Chapter 6 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. First, polynomial interpolation is carried out using both the monomial basis and the Lagrange-interpolation formalism, sped up via the barycentric formula. This includes a derivation of the error and an emphasis on using unequally spaced points (Chebyshev nodes). Second, cubic-spline interpolation is introduced. Third, a section is dedicated to trigonometric interpolation, carefully working through the conventions and formalism needed to implement one of the most successful algorithms ever, the fast Fourier transform (FFT). Fourth, the topic of linear least-squares fitting is tackled, including the general formalism of the normal equations. The second edition includes a substantive new section on statistical inference, covering both frequentist and Bayesian approaches to linear regression. Nonlinear least-squares fitting is covered next, including the Gauss-Newton method and artificial neural networks. The chapter is rounded out by a physics project, on the experimental verification of the Stefan-Boltzmann law, and a problem set. In addition to providing a historical background on black-body radiation, the physics project shows an example of nonlinear least-squares fitting.

The story of the discovery of quantisation and quanta begins with the numerous problems facing physicists at the end of the nineteenth century. A number of experimental results did not fit naturally into the scheme of classical physics. Among the most important of these was the spectrum of black-body radiation, which was being determined much more precisely experimentally in the last decade of the nineteenth century and which had to be explained theoretically. Important clues came from the pioneering studies of the origin of the Stefan-Boltzmann law and Wien's displacement law, the latter involving the clever use of dimensional methods.