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

Mitsunoai Makino, Shinichi Oishi, Masahide Kashiwagi, Kazuo Horiuchi

    Research output: Contribution to journalArticle

    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 languageEnglish
    Pages (from-to)90-100
    Number of pages11
    JournalElectronics and Communications in Japan, Part III: Fundamental Electronic Science (English translation of Denshi Tsushin Gakkai Ronbunshi)
    Volume74
    Issue number11
    Publication statusPublished - 1991 Nov

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    Computational complexity
    Networks (circuits)
    Newton-Raphson method
    Error analysis
    Mathematical operators

    ASJC Scopus subject areas

    • Electrical and Electronic Engineering

    Cite this

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    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.",
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    AU - Kashiwagi, Masahide

    AU - Horiuchi, Kazuo

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