{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T21:54:12Z","timestamp":1775858052598,"version":"3.50.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,1,29]],"date-time":"2021-01-29T00:00:00Z","timestamp":1611878400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,1,29]],"date-time":"2021-01-29T00:00:00Z","timestamp":1611878400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2021,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The recently developed new algorithm for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization opens new possibilities to apply Taylor series integration methods. In this paper, we show how corresponding projected explicit and implicit Taylor series methods can be adapted to DAEs of arbitrary index. Owing to our formulation as a projected optimization problem constrained by the derivative array, no explicit description of the inherent dynamics is necessary, and various Taylor integration schemes can be defined in a general framework. In particular, we address higher-order Pad\u00e9 methods that stand out due to their stability. We further discuss several aspects of our prototype implemented in Python using Automatic Differentiation. The methods have been successfully tested on examples arising from multibody systems simulation and a higher-index DAE benchmark arising from servo-constraint problems.<\/jats:p>","DOI":"10.1007\/s11075-020-01051-z","type":"journal-article","created":{"date-parts":[[2021,1,29]],"date-time":"2021-01-29T19:03:17Z","timestamp":1611946997000},"page":"615-646","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["Projected explicit and implicit Taylor series methods for DAEs"],"prefix":"10.1007","volume":"88","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1478-075X","authenticated-orcid":false,"given":"Diana","family":"Est\u00e9vez Schwarz","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ren\u00e9","family":"Lamour","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,1,29]]},"reference":[{"key":"1051_CR1","volume-title":"Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables","author":"M Abramowitz","year":"1972","unstructured":"Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas Graphs and Mathematical Tables. Dover, New York (1972)"},{"issue":"2-4","key":"1051_CR2","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/S0898-1221(97)00117-X","volume":"34","author":"PG Akishin","year":"1997","unstructured":"Akishin, P.G., Puzynin, I.V., Vinitsky, S.I.: A hybrid numerical method for analysis of dynamics of the classical Hamiltonian systems. Comput. Math. Appl. 34(2-4), 45\u201373 (1997)","journal-title":"Comput. Math. Appl."},{"issue":"2","key":"1051_CR3","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1016\/j.amc.2004.02.015","volume":"163","author":"R Barrio","year":"2005","unstructured":"Barrio, R.: Performance of the Taylor series method for ODEs\/DAEs. Appl. Math. Comput. 163(2), 525\u2013545 (2005)","journal-title":"Appl. Math. Comput."},{"key":"1051_CR4","first-page":"14","volume-title":"Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Unabridged, corr. republ. Classics in Applied Mathematics","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. Unabridged, corr. republ. Classics in Applied Mathematics, p 14. SIAM Society for Industrial and Applied Mathematics, Philadelphia (1996)"},{"key":"1051_CR5","doi-asserted-by":"publisher","first-page":"334","DOI":"10.1137\/0906024","volume":"6","author":"SL Campbell","year":"1985","unstructured":"Campbell, S.L.: The numerical solution of higher index linear time varying singular systems of differential equations. SIAM J. Sci. Stat. Comput. 6, 334\u2013348 (1985)","journal-title":"SIAM J. Sci. Stat. Comput."},{"issue":"2","key":"1051_CR6","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1007\/s002110050165","volume":"72","author":"SL Campbell","year":"1995","unstructured":"Campbell, S.L., Gear, C.W.: The index of general nonlinear DAEs. Numer. Math. 72(2), 173\u2013196 (1995)","journal-title":"Numer. Math."},{"key":"1051_CR7","doi-asserted-by":"crossref","unstructured":"Corliss, G.F., Griewank, A., Henneberger, P., Kirlinger, G., Potra, F.A., Stetter, H.J.: High-order stiff ODE solvers via automatic differentiation and rational prediction. In: Vulkov, L., Wa\u015bniewski, J., Yalamov, P. (eds.) Numerical Analysis and Its Applications. WNAA 1996. Lecture Notes in Computer Science, vol. 1196, pp 114\u2013124 (1997)","DOI":"10.1007\/3-540-62598-4_85"},{"key":"1051_CR8","volume-title":"Numerical mathematics 2. Ordinary differential equations. (Numerische Mathematik 2 Gew\u00f6hnliche Differentialgleichungen.) 4th revised and augmented ed.","author":"P Deuflhard","year":"2013","unstructured":"Deuflhard, P., Bornemann, F.: Numerical mathematics 2. Ordinary differential equations. (Numerische Mathematik 2 Gew\u00f6hnliche Differentialgleichungen.) 4th revised and augmented ed. de Gruyter Studium, Berlin (2013)"},{"key":"1051_CR9","unstructured":"Dimova, S.N., Hristov, I.G., Hristova, R.D., Puzynin, I.V., Puzynina, T.P., Sharipov, Z.A., Shegunov, N.G., Tukhliev, Z.K.: Combined explicit-implicit Taylor Series Methods. In: Proceedings of the VIII International Conference \u201cDistributed Computing and Grid-technologies in Science and Education\u201d (GRID 2018), Dubna, Moscow region, Russia, September 10 -14 (2018)"},{"issue":"1","key":"1051_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1002\/cta.168","volume":"30","author":"D Est\u00e9vez Schwarz","year":"2002","unstructured":"Est\u00e9vez Schwarz, D.: A step-by-step approach to compute a consistent initialization for the MNA. Int. J. Circuit Theory Appl. 30(1), 1\u20136 (2002)","journal-title":"Int. J. Circuit Theory Appl."},{"issue":"4","key":"1051_CR11","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1007\/s11075-009-9304-1","volume":"52","author":"D Est\u00e9vez Schwarz","year":"2009","unstructured":"Est\u00e9vez Schwarz, D.: Consistent initialization for DAEs in Hessenberg form. Numer. Algorithms 52(4), 629\u2013648 (2009)","journal-title":"Numer. Algorithms"},{"key":"1051_CR12","unstructured":"Est\u00e9vez Schwarz, D., Lamour, R.: InitDAE\u2019s documentation. Available from: https:\/\/www.mathematik.hu-berlin.de\/~lamour\/software\/python\/InitDAE\/html\/"},{"key":"1051_CR13","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1016\/j.cam.2013.09.018","volume":"262","author":"D Est\u00e9vez Schwarz","year":"2014","unstructured":"Est\u00e9vez Schwarz, D., Lamour, R.: Projector based integration of DAEs with the Taylor series method using automatic differentiation. J. Comput. Appl Math. 262, 62\u201372 (2014)","journal-title":"J. Comput. Appl Math."},{"issue":"2","key":"1051_CR14","doi-asserted-by":"publisher","first-page":"535","DOI":"10.1007\/s11075-016-0107-x","volume":"73","author":"D Est\u00e9vez Schwarz","year":"2016","unstructured":"Est\u00e9vez Schwarz, D., Lamour, R.: A new projector based decoupling of linear DAEs for monitoring singularities. Numer. Algorithms 73(2), 535\u2013565 (2016)","journal-title":"Numer. Algorithms"},{"key":"1051_CR15","volume-title":"Consistent initialization for higher-index DAEs using a projector based minimum-norm specification. Technical Report 1","author":"D Est\u00e9vez Schwarz","year":"2016","unstructured":"Est\u00e9vez Schwarz, D., Lamour, R.: Consistent initialization for higher-index DAEs using a projector based minimum-norm specification. Technical Report 1. Institut f\u00fcr Mathematik, Humboldt-Universit\u00e4t zu Berlin (2016)"},{"issue":"2","key":"1051_CR16","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1007\/s11075-017-0379-9","volume":"78","author":"D Est\u00e9vez Schwarz","year":"2018","unstructured":"Est\u00e9vez Schwarz, D., Lamour, R.: A new approach for computing consistent initial values and Taylor coefficients for DAEs using projector-based constrained optimization. Numer. Algorithms 78(2), 355\u2013377 (2018)","journal-title":"Numer. Algorithms"},{"key":"1051_CR17","doi-asserted-by":"publisher","unstructured":"Est\u00e9vez Schwarz, D., Lamour, R.: InitDAE: Computation of consistent values, index determination and diagnosis of singularities of DAEs using automatic differentiation in Python. Journal of Computational and Applied Mathematics. https:\/\/doi.org\/10.1016\/j.cam.2019.112486 (2019)","DOI":"10.1016\/j.cam.2019.112486"},{"key":"1051_CR18","doi-asserted-by":"crossref","unstructured":"Est\u00e9vez Schwarz, D., Lamour, R.: A projector based decoupling of DAEs obtained from the derivative array. In: Progress in Differential-Algebraic Equations II, Differential-Algebraic Equations Forum (DAE-F) (2020)","DOI":"10.1007\/978-3-030-53905-4_1"},{"key":"1051_CR19","doi-asserted-by":"crossref","unstructured":"Est\u00e9vez Schwarz, D., Lamour, R., M\u00e4rz, R.: Singularities of the Robotic Arm DAE. Progress in Differential-Algebraic Equations II, Differential-Algebraic Equations Forum (DAE-F) (2020)","DOI":"10.1007\/978-3-030-53905-4_14"},{"key":"1051_CR20","doi-asserted-by":"crossref","unstructured":"Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II. Springer (1996)","DOI":"10.1007\/978-3-642-05221-7"},{"key":"1051_CR21","unstructured":"Kirlinger, G., Corliss, G.F.: On implicit Taylor series methods for stiff ODEs. In: Computer Arithmetic and Enclosure Methods. Proceedings of the 3rd International IMACS-GAMM Symposium on Computer Arithmetic and Scientific Computing (SCAN-91), Oldenburg, Germany, 1-4 October 1991, pp 371\u2013379, Amsterdam (1992)"},{"key":"1051_CR22","doi-asserted-by":"publisher","DOI":"10.4171\/017","volume-title":"Differential-Algebraic Equations - Analysis and Numerical Solution","author":"P Kunkel","year":"2006","unstructured":"Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations - Analysis and Numerical Solution. EMS Publishing House, Z\u00fcrich (2006)"},{"key":"1051_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-27555-5","volume-title":"Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum, vol. 1","author":"R Lamour","year":"2013","unstructured":"Lamour, R., M\u00e4rz, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum, vol. 1. Springer, Berlin (2013)"},{"key":"1051_CR24","unstructured":"Mazzia, F., Magherini, C.: Test set for initial value problems, release 2.4. Technical report, Department of Mathematics, University of Bari and INdAM, Research Unit of Bari, February 2008. Available from: http:\/\/pitagora.dm.uniba.it\/~testset"},{"key":"1051_CR25","volume-title":"Applications of Differential-Algebraic Equations: Examples and Benchmarks, chapter Open-loop Control of Underactuated Mechanical Systems Using Servo-constraints: Analysis and Some Examples. Differential-Algebraic Equations Forum","author":"S Otto","year":"2019","unstructured":"Otto, S., Seifried, R.: Applications of Differential-Algebraic Equations: Examples and Benchmarks, chapter Open-loop Control of Underactuated Mechanical Systems Using Servo-constraints: Analysis and Some Examples. Differential-Algebraic Equations Forum. Springer, Cham (2019)"},{"issue":"1\u20134","key":"1051_CR26","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1023\/A:1019150322187","volume":"19","author":"JD Pryce","year":"1998","unstructured":"Pryce, J.D.: Solving high-index DAEs by Taylor series. Numer. Algorithms 19(1\u20134), 195\u2013211 (1998)","journal-title":"Numer. Algorithms"},{"issue":"4\u20136","key":"1051_CR27","doi-asserted-by":"publisher","first-page":"729","DOI":"10.1080\/10556788.2018.1428605","volume":"33","author":"JD Pryce","year":"2018","unstructured":"Pryce, J.D., Nedialkov, N.S., Tan, G., Li, X.: How AD can help solve differential-algebraic equations. Optim. Methods Softw. 33(4\u20136), 729\u2013749 (2018)","journal-title":"Optim. Methods Softw."},{"key":"1051_CR28","doi-asserted-by":"publisher","DOI":"10.1142\/6746","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":"1051_CR29","unstructured":"Scott, J.R., Solving, ODE: Initial Value Problems with Implicit Taylor Series Methods. Technical report NASA\/TM-2000-209400 (2000)"},{"key":"1051_CR30","doi-asserted-by":"publisher","first-page":"113","DOI":"10.5194\/ms-4-113-2013","volume":"4","author":"R Seifried","year":"2013","unstructured":"Seifried, R., Blajer, W.: Analysis of servo-constraint problems for underactuated multibody systems. Mech. Sci. 4, 113\u2013129 (2013)","journal-title":"Mech. Sci."},{"issue":"5","key":"1051_CR31","doi-asserted-by":"publisher","first-page":"334","DOI":"10.1016\/j.jocs.2011.10.007","volume":"4","author":"SF Walter","year":"2013","unstructured":"Walter, S.F., Lehmann, L.: Algorithmic differentiation in Python with AlgoPy. J. Comput. Sci. 4(5), 334\u2013344 (2013)","journal-title":"J. Comput. Sci."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-01051-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-020-01051-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-020-01051-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,29]],"date-time":"2023-10-29T17:33:53Z","timestamp":1698600833000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-020-01051-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,29]]},"references-count":31,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,10]]}},"alternative-id":["1051"],"URL":"https:\/\/doi.org\/10.1007\/s11075-020-01051-z","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,29]]},"assertion":[{"value":"3 October 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 November 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 January 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 February 2021","order":4,"name":"change_date","label":"Change Date","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Update","order":5,"name":"change_type","label":"Change Type","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"Family name presentation of the 1st Author","order":6,"name":"change_details","label":"Change Details","group":{"name":"ArticleHistory","label":"Article History"}}]}}