*Michael Sielemann*: Deutsches Zentrum für Luft- und Raumfahrt, Robotics and Mechatronics Center, System Dynamics and Control, Germany

**Keywords:**Initialization; DAE; homotopy; nonlinear equations

## Proceedings of the 9th International MODELICA Conference; September 3-5; 2012; Munich; Germany

[2] J. E. Dennis and R. B. Schnabel. Numerical methods for unconstrained optimization and nonlinear equations. SIAM Classics in Applied Mathematics; 1996. doi: 10.1137/1.9781611971200.

[3] P. Deuflhard. Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms. Springer Verlag; 2004.

[4] P. Dimo. Nodal analysis of power systems. Taylor & Francis; 1975.

[5] R. Duffin. Nonlinear networks lIa. B. Am. Math. Soc.; 53:963–971; 1947. doi: 10.1090/S0002-9904-1947-08917-5.

[6] H. Elmqvist. A Structured Model Language for Large Continuous Systems. PhD thesis; Lund University; Department of Automatic Control; Sweden; May 1978.

[7] M. Green and R. Melville. Sufficient conditions for finding multiple operating points of dc circuits using continuation methods. In IEEE International Symposium on Circuits and Systems; pages 117–120; Seattle; 1995.

[8] M. A. Heroux; R. A. Bartlett; V. E. Howle; R. J. Hoekstra; J. J. Hu; T. G. Kolda; R. B. Lehoucq; K. R. Long; R. P. Pawlowski; E. T. Phipps; A. G. Salinger; H. K. Thornquist; R. S. Tuminaro; J. M. Willenbring; A. Williams; and K. S. Stanley. An overview of the Trilinos project. Acm. T. Math. Software.; 31(3):397–423; 2005. doi: 10.1145/1089014.1089021.

[9] Y. Inoue. A practical algorithm for DC operatingpoint analysis of large-scale circuits. Electronics and Communications in Japan (Part III: Fundamental Electronic Science); 77(10):49–62; 1994. doi: 10.1002/ecjc.4430771005.

[10] C. T. Kelley. Solving nonlinear equations with Newton’s method. SIAM Classics in Applied Mathematics; 2003. doi: 10.1137/1.9780898718898.

[11] W. Mathis; L. Trajkovic; M. Koch; and U. Feldmann. Parameter embedding methods for finding DC operating points of transistor circuits. In Third international specialist workshop on Nonlinear Dynamics of Electronic Systems; NDES 1995; pages 147–150; Dublin; Ireland; July 1995.

[12] S. Mattsson; H. Elmqvist; M. Otter; and H. Olsson. Initialization of hybrid differential-algebraic equations in Modelica 2.0. In Proceedings of the Second International Modelica Conference; 2002.

[13] R. Melville; S. Moinian; P. Feldmann; and L. Watson. Sframe: An efficient system for detailed DC simulation of bipolar analog integrated circuits using continuation methods. Analog. Integr. Circ. S.; 3(3):163–180; 1993. doi: 10.1007/BF01239359.

[14] R. C. Melville; L. Trajkovic; S.-C. Fang; and L. T. Watson. Artificial parameter homotopy methods for the DC operating point problem. IEEE T. Comput. Aid. D.; 12(6):861–877; June 1993. doi: 10.1109/43.229761.

[15] J. J. Moré; B. S. Garbow; and K. E. Hillstrom. User guide for MINPACK-1. Technical Report ANL-80-74; Argonne National Laboratory; 1980.

[16] J. Roychowdhury and R. Melville. Delivering global DC convergence for large mixedsignal circuits via homotopy/continuation methods. IEEE T. Comput. Aid. D.; 25(1):66–78; January 2006. doi: 10.1109/TCAD.2005.852461.

[17] J. S. Roychowdhury and R. C. Melville. Homotopy techniques for obtaining a DC solution of large-scale mos circuits. In Proceedings of the 33rd Design Automation Conference; pages 286–291; 1996.

[18] M. Sielemann. Device-Oriented Modeling and Simulation in Aircraft Energy Systems Design. PhD thesis; Technical University of Hamburg-Harburg; Institute of Thermo-Fluid Dynamics; 2012.

[19] M. Sielemann; F. Casella; M. Otter; C. Clauss; J. Eborn; S. Mattsson; and H. Olsson. Robust initialization of differential-algebraic equations using homotopy. In Proceedings of Eighth International Modelica Conference; Dresden; Germany; March 2011.

[20] M. Sielemann and G. Schmitz. A quantitative metric for robustness of nonlinear algebraic equation solvers. Math. Comput. Simulat.; 81(12):2673–2687; 2011. doi: 10.1016/j.matcom.2011.05.010.

[21] R. Tarjan. Depth-first search and linear graph algorithms. SIAM J. Comput.; 1:146–160; 1972. doi: 10.1137/0201010.

[22] L. Trajkovic and W. Mathis. Parameter embedding methods for finding DC operating points: formulation and implementation. In 1995 International Symposium on Nonlinear Theory and its Applications; NOLTA 1995; pages 1159–1164; Las Vegas NE; USA; December 1995.

[23] L. Trajkovic; R. Melville; and S.-C. Fang. Passivity and no-gain properties establish global convergence of a homotopy method for DC operating points. In IEEE International Symposium on Circuits and Systems; volume 2; pages 914–917; May 1990. doi: 10.1109/ISCAS.1990.112242.

[24] L. Trajkovic; R. C. Melville; and S.-C. Fang. Finding DC operating points of transistor circuits using homotopy methods. In Proc. IEEE Int Circuits and Systems Sympoisum; pages 758–761; 1991.

[25] L. Trajkovic; R. C. Melville; and S.-C. Fang. Improving DC convergence in a circuit simulator using a homotopy method. In Proc. Custom Integrated Circuits Conf. the IEEE 1991; 1991.

[26] L. T. Watson. Globally convergent homotopy methods: A tutorial. Appl. Math. Comput.; 31:369–396; May 1989. doi: 10.1016/0096-3003(89)90129-X.

[27] L. T. Watson. Probability-one homotopies in computational science. J. Comput. Appl. Math.; 140:785–807; 2002. doi: 10.1016/S0377-0427(01)00473-3.

[28] A. N. Willson Jr. The no-gain property for networks containing three-terminal elements. IEEE T. Circuits. Syst.; 22(8):678–687; August 1975. doi: 10.1109/TCS.1975.1084110.

[29] K. Yamamura; T. Sekiguchi; and Y. Inoue. A fixed-point homotopy method for solving modified nodal equations. Circuits and Systems I: Fundamental Theory and Applications; IEEE Transactions on; 46(6):654–665; 1999.