6 Polynomial approximation and interpolation

A polynomial is the simplest mathematical object malleable enough to present the concepts of approximation and interpolation well. You have an approximation when you have a complex mathematical object and want to represent it approximately with a simpler one, such as when you want to find a linear function that represents 1000 data points or when you want to emulate a trigonometric function using only multiplications and additions.

The approximation is important in data representation when we want to show a general trend behind some data, and we’re fine with the approximating function missing the actual data points. But in a special case of approximation called interpolation, the approximating function goes through all the data points precisely. Interpolation is often used in descriptive geometry for building curves and surfaces. In chapter 8, we’ll learn to build surfaces using polynomial interpolation.

Moreover, understanding polynomials and polynomial interpolation is a necessary step toward understanding polynomial splines, which we’ll discuss in chapter 7, and splines in general are staples of computer graphics and computer-aided design.

6.1 What are polynomials?

6.1.1 Axis intersections and roots of polynomial equations

6.1.2 Polynomial derivatives

6.1.3 Section 6.1 summary

6.2 Polynomial approximation

6.2.1 Maclaurin and Taylor series

6.2.2 The method of least squares

6.2.3 Practical example: Showing a trend with approximation

6.2.4 Section 6.2 summary

6.3 Polynomial interpolation

6.3.1 Using Vandermonde matrix to get the interpolating polynomial

6.3.2 What limits polynomial interpolation application to small data only?

6.3.3 How to lessen unwanted oscillations