# Blog

## Archive

Show me a random blog post**2019**

**2018**

**2017**

**2016**

**2015**

**2014**

**2013**

**2012**

## Tags

propositional calculus mathslogicbot twitter rhombicuboctahedron sorting the aperiodical bodmas data dataset plastic ratio harriss spiral pizza cutting people maths programming countdown final fantasy hexapawn chebyshev weather station latex london underground hats game show probability reuleaux polygons realhats curvature sound palindromes aperiodical big internet math-off draughts error bars craft frobel braiding pac-man bubble bobble reddit manchester science festival triangles arithmetic puzzles a gamut of games polynomials games golden spiral misleading statistics fractals pythagoras european cup accuracy probability royal baby raspberry pi mathsteroids go christmas card menace binary rugby machine learning cross stitch gerry anderson interpolation trigonometry folding tube maps php javascript inline code wool approximation radio 4 news coins sport geometry dates nine men's morris noughts and crosses national lottery game of life oeis chess football light matt parker books tennis platonic solids asteroids london dragon curves world cup estimation map projections captain scarlet speed statistics logic manchester stickers python folding paper video games martin gardner electromagnetic field golden ratio flexagons graph theory christmas ternary chalkdust magazine**2018-09-13**

## Runge's Phenomenon

This is a post I wrote for round 2 of The Aperiodical's Big Internet Math-Off 2018. As I went out in round 1 of the Big Math-Off, you got to read about the real projective plane instead of this.

Polynomials are very nice functions: they're easy to integrate and differentiate, it's quick to calculate their value at points, and they're generally friendly to deal with. Because of this, it can often be useful to find a polynomial that closely approximates a more complicated function.

Imagine a function defined for \(x\) between -1 and 1. Pick \(n-1\) points that lie on the function. There is a unique degree \(n\) polynomial (a polynomial whose highest power of \(x\) is \(x^n\)) that passes through these points. This polynomial is called an

*interpolating polynomial*, and it sounds like it ought to be a pretty good approximation of the function.So let's try taking points on a function at equally spaces values of \(x\), and try to approximate the function:

$$f(x)=\frac1{1+25x^2}$$
I'm sure you'll agree that these approximations are pretty terrible, and they get worse as more points are added. The high error towards 1 and -1 is called Runge's phenomenon, and was discovered in 1901 by Carl David Tolmé Runge.

All hope of finding a good polynomial approximation is not lost, however: by choosing the points more carefully, it's possible to avoid Runge's phenomenon. Chebyshev points (named after Pafnuty Chebyshev) are defined by taking the \(x\) co-ordinate of equally spaced points on a circle.

The following GIF shows interpolating polynomials of the same function as before using Chebyshev points.

Nice, we've found a polynomial that closely approximates the function... But I guess you're now wondering how well the Chebyshev interpolation will approximate other functions. To find out, let's try it out on the votes over time of my first round Big Internet Math-Off match.

The graphs below show the results of the match over time interpolated using 16 uniform points (left) and 16 Chebyshev points (right). You can see that the uniform interpolation is all over the place, but the Chebyshev interpolation is very close the the actual results.

But maybe you still want to see how good Chebyshev interpolation is for a function of your choice... To help you find out, I've written @RungeBot, a Twitter bot that can compare interpolations with equispaced and Chebyshev points. Just tweet it a function, and it'll show you how bad Runge's phenomenon is for that function, and how much better Chebysheb points are.

For example, if you were to tweet @RungeBot f(x)=abs(x), then RungeBot would reply: Here's your function interpolated using 17 equally spaced points (blue) and 17 Chebyshev points (red). For your function, Runge's phenomenon is terrible.

A list of constants and functions that RungeBot understands can be found here.

### Similar posts

Mathsteroids | realhats | Harriss and other spirals | Christmas (2018) is over |

### Comments

Comments in green were written by me. Comments in blue were not written by me.

**
**

**© Matthew Scroggs 2019**

Add a Comment