[jason_s]

While I'm glad to see the OP got a good minimax solution at the end, it seems like the article missed clarifying one of the key points: error waveforms over a specified interval are critical, and if you don't see the characteristic minimax-like wiggle, you're wasting easy opportunity for improvement.

Taylor series in general are a poor choice, and Pade approximants of Taylor series are equally poor. If you're going to use Pade approximants, they should be of the original function.

I prefer Chebyshev approximation: https://www.embeddedrelated.com/showarticle/152.php which is often close enough to the more complicated Remez algorithm.

[rockmeamedee]

I had no idea, but this "wiggle" is required for an optimal approximation, it's called the "equioscillation property" [https://en.wikipedia.org/wiki/Equioscillation_theorem].

For a polynomial P (of degree n) to approximate a function F on the real numbers with minimal absolute error, the max error value of |P - F| needs to be hit multiple times, (n+2 times to be precise). You need to have the polynomial "wiggle" back and forth between the top of the error bound and the bottom.

And even more surprisingly, this is a necessary _and sufficient_! condition for optimality. If you find a polynomial whose error alternates and it hits its max error bound n+2 times, you know that no other polynomial of degree n can do better, that is the best error bound you can get for degree n.

Very cool!

[ogogmad]

Chebyshev polynomials cos(n arcos(x)) provide one of the proofs that every continuous function f:[0,1]->R can be uniformly approximated by polynomial functions. Bernstein polynomials provide a shorter proof, but perhaps not the best numerical method: https://en.wikipedia.org/wiki/Bernstein_polynomial#See_also

[AlotOfReading]

Those don't guarantee that that they can be well approximated by a polynomial of degree N though, like we have here. You can apply Jackson's inequality to calculate a maximum error bound, but the epsilon for degree 5 is pretty atrocious.

[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.