The core step of point difference method is to construct point difference function. For each equation in the nonlinear equations, we can rewrite it as a linear function about the variable to be solved, and then subtract this linear function from the adjacent linear function to get a new linear function, that is, the point difference function. The point difference function value is the error of the original nonlinear equations at the current point.
Through the point difference function, we can transform the original nonlinear equations into a series of linear equations. The coefficient matrix and constant terms of these linear equations are known, so we can directly solve these linear equations and get the approximate solution of the original nonlinear equations.
The advantage of point difference method is that the calculation process is simple, only simple addition, subtraction, multiplication and division operations are needed, the convergence speed is fast, and a better approximate solution can be obtained quickly. However, the shortcoming of the point difference method is also obvious, that is, only the local optimal solution can be obtained, but the global optimal solution cannot be guaranteed. In addition, the point difference method is very sensitive to the selection of initial values, and different initial values may lead to completely different results.
Generally speaking, the point difference method is a very practical method for solving nonlinear equations, which is widely used in engineering, science and mathematics. However, due to its shortcomings, it is necessary to choose the initial value carefully when using the point difference method, and it may be necessary to combine other methods to get more accurate results.