Question

S. Questions for nest time How do we generalize to n-point Gausian quadrature? Is this going to be useful for integrals that aren't from -1 to 1? How good is this method?

Answer 1

Given: To generation the n-point Gaussian quadrature integral^b_a w(x) f(x) dx = sigma^N_j = 1 w_j f(x_j) = sigma^N_j = 1 delta_ij w_j = w_i > 0 where w(x) is the non-negative weight function ad f(x) is the quadrature polynomial Now the intervals [a, b] are generally converted into [-1, 1] by integral^b_a f(x) dx = b - a/2 integral^1_-1 f(b - a/2 x + a + b/2) dx or integral^b_a f(x) dx aymp b - a/2 sigma^N_i = 1 w_i f(b - a/2 x_i + a + b/2) actually this method is constructed to yield an exact result for polynomials of deg 2n - 1 or less by a suitable choice of the nodes x_i and weights w_i for i = 1, 2, n so it is a suitable method for approximating the integral