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 :