Improved spherical continuation algorithm with application to the double-bounded homotopy (DBH)
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Improved spherical continuation algorithm with application to the double-bounded homotopy (DBH) D. Torres-Muñoz · H. Vazquez-Leal · L. Hernandez-Martinez · A. Sarmiento-Reyes
Received: 27 May 2013 / Accepted: 30 May 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract The homotopy continuation methods are useful tools for finding multiple solutions of nonlinear problems. An important issue of this kind of method is the correct implementation of the path-following techniques used to trace the homotopy trajectory. Therefore, in this work we propose a modification of the spherical algorithm to successfully trace the closed paths of a DBH homotopy. The proposed methodology is depicted with three examples. Finally, a comparison of the results with a standard path-following technique is presented and discussed. Keywords
Homotopy continuation methods · Multiple operating points
Mathematics Subject Classification
34A12 · 34A34
1 Introduction The DC analysis is an important task for analyzing electrical circuits. This analysis consists of solving a system of nonlinear algebraic equations (NAEs) formulated using the Communicated by Cristina Turner. D. Torres-Muñoz (B) · L. Hernandez-Martinez · A. Sarmiento-Reyes National Institute for Astrophysics, Optics and Electronics, Sta. María Tonantzintla, Puebla, México e-mail: [email protected]; [email protected] L. Hernandez-Martinez e-mail: [email protected] A. Sarmiento-Reyes e-mail: [email protected] H. Vazquez-Leal School of Electronic Instrumentation and Atmospheric Sciences, University of Veracruz, Xalapa, Veracruz, México e-mail: [email protected]
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Kirchhoff laws (Vazquez-Leal et al. 2011; Melville et al. 1993; Vazquez-Leal et al. 2013). Newton–Raphson (NR) is the most employed method to solve the NAEs due to the quadratic convergence rate (Ogrodzki 1994). Nonetheless, NR may fail to converge to a solution unless the initial estimation point is close enough to the solution. Therefore, homotopy continuation methods (HCM) are proposed as an alternative to the NR method. What is more, HCM methods are capable of locating multiple operating points, which have applications in the analysis of multistable circuits. The basic idea of the HCM methods is to embed a homotopy parameter λ into the NAES, yielding to a continuous deformation from a trivial state to the nonlinear state (Melville et al. 1993; Watson et al. 1997; Dyes et al. 1999; Vazquez-Leal et al. 2013, 2012). Such procedure can be represented by H : R n+1 −→ R n , x ∈ R n and λ ∈ [0, 1],
(1)
where n is the number of variables in the system and x represents circuit electrical variables. The last equation satisfies the following conditions: (1) If λ = 0, then H (x, 0) = 0 has a trivial or known solution. (2) If λ = 1, then H (x, 1) = F(x) has the solution of the original NAEs. Several homotopy formulations have been reported as the Newton homotopy (NH) (Wu 2006), the fixed point homotopy (FPH) (Yamamura et al. 1999), multiparameter homotopy (Wolf and Sander
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