I have an algorithm that takes 5 input parameters $r, i, \Omega, \omega, f$ and returns two outputs $X$ and $Y$. I happen to know the errors $\Delta X$, $\Delta Y$ associated to $X$ and $Y$ (which are found with a completely different and independent algorithm), and I want to determine the errors $\Delta r, \Delta i, \Delta \Omega, \Delta \omega, \Delta f$ associated to my 5 input parameters. The problem is that the relations between $X$ and $Y$ and the inputs are non-linear. The expression is :
\begin{equation} \begin{pmatrix} X \\ Y \end{pmatrix} = \begin{pmatrix} \cos{\Omega} & -\sin{\Omega} \\ \sin{\Omega} & \cos{\Omega} \\ \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 0 & \cos{i} \\ \end{pmatrix} \begin{pmatrix} \cos{\omega} & -\sin{\omega} \\ \sin{\omega} & \cos{\omega} \\ \end{pmatrix} \begin{pmatrix} r\cos f \\ r\sin f \end{pmatrix} \\ \\\\ \end{equation}
which also writes :
\begin{equation} \begin{matrix} X = r (\cos\Omega\cos(\omega +f)-\sin(\Omega)\sin(\omega +f)\cos{i}) \\ Y = r (\sin\Omega\cos(\omega +f)+\cos(\Omega)\sin(\omega +f)\cos{i}) \end{matrix} \end{equation}
As $X$ and $Y$ are not correlated, I've tried to determine the errors with the following quadratic error propagation equation :
\begin{equation} \sigma^2_{\mu} \approx \left| \frac{\partial f_{\mu}}{\partial X}\right| ^2\sigma^2_X+\left| \frac{\partial f_{\mu}}{\partial Y}\right|^2\sigma^2_Y \end{equation}
where $\mu$ is one of my orbital parameters. But for that, I have to establish the function $f_{\mu}$ that depends only on $X$ and $Y$ for each of the 5 input parameters but that is impossible. If I try to write such a function I always end-up with one input parameter depending on $X, Y$ and the other inputs parameters...
I guess this isn't the right approach to tackle such sort of problems so I tried to look at what is called "Backward Error Analysis" but complete and detailed methods are nowhere to be found... Does someone know how this works ? Is it simply a matter of applying independent perturbations on the inputs and seeing which perturbation yields output errors close to $\Delta X$, $\Delta Y$ ?
Thank you very much for your answers !