Comparative analysis of numerical integration techniques

When integrating numerically, if the integrand can be expressed exactly as a polynomial of degree n, over a finite interval; then either Simpson\u27s rule, Romberg integration, Legendre-Gauss or Jacobi-Gauss quadrature formulas provide good results. However, if the integrand can not be expressed exactly as an nth degree polynomial, then perhaps it can be expressed as a function f(x) divided by √1-x 2, or as a function g(x) times (1-x)α (l+x)ß , where α and ß are some real numbers \u3e1, or as a function h(x) times one. If this is the case then the Chebyshev-Gauss, Jacobi-Gauss, and Legendre- Gauss quadrature are respectively quite useful. If the integrand can not be expressed as f(x)/ √1-x 2 or as g(x)·(1-x) α ·(1+x) ß or as h(x) ·(1) then the Romberg method should be used. If the interval of integration is [0,∞] or [-∞,∞], then the Laguerre-Gauss and the Hermite-Gauss methods respectively are generally quite useful. The results of this study indicate that the quadrature formula to use in a given situation is dependent upon the interval of integration and the integrand. However, the results also indicate certain guide lines for choosing the type of quadrature formula to use in a given situation --Abstract, page iii