Finding the Largest (or smallest) Value of a Function - given some constant symmetric errors

[erobz]

Just wondering if there is a mathematical way to find which sign ($\pm$) to take on symmetric measured error in a function $f$ of some variables. An example, lets say formulaically we find that $f = k x$, with $k>0$ we measure $x$ and append some symmetric error $\pm \epsilon_x$. So we say:

$$ (f + \epsilon_f) - f = k( x + (\pm \epsilon_x) ) - kx $$

$$ \implies ( \epsilon_f ) = k (\pm \epsilon_x) $$

So by inspection if we want to increase $f$ we don't want its change to be negative, thus we select $+\epsilon_x$. And visa-versa if we wish to find the smallest $f$.

Now, lets increase the complexity of $f$ with more measured variables that have symmetric error, for example:

$$f = \frac{kx}{y+1}$$

$$ \implies \epsilon_f = \frac{k(\pm \epsilon_x) ( y+1)-kx (\pm \epsilon_y) }{(y+1)^2+(y+1)(\pm \epsilon_y)}$$

Now I can still reason out this one, if we want $f$ to be its largest value we make numerator largest and denominator smallest:

$$ \implies \epsilon_f = \frac{k(+ \epsilon_x) ( y+1)-kx (- \epsilon_y) }{(y+1)^2+(y+1)(- \epsilon_y)}$$

What do you do if the function is complex enough such that it's not at all clear what combination will produce the upper/lower bound for the function?

Checking every sign combination by brute force will work, but it feels like an optimization of some kind.

 

[pasmith]

There isn't much guesswork. To first order we have $$\epsilon_f = \frac{\partial f}{\partial x}\epsilon_x + \frac{\partial f}{\partial y}\epsilon_y$$ To maximize this, we want $\epsilon_x$ to be positive if $\frac{\partial f}{\partial x}$ is positive and negative if $\frac{\partial f}{\partial x}$ is negative, etc.

 

[erobz]

There isn't much guesswork. To first order we have $$\epsilon_f = \frac{\partial f}{\partial x}\epsilon_x + \frac{\partial f}{\partial y}\epsilon_y$$ To maximize this, we want $\epsilon_x$ to be positive if $\frac{\partial f}{\partial x}$ is positive and negative if $\frac{\partial f}{\partial x}$ is negative, etc.

Thats propbably more efficient than checking each combination.

So you just look at the first order change, and that will always correctly decide how to arrange the signs to get a minimum or maximum. So for a minimum we choose opposite signs of the partial derivative.

Then you just plug in the accordingly signed errors into the actual function and we are good to go?

You say to first order, are there caveats where the higher order derivatives will bungle this up?

 

[erobz]

Another thing, what if the function had a peak around the measured variable. Imagine our measurement is on 1 side of a peak, we evaluate this expression, it tells us to select the positive error. However, when I input the finite error, I could end up lower than I would have otherwise if it crosses the peak.

Does evaluating the derivative $\left. \frac{\partial f }{\partial x } \right|_{x+\epsilon_x}$ cover the possible sign change?