In this paper, a $(u+1)\times v$ horn torus resistor network with a special boundary is researched. According to Kirchhoff's law and the recursion-transform method, a model of the resistor network is established by the voltage $V$ and a perturbed tridiagonal Toeplitz matrix. We obtain the exact potential formula of a horn torus resistor network. First, the orthogonal matrix transformation is constructed to obtain the eigenvalues and eigenvectors of this perturbed tridiagonal Toeplitz matrix; second, the solution of the node voltage is given by using the famous fifth kind of discrete sine transform (DST-V). We introduce Chebyshev polynomials to represent the exact potential formula. In addition, the equivalent resistance formulae in special cases are given and displayed by a three-dimensional dynamic view. Finally, a fast algorithm of computing potential is proposed by using the mathematical model, famous DST-V, and fast matrix-vector multiplication. The exact potential formula and the proposed fast algorithm realize large-scale fast and efficient operation for a $(u+1)\times v$ horn torus resistor network, respectively.

Resistor network is widely used. Many potential formulae of resistor networks have been solved accurately, but the scale of data is limited by manual calculation, and numerical simulation has become the trend of large-scale operation. This paper improves the general solution of potential formula for an $$m\times n$$m × n globe network. Chebyshev polynomials are introduced to represent new potential formula of a globe network. Compared with the original potential formula, it saves time to calculate the potential. In addition, an algorithm for computing potential by the famous second type of discrete cosine transform (DCT-II) is also proposed. It is the first time to be used for machine calculation. Moreover, it greatly increases the efficiency of computing potential. In the application of this new potential formula, the equivalent resistance formulae in special cases are given and displayed by three-dimensional dynamic view. The new potential formulae and the proposed fast algorithm realize large-scale operation for resistor networks.