Mathematical Programming

, Volume 121, Issue 1, pp 105–121

Index minimization of differential-algebraic equations in hybrid analysis for circuit simulation

FULL LENGTH PAPER Series A

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.

Keywords

Circuit simulation DAE Index Matrix pencil 

Mathematics Subject Classification (2000)

15A22 34A09 65L80 68Q25 

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References

  1. 1.
    Amari S.: Topological foundations of Kron’s tearing of electric networks. RAAG Mem. 3(F-VI), 322–350 (1962)Google Scholar
  2. 2.
    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)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Branin F.H.: The relation between Kron’s method and the classical methods of network analysis. Matrix Tensor Q. 12, 69–115 (1962)Google Scholar
  4. 4.
    Brenan K.E., Campbell S.L., Petzold L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, 2nd edn. SIAM, Philadelphia (1996)MATHGoogle Scholar
  5. 5.
    Bujakiewicz, P.: Maximum Weighted Matching for High Index Differential Algebraic Equations. Doctor’s dissertation, Delft University of Technology (1994)Google Scholar
  6. 6.
    Campbell S.L., Gear C.W.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    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)
  8. 8.
    Gantmacher F.R.: The Theory of Matrices. Chelsea, New York (1959)MATHGoogle Scholar
  9. 9.
    Gear C.W.: Simultaneous numerical solution of differential-algebraic equations. IEEE Trans. Circ. Theory 18, 89–95 (1971)CrossRefGoogle Scholar
  10. 10.
    Günther M., Rentrop P.: The differential-algebraic index concept in electric circuit simulation. Z. Angew. Math. Mech. 76(Suppl 1), 91–94 (1996)MATHGoogle Scholar
  11. 11.
    Hairer E., Wanner G.: Solving Ordinary Differential Equations II, 2nd edn. Springer, Berlin (1996)MATHGoogle Scholar
  12. 12.
    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)MathSciNetGoogle Scholar
  13. 13.
    Iri M.: Applications of Matroid Theory. Mathematical Programming—The State of the Art, pp. 158–. Springer, Berlin (1983)Google Scholar
  14. 14.
    Iwata S.: Computing the maximum degree of minors in matrix pencils via combinatorial relaxation. Algorithmica 36, 331–341 (2003)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Iwata, S., Takamatsu, M.: Computing the degrees of all cofactors in mixed polynomial matrices. SIAM J. Discrete Math. (2008, in press)Google Scholar
  16. 16.
    Kishi G., Kajitani Y.: Maximally distinct trees in a linear graph (in Japanese). Trans. Inst. Electron. Commun. Eng. Jpn. 51A, 196–203 (1968)MathSciNetGoogle Scholar
  17. 17.
    Kron G.: Tensor Analysis of Networks. Wiley, New York (1939)Google Scholar
  18. 18.
    Murota K.: Matrices and Matroids for Systems Analysis. Springer, Berlin (2000)MATHGoogle Scholar
  19. 19.
    Narayanan H.: Submodular Functions and Electrical Networks. Elsevier, Amsterdam (1997)MATHGoogle Scholar
  20. 20.
    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)MathSciNetGoogle Scholar
  21. 21.
    Recski A.: Matroid Theory and Its Applications in Electric Network Theory and in Statics. Springer, Berlin (1989)Google Scholar
  22. 22.
    Schulz, S.: Four Lectures on Differential-Algebraic Equations. Technical Report 497, The University of Auckland, New Zealand (2003)Google Scholar
  23. 23.
    Schwarz D.E., Tischendorf C.: Structural analysis of electric circuits and consequences for MNA. Int. J. Circ. Theory Appl. 28, 131–162 (2000)MATHCrossRefGoogle Scholar
  24. 24.
    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)Google Scholar
  25. 25.
    Tischendorf C.: Topological index calculation of differential-algebraic equations in circuit simulation. Surv. Math. Ind. 8, 187–199 (1999)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Research Institute for Mathematical SciencesKyoto UniversityKyotoJapan
  2. 2.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan

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