Mathquery

Three_Point Gauss_Legendre Rule:        We have from  Gauss Quadrature Rule that   Rₙ₊₁(f) = w₀f(x₀) + w₁d(x₁) + ......+wₙf(xₙ) ...(1)    ⇒R₃(f) = w₀f(x₀) + w₁f(x₁) + w₂f(x₂) .....(2)                         1 Again , I(f) = ∫ f(x)dx .........(3)                        -1 Now , I(f) is given by      I(f) = R₃(f) + E₃(f) ...........(4)       where E₃(f) is its error . 

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