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To benefit from the performance gained by using reduced-order models, the computation of these models itself must be done with a reasonable use of resources. In this paper, we consider the case of medium-scale dense discrete-time systems and compare the performance of different numerical methods for the implementation of two basic model reduction techniques. Therefore, we give an overview of the considered model reduction methods and of the techniques used in underlying implementations. The outlined methods are then compared with established implementations in several numerical examples in terms of accuracy and performance.<\/jats:p>","DOI":"10.1515\/auto-2021-0035","type":"journal-article","created":{"date-parts":[[2021,8,11]],"date-time":"2021-08-11T21:56:28Z","timestamp":1628718988000},"page":"683-694","source":"Crossref","is-referenced-by-count":0,"title":["A comparison of numerical methods for model reduction of dense discrete-time systems"],"prefix":"10.1515","volume":"69","author":[{"given":"Robert","family":"Jendersie","sequence":"first","affiliation":[{"name":"Faculty of Computer Science , Otto von Guericke University , Universit\u00e4tsplatz 2 , Magdeburg , Germany"}]},{"given":"Steffen W.\u2009R.","family":"Werner","sequence":"additional","affiliation":[{"name":"Computational Methods in Systems and Control Theory , 28307 Max Planck Institute for Dynamics of Complex Technical Systems , Sandtorstra\u00dfe 1 , Magdeburg , Germany"}]}],"member":"374","published-online":{"date-parts":[[2021,8,10]]},"reference":[{"key":"2023033110334178739_j_auto-2021-0035_ref_001","doi-asserted-by":"crossref","unstructured":"A.\u2009C. Antoulas. Approximation of Large-Scale Dynamical Systems, volume 6 of Adv. Des. Control. SIAM Publications, Philadelphia, PA, 2005. doi:10.1137\/1.9780898718713.","DOI":"10.1137\/1.9780898718713"},{"key":"2023033110334178739_j_auto-2021-0035_ref_002","doi-asserted-by":"crossref","unstructured":"W.\u2009F. Arnold and A.\u2009J. Laub. Generalized eigenproblem algorithms and software for algebraic Riccati equations. Proc. IEEE, 72(12):1746\u20131754, 1984. doi:10.1109\/PROC.1984.13083.","DOI":"10.1109\/PROC.1984.13083"},{"key":"2023033110334178739_j_auto-2021-0035_ref_003","doi-asserted-by":"crossref","unstructured":"Z. Bai, J. Demmel and M. Gu. An inverse free parallel spectral divide and conquer algorithm for nonsymmetric eigenproblems. Numer. Math., 76(3):279\u2013308, 1997. doi:10.1007\/s002110050264.","DOI":"10.1007\/s002110050264"},{"key":"2023033110334178739_j_auto-2021-0035_ref_004","doi-asserted-by":"crossref","unstructured":"Y. Bar-Ness and A. Halbersberg. Solution of the singular discrete regulator problem using eigenvector methods. Internat. J. Control, 31(4):615\u2013625, 1980. doi:10.1080\/00207178008961069.","DOI":"10.1080\/00207178008961069"},{"key":"2023033110334178739_j_auto-2021-0035_ref_005","doi-asserted-by":"crossref","unstructured":"A.\u2009Y. Barraud. A numerical algorithm to solve ATXA-X=Q. In IEEE Conference on Decision and Control including the 16th Symposium on Adaptive Processes and A Special Symposium on Fuzzy Set Theory and Applications, pages 420\u2013423, 1977. doi:10.1109\/CDC.1977.271607.","DOI":"10.1109\/CDC.1977.271607"},{"key":"2023033110334178739_j_auto-2021-0035_ref_006","doi-asserted-by":"crossref","unstructured":"R.\u2009H. Bartels and G.\u2009W. Stewart. Solution of the matrix equation \nA\nX\n+\nX\nB\n=\nCAX+XB=C: Algorithm 432. Comm. ACM, 15:820\u2013826, 1972. doi:10.1145\/361573.361582.","DOI":"10.1145\/361573.361582"},{"key":"2023033110334178739_j_auto-2021-0035_ref_007","unstructured":"P. Benner. Contributions to the Numerical Solution of Algebraic Riccati Equations and Related Eigenvalue Problems. Dissertation, Fakult\u00e4t f\u00fcr Mathematik, TU Chemnitz\u2013Zwickau, Chemnitz, Germany, 1997."},{"key":"2023033110334178739_j_auto-2021-0035_ref_008","unstructured":"P. Benner. Factorized solution of Sylvester equations with applications in control. In Proc. Intl. Symp. Math. Theory Networks and Syst. MTNS 2004, 2004."},{"key":"2023033110334178739_j_auto-2021-0035_ref_009","doi-asserted-by":"crossref","unstructured":"P. Benner. Partial stabilization of descriptor systems using spectral projectors. In P. Van Dooren, S.\u2009P. Bhattacharyya, R.\u2009H. Chan, V. Olshevsky and A. Routray, editors, Numerical Linear Algebra in Signals, Systems and Control, volume 80 of Lect. Notes Electr. Eng., pages 55\u201376. Springer Netherlands, 2011. doi:10.1007\/978-94-007-0602-6_3.","DOI":"10.1007\/978-94-007-0602-6_3"},{"key":"2023033110334178739_j_auto-2021-0035_ref_010","unstructured":"P. Benner, J.\u2009M. Claver and E.\u2009S. Quintana-Ort\u00ed. Efficient solution of coupled Lyapunov equations via matrix sign function iteration. In Proc. 3rd Portuguese Conf. on Automatic Control CONTROLO\u201998, Coimbra, pages 205\u2013210, 1998."},{"key":"2023033110334178739_j_auto-2021-0035_ref_011","doi-asserted-by":"crossref","unstructured":"P. Benner, P. Ezzatti, Quintana-Ort\u00ed E.\u2009S. and A. Rem\u00f3n. A factored variant of the Newton iteration for the solution of algebraic Riccati equations via the matrix sign function. Numer. Algorithms, 66(2):363\u2013377, 2014. doi:10.1007\/s11075-013-9739-2.","DOI":"10.1007\/s11075-013-9739-2"},{"key":"2023033110334178739_j_auto-2021-0035_ref_012","doi-asserted-by":"crossref","unstructured":"P. Benner, V. Mehrmann, V. Sima, S. Van Huffel and A. Varga. SLICOT \u2014 A subroutine library in systems and control theory. In B.\u2009N. Datta, editor, Applied and Computational Control, Signals, and Circuits, volume 1, chapter 10, pages 499\u2013539. Birkh\u00e4user, Boston, MA, 1999. doi:10.1007\/978-1-4612-0571-5_10.","DOI":"10.1007\/978-1-4612-0571-5_10"},{"key":"2023033110334178739_j_auto-2021-0035_ref_013","doi-asserted-by":"crossref","unstructured":"P. Benner, E.\u2009S. Quintana-Ort\u00ed and G. Quintana-Ort\u00ed. Parallel algorithms for model reduction of discrete-time systems. Internat. J. Systems Sci., 34(5):319\u2013333, 2003. doi:10.1080\/0020772031000158564.","DOI":"10.1080\/0020772031000158564"},{"key":"2023033110334178739_j_auto-2021-0035_ref_014","doi-asserted-by":"crossref","unstructured":"P. Benner, E.\u2009S. Quintana-Ort\u00ed and G. Quintana-Ort\u00ed. Numerical solution of discrete stable linear matrix equations on multicomputers. 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Technical Report TR\/PA\/92\/86, CERFACS, Toulouse Cedex, France, 1992."},{"key":"2023033110334178739_j_auto-2021-0035_ref_021","doi-asserted-by":"crossref","unstructured":"A.\u2009H. Freedman, K.\u2009M. Portier and M.\u2009E. Sunquist. Life history analysis for black bears (Ursus americanus)in a changing demographic landscape. Ecological Modelling, 167(1\u20132):47\u201364, 2003. doi:10.1016\/S0304-3800(03)00171-6.","DOI":"10.1016\/S0304-3800(03)00171-6"},{"key":"2023033110334178739_j_auto-2021-0035_ref_022","unstructured":"G.\u2009H. Golub and C.\u2009F. Van Loan. Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, fourth edition, 2013."},{"key":"2023033110334178739_j_auto-2021-0035_ref_023","doi-asserted-by":"crossref","unstructured":"G. Gu. Discrete-Time Linear Systems. 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Twelfth Asilomar Conf. on Circuits, Systems and Computers, November 6\u20138, pages 705\u2013714, 1978."},{"key":"2023033110334178739_j_auto-2021-0035_ref_030","doi-asserted-by":"crossref","unstructured":"J.\u2009N. Kutz, S.\u2009L. Brunton, B.\u2009W. Brunton and J.\u2009L. Proctor. Dynamic Mode Decomposition: Data-Driven Modeling of Complex Systems. Society of Industrial and Applied Mathematics, Philadelphia, USA, 2016. doi:10.1137\/1.9781611974508.","DOI":"10.1137\/1.9781611974508"},{"key":"2023033110334178739_j_auto-2021-0035_ref_031","doi-asserted-by":"crossref","unstructured":"M. Lehner and P. Eberhard. A two-step approach for model reduction in flexible multibody dynamics. Multibody Syst. Dyn., 17(2-3):157\u2013176, 2007. doi:10.1007\/s11044-007-9039-5.","DOI":"10.1007\/s11044-007-9039-5"},{"key":"2023033110334178739_j_auto-2021-0035_ref_032","doi-asserted-by":"crossref","unstructured":"B.\u2009C. Moore. Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control, AC\u201326(1):17\u201332, 1981. doi:10.1109\/TAC.1981.1102568.","DOI":"10.1109\/TAC.1981.1102568"},{"key":"2023033110334178739_j_auto-2021-0035_ref_033","doi-asserted-by":"crossref","unstructured":"M.\u2009R. Opmeer and F. Curtain. Linear quadratic gaussian balancing for discrete-time infinite-dimensional linear systems. SIAM J. Control Optim., 43(4):1196\u20131221, 2004. doi:10.1137\/S0363012903431189.","DOI":"10.1137\/S0363012903431189"},{"key":"2023033110334178739_j_auto-2021-0035_ref_034","doi-asserted-by":"crossref","unstructured":"T. Pappas, A.\u2009J. Laub and N.\u2009R. Sandell. On the numerical solution of the discrete-time algebraic Riccati equation. IEEE Trans. Autom. Control, 25(4):631\u2013641, 1980. doi:10.1109\/TAC.1980.1102434.","DOI":"10.1109\/TAC.1980.1102434"},{"key":"2023033110334178739_j_auto-2021-0035_ref_035","unstructured":"T. Penzl. A cyclic low rank Smith method for large, sparse Lyapunov equations with applications in model reduction and optimal control. Technical Report SFB393\/98-6, Fakult\u00e4t f\u00fcr Mathematik, TU Chemnitz, Chemnitz, Germany, 1998."},{"key":"2023033110334178739_j_auto-2021-0035_ref_036","doi-asserted-by":"crossref","unstructured":"J.\u2009D. Roberts. Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. Internat. J. Control, 32(4):677\u2013687, 1980. (Reprint of Technical Report No.\u2009TR-13, CUED\/B-Control, Cambridge University, Engineering Department, 1971). doi:10.1080\/00207178008922881.","DOI":"10.1080\/00207178008922881"},{"key":"2023033110334178739_j_auto-2021-0035_ref_037","doi-asserted-by":"crossref","unstructured":"P.\u2009J. Schmid. Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech., 656:5\u201328, 2010. doi:10.1017\/S0022112010001217.","DOI":"10.1017\/S0022112010001217"},{"key":"2023033110334178739_j_auto-2021-0035_ref_038","doi-asserted-by":"crossref","unstructured":"V. Simoncini. Computational methods for linear matrix equations. SIAM Rev., 38(3):377\u2013441, 2016. doi:10.1137\/130912839.","DOI":"10.1137\/130912839"},{"key":"2023033110334178739_j_auto-2021-0035_ref_039","doi-asserted-by":"crossref","unstructured":"R.\u2009A. Smith. Matrix equation \nX\nA\n+\nB\nX\n=\nCXA+BX=C. SIAM J. Appl. Math., 16(1):198\u2013201, 1968. doi:10.1137\/0116017.","DOI":"10.1137\/0116017"},{"key":"2023033110334178739_j_auto-2021-0035_ref_040","doi-asserted-by":"crossref","unstructured":"A. Varga. Enhanced modal approach for model reduction. Math. Model. Syst., 1(2):91\u2013105, 1995. doi:10.1080\/13873959508837010.","DOI":"10.1080\/13873959508837010"},{"key":"2023033110334178739_j_auto-2021-0035_ref_041","doi-asserted-by":"crossref","unstructured":"E.\u2009L. Wachspress. Iterative solution of the Lyapunov matrix equation. Appl. Math. Letters, 107:87\u201390, 1988. doi:10.1016\/0893-9659(88)90183-8.","DOI":"10.1016\/0893-9659(88)90183-8"},{"key":"2023033110334178739_j_auto-2021-0035_ref_042","doi-asserted-by":"crossref","unstructured":"B. Zhou, J. Lam and G.-R. Duan. On Smith-type iterative algorithms for the Stein matrix equation. Appl. Math. 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