Abstract
Modern modeling approaches for circuit analysis 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. The index of the DAE arising from the modified nodal analysis (MNA) is determined uniquely by the structure of the circuit. Instead, we consider a broader class of analysis method called the hybrid analysis. For linear time-invariant electric circuits, we devise a combinatorial algorithm for finding an optimal hybrid analysis in which the index of the DAE to be solved attains the minimum. The optimal hybrid analysis often results in a DAE with lower index than MNA.
Similar content being viewed by others
References
Amari S.: Topological foundations of Kron’s tearing of electric networks. RAAG Mem. 3(F-VI), 322–350 (1962)
Aspvall B., Plass M.F., Tarjan R.E.: A linear-time algorithm for testing the truth of certain qualified Boolean formulas. Inf. Process. Lett. 8, 121–123 (1979)
Branin F.H.: The relation between Kron’s method and the classical methods of network analysis. Matrix Tensor Q. 12, 69–115 (1962)
Brenan K.E., Campbell S.L., Petzold L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 2nd edn. SIAM, Philadelphia (1996)
Bujakiewicz, P.: Maximum Weighted Matching for High Index Differential Algebraic Equations. Doctor’s dissertation, Delft University of Technology (1994)
Campbell S.L., Gear C.W.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)
Emoto, K., Matsuoka, Y.: VIAP: degree of subdeterminant of mixed polynomial matrix. http://www.sr3.t.u-tokyo.ac.jp/research/CCF/ccf.html (2004)
Gantmacher F.R.: The Theory of Matrices. Chelsea, New York (1959)
Gear C.W.: Simultaneous numerical solution of differential-algebraic equations. IEEE Trans. Circ. Theory 18, 89–95 (1971)
Günther M., Rentrop P.: The differential-algebraic index concept in electric circuit simulation. Z. Angew. Math. Mech. 76(Suppl 1), 91–94 (1996)
Hairer E., Wanner G.: Solving Ordinary Differential Equations II, 2nd edn. Springer, Berlin (1996)
Iri M.: A min-max theorem for the ranks and term-ranks of a class of matrices: an algebraic approach to the problem of the topological degrees of freedom of a network (in Japanese). Trans. Inst. Electron. Commun. Eng. Jpn. 51A, 180–187 (1968)
Iri M.: Applications of Matroid Theory. Mathematical Programming—The State of the Art, pp. 158–. Springer, Berlin (1983)
Iwata S.: Computing the maximum degree of minors in matrix pencils via combinatorial relaxation. Algorithmica 36, 331–341 (2003)
Iwata, S., Takamatsu, M.: Computing the degrees of all cofactors in mixed polynomial matrices. SIAM J. Discrete Math. (2008, in press)
Kishi G., Kajitani Y.: Maximally distinct trees in a linear graph (in Japanese). Trans. Inst. Electron. Commun. Eng. Jpn. 51A, 196–203 (1968)
Kron G.: Tensor Analysis of Networks. Wiley, New York (1939)
Murota K.: Matrices and Matroids for Systems Analysis. Springer, Berlin (2000)
Narayanan H.: Submodular Functions and Electrical Networks. Elsevier, Amsterdam (1997)
Ohtsuki T., Ishizaki Y., Watanabe H.: Network analysis and topological degrees of freedom (in Japanese). Trans. Inst. Electron. Commun. Eng. Jpn. 51A, 238–245 (1968)
Recski A.: Matroid Theory and Its Applications in Electric Network Theory and in Statics. Springer, Berlin (1989)
Schulz, S.: Four Lectures on Differential-Algebraic Equations. Technical Report 497, The University of Auckland, New Zealand (2003)
Schwarz D.E., Tischendorf C.: Structural analysis of electric circuits and consequences for MNA. Int. J. Circ. Theory Appl. 28, 131–162 (2000)
Takamatsu, M., Iwata, S.: Index characterization of differential-algebraic equations in hybrid analysis for circuit simulation. METR 2008-10, Department of Mathematical Informatics, University of Tokyo (2008)
Tischendorf C.: Topological index calculation of differential-algebraic equations in circuit simulation. Surv. Math. Ind. 8, 187–199 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Iwata, S., Takamatsu, M. Index minimization of differential-algebraic equations in hybrid analysis for circuit simulation. Math. Program. 121, 105–121 (2010). https://doi.org/10.1007/s10107-008-0227-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-008-0227-8