31 5.5 Application: Stirling’s Formula Stirling’s formula is an interesting and useful way to approximate the factorial function, n!, for large values of n. Use Stirling ’ s formula to show that for all x. Example
32 5.6 Gaussian Quadrature Gaussian quadrature is a very powerful tool for approximating integrals. The quadrature rules are all base on special values of weights and abscissas (called Gauss points) The quadrature rule is written in the form weights Gauss points
36 Discussion The high accuracy of Gaussian quadrature then comes from the fact that it integrates very-high-degree polynomials exactly. We should choose N=2n-1, because a polynomial of degree 2n-1 has 2n coefficients, and thus the number of unknowns ( n weights plus n abscissas) equals the number of equations. Taking N=2n will yield a contradiction.
48 5.7 Extrapolation Methods One of the most important ideas in computational mathematics is— We can take the information from a few approximations and Use that to both estimate the error in the approximation and generate a significantly improved approximation In this section we will embark on a more detailed study of some of these ideas.