The study proposed an innovative path planning algorithm based on the potential function of a special case of the cobweb resistor network, addressing the path planning problem in globe environments with obstacles. For the non-regular $m\times n$ cobweb resistor network with arbitrary longitude, we found that by introducing Chebyshev polynomial of the second class, the precise equivalent resistance formulas could be optimized effectively. Compared with the original formula, optimized equivalent resistance formulas significantly reduced the time cost in large-scale data calculations. Furthermore, we have plotted 3D views of the equivalent resistance formulas for several special cases and conducted simulation experiments on the computational efficiency of the original and optimized formulas at different data scales, verifying the superiority of the optimized formulas. These findings provided new perspectives and tools for the computation of resistor networks and the design of path planning algorithms.

In this paper, we focused on solving the perturbed four-banded linear system derived from the traffic process associated with a Markovian queueing model. Utilizing the spectral decomposition of circulant and skew circulant matrices, we computed the product of Toeplitz inversion and a vector, leading to a decomposition algorithm for perturbed four-banded linear systems. This decomposed Toeplitz system features multiple right-hand terms, significantly reducing computational complexity through Toeplitz inversion. Additionally, we introduced an algorithm based on banded LU decomposition, resulting in a banded linear system with multiple right-hand terms, where the sparsity of the banded LU decomposition is pivotal. To evaluate the algorithm's performance, we presented two examples in numerical simulations.