Introduction

The inspiration for this text grew out of a simple question that emerged over a number of years of teaching math to Middle School, High School and College students.

Practically speaking, what is the origin of a particular polynomial?

So much time is spent analyzing, factoring, simplifying and graphing polynomials that it is easy to lose sight of the fact that polynomials have a wealth of practical uses. Exploring the techniques of interpolating data allows us to view the development and birth of a polynomial. This text is focused on laying a foundation for understanding and applying several common forms of polynomial interpolation. The principal goals of the text are:

1. Breakdown the process of developing polynomials to demonstrate and give the student a feel for the process and meaning of developing estimates of the trend (s) a collection of data may represent.
2. Introduce basic matrix algebra to assist students with understanding the process without getting bogged down in purely manual calculations. Some manual calculations have been included, however, to assist with understanding the concept.
3. Assist students in building a basic foundation allowing them to add additional techniques, of which there are many, not covered in this text.

What this text is not:

It is not a comprehensive survey of interpolation techniques.

The techniques presented are ones the author believes will provide a basic understanding of polynomial interpolation that students can build upon. There are many flavors and sub flavors of interpolation and I encourage students who are interested to check them out.

It is not a lesson in using interpolation apps:

Quite the opposite. By engaging in exercising calculations, the student is better equipped to understand how and why these techniques work.

What is polynomial interpolation?

We experience information in discrete ways.

Typically, it comes from measurements or observations. However, what we often want to do is look at a continuous process the data represents; all at once or at least at any point we choose. While we cannot represent a continuous process with a single number we can do so with an equation. Graphically this equation could be a single point (not usually that interesting). A straight line (degree one polynomial), a curved line (degree two polynomial) that we often call a quadratic equation or parabola; or some higher degree that graphically, often begins to look like a wave repeating itself.

When dealing with data the specific numbers always represent a snapshot. For example, if we measure rainfall and wind speed each day for a year, we have a collection of data points that compare rainfall to wind speed. We might ask if there is a relationship between wind speed and rainfall. For example, hurricane force winds are usually accompanied by heavy rainfall. It would be nice to develop an equation that can predict rainfall when high winds are expected. Normally someone analyzing this data would plot the points on the x, y coordinate plane. In this text, the sample data used to illustrate the various interpolation methods will be plotted in this way.

Can the math stand alone? Most certainly not. The challenge for someone utilizing interpolation techniques is to apply expertise and experience to determine the most appropriate polynomial structure. In other words, is the model most likely to accurately (or at least reasonably) produce useful estimated values? This is what I mean by the “Art” of polynomial interpolation.

Interpolation uses a known set of independent and dependent values to estimate other dependent values, typically along a continuous line represented by a polynomial. Technically if you use the model to identify additional data points outside of the range of the given points this is known as extrapolation. Our focus will be on interpolation within the given range.

Adaptations of the techniques we explore have been used in pre-computer times to generate tables of trig or log values used in applications such as navigation. Nowadays they are adapted for use by computers and calculators and they are an important part of the tool kit researchers use to predict future events such as emerging storms tracks, climate change, political elections, changing demographics, spread of disease, and so forth.

We will explore five Interpolation techniques: Elimination (Substitution), Newton’s Divided Difference, Splines, Least Squares and Taylor Series.