TITLE: Determine $x$-value given 3 points on a curve (Quadratic approximation)
QUESTION [1 upvotes]: I have 3 points ($x_0$, $y_0$), ($x_1$, $y_1$) and ($x_2$, $y_2$) that lie on a monotonically-increasing (asymptotic) curve (whose function is not known a priori).
The only unknown value is $x_2$, which corresponds to reaching an asymptote (in the limit sense) of $y_2$.
My question is: Is there a straightforward numeric way to determine $x_2$? My eventual goal is to proceed iteratively until convergence is reached.
I am familiar with numerical root-finding algorithms, of which Muller's method crossed my mind, since it relies on three known points to construct a parabola that interpolates the points. Apart from this, my problem is different in that I wish to know the value of $x_2$ that gives $f(x_2) = y_2$.
Any thoughts on a sound approach?

REPLY [0 votes]: There is a very simple function that can be made to fit the data exactly.
$$y=a+\frac b{x+c}$$
$a=y_2$ to reach the asymptote. To find $b$ and $c$, rearrange the equation to be linear in them and solve:
$$(y_0-y_2)(x_0+c)=(y_1-y_2)(x_1+c)=b\tag1$$
$$c=\frac{x_0(y_2-y_0)+x_1(y_1-y_2)}{y_0-y_1}$$
where the value obtained for $c$ is back-substituted into $(1)$ to find $b$. Note that $b$ will always be negative, since the function is monotonically increasing.
Now suppose $x_2$ is to be found which makes the difference between the function value there and $y_2$ less than some constant $\varepsilon>0$. This entails solving
$$y_2-\left(y_2+\frac b{x_2+c}\right)<\varepsilon$$
This is also easy:
$$-\frac b{x_2+c}<\varepsilon$$
$$x_2>-\frac b\varepsilon-c$$
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