Fast algorithm and new potential formula represented by Chebyshev polynomials for an m × n globe network

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 × 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. With the development of natural science, people have encountered various new problems. According to the com- plexity of the problem, solutions emerge in endlessly. Researches show that a host of problems can be solved by establishing resistor network model 1–10 and neural network model 11–17 . In the past few years, many results have been achieved in the study of resistor networks. For example, the establishment of graph theory, the researches on Laplacian matrix ( LM ) method of resistor network, infinite network, finite network, and corner-to-corner resistance and so on 7–10,18–31 . Shi 11,12 et al. studied a novel discrete-time recurrent neural network. Liu 13 et al. and Sun 14 et al. proposed different types of the zeroing neural network. And Jin 15–17 et al. studied an innovative control theory stimulated gradient neural network algorithm. In recent years

With the development of natural science, people have encountered various new problems. According to the complexity of the problem, solutions emerge in endlessly. Researches show that a host of problems can be solved by establishing resistor network model [1][2][3][4][5][6][7][8][9][10] and neural network model [11][12][13][14][15][16][17] . In the past few years, many results have been achieved in the study of resistor networks. For example, the establishment of graph theory, the researches on Laplacian matrix (LM) method of resistor network, infinite network, finite network, and corner-to-corner resistance and so on 7-10,18-31 . Shi 11,12 et al. studied a novel discrete-time recurrent neural network. Liu 13  In recent years,  proposed the Recursion-Transform (RT) method which is different from the Laplacian matrix method. It only depends on one matrix and one direction, and the calculation is simpler. In 2013, Tan 32 studied the problem of two-point resistance on cobweb with a 2r boundary, which has never been solved before. Under finite and infinite conditions, the m × n cobweb general formula of resistance between any two nodes is proposed. In 2014, Tan 36,38 made a new breakthrough and introduced the equivalent resistance of the globe network and fan network model by RT method. Shortly after that many researches on resistor networks are based on RT method. RT method needs eigenvalues of a tridiagonal matrix to represent the potential formula. At present, there have been many results on tridiagonal matrices [45][46][47][48][49][50][51] , which are also widely used. It can be said that it is a powerful tool to solve the resistor network [33][34][35][36][37][38][39][40][41][42][43] .
In 2018, Tan 44 proposed an m × n globe network, as shown in Fig. 1. The resistances in the latitude and longitude directions are r and r 0 , respectively, where m and n are the resistance numbers along the latitude and longitude directions. The nodes of the network are represented by coordinates {x, y} . Considering O 0 = 0 as the origin of the coordinate system, the potential of node d(x, y) is U m×n (x, y) as shown in Fig. 2. The potential of any node in the m × n globe network is as follows

New formula of potential represented by Chebyshev polynomials
Since the potential formula (1) contains exponential function operation, it is less efficient to calculate potential of globe resistor networks by computer. The purpose of this section is to give an improved potential formula of the globe network 44 , i.e. a new formula of potential expressed by Chebyshev polynomial of the second class is given 53 . it saves time to calculate the potential by computer. Let the current J from d 1 (x 1 , y 1 ) to d 2 (x 2 , y 2 ) , the potential formula of two nodes in an m × n resistor network can be written as where In particular, the input and output points can be one or more, so formula (5) applies to all coordinate points (x k , y k ) (0 ≤ k ≤ n) . Therefore, p can also be y 3 , y 4 , . . . , y k .
Suppose that O 0 = 0 , then calculating the node potential between any two points by Ohm's law can be described as x is denoted by the current in the vertical direction. Base on formula (5), using the visualization of data to realize the dynamic visualization of the potential change. With the change of the current input and output point change, the potential change of any two nodes is shown in Fig. 3.

Three terms recurrence sequence and discrete cosine transform
In this section, in order to improve the actual performance and realize the fast algorithm, the three terms recurrence sequence represented by Chebyshev polynomial of the second class and two types of discrete cosine transform are introduced. www.nature.com/scientificreports/ The three terms recurrence sequence is defined by the following conditions: where k ∈ N, k ≥ 2, A, B, d, q ∈ C , N is the set of all natural numbers and C is the set of all complex numbers. The three terms recurrence sequence 52 represented by Chebyshev polynomial of the second class is where is the Chebyshev polynomial of the second class 53 .
In order to obtain a fast numerical algorithm for computing the potential, the orthogonal diagonalization of the matrix B m is given.
Then the eigenvalues t 1 , . . . , t m of B m are given by and the corresponding eigenvectors Obviously, the matrix C III m is the famous third type of discrete cosine transform (DCT-III) [54][55][56][57] . C III m is an orthogonal matrix and the inverse of C III m is actually C II m , i.e.
where the orthogonal matrix C II m is the famous second type of discrete cosine transform (DCT-II) 54 Based on the Chebyshev polynomial of the second class, the general solution formula of current is rewrited . When j = 1 , the following formula is obtained When j ≥ 2 , the following formula is obtained where γ (j) , and ς 2,j = 2 sin( ).

Displaying of some special and interesting potential formulae
Since formula (5) is a general potential conclusion of a globe network including all cases, some special conditions in formula (5) and a series of fascinating results under various parameters will be displayed. In the following, assuming that the potential reference at point O(0, 0) is U(0, 0) = 0. k is the same as Eq. (9). When m = n = 90, J = 10, x 1 = y 1 = 50, x 2 = y 2 = 0, and r 0 = r = 1, the following formula is obtained

Special 2.
Consider an arbitrary m × n globe network as shown in Fig. (1). Assume the electric current J outflow the network from the pole d 2 (x 2 , y 2 ) , and the current J input at the node d 1 (x 1 , y 1 ) = O(0, 0) , the potential of an arbitrary node d(x, y) can be written as where γ (j) x s ,x k is defined in Eq. (6), and U (j) k is the same as Eq. (9). In the network with m = 90 and n = 90 , the current flows in from (x 1 , y 1 )(x 1 = 0, y 1 = 0) and out from (x 2 , y 2 )(x 2 = 30, y 2 = 30) . Let r 0 = r = 1 and J = 10 , the formula is deduced.
where t j and U ) sin(  Special 4. When we inject current J at node d 1 (x 1 , y 1 ) and exit the current J at node d 2 (x 1 , y 2 )(x 2 = x 1 ) , the potential of an arbitrary node d(x, y) is where C k,j , U  Fig. 7 by Matlab. , k and C k,j are equal to Eq. (6), Eq. (9) and Eq. (7), respectively. When m = n = 90, J = 10, y 1 = y 2 = x 1 = 50, x 2 = 30, and r 0 = r = 1, the following formula is obtained  And a 3D dynamic view is shown in Fig. 9 by Matlab.

Fast numerical algorithm for computing potential
In order to realize fast calculation of potential for large-scale resistor networks. In this section, by summarizing the previous discussion and analysis, a fast numerical algorithm of computing potential by the Eqs. www.nature.com/scientificreports/ As is well known, the complexity of tridiagonal matrix-vector multiplication is O(n), which is the same as Algorithm 1. Moreover, one DCT-II needs 2n log 2 n + O(n) real arithmetic operations 57,58 . So the complexity of Algorithm 2 is 4n log 2 n + O(n) consist of two DCT-II and Algorithm 1.
According to the above two algorithms, two examples are used to vividly show the computational efficiency for the large scale globe resistor networks.
Conclusions. This paper achieved a series of improved exact potential formulae in an m × n globe network by the RT-I method. Chebyshev polynomial of the second kind is introduced to improve the potential formula of the globe network 44 . Some applications of the new potential formula of the globe network are presented, such as some special and interesting potential formulae are given in Eqs. (33), (38), (41), (43), (46) and (48), respectively. The image numerical simulation using matlab has produced many interesting 3D dynamic views. Finally, we also put forward a fast numerical algorithm by the famous second type of discrete cosine transform, which can realize fast calculation of potential for large-scale resistor networks. Furthermore, by using our research ideas of resistor network, we can also explore neural networks [11][12][13][14][15][16][17] . That will be our next step.