7 - Integrals

Summary

Chapter 7 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. Newton-Cotes integration methods are first studied ad hoc, via Taylor expansions and, second, building on the interpolation machinery of the previous chapter. Standard techniques like the trapezoid rule and Simpson’s rule are introduced, including the Euler-Maclaurin summation formula. The error behavior is employed to produce an adaptive-integration routine and also, separately, to introduce the topic of Romberg integration. The theme of integration from interpolation continues, when Gauss-Legendre quadrature is explicitly derived, including the integration abscissas, weights, and error behavior. Emphasis is placed on analytic manipulations that can help the numerical evaluation of integrals. The chapter then turns to Monte Carlo, namely stochastic integration: this is painstakingly introduced for one-dimensional problems, and then generalized to the real-world problem of multidimensional integration. The chapter is rounded out by a physics project, on variational Monte Carlo for many-particle quantum mechanics, and a problem set.