First of all, the most common method is dichotomy to evaluate. The main concern here is accuracy. There is dichotomy to evaluate, but this method sometimes fails to meet the time complexity requirements given by the topic, so a new method is needed to evaluate.

This is Newton's iterative method:

All universities here should know that a function f (x) = x 3-y can be drawn on a coordinate system.

Just find a point on the curve (why just find it? According to the fact that the tangent is the approximation of the curve near the tangent point, you should find it near the root point, but obviously we don't know where the root point is yet. Make a tangent, and the root of the tangent (that is, the intersection with the X axis) is still a certain distance from the root of the curve. Newton and Laveson thought, it doesn't matter, let's start from the root of this tangent, make a vertical line, intersect the curve at point B, and continue to repeat the work just now:

As I said before, point B is closer to the root of the curve than point A. Newton and La Wei Sen were very excited and continued to repeat the work just now:

After many iterations, it will get closer and closer to the root of the curve (there are 50 iterations in the figure below, and even after countless iterations, it will only get closer and closer to the root of the curve. In mathematical terms, iteration converges):

As we all know, f'(x) is the derivative of f(x) and the tangent equation of a certain point.

Newton's method of approaching roots according to the above method can be summarized as follows.

Therefore, when seeking the cube root, we can assume that

Finding the cube root of y is the value of x when f(x)=0.

According to the above formula, we can get

Find the root of the formula here, and you can write the solution of the cube root.

Reference:

Newton iteration method