{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T11:51:20Z","timestamp":1759837880748},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"2-4","license":[{"start":{"date-parts":[[2010,6,11]],"date-time":"2010-06-11T00:00:00Z","timestamp":1276214400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2011,11]]},"DOI":"10.1007\/s10444-010-9156-1","type":"journal-article","created":{"date-parts":[[2010,6,10]],"date-time":"2010-06-10T01:03:41Z","timestamp":1276131821000},"page":"281-322","source":"Crossref","is-referenced-by-count":13,"title":["QR methods and error analysis for computing Lyapunov and Sacker\u2013Sell spectral intervals for linear differential-algebraic equations"],"prefix":"10.1007","volume":"35","author":[{"given":"Vu Hoang","family":"Linh","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Volker","family":"Mehrmann","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Erik S.","family":"Van Vleck","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2010,6,11]]},"reference":[{"key":"9156_CR1","volume-title":"Trans. Math. Monographs, vol. 146","author":"LYa Adrianova","year":"1995","unstructured":"Adrianova, L.Ya.: Introduction to linear systems of differential equations. In: Trans. Math. Monographs, vol. 146. AMS, Providence (1995)"},{"key":"9156_CR2","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971392","volume-title":"Computer Methods for Ordinary Differential and Differential-Algebraic Equations","author":"UM Ascher","year":"1998","unstructured":"Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential and Differential-Algebraic Equations. SIAM, Philadelphia (1998)"},{"key":"9156_CR3","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.apnum.2004.08.015","volume":"53","author":"K Balla","year":"2005","unstructured":"Balla, K., Linh, V.H.: Adjoint pairs of differential-algebraic equations and Hamiltonian systems. Appl. Numer. Math. 53, 131\u2013148 (2005)","journal-title":"Appl. Numer. Math."},{"key":"9156_CR4","first-page":"9\u201320, 21\u201330","volume":"15","author":"G Benettin","year":"1980","unstructured":"Benettin, G., Galgani, G., Giorgilli, L., Strelcyn, J.M.: Lyapunov exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part I: theory and part II: numerical applications. Meccanica 15, 9\u201320, 21\u201330 (1980)","journal-title":"Meccanica"},{"key":"9156_CR5","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1007\/s00211-006-0001-x","volume":"103","author":"P Benner","year":"2006","unstructured":"Benner, P., Byers, R.: An arithmetic for matrix pencils: theory and new algorithms. Numer. Math. 103, 539\u2013573 (2006)","journal-title":"Numer. Math."},{"key":"9156_CR6","first-page":"284","volume":"144","author":"P Bohl","year":"1913","unstructured":"Bohl, P.: \u00dcber Differentialungleichungen. J. f. d. Reine und Angew. Math. 144, 284\u2013313 (1913)","journal-title":"J. f. d. Reine und Angew. Math."},{"key":"9156_CR7","volume-title":"Numerical Solution of Initial-Value Problems in Differential Algebraic Equations","author":"KE Brenan","year":"1996","unstructured":"Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential Algebraic Equations, 2nd edn. SIAM, Philadelphia (1996)","edition":"2"},{"key":"9156_CR8","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/S0167-2789(01)00283-4","volume":"156","author":"TJ Bridges","year":"2001","unstructured":"Bridges, T.J., Reich, S.: Computing Lyapunov exponents on a Stiefel manifold. Physica, D 156, 219\u2013238 (2001)","journal-title":"Physica, D"},{"key":"9156_CR9","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1007\/BF00952257","volume":"46","author":"SL Campbell","year":"1995","unstructured":"Campbell, S.L.: Linearization of DAE\u2019s along trajectories. Z. Angew. Math. Phys. 46, 70\u201384 (1995)","journal-title":"Z. Angew. Math. Phys."},{"key":"9156_CR10","doi-asserted-by":"crossref","first-page":"772","DOI":"10.1137\/S0895479899353622","volume":"22","author":"J-L Chern","year":"2000","unstructured":"Chern, J.-L., Dieci, L.: Smoothness and periodicity of some matrix decompositions. SIAM J. Matrix Anal. Appl. 22, 772\u2013792 (2000)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9156_CR11","doi-asserted-by":"crossref","first-page":"2078","DOI":"10.1016\/j.jde.2008.07.016","volume":"245","author":"CJ Chyan","year":"2008","unstructured":"Chyan, C.J., Du, N.H., Linh, V.H.: On data-dependence of exponential stability and the stability radii for linear time-varying differential-algebraic systems. J. Diff. Equ. 245, 2078\u20132102 (2008)","journal-title":"J. Diff. Equ."},{"key":"9156_CR12","first-page":"73","volume":"28","author":"ND Cong","year":"2003","unstructured":"Cong, N.D., Nam, H.: Lyapunov\u2019s inequality for linear differential algebraic equation. Acta Math. Vietnam. 28, 73\u201388 (2003)","journal-title":"Acta Math. Vietnam."},{"key":"9156_CR13","first-page":"1","volume":"29","author":"ND Cong","year":"2004","unstructured":"Cong, N.D., Nam, H.: Lyapunov regularity of linear differential algebraic equations of index 1. Acta Math. Vietnam. 29, 1\u201321 (2004)","journal-title":"Acta Math. Vietnam."},{"key":"9156_CR14","volume-title":"Stability of Solutions of Differential Equations in Banach Spaces","author":"JL Daleckii","year":"1974","unstructured":"Daleckii, J.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Spaces. American Mathematical Society, Providence (1974)"},{"key":"9156_CR15","doi-asserted-by":"crossref","first-page":"800","DOI":"10.1137\/S0895479897330182","volume":"20","author":"L Dieci","year":"1999","unstructured":"Dieci, L., Eirola, T.: On smooth decompositions of matrices. SIAM J. Matrix Anal. Appl. 20, 800\u2013819 (1999)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9156_CR16","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1137\/0731014","volume":"31","author":"L Dieci","year":"1994","unstructured":"Dieci, L., Russell, R.D., Van Vleck, E.S.: Unitary integrators and applications to continuous orthonormalization techniques. SIAM J. Numer. Anal. 31, 261\u2013281 (1994)","journal-title":"SIAM J. Numer. Anal."},{"key":"9156_CR17","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1137\/S0036142993247311","volume":"34","author":"L Dieci","year":"1997","unstructured":"Dieci, L., Russell, R.D., Van Vleck, E.S.: On the computation of Lyapunov exponents for continuous dynamical systems. SIAM J. Numer. Anal. 34, 402\u2013423 (1997)","journal-title":"SIAM J. Numer. Anal."},{"key":"9156_CR18","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1016\/0168-9274(95)00033-Q","volume":"17","author":"L Dieci","year":"1995","unstructured":"Dieci, L., Van Vleck, E.S.: Computation of a few Lyapunov exponents for continuous and discrete dynamical systems. Appl. Numer. Math. 17, 275\u2013291 (1995)","journal-title":"Appl. Numer. Math."},{"key":"9156_CR19","first-page":"197","volume-title":"Collected Lectures on the Preservation of Stability Under Discretization (Fort Collins, CO, 2001)","author":"L Dieci","year":"2002","unstructured":"Dieci, L., Van Vleck, E.S.: Lyapunov and other spectra: a survey. In: Collected Lectures on the Preservation of Stability Under Discretization (Fort Collins, CO, 2001), pp. 197\u2013218. SIAM, Philadelphia (2002)"},{"key":"9156_CR20","doi-asserted-by":"crossref","first-page":"516","DOI":"10.1137\/S0036142901392304","volume":"40","author":"L Dieci","year":"2002","unstructured":"Dieci, L., Van Vleck, E.S.: Lyapunov spectral intervals: theory and computation. SIAM J. Numer. Anal. 40, 516\u2013542 (2002)","journal-title":"SIAM J. Numer. Anal."},{"key":"9156_CR21","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1007\/s00211-005-0644-z","volume":"101","author":"L Dieci","year":"2005","unstructured":"Dieci, L., Van Vleck, E.S.: On the error in computing Lyapunov exponents by QR methods. Numer. Math. 101, 619\u2013642 (2005)","journal-title":"Numer. Math."},{"key":"9156_CR22","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1007\/s10884-006-9024-3","volume":"18","author":"L Dieci","year":"2006","unstructured":"Dieci, L., Van Vleck, E.S.: Perturbation theory for approximation of Lyapunov exponents by QR methods. J. Dyn. Diff. Equ. 18, 815\u2013842 (2006)","journal-title":"J. Dyn. Diff. Equ."},{"key":"9156_CR23","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/s10884-006-9030-5","volume":"19","author":"L Dieci","year":"2007","unstructured":"Dieci, L., Van Vleck, E.S.: Lyapunov and Sacker\u2013Sell spectral intervals. J. Dyn. Diff. Equ. 19, 265\u2013293 (2007)","journal-title":"J. Dyn. Diff. Equ."},{"key":"9156_CR24","doi-asserted-by":"crossref","first-page":"1166","DOI":"10.1137\/06067818X","volume":"46","author":"L Dieci","year":"2008","unstructured":"Dieci, L., Van Vleck, E.S.: On the error in QR integration. SIAM J. Numer. Anal. 46, 1166\u20131189 (2008)","journal-title":"SIAM J. Numer. Anal."},{"issue":"12","key":"9156_CR25","doi-asserted-by":"crossref","first-page":"2869","DOI":"10.1002\/aic.690481214","volume":"48","author":"M Diehl","year":"2002","unstructured":"Diehl, M., Leineweber, D.B., Sch\u00e4fer, A., Bock, H.G., Schl\u00f6der, J.P.: Optimization of multiple-fraction batch distillation with recycled waste cuts. AIChE J. 48(12), 2869\u20132874 (2002)","journal-title":"AIChE J."},{"key":"9156_CR26","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/978-3-662-04331-8_20","volume-title":"Online Optimization of Large Scale Systems: State of the Art","author":"M Diehl","year":"2001","unstructured":"Diehl, M., Uslu, I., Findeisen, R., Schwarzkopf, S., Allg\u00f6wer, F., Bock, H.G., B\u00fcrner, T., Gilles, E.D., Kienle, A., Schl\u00f6der, J.P., Stein, E.: Real-time optimization for large scale processes: nonlinear model predictive control of a high purity distillation column. In: Gr\u00f6tschel, M., Krumke, S.O., Rambau, J. (eds.) Online Optimization of Large Scale Systems: State of the Art, pp. 363\u2013384. Springer, Heidelberg (2001)"},{"key":"9156_CR27","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-663-09828-7","volume-title":"Numerical Methods in Multibody Systems","author":"E Eich-Soellner","year":"1998","unstructured":"Eich-Soellner, E., F\u00fchrer, C.: Numerical Methods in Multibody Systems. Teubner, Stuttgart (1998)"},{"key":"9156_CR28","doi-asserted-by":"crossref","first-page":"875","DOI":"10.1143\/PTP.83.875","volume":"83","author":"K Geist","year":"1990","unstructured":"Geist, K., Parlitz, U., Lauterborn, W.: Comparison of different methods for computing Lyapunov exponents. Prog. Theoret. Phys. 83, 875\u2013893 (1990)","journal-title":"Prog. Theoret. Phys."},{"key":"9156_CR29","volume-title":"Matrix Computations","author":"GH Golub","year":"1996","unstructured":"Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)","edition":"3"},{"key":"9156_CR30","doi-asserted-by":"crossref","first-page":"213","DOI":"10.1016\/0167-2789(87)90076-5","volume":"24","author":"JM Greene","year":"1983","unstructured":"Greene, J.M., Kim, J.-S.: The calculation of Lyapunov spectra. Physica D 24, 213\u2013225 (1983)","journal-title":"Physica D"},{"key":"9156_CR31","volume-title":"Differential-Algebraic Equations and their Numerical Treatment","author":"E Griepentrog","year":"1986","unstructured":"Griepentrog, E., M\u00e4rz, R.: Differential-Algebraic Equations and their Numerical Treatment. Teubner, Leipzig (1986)"},{"key":"9156_CR32","first-page":"97","volume":"8","author":"M G\u00fcnther","year":"1999","unstructured":"G\u00fcnther, M., Feldmann, U.: CAD-based electric-circuit modeling in industry I. Mathematical structure and index of network equations. Surv. Math. Ind. 8, 97\u2013129 (1999)","journal-title":"Surv. Math. Ind."},{"key":"9156_CR33","first-page":"131","volume":"8","author":"M G\u00fcnther","year":"1999","unstructured":"G\u00fcnther, M., Feldmann, U.: CAD-based electric-circuit modeling in industry II. Impact of circuit configurations and parameters. Surv. Math. Ind. 8, 131\u2013157 (1999)","journal-title":"Surv. Math. Ind."},{"key":"9156_CR34","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-05221-7","volume-title":"Solving Ordinary Differential Equations II: Stiff and Differential\u2013Algebraic Problems","author":"E Hairer","year":"1996","unstructured":"Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential\u2013Algebraic Problems, 2nd edn. Springer, Berlin (1996)","edition":"2"},{"key":"9156_CR35","first-page":"359","volume":"13","author":"P Kunkel","year":"2005","unstructured":"Kunkel, P., Mehrmann, V.: Characterization of classes of singular linear differential-algebraic equations. Electr. J. Lin. Alg. 13, 359\u2013386 (2005)","journal-title":"Electr. J. Lin. Alg."},{"key":"9156_CR36","doi-asserted-by":"crossref","DOI":"10.4171\/017","volume-title":"Differential\u2013Algebraic Equations. Analysis and Numerical Solution","author":"P Kunkel","year":"2006","unstructured":"Kunkel, P., Mehrmann, V.: Differential\u2013Algebraic Equations. Analysis and Numerical Solution. EMS, Z\u00fcrich (2006)"},{"key":"9156_CR37","first-page":"385","volume":"26","author":"P Kunkel","year":"2007","unstructured":"Kunkel, P., Mehrmann, V.: Stability properties of differential-algebraic equations and spin-stabilized discretizations. Electr. Trans. Numer. Anal. 26, 385\u2013420 (2007)","journal-title":"Electr. Trans. Numer. Anal."},{"key":"9156_CR38","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1137\/S1064827595286347","volume":"18","author":"P Kunkel","year":"1997","unstructured":"Kunkel, P., Mehrmann, V., Rath, W., Weickert, J.: A new software package for linear differential-algebraic equations. SIAM J. Sci. Comput. 18, 115\u2013138 (1997)","journal-title":"SIAM J. Sci. Comput."},{"key":"9156_CR39","unstructured":"Kunkel, P., Mehrmann, V., Seidel, S.: A MATLAB package for the numerical solution of general nonlinear differential-algebraic equations. Technical Report 16\/2005, Institut f\u00fcr Mathematik, TU Berlin, Berlin, Germany (2005). http:\/\/www.math.tu-berlin.de\/preprints\/"},{"key":"9156_CR40","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s10884-009-9128-7","volume":"21","author":"VH Linh","year":"2009","unstructured":"Linh, V.H., Mehrmann, V.: Lyapunov, Bohl and Sacker\u2013Sell spectral intervals for differential-algebraic equations. J. Dyn. Diff. Equ. 21, 153\u2013194 (2009)","journal-title":"J. Dyn. Diff. Equ."},{"issue":"1992","key":"9156_CR41","first-page":"521","volume":"55","author":"AM Lyapunov","year":"1907","unstructured":"Lyapunov, A.M.: The general problem of the stability of motion. Translated by A. T. Fuller from Edouard Davaux\u2019s French translation (1907) of the 1892 Russian original. Int. J. Control, 55, 521\u2013790 (1992)","journal-title":"Int. J. Control"},{"key":"9156_CR42","volume-title":"CRC Handbook of Dynamic System Modeling","author":"M Otter","year":"2006","unstructured":"Otter, M., Elmqvist, H., Mattson, S.E.: Multi-domain modeling with modelica. In: Fishwick, P. (ed.) CRC Handbook of Dynamic System Modeling. CRC, Boca Raton (2006)"},{"key":"9156_CR43","first-page":"183","volume-title":"Handbook of Numerical Analysis, vol. VIII","author":"PJ Rabier","year":"2002","unstructured":"Rabier, P.J., Rheinboldt, W.C.: Theoretical and numerical analysis of differential-algebraic equations. IN: Handbook of Numerical Analysis, vol. VIII, pp. 183\u2013540. North-Holland, Amsterdam (2002)"},{"key":"9156_CR44","doi-asserted-by":"crossref","DOI":"10.1142\/9789812791818","volume-title":"Differential-algebraic Systems. Analytical Aspects and Circuit Applications","author":"R Riaza","year":"2008","unstructured":"Riaza, R.: Differential-algebraic Systems. Analytical Aspects and Circuit Applications. World Scientific, Hackensack (2008)"},{"key":"9156_CR45","doi-asserted-by":"crossref","first-page":"2226","DOI":"10.1137\/08072557X","volume":"41","author":"R Riaza","year":"2010","unstructured":"Riaza, R.: Stability loss in quasilinear DAEs by divergence of a pencil eigenvalue. SIAM J. Math. Anal. 41, 2226\u20132245 (2010)","journal-title":"SIAM J. Math. Anal."},{"key":"9156_CR46","doi-asserted-by":"crossref","first-page":"320","DOI":"10.1016\/0022-0396(78)90057-8","volume":"27","author":"RJ Sacker","year":"1978","unstructured":"Sacker, R.J., Sell, G.R.: A spectral theory for linear differential systems. J. Diff. Equ. 27, 320\u2013358 (1978)","journal-title":"J. Diff. Equ."},{"key":"9156_CR47","doi-asserted-by":"crossref","first-page":"509","DOI":"10.1137\/0714030","volume":"14","author":"GW Stewart","year":"1977","unstructured":"Stewart, G.W.: Perturbation bounds for the $\\mathit{QR}$ factorization of a matrix. SIAM J. Numer. Anal. 14, 509\u2013518 (1977)","journal-title":"SIAM J. Numer. Anal."},{"key":"9156_CR48","doi-asserted-by":"crossref","first-page":"1775","DOI":"10.1137\/090761562","volume":"31","author":"ES Vleck Van","year":"2010","unstructured":"Van Vleck, E.S.: On the error in the product $\\mathit{QR}$ decomposition. SIAM J. Matrix Anal. Appl. 31, 1775\u20131791 (2010)","journal-title":"SIAM J. Matrix Anal. Appl."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-010-9156-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10444-010-9156-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-010-9156-1","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,29]],"date-time":"2019-05-29T13:08:13Z","timestamp":1559135293000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10444-010-9156-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,6,11]]},"references-count":48,"journal-issue":{"issue":"2-4","published-print":{"date-parts":[[2011,11]]}},"alternative-id":["9156"],"URL":"https:\/\/doi.org\/10.1007\/s10444-010-9156-1","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,6,11]]}}}