# How to show the relative error of $\frac{x_A}{y_A}$

First, this is how the relative error of $x_Ay_A$ (approximated errors) is computed as compared to $x_Ty_T$ (true errors) -

$\displaystyle Rel(x_Ay_A) = \frac{x_Ty_T - x_Ay_A}{x_Ty_T}$,

Letting $x_T = x_A + \epsilon$, and $y_T = y_A + \eta$, then

$\displaystyle Rel(x_Ay_A) = \frac{x_Ty_T - (x_T - \epsilon)(xy_T - \eta)}{x_Ty_T}$

$\displaystyle = \frac{\eta x_T + \epsilon y_T - \eta \epsilon}{x_Ty_T}$

$\displaystyle = \frac{\epsilon}{x_T} +\frac{\eta}{y_T} - \frac{\eta}{x_T} \frac{\epsilon}{y_T}$

$\displaystyle = Rel(x_A)+Rel(y_A)-Rel(x_A)Rel(y_A)$

When both $Rel(x_A)$ and $Rel(y_A)$ are small in size when compared with 1, $Rel(x_A y_A) \approx Rel(x_A)+Rel(y_A)$.

Given this information, I am asked to show that $Rel(\frac{x_A}{y_A}) = \frac{Rel(x_A)-Rel(y_A)}{1-Rel(y_A)} \approx Rel(x_A)-Rel(y_A)$.

Can this be done following the same general method in the example of finding the relative error of multiplication, e.g. substituting values for $x_A$ and $y_A$? I seem to get rather tripped up while doing that, such as the fraction becoming very complicated, etc. - though perhaps it's just algebra mistakes!

I found this post, asking essentially the exact same question - however, the accepted answer uses a method not shown in the book we are using, or any concepts we've gone over in class time, so I assume there is a way to do it which is similar to the given example, instead?

## 1 Answer

The method in the answer you linked is the same as your example for $Rel(x_Ay_A)$, just it deals only in relative errors ($e_X$ and $e_Y$), and you are using absolute errors ($\epsilon$ and $\eta$). Write $x_A$ in terms of $x_T$, and $y_A$ in terms of $y_T$, then evaluate $Rel(x_A/y_A)$:

\begin{align} Rel(x_A/y_A)&=\frac{x_T/y_T-\frac{x_T-\epsilon}{y_T-\eta}}{x_T/y_T} \\ &=1-\frac{y_T(x_T-\epsilon)}{x_T(y_T-\eta)} \\ &=1-\frac{y_T}{y_T-\eta}+\frac{\epsilon}{x_T(1-\eta/y_t)} \\ &\approx 1-\left[1+\frac{\eta}{y_T}\right]+\frac{\epsilon}{x_T}\left[1+\frac{\eta}{y_T}\right] \\ %&=-\frac{\eta}{y_T}+\frac{\epsilon}{x_T}+\frac{\eta}{y_T}\frac{\epsilon}{x_T} \\ %&\approx-\frac{\eta}{y_T}+\frac{\epsilon}{x_T} \end{align} using Taylor series expansions around $\eta=0$ and $\epsilon=0$ to expand the fractions as series'.

I think you should be able to finish it off.

• Very helpful. Thanks. Feb 18, 2015 at 5:00