Lecture 31 Summary

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New topic: numerical integration (numerical quadrature). Began by basic definition of the problem (in 1d) and differences from general ODE problems. Then gave trapezoidal quadrature rule, and simple argument why the error generally decreases with the square of the number of function evaluations.

Showed numerical experiment (see handout) demonstrating that sometimes the trapezoidal rule can do much better than this: it can even have exponential convergence with the number of points! To understand this at a deeper level, I analyze the problem using Fourier series (see handout), and show that the error in the trapezoidal rule is directly related to the convergence rate of the Fourier series. Claimed (without proof for now) that this convergence rate is related to the smoothness of the periodic extension of the function, and in fact an analytic periodic function has Fourier coefficients that vanish exponentially fast, and thus the trapezoidal rule converges exponentially in that case.