In this chapter, we have explored two basic problems in the field of approximation theory: interpolation and approximation in the sense of least squares. We learned that there are three different modes to approach solutions to these problems in SciPy:
- A procedural mode, that offers quick numerical solutions in the form of
ndarrays. - A functional mode that offers
numpyfunctions as the output. - An object-oriented mode, with great flexibility through different classes and their methods. We use this mode when we require from our solutions extra information (such as information about roots, coefficients, knots, and errors), or related objects (such as the representation of derivatives or antiderivatives).
We explored in detail all the different implementations for the interpolation coded in the scipy.interpolate module, and learned in particular that those related to splines are wrappers of several routines in the Fortran library FITPACK.
In the case of linear approximations in the least squares sense, we learned that we may achieve solutions either through systems of linear equations (by means of techniques from the previous chapter), or in the case of spline approximation, through wrappers to Fortran routines in the FITPACK library. All of these functions are coded in the scipy.interpolate module.
For nonlinear approximation in the least squares sense, we found two variants of the Levenberg-Marquardt iterative algorithm coded in the scipy.optimize module. These are in turn calls to the Fortran routines LMDER and LMDIF from the library MINPACK.
In the next chapter, we will master techniques and applications of differentiation and integration.