The
Gauss Quadrature method will find the value of an integral to order
2n+1 by evaluating a polynomial of degree
n only, but at judiciously selected points. Is is also known as
Gauss-Legendre Quadrature.
The derivation involves expanding the integrand in a
Lagrange polynomial plus an error term. Next, this correction term is expressed in terms of
Legendre polynomials, which are orthogonal in the range [-1,1]. It is seen the error term will only vanish if samples are taken at the roots of Legendre polynomials. Eventually, the roots are tranformed from [-1,1] to the interval given.
There is a
program for this section, too. The following table gives sampling points, referred to [-1,1] and the corresponding weights for order five.
| Sampling point | weight |
| 0 | 0.5688889 |
| +0.5384693 | 0.4786287 |
| +0.9061798 | 0.2369268 |
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