Modern modeling approaches for circuit simulation such as the modified nodal analysis\u00a0(MNA) lead to differential-algebraic equations (DAEs). The index of a DAE is a measure of the degree of numerical difficulty. In general, the higher the index is, the more difficult it is to solve the DAE.<\/p>\n
In this paper, we consider a broader class of analysis methods called the hybrid analysis. For nonlinear time-varying circuits with general dependent sources, we give a structural characterization of the tractability index of DAEs arising from the hybrid analysis. This enables us to determine the tractability index efficiently, which helps to avoid solving higher index DAEs in circuit simulation.<\/p>","DOI":"10.1090\/s0025-5718-2011-02558-5","type":"journal-article","created":{"date-parts":[[2011,11,8]],"date-time":"2011-11-08T13:38:34Z","timestamp":1320759514000},"page":"923-939","source":"Crossref","is-referenced-by-count":13,"title":["Tractability index of hybrid equations for circuit simulation"],"prefix":"10.1090","volume":"81","author":[{"given":"Satoru","family":"Iwata","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mizuyo","family":"Takamatsu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Caren","family":"Tischendorf","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,11,8]]},"reference":[{"key":"1","unstructured":"S. Amari, Topological foundations of Kron\u2019s tearing of electric networks, RAAG Memoirs 3 (1962), 322\u2013350."},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"783","DOI":"10.4171\/ZAA\/1108","article-title":"A unified approach to linear differential algebraic equations and their adjoints","volume":"21","author":"Balla, K.","year":"2002","journal-title":"Z. Anal. Anwendungen","ISSN":"https:\/\/id.crossref.org\/issn\/0232-2064","issn-type":"print"},{"key":"3","series-title":"CMS Books in Mathematics\/Ouvrages de Math\\'{e}matiques de la SMC","isbn-type":"print","volume-title":"Generalized inverses","volume":"15","author":"Ben-Israel, Adi","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0387002936","edition":"2"},{"key":"4","unstructured":"F. H. Branin, The relation between Kron\u2019s method and the classical methods of network analysis, The Matrix and Tensor Quarterly 12 (1962), 69\u2013115."},{"key":"5","series-title":"Classics in Applied Mathematics","isbn-type":"print","volume-title":"Numerical solution of initial-value problems in differential-algebraic equations","volume":"14","author":"Brenan, K. E.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0898713536"},{"key":"6","doi-asserted-by":"crossref","unstructured":"P. R. Bryant, The order of complexity of electrical networks, Proceedings of the Institution of Electrical Engineers, Part C 106 (1959), 174\u2013188.","DOI":"10.1049\/pi-c.1959.0031"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1007\/s002110050165","article-title":"The index of general nonlinear DAEs","volume":"72","author":"Campbell, Stephen L.","year":"1995","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"11","key":"8","doi-asserted-by":"publisher","first-page":"1059","DOI":"10.1109\/TCS.1980.1084745","article-title":"Dynamic nonlinear networks: state-of-the-art","volume":"27","author":"Chua, Leon O.","year":"1980","journal-title":"IEEE Trans. Circuits and Systems","ISSN":"https:\/\/id.crossref.org\/issn\/0098-4094","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","unstructured":"D. Est\u00e9vez Schwarz and C. Tischendorf, Structural analysis of electric circuits and consequences for MNA, International Journal of Circuit Theory and Applications 28 (2000), 131\u2013162.","DOI":"10.1002\/(SICI)1097-007X(200003\/04)28:2<131::AID-CTA100>3.0.CO;2-W"},{"key":"10","doi-asserted-by":"crossref","unstructured":"W. Fischer, Equivalent circuit and gain of MOS field effect transistors, Solid-State Electronics 9 (1966), 71\u201381.","DOI":"10.1016\/0038-1101(66)90026-8"},{"key":"11","series-title":"Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]","isbn-type":"print","volume-title":"Differential-algebraic equations and their numerical treatment","volume":"88","author":"Griepentrog, Eberhard","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/3322003434"},{"key":"12","doi-asserted-by":"crossref","unstructured":"M. G\u00fcnther and P. Rentrop, The differential-algebraic index concept in electric circuit simulation, Zeitschrift f\u00fcr angewandte Mathematik und Mechanik 76, supplement 1 (1996), 91\u201394.","DOI":"10.1007\/978-3-322-82967-2_34"},{"key":"13","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-05221-7","volume-title":"Solving ordinary differential equations. II","volume":"14","author":"Hairer, E.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/3540604529","edition":"2"},{"issue":"2-3","key":"14","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1016\/S0168-9274(02)00215-5","article-title":"Stability preserving integration of index-1 DAEs","volume":"45","author":"Higueras, I.","year":"2003","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"key":"15","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511810817","volume-title":"Matrix analysis","author":"Horn, Roger A.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0521305861"},{"issue":"5","key":"16","first-page":"18","article-title":"A min-max theorem for the ranks and term-ranks of a class of matrices\u2014An algebraic approach to the problem of the topological degrees of freedom of a network","volume":"51","author":"Iri, Masao","year":"1968","journal-title":"Electron. Comm. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0424-8368","issn-type":"print"},{"key":"17","isbn-type":"print","first-page":"158","article-title":"Applications of matroid theory","author":"Iri, M.","year":"1983","ISBN":"https:\/\/id.crossref.org\/isbn\/3540120823"},{"issue":"1","key":"18","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1007\/s10107-008-0227-8","article-title":"Index minimization of differential-algebraic equations in hybrid analysis for circuit simulation","volume":"121","author":"Iwata, Satoru","year":"2010","journal-title":"Math. Program.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5610","issn-type":"print"},{"key":"19","doi-asserted-by":"crossref","unstructured":"S. Iwata, M. Takamatsu, and C. Tischendorf, Hybrid analysis of nonlinear time-varying circuits providing DAEs with index at most one, Scientific Computing in Electrical Engineering SCEE 2008 (J. Roos and L. R. J. Costa, eds.), Mathematics in Industry, vol. 14, Springer, 2010, pp. 151\u2013158.","DOI":"10.1007\/978-3-642-12294-1_20"},{"issue":"5","key":"20","first-page":"35","article-title":"Maximally distinct trees in a linear graph","volume":"51","author":"Kishi, Genya","year":"1968","journal-title":"Electron. Comm. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0424-8368","issn-type":"print"},{"key":"21","unstructured":"G. Kron, Tensor Analysis of Networks, John Wiley and Sons, New York, 1939."},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1016\/0377-0427(94)90080-9","article-title":"Canonical forms for linear differential-algebraic equations with variable coefficients","volume":"56","author":"Kunkel, Peter","year":"1994","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"9","key":"23","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1002\/zamm.200310127","article-title":"Index reduction for differential-algebraic equations by minimal extension","volume":"84","author":"Kunkel, Peter","year":"2004","journal-title":"ZAMM Z. Angew. Math. Mech.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-2267","issn-type":"print"},{"key":"24","isbn-type":"print","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1017\/s0962492900002269","article-title":"Numerical methods for differential algebraic equations","author":"M\u00e4rz, Roswitha","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0521410266"},{"key":"25","unstructured":"\\bysame, Nonlinear differential-algebraic equations with properly formulated leading term, Tech. Report 01-3, Department of Mathematics, Humboldt-Universit\u00e4t zu Berlin, 2001, http:\/\/www.mathematik.hu-berlin.de\/publ\/pre\/2001\/P-01-3.ps."},{"issue":"3-4","key":"26","doi-asserted-by":"publisher","first-page":"308","DOI":"10.1007\/BF03322858","article-title":"The index of linear differential algebraic equations with properly stated leading terms","volume":"42","author":"M\u00e4rz, R.","year":"2002","journal-title":"Results Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1422-6383","issn-type":"print"},{"issue":"2","key":"27","doi-asserted-by":"publisher","first-page":"1279","DOI":"10.1016\/j.jmaa.2005.11.038","article-title":"Linear differential-algebraic equations with properly stated leading term: regular points","volume":"323","author":"M\u00e4rz, Roswitha","year":"2006","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"3","key":"28","doi-asserted-by":"publisher","first-page":"677","DOI":"10.1137\/0914043","article-title":"Index reduction in differential-algebraic equations using dummy derivatives","volume":"14","author":"Mattsson, Sven Erik","year":"1993","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"29","series-title":"Annals of Discrete Mathematics","isbn-type":"print","volume-title":"Submodular functions and electrical networks","volume":"54","author":"Narayanan, H.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0444825231"},{"issue":"6","key":"30","first-page":"33","article-title":"Network analysis and topological degrees of freedom","volume":"51","author":"Ohtsuki, Tatsuo","year":"1968","journal-title":"Electron. Comm. Japan","ISSN":"https:\/\/id.crossref.org\/issn\/0424-8368","issn-type":"print"},{"key":"31","unstructured":"J. M. Rabaey, The spice page, \\url{http:\/\/bwrc.eecs.berkeley.edu\/Classes\/icbook\/SPICE\/}."},{"key":"32","doi-asserted-by":"crossref","unstructured":"A. Recski, Matroid Theory and Its Applications in Electric Network Theory and in Statics, Springer-Verlag, Berlin, 1989.","DOI":"10.1007\/978-3-662-22143-3"},{"key":"33","doi-asserted-by":"crossref","unstructured":"G. Rei\u00dfig, The index of the standard circuit equations of passive RLCTG-networks does not exceed 2, Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (ISCAS \u201998) 3 (1998), 419\u2013422.","DOI":"10.1109\/ISCAS.1998.704039"},{"key":"34","doi-asserted-by":"crossref","unstructured":"\\bysame, Extension of the normal tree method, International Journal of Circuit Theory and Applications 27 (1999), 241\u2013265.","DOI":"10.1002\/(SICI)1097-007X(199903\/04)27:2<241::AID-CTA62>3.0.CO;2-8"},{"issue":"168","key":"35","doi-asserted-by":"publisher","first-page":"473","DOI":"10.2307\/2008288","article-title":"Differential-algebraic systems as differential equations on manifolds","volume":"43","author":"Rheinboldt, Werner C.","year":"1984","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"36","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1142\/6746","volume-title":"Differential-algebraic systems","author":"Riaza, Ricardo","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9789812791801"},{"issue":"1","key":"37","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1023\/B:ACAP.0000045308.01276.41","article-title":"Linear index-1 DAEs: regular and singular problems","volume":"84","author":"Riaza, Ricardo","year":"2004","journal-title":"Acta Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0167-8019","issn-type":"print"},{"key":"38","doi-asserted-by":"crossref","unstructured":"M. Takamatsu and S. Iwata, Index characterization of differential-algebraic equations in hybrid analysis for circuit simulation, International Journal of Circuit Theory and Applications 38 (2010), 419\u2013440.","DOI":"10.1002\/cta.577"},{"issue":"3-4","key":"39","first-page":"187","article-title":"Topological index calculation of differential-algebraic equations in circuit simulation","volume":"8","author":"Tischendorf, Caren","year":"1999","journal-title":"Surveys Math. Indust.","ISSN":"https:\/\/id.crossref.org\/issn\/0938-1953","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2012-81-278\/S0025-5718-2011-02558-5\/S0025-5718-2011-02558-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2012-81-278\/S0025-5718-2011-02558-5\/S0025-5718-2011-02558-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:09:18Z","timestamp":1776791358000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2012-81-278\/S0025-5718-2011-02558-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,11,8]]},"references-count":39,"journal-issue":{"issue":"278","published-print":{"date-parts":[[2012,4]]}},"alternative-id":["S0025-5718-2011-02558-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02558-5","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2011,11,8]]}}}