Differentiation and Integration

This chapter develops numerical methods for computing derivatives and integrals. Numerical differentiation of polynomials can be performed by synthetic division, or through special properties of trigonometric polynomials or orthogonal polynomials. For derivatives of more general functions, finite differences lead to difficulties with rounding errors that can be largely overcome by clever post-processing, such as Richardson extrapolation. Integration is a more complicated topic. The Lebesgue integral is related to Monte Carlo methods, and Riemann sums are improved by trapezoidal and midpoint rules. Analysis of the errors leads to the Euler-MacLaurin formula. Various polynomial interpolation techniques lead to specialized numerical integration methods. The chapter ends with discussions of tricks for difficult integrals, adaptive quadrature, and integration in multiple dimensions.

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