Two_Point Gauss _Legender Rule

Two_Point Gauss_Legender Rule:

     We know from Gauss Quadrature Rule that 

   Rₙ₊₁(f) = w₀f(x₀) + w₁f(x₁)+....+wₙf(xₙ) .....(1)

⇒R₂(f) = w₀f(x₀) + w₁f(x₁) ......(2)

                        1

Again , I(f) = ∫ f(x) dx .........(3)

                      -1

But I(f) is also given by 

  I(f) = R₂(f) + E₂(f) .........(4)

          where E₂(f) is its error .

The four unknowns w₀,w₁,x₀,x₁ are determined by using equation (2) exact for the monomials 1,x,x²,x³

i.e  E₂(xⁱ) = 0 ,  i= 0,1,2,3      ........(5)

Now equation (4) can be written as 

 1

 ∫ f(x) dx = w₀f(x₀) + w₁f(x₁) + E₂(xⁱ).....(6)

-1

For f(x)= 1 :

         We get from equation (6) 

    1

    ∫ 1 dx =w₀.1 + w₁.1 + 0

   -1

  ⇒2 = w₀ + w₁ 

For f(x) = x :

        we get from equation(6)

    1

    ∫ x dx = w₀x₀ + w₁x₁ + 0 

   -1

  ⇒0 = w₀x₀ + w₁x₁ 

For f(x) = x² :

        we get from equation(6) 

    1

    ∫ x² dx = w₀x₀² + w₁x₁² + 0

   -1

 ⇒2/3 = w₀x₀² + w₁x₁² 

For f(x) = x³ : 

       We get from equation(6) 

    1

    ∫ x³ dx = w₀x₀³ + w₁x₁³ + 0

  -1

 ⇒ 0 = w₀x₀³ + w₁x₁³ 

So we get 

  [ w₀+w₁ = 2

   w₀x₀ + w₁x₁ = 0 

   w₀x₀² + w₁x₁² = 2/3 

   w₀x₀³ + w₁x₁³ = 0  ] .........(7) 

Equation(7) is satisfied if 

  w₀= w₁ = 1 , x₀= 1/√(3) , x₁ = -1/√(3)

Hence the two_ point Gauss _ Legender rule is given by 

    R₂(f) = f[-1/√(3)] + f[1/√(3)]

Hence Proved .