Rate Of Convergence Of Newton_Raphson Method

Rate Of Convergence Of Newton_Raphson Method :

Definition Of Rate Of Convergence :

         Rate of convergence is the speed of a convergent sequence to approach its limit point. 

     General equation for rate of convergence is

          |εₙ₊₁|≤ c |εₙ|ᵖ 

where   c is a constant ,

      p= degree of rate of convergence 

        Using this definition rate of convergence of Newton_ Raphson method can be calculated .

 Calculation Of Rate Of Convergence Of Newton_Raphson Method :

         The Newton_Raphson method in equation 

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

  Let ξ be the exact root of f(x) = 0 

                                 (   i.e f(ξ) = 0)

     Let xₙ,xₙ₊₁ be the approximate value at nth and (n+1)th iteration respectively . 

  Similarly εₙ,εₙ₊₁ be the error at nth and (n+1)th iteration respectively .

             εₙ + ξ = xₙ 

            εₙ₊₁ + ξ = xₙ₊₁ 

Substituting εₙ , εₙ₊₁ in equation (1) 

 ⇒  εₙ₊₁ + ξ = εₙ + ξ - f(εₙ+ξ)/f'(εₙ+ξ) 

 ⇒  εₙ₊₁= 

       εₙ - [f(ξ)+εₙf'(ξ)+εₙ/2 f"(ξ)]/[f(ξ)+εₙf"(ξ)]

(Neglecting the higher order derivatives in the Taylor Series )

 Since ξ is exact root of f(x) = 0

           ⇒f(ξ) = 0

⇒εₙ₊₁ = εₙ - [εₙf(ξ)+εₙ/2 f"(ξ)]/[f(ξ)+εₙf"(ξ)]

⇒εₙ₊₁ = εₙ 

 - f'(ξ)[εₙ+εₙf"(ξ)/2f'(ξ)] / f'(ξ)[1+εₙf"(ξ)/f'(ξ)]

⇒εₙ₊₁ = 

      εₙ - [εₙ + εₙf"(ξ)/2f'(ξ)][1 + εₙf"(ξ)/f'(ξ)]⁻¹

Using the property (1+x)⁻¹ = 1-x+x²-x³+....

 ⇒εₙ₊₁ = 

        εₙ - [εₙ + εₙf"(ξ)/2f'(ξ)][1 - εₙf"(ξ)/f'(ξ)]

           (Neglecting higher degree terms)

⇒εₙ₊₁ = 

     εₙ - [εₙ+εₙ²f"(ξ)/f'(ξ) + εₙ²f"(ξ)/2f'(ξ)

                - εₙ³f"(ξ)f"(ξ)/f(ξ)² ]

Neglecting higher order derivatives 

⇒εₙ₊₁ = εₙ - εₙ + εₙ²f"(ξ)/f'(ξ) - εₙ²f"(ξ)/2f'(ξ)

⇒εₙ₊₁ = εₙ²f"(ξ)/2f'(ξ) 

⇒|εₙ₊₁| ≤ |c εₙ²|

⇒|εₙ₊₁| ≤ c |εₙ|² ...........(2)

Now comparing equation (2) with equation 

              |εₙ₊₁| ≤ c |εₙ|ᵖ

which is the general equation for rate of convergence . 

   we get p= 2

∴ Newton_Raphson has rate of convergence p=2 or N_R method has quadratic rate of convergence .