Trefethen, L.N.: Computation of pseudospectra. Acta Numer. 8, 247-295 (1999) · Zbl 0945.65039 [29] Trefethen, L.N., Embree, M.: Spectra and Pseudospectra. The behavior of nonnormal matrices and operators. Princeton University Press, Princeton (2005) · Zbl 1085.15009 [30] Wagenknecht, T., Agarwal, J.: Structured pseudospectra in structural engineering. Intern. J. Numer. Methods Eng. 64(13), 1735-1751 (2005) · Zbl 1113.74433 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.