Newton Raphson Method

Newton Raphson Method :

       Let x₀ be an approximate root of the equation f(x) = 0 . 

  If x₁= x₀+h be the exact root , then f(x) = 0

 ∴ Expanding f(x₀+h) by Taylor's series 

we have 

   f(x₀+h) = f(x₀) + hf'(x₀) + h²/2 f"(x₀) +...... = 0

Since h is small , neglecting h² and higher power of h , we get 

        f(x₀) + hf'(x₀) = 0

 ⇒    h = -f(x₀)/f'(x₀) ..........(1)

Similarly starting with x₁ , a still better approximation x₂ is given by 

    x₂ = x₁ - f(x₁)/f'(x₁) 

In general , 

   xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ)       .........(2)

                where n = 0,1,2.......

Which is known as                                 Newton Raphson formula or Newton's iteration formula .

For Example : 

         Use Newton Rapson method to deduce the iterative procedure 

     xₙ₊₁ = 1/2 (xₙ + a/xₙ) for evaluating √(a) as the solution of the equation x²-a = 0

 Solution : 

             Here f(x) = x² - a 

                 ⇒f'(x) = 2x 

So , xₙ₊₁ = xₙ - f(xₙ)/f'(xₙ) 

    ⇒ xₙ₊₁ = xₙ - (xₙ²-a)/2xₙ

i.e xₙ₊₁ = 1/2 (xₙ + a/xₙ) ,  n= 0,1,2.........