Re: What has become of John Maddock's implementation of the Remez algorithm for Chebychev approximation?

From: John Maddock (jz.maddock_at_[hidden])
Date: 2021-06-16 17:07:31

On 16/06/2021 14:03, Joachim Wuttke via Boost wrote:
> Dear John, dear Bjorn,
>
> thank you very much for your answers.
>
> If I understand correctly, function 
> chebyshev_transform::chebyshev_transform
> solves the minimax approximation problem in a completely different way 
> than
> the Remez algorithm, but results should be of similar quality?

Exactly.  Depending on the method used and your input conditions you 
will end up with subtly different results, but should end up with a set 
of coefficients that are "in the zone".  The actual minima can be broad 
and bumpy if you see what I mean.

Be aware that even smooth functions can be resistant to decent 
approximations - for example I have never managed to generate any for 
the elliptic integrals, and haven't heard of any one else succeeding 
either (I haven't looked in a while though), even though the functions 
are smooth and parabola like.

Whatever method you use, you will need an accurate implementation of the 
function (which may be super slow - that doesn't matter at this stage - 
and perhaps calculated at extended precision via numeric integration or 
whatever).  You may also need to "divide out" whatever the function 
converges to at it's limits.  Some more info here: 
https://www.boost.org/doc/libs/1_76_0/libs/math/doc/html/math_toolkit/remez.html#math_toolkit.remez.remez_practical

Good luck ;)

John.

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