Need an algorithm to calculate a function

[Mentz114]

I'm trying to calculate a table of x vs t, with x as the dependent variable from
this formula

$$t + C = \frac{1}{2}(x\sqrt( 1 - x^2) + arcsin(x) )$$

C=0.4783 is given when t=0 and x = 1/2

I thought it would be simple but my code is giving nonsensical results, viz.
a straight line ! I do get the correct answer when t =0 (1/2) and I can't find a fault in my code. My code also gives a straight line for

t = arcsin(x) + 0.5 which obviously wrong, since x = sin( t - 1/2).

Has anyone got a general algorithm for this ? I'm using simplex but I
haven't tried Newton-Ralphson

This is related to the 'Interesting Oscillator Potential' topic below.

 

[arildno]

You might use power series inversion:
1. Introduce the "tiny" variable $\epsilon=x-\frac{1}{2}$
and find the first few Taylor series terms of the right-hand side about $\epsilon=0$
You have now effectively found t as a power series in $\epsilon$

2. Assume that this power series is invertible, i.e, it exists a function:
$$\epsilon(t)=\sum_{i=0}^{\infty}a_{i}t^{i}$$

If you can find the coefficients $a_{i}$ you're finished!

3.Insert this power series on the epsilon places, and collect terms of equal power in t. (For each finite power of t, only a finite number of terms need to be collected.*)

4. Different powers of t are linearly independent functions, hence all coefficients of the powers must equal zero. This demand of zeroes furnishes you with the equations to determine the coefficients $a_{i}$

*EDIT:
This requires that $a_{0}=0$.
This, however, holds, since $\epsilon=0$ when t=0.

 

[Mentz114]

Thanks, arildno. Food for thought.