Re: trying to investigate and find the sensitivity of function
akerman said:
I have a question about function F(x) = sqrt(x)
I found that it has Kappa value equal to 1/2
I am not too sure what happens if when x = 0 is it just a minimum?
But now I am trying to investigate and find the sensitivity of F(x) to errors in x when we use x+ϵ, where ϵ is small.
Welcome to MHB, akerman! :)
I am assuming that with a Kappa value you mean the condition number.
If so then we have:
$$\frac{|\Delta y|}{|y|} = \kappa \frac{|\Delta x|}{|x|}$$
where $y=F(x)$, where $\Delta x$ is the error in $x$, and $\Delta y$ is the error in $y$.
In other words: $\kappa$ gives the amplification of the relative error.
At $x=0$, $x$ has an infinite relative error (we're dividing by zero), so $\kappa$ is not defined there.
Since $F(0)=0$, the relative error of $y$ is also infinite.
We can calculate $\kappa$ with:
$$\kappa = \left| \frac{x \cdot F'(x)}{F(x)} \right|$$
This stems from the first order Taylor approximation:
$$F(x + \Delta x) \approx F(x) + \Delta x \cdot F'(x)$$
Since we are using an approximation, the end result is also an approximation.
In particular for points where $F'(x)$ or $F''(x)$ are not defined, the relationship will break down.
This is in particular the case for your function at $x=0$.