Newton’s method

In this post I will shortly explain what Newton’s method is and how it works.

What is it:

• Newton’s method is an algorithm to find zeropoints of a function
• Because you cannot always calculate the exact zeropoint we can use this algorithm to approximate the zeros of the function

How does it work:

1. Start at a random point A of the function
2. Calculate the tangent T at this point A
3. Calculate the intersection between the x-axis and the tangent T
4. Take the x-value of the intersection point  ( x-axis and the tangent T ), calculate the y-value ( f(x) )  → new Point B at (x|y)

• Repeat the steps 1- 4 (like 500 times) → approximation of the zeropoint

Important sidenote:
You should try to start with a nearby point to the actual zeropoint, because it might be that you would get a solution which is just wrong. This might be the case if you would get to a point with the slope of a tangent being 0, which does not have one intersection point with the x-axis (saddle point).

Interesting fact – Approximate square root numbers:
With this approximation we can also calculate/approximate square roots, like √2 (irrational number).
To calculate the square root of 2 with Newton’s method we would first create a function (polynomial) where f(√2) = 0 (zeropoint at √2).

How to achive this:

1. x = √2| ^2
2. x² = 2 | – 2
3. x² – 2 = 0

Step by step math for Newton’s method:

1. Start at point A and calculate the slope
2. linear function is y=m*x+b → we got m (the calculated slope) and the point A throught which the line goes → calculate b: b=y-(m*x)
3. To calculate the intersection of the tangent and the x-axis set y to 0: 0=m*x+b → we got m and b, so we need to calculate x → x = (y-b)/m = (0-b)/m
4. Calculate the new point B of the function with f(x) and repeat the steps

Python code example to approximate √2 with Newton’s method: (github.com/Heuristic-Analyst/…)

def newtons_method(iterations:int):
    x = 2

    for i in range(iterations):
        y = function_one(x)
        m = function_one_derivative(x)
        b = y-(m*x)
        x=(0-b)/m
    return x


def function_one(x:float):
    return x**2-2


def function_one_derivative(x:float):
    return 2*x

print(newtons_method(5))
>> Output: 1.414213562373095