TY - JOUR

T1 - Computational complexity of the homotopy method for calculating solutions of strongly monotonic resistive circuit equations

AU - Makino, Mitsunoai

AU - Oishi, Shin'ichi

AU - Kashiwagi, Masahide

AU - Horiuchi, Kazuo

PY - 1991

Y1 - 1991

N2 - A priori estimation is presented for a computational complexity of the homotopy method applied to a certain class of hybrid equations for nonlinear strongly monotonic resistive circuits. First, an explanation is given as to why a computational complexity of the homotopy method cannot be a priori estimated for calculating solutions of hybrid equations in general. In this paper, the homotopy algorithm is considered in which a numerical path‐following algorithm is executed based on the simplified Newton method. Then by introducing Urabe's theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, it is shown that a computational complexity of the algorithm can be a priori estimated when applied to a certain class of hybrid equations for nonlinear strongly monotonic resistive circuits whose domains are bounded. This paper considers two types of path‐following algorithms: one with a numerical error estimation in the domain of a nonlinear operator; and one with a numerical error estimation in the range of the operator.

AB - A priori estimation is presented for a computational complexity of the homotopy method applied to a certain class of hybrid equations for nonlinear strongly monotonic resistive circuits. First, an explanation is given as to why a computational complexity of the homotopy method cannot be a priori estimated for calculating solutions of hybrid equations in general. In this paper, the homotopy algorithm is considered in which a numerical path‐following algorithm is executed based on the simplified Newton method. Then by introducing Urabe's theorem, which gives a sufficient condition guaranteeing the convergence of the simplified Newton method, it is shown that a computational complexity of the algorithm can be a priori estimated when applied to a certain class of hybrid equations for nonlinear strongly monotonic resistive circuits whose domains are bounded. This paper considers two types of path‐following algorithms: one with a numerical error estimation in the domain of a nonlinear operator; and one with a numerical error estimation in the range of the operator.

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U2 - 10.1002/ecjc.4430741109

DO - 10.1002/ecjc.4430741109

M3 - Article

AN - SCOPUS:0026259454

SN - 1042-0967

VL - 74

SP - 90

EP - 100

JO - Electronics and Communications in Japan (Part III: Fundamental Electronic Science)

JF - Electronics and Communications in Japan (Part III: Fundamental Electronic Science)

IS - 11

ER -