6 Polynomial approximation and interpolation

A polynomial is the simplest mathematical object malleable enough to present the concepts of approximation and interpolation well.

The approximation is when you have a complex mathematical object and you want to represent it approximately with a simpler one. For instance, when you want to find a linear function that represents a thousand 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 there is also a special case of approximation, called interpolation, and it’s when the approximating function goes through all the data points precisely. Interpolation is often used in descriptive geometry for building curves and surfaces. For instance, in chapter 8 we will learn to build surfaces using polynomial interpolation.

6.1 Axis intersections and roots of polynomial equations

6.1.1 Polynomial derivatives

6.1.2 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 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 the unwanted oscillations?

6.3.4 Lagrange interpolation. Simple, genius, unnecessary

6.3.5 Practical example: showing the trend with interpolation

6.3.6 Summary

6.4 Practical example: showing a trend with a combination of approximation, interpolation, and extrapolation

6.4.1 The problem

6.4.2 The solution

6.4.3 Summary

6.5 Summary

6.6 Exercises