Numerical Integration

Abstract

1. Numerical integration: basic rules 1 1.1. General formulas for the error on numerical quadrature 1 1.2. Particular cases 3 2. A birds’-eye view on interpolatory quadrature rules 5 3. Composite rules 6 3.1. Composite trapezoidal rule 6 4. The Romberg integration 9 4.1. Composite Simpson’s rule 9 4.2. Obtaining composite Simpson’s rule by Richardson extrapolation 10 5. Adaptive Simpson’s rule 11 6. Gaussian quadrature 14 6.1. Maximizing the degree of exactness 15 6.2. Orthogonal polynomials 16 6.3. Three Term Recurrence Relationships 18 6.4. The exactness of Gaussian quadrature 20 6.5. Error estimate 22 6.6. Example: Gauss-Chebyshev quadrature 22 6.7. A naive way for finding nodes and weights 23 6.8. Golub-Welsch algorithm for finding nodes and weights 24