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 wrote @RungeBot, a Twitter bot that can compare interpolations with equispaced and Chebyshev points. Since first publishing this post, Twitter's API changes broke @RungeBot, but it lives on on Mathstodon: @RungeBot@mathstodon.xyz. 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 toot "@RungeBot@mathstodon.xyz 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.
×1      ×1      ×1      ×1      ×1
(Click on one of these icons to react to this blog post)

You might also enjoy...

Comments

Comments in green were written by me. Comments in blue were not written by me.
Hi Matthew, I really like your post. Is there a benefit of using chebyshev spaced polynomial interpolation rather than OLS polynomial regression when it comes to real world data? It is clear to me, that if you have a symmetric function your approach is superior in capturing the center data point. But in my understanding in your vote-example a regression minimizing the residuals would be preferrable in minimizing the error. Or do I miss something?
Benedikt
                 Reply
 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> <logo>
To prove you are not a spam bot, please type "htdiw" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2025 

Jan 2025

Christmas (2024) is over
Friendly squares
 2024 
▼ show ▼
 2023 
▼ show ▼
 2022 
▼ show ▼
 2021 
▼ show ▼
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

electromagnetic field bots stirling numbers dataset crossnumber pascal's triangle latex computational complexity friendly squares a gamut of games dragon curves interpolation talking maths in public cross stitch london regular expressions triangles logo bubble bobble mean finite group javascript asteroids game show probability gerry anderson royal institution map projections harriss spiral wave scattering mathslogicbot palindromes correlation geogebra machine learning matrix multiplication hats the aperiodical pac-man manchester gaussian elimination gather town books data visualisation graphs signorini conditions dinosaurs puzzles numerical analysis ucl folding paper fence posts errors royal baby rhombicuboctahedron chalkdust magazine coins football frobel polynomials fractals sport hyperbolic surfaces crochet matrix of minors logic sound european cup geometry reuleaux polygons raspberry pi bempp anscombe's quartet christmas card approximation hannah fry noughts and crosses christmas php standard deviation rugby databet quadrilaterals london underground hexapawn mathsteroids inline code arithmetic tennis pi approximation day realhats live stream finite element method youtube phd oeis menace zines fonts preconditioning accuracy estimation platonic solids nine men's morris error bars national lottery exponential growth probability countdown simultaneous equations recursion bodmas matrix of cofactors logs weak imposition data turtles numbers sobolev spaces inverse matrices big internet math-off reddit edinburgh people maths wool folding tube maps sorting trigonometry go cambridge determinants programming video games tmip binary world cup speed boundary element methods manchester science festival stickers propositional calculus matt parker dates ternary final fantasy convergence chess python radio 4 light misleading statistics statistics captain scarlet draughts guest posts graph theory kings game of life squares matrices golden spiral mathsjam news pythagoras games runge's phenomenon weather station advent calendar braiding plastic ratio golden ratio datasaurus dozen curvature pi pizza cutting 24 hour maths flexagons craft newcastle martin gardner chebyshev

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2025