Newton's method, also known as Newton-Raphson method, is a method proposed by Newton in17th century to approximately solve equations in real number domain and complex number domain. Most equations have no formula for finding the root, so it is difficult or even impossible to find the exact root, so it is particularly important to find the approximate root of the equation. Methods The first few terms of Taylor series of function f(x) were used to find the root of equation f(x) = 0. Newton iteration method is one of the important methods to find the roots of equations. Its greatest advantage is that it has square convergence near the single root of the equation f(x) = 0, and it can also be used to find the multiple roots and complex roots of the equation. In addition, this method is widely used in computer programming.

Let r be the root of f(x) = 0, and choose x0 as the initial approximation of r, so that the tangent L of the curve y = f(x) passes through points (x0, f(x0)). The equation of l is y = f(x0)+f'(x0)(x-x0). Find the abscissa of the intersection of l and x axis. The intersection (x 1, f(x 1)) is the tangent of the curve y = f(x), and the abscissa of the intersection of the tangent and the x axis is x2 = x1-f (x1)/f' (x1. Repeat the above process to get the approximate sequence of R, where x (n+ 1) = x (n)-f (x (n))/f' (x (n)), which is called the n+1approximation of R. The above formula is called Newton iterative formula.

Newton method for solving the nonlinear equation f(x)=0 is an approximate method to linearize the nonlinear equation. Expand f(x) into Taylor series f (x) = f (x0)+(x-x0) f' (x0)+(x-x0) 2 * f'' (x0)/2! +… Take the approximate equation whose linear part is the nonlinear equation f(x) = 0, that is, the first two terms of Taylor expansion, then f (x0)+f' (x0) (x-x0) = 0 Let f'(x0)≠0, and the solution is X 1 = x0.

Note: Because I did the problem twice at most, you appeared here three times. In fact, this problem is very complicated and difficult to solve. The above knowledge can be used for reference.