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Sulaiman Khan
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Newton-Raphson Method

The Newton-Raphson method is a powerful numerical technique for finding the roots of a real-valued function. It is based on iteratively refining guesses for the root by using the function's derivative.


Given a function:

f(x)=0f(x) = 0
xn+1=xn−\ racf(xn)f′(xn)x_{n+1} = x_n - \ rac{f(x_n)}{f'(x_n)}
Where :
  • xnx_n is the current guess.
  • f(xn)f(x_n) is the value of the function at xnx_n.
  • f′(xn)f'(x_n) is the derivative of the function at xnx_n.

How the Newton-Raphson Method Works

  1. Initial Guess: Start with an initial guess x0x_0.
  2. Iterative Formula: Compute the next approximation using:
    xn+1=xn−\ racf(xn)f′(xn)x_{n+1} = x_n - \ rac{f(x_n)}{f'(x_n)}
  3. Repeat: Continue iterating until the result converges to a desired accuracy.

Example

Find the square root :

f(x)=x2−2f(x) = x^2 - 2

Initial guess: x0=1x_0 = 1

Let's solve f(x)=x2−2=0 f(x) = x^2 - 2 = 0 using the Newton-Raphson method, which seeks to find the square.

  1. Function: f(x)=x2−2 f(x) = x^2 - 2
  2. Derivative: f′(x)=2x f'(x) = 2x
  3. Initial guess: x0=1 x_0 = 1

First Iteration

x1=x0−\ rac(12−2)2⋅1=1.5x_1 = x_0 - \ rac{(1^2 - 2)}{2 \cdot 1} = 1.5

Second Iteration

x2=1.5−\ rac(1.52−2)2⋅1.5=1.4167x_2 = 1.5 - \ rac{(1.5^2 - 2)}{2 \cdot 1.5} = 1.4167

Third Iteration

x3=1.4167−\ rac(1.41672−2)2⋅1.4167=1.4142x_3 = 1.4167 - \ rac{(1.4167^2 - 2)}{2 \cdot 1.4167} = 1.4142

After three iterations, the approximate root is x≈1.4142x \approx 1.4142, which is close to 2\sqrt{2}.

Conclusion

The Newton-Raphson Method is an efficient way to find the roots of a function, especially when the function is smooth and its derivative is known.

Newton-Raphson Method