Evaluating the inverse of a function

Question

I need to evaluate the inverse of this function in the domain [0,1):

There is an analytic inverse (which I won't bother putting here as it's quite big), but it only works for a limited range of the constant A, i.e. if A>0.385f(x). Then, I thought of two possible solutions:

Use Newton-Raphson method (or any other root-finding method) to evaluate the x for each f(x);
Approximate the inverse with a sigmoid function and determine the parameters through least-squares.

I am tending towards (2), since it seems simple enough and doesn't slow down my code, as (1) might, but I'm insecure about the validity of the approximation. I'd also appreciate other suggestions.

My code is in C++, if there is some built-in way of solving my problem.

I think that this question is completely on-topic here, in the same way that other questions in the function-approximation tag are. — Pascal Cuoq, Aug 01 '18 at 11:52
My program isn't on Mathematica, I only used it to plot the function. — Renan Nobuyuki Hirayama, Aug 01 '18 at 11:53

score 1 · Accepted Answer · edited Aug 01 '18 at 13:41

1

Although both proposed methods you suggested are interesting for more general cases, this function can be inverted analytically.

In Mathematica you probably inverted it and got this "wrong" answer:

"wrong" answer

The correct real answer is the absolute value of this due to the squareroot.

Here is a plot to illustrate this "right" answer works:

"right" answer

edited Aug 01 '18 at 13:41

Paul R

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answered Aug 01 '18 at 13:04

Sorry but I couldn't post pictures due to my reputation in this stack! – Aug 01 '18 at 13:05

score 0 · Answer 2 · answered Aug 01 '18 at 12:16

0

Let y = Ax/(1-x**2)**(1/3), then y**3 = A**3*x**3/(1-x**2), then A**3*x**3 = y**3*(1-x**2). This is a cubic equation. Solve it, and you'll find x(y).

answered Aug 01 '18 at 12:16

user31264

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This is exactly the inverse I mentioned, but it contains a term sqrt(27A⁴x² - 4A²x⁴), which limits the values of A that I can use (not a possibility). – Renan Nobuyuki Hirayama Aug 01 '18 at 12:36
How does the term limit the values of A you can use? – user31264 Aug 01 '18 at 20:59

Evaluating the inverse of a function

2 Answers2