Explicit potential function and fast algorithm for computing potentials in [formula omitted] conic surface resistor network

Abstract

Resistor network research is of great importance, yet many resistor networks and their large-scale fast computations have not received sufficient attention. This paper proposes a new resistor network with idiosyncratic shape, i.e., a α×β conic surface (CS) resistor network that resembles the upper part of a three-dimensional Dirac function. Utilizing the Recursion Transform (RT-V) method of Tan, a recursive matrix equation model is constructed based on Kirchhoff’s law and nodal voltages, which contains the modified tridiagonal Toeplitz matrix. By using the orthogonal matrix transformation, the eigenvalues and eigenvectors of the modified tridiagonal Toeplitz are obtained. The discrete sine transform of the fourth type (DST−IV) is utilized to solve node voltages, while the explicit potential function is represented by the Chebyshev polynomials of the second kind. In addition, explicit potential functions for some special cases are provided, and the potential distribution is illustrated using dynamic three-dimensional graph. To achieve a rapid calculation of the potential, a fast algorithm based on the multiplication of DST-IV with a vector is proposed. In the end, analysis of computational efficiency for the explicit potential function and the fast algorithm are shown.