Comment by zackmorris

Thanks so much for that!

I've been struggling to curve fit an aerodynamics equation relating Mach number to rocket nozzle exit/entrance area ratio for quite some time. It's a 5th or 6th degree polynomial whose inverse doesn't have a closed-form solution:

https://en.wikipedia.org/wiki/Abel%E2%80%93Ruffini_theorem

But I was able to use a Chebyshev fit that is within a few percent accurate at 3rd degree, and is effectively identical at 4th degree or higher. And 4th degree (quartic) polynomials do have a closed-form solution for their inverse. That lets me step up to higher abstractions for mass flow rate, power, etc without having to resort to tables. At most, I might need to use piecewise-smooth sections, which are far easier to work with since they can just be dropped into spreadsheets, used for derivatives/integrals, etc.

Anyway, I also discovered (ok AI mentioned) that the Chebyshev approximation is based on the discrete cosine transform (DCT):

https://en.wikipedia.org/wiki/Discrete_Chebyshev_transform#R...

https://en.wikipedia.org/wiki/Discrete_cosine_transform#Appl...

That's why it's particularly good at curve-fitting in just 2 or 3 terms. Which is why they use the DCT for image compression in JPG etc:

https://www.mathworks.com/help/images/discrete-cosine-transf...

The secret sauce is that Chebyshev approximation spreads the error as ripples across the function, rather than at its edges like with Taylor series approximation. That helps it fit more intricate curves and arbitrary data points, as well as mesh better with neighboring approximations.