# Newton-Raphson Method: This is a fixed-point estimation method.

Method 2: Newton Raphson

Newton Raphson

Newton-Raphson Method:

This is a fixed-point estimation method. The estimate starts at x 1,for a root of f(x) = 0. A tangent is then draw to the curve y = f(x) at the point (x 1, f(x 1)). The point at which the tangent cuts the x-axis then gives the next approximation for the root, and the process is repeated.

I am going to use the equation y = x³ - 3x + 1.

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As you can see there are three roots in this graph, they are in the interval [-2, -1]

[0, 1]

[1, 2]

The gradient for the tangent to the curve at (x 1, f(x 1)) is f’(x 1) (meaning dy/ dx for x). The equation of the tangent is: y-y1 = m(x-x1). Therefore y-f(x1) = f’(x1) [x-x1]. This tangent passes through the point (x2, 0). Carrying on with this process, this will get closer and closer to the tangent. But there is a general formula for this process:

x n+1 = x n – f(Xn)/ f’(Xn)

Returning to my function: f(x) = x³ – 3x +1

dy/ dx = f’(x) = 3x – 3

The Newton-Raphson formula becomes:

x n+1 = x n - x n³ - 3x + 1

3x n² - 3

I am now going to use this formula to search for the root in the interval [-2 , -1]

Let x 1 = -2

So x 2 = -2 – (-2³ - 3(-2) +1)

3(-2)² - 3

= -2 – (- 1/ 9)

= -1.888888889 (-1 8/ 9)

Method 2: Newton-Raphson method

Fig1. Shows the function f(x) = 1/3x 3 – 5x + 1.4

Differentiating the function we get: f(x)` ...