Question

3. Recall that the three-point Gaassian quadrature rule is with and the corresponding weight are (A). Show that this Gaussian quadrature rule is exact for polynoinials 1,エ·エ2,?,24, X5. (B). Use the transform to write the corresponding Gaussian quadrature formula

Answer 1

Given formula integral^1_- 1 f (x) dx = f (x_0) w_0 + f (x_1)w_1 + f (x_2) w_2 x_0 = -squareroot 3/5 x_1 = 0. x_2 = squareroot 3/5 w_0 = 5/9 w_1 = 8/9 w_2 = 5/9 let f (x) = 1 i) integral^1_- 1 dx = 2 f (x) = f (x_1) = f (x_3) = 1 f (x_0) w_0 + f (x_1) w_1 + f (x_2)w_2 = 5/9 + 8/9 + 5/9 = 2 (i) f (x) = x integral^1_- 1 x dx = x^2/2 integral^1_- 1 = 1/2 - 1/2 = 0 f (x_0) = -squareroot 3/5 f(x_1) = 0 f (x_2) = squareroot 3/5 w_0 f (x_0) + w_1 f (x_1) + w_2 f (x_2) = -5/9 squareroot 3/5 + 0 8/9 + squareroot 3/5 5/9 = 0 implies integral^1_- 1 f (x) dx = sigma^2_i = 0 f (x_i) w_i