mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

 2018-09-13 
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 spaced values of \(x\), and try to approximate the function:
$$f(x)=\frac1{1+25x^2}$$
Polynomial interpolations of \(\displaystyle f(x)=\frac1{1+25x^2}\) using equally spaced points
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.
Eight Chebyshev points
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.
Scroggs vs Parker, 6-8 July 2018
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.
Scroggs vs Parker, 6-8 July 2018, approximated using uniform points (left) and Chebyshev points (right)
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.
A list of constants and functions that RungeBot understands can be found here.

Similar posts

Big Internet Math-Off stickers 2019
Mathsteroids
realhats
PhD thesis, chapter 5

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "t" then "h" then "e" then "o" then "r" then "e" then "m" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Feb 2020

PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

ternary christmas card statistics palindromes interpolation hexapawn hannah fry numerical analysis matt parker raspberry pi tmip logic programming cambridge pizza cutting map projections geometry national lottery simultaneous equations realhats cross stitch weak imposition game show probability golden ratio martin gardner accuracy reuleaux polygons craft estimation sport speed london underground wave scattering wool computational complexity weather station books pythagoras frobel determinants folding paper european cup braiding chess coins trigonometry big internet math-off royal baby inline code captain scarlet dragon curves people maths harriss spiral countdown electromagnetic field sound bubble bobble final fantasy manchester nine men's morris error bars matrix of minors stickers light mathslogicbot fractals talking maths in public gerry anderson golden spiral propositional calculus ucl chalkdust magazine graph theory preconditioning asteroids news twitter chebyshev tennis polynomials game of life manchester science festival sobolev spaces phd machine learning gaussian elimination probability games plastic ratio football latex flexagons php reddit matrices binary python inverse matrices approximation matrix multiplication radio 4 data royal institution video games mathsjam boundary element methods a gamut of games draughts go matrix of cofactors finite element method puzzles folding tube maps noughts and crosses platonic solids rhombicuboctahedron bodmas hats dates mathsteroids christmas rugby world cup javascript advent calendar the aperiodical sorting data visualisation pac-man signorini conditions bempp curvature triangles menace misleading statistics dataset arithmetic london oeis

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020