### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 90-100 |

Number of pages | 11 |

Journal | Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi) |

Volume | 74 |

Issue number | 11 |

Publication status | Published - 1991 Nov |

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### ASJC Scopus subject areas

- Electrical and Electronic Engineering

### Cite this

*Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)*,

*74*(11), 90-100.

**Computational complexity of the homotopy method for calculating solutions of strongly monotonic resistive circuit equations.** / Makino, Mitsunoai; Oishi, Shinichi; Kashiwagi, Masahide; Horiuchi, Kazuo.

Research output: Contribution to journal › Article

*Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)*, vol. 74, no. 11, pp. 90-100.

}

TY - JOUR

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

AU - Makino, Mitsunoai

AU - Oishi, Shinichi

AU - Kashiwagi, Masahide

AU - Horiuchi, Kazuo

PY - 1991/11

Y1 - 1991/11

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.

UR - http://www.scopus.com/inward/record.url?scp=0026259454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026259454&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0026259454

VL - 74

SP - 90

EP - 100

JO - Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)

JF - Electronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)

SN - 1042-0967

IS - 11

ER -