Practically all the different areas of numerical analysis are contemplated in some SciPy module. For example, in order to compute values of special functions, we use the scipy.special module. The scipy.interpolate module takes care of interpolation, extrapolation, and regression. For optimization, we have the
scipy.optimize module, and finally, we have the scipy.integrate module for numerical evaluation of integrals. This last module serves as the interface to perform numerical solutions of ordinary differential equations as well.
Thus, in this chapter, we will first extensively explore how to use SciPy to numerically evaluate the special functions that are commonly found in the field of mathematical physics. Then, we will discuss the modules available in SciPy to tackle regression, interpolation, and optimization problems.
The chapter ends with a solution of the chaotic Lorenz system as an illustration of the capabilities included in SciPy to find numerical solutions of ordinary differential equations. The corresponding IPython Notebook will help you to try the functionalities of the modules involved in the computations and to modify each illustrative example according to your specific needs.
The scipy.special module contains numerically stable definitions of useful functions. Most often, the straightforward evaluation of a function at a single value is not very efficient. For instance, we would rather use a Horner scheme (http://en.wikipedia.org/wiki/Horner%27s_method) to find the value of a polynomial at a point than use the raw formula. The NumPy and SciPy modules ensure that this optimization is always guaranteed with the definition of all its functions, whether by means of Horner schemes or with more advanced techniques.