An optimized potential formula for the $m\times n$ apple surface network has been introduced in this paper. Compared with the original potential formula, this method significantly enhances the efficiency required for rapid and large-scale numerical simulations. Based on the optimized potential function, we proposed a metaheuristic algorithm suitable for apple surface environment path planning. Chebyshev polynomials of the first class were employed to represent the potential function. Subsequently, a fast algorithm for calculating the potential utilizing the first kind of discrete sine transform (DST-I) was devised. We proposed potential formulas for several cases to visually present the distribution of the potential and illustrated them using three-dimensional graphs. We also 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. Experiments were conducted to analyze the efficiency and availability of various techniques for computing potential.

In this paper, fast numerical methods for solving the real quasi-symmetric Toeplitz linear system are studied in two stages. First, based on an order-reduction algorithm and the factorization of Toeplitz matrix inversion, a sequence of linear systems with a constant symmetric Toeplitz matrix are solved. Second, two new fast algorithms are employed to solve the real quasi-symmetric Toeplitz linear system. Furthermore, we show a fast algorithm for quasi-symmetric Toeplitz matrix-vector multiplication. In addition, the stability analysis of the splitting symmetric Toeplitz inversion is discussed. In mathematical or engineering problems, the proposed algorithms are extraordinarily effective for solving a sequence of linear systems with a constant symmetric Toeplitz matrix. Fast matrix-vector multiplication and a quasi-symmetric Toeplitz linear solver are proven to be suitable for image encryption and decryption.

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.