CS3414 Afterclass Notes --- 17 June, 2002

Numerical Integration (Chapter 5)
  1. Simple quadrature rules

    1. Midpoint rule: see handout

    2. Trapezoid rule: see handout

    3. Simpson's rule: see handout

    4. Gaussian quadrature rules

      • Idea: improve accuracy by choosing quadrature points carefully.

      • Facts: (1) with k equally spaced points, you can get polynomial degree k if k is odd, k-1 if k is even. (2) with k Gauss points, can get polynomial degree 2k-1.

      • Example. Suppose we want , , , such that is exact for all on [-1,1].

        So, require

        System has a unique solution:

      • Remarks:

        • Derivation of Gauss rules for any k involves roots of orthogonal polynomials as the points.

        • Points and weights are tabulated for many standard cases.

        • Points are different for each value of k, but there are combinations of Gauss rules that allow you to reuse function values as points are added.

  2. Quadrature algorithms