chapter six

6 Polynomial approximation and interpolation

This chapter covers

Polynomials and their properties.
Using polynomial interpolation and approximation to describe continuous phenomena.
Understanding power series and the balance between computation time and approximation accuracy.
Circumventing the limitations of polynomials for data representation.

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

You can often use polynomial approximations when you have a complex mathematical object and you want to represent it approximately as a simpler one. When you have a thousand data points you might want to approximate them as a single linear function. Such a scenario is called linear approximation or linear regression. Another example is when you have a complex trigonometric function, and you want to compute it using only multiplications and additions of numbers. This is usually done with power series.

When approximating we exchange accuracy for performance in computations. In data representation, we approximate to show a trend even though the actual data may be noisy and not fit the trend perfectly. The special case of precise approximation is called interpolation is also used in descriptive geometry for building curves and surfaces.

6.1 What are polynomials?

6.1.1 Axis intersections and roots of polynomial equations

6.1.2 Odd and even polynomials

6.1.3 Polynomial derivatives

6.1.4 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 and interpolation