Approximate implicit function on real line

[stevengj]

@Tamas_Papp

It is easy to find the solution numerically (eg with Newton’s method). But I need to evaluate it a gazillion times for a given set of a_i, b_i, so I want to speed up the calculation. Currently it is taking up 70% of my runtime […]
I would solve for y(x) on some grid, approximate that, and from then on look up values from a quick approximation.

See also: Approximate inverse of a definite integral - #3 by stevengj

For a smooth function, you can get exponential convergence on a finite interval by evaluating on a Chebyshev grid and then interpolating with a polynomial, ala ApproxFun.jl or FastChebInterp.jl (as in the above post).

But I guess you have already thought about Chebyshev polynomials?

@Tamas_Papp

If I had lim x→±∞ y(x)=0 or some constant I would know how to approximate it with Chebyshev polynomials (appropriately transformed to R) But the asymptotes are giving me a hard time.

Just subtract off the asymptotes and approximate the what’s left? Oh, I see that you already thought of this:

@Tamas_Papp

It would be great to have a nice (= cheap, smooth, …) function g(x) that fits the asymptotes, then I would approximate y(x) - g(x).

@Tamas_Papp

The approximation should be AD friendly but that should not be difficult.

FastChebInterp.jl should be AD-friendly (it works with ForwardDiff, and it also hooks into ChainRules.jl to expose its own optimized derivatives to AD).