# Error propagation - implicit functions

I have a little problem that I should solve quickly and I'm a little bit on pressure, so that any help/tip would be of great help.

I have two nonlinear equations with two unknown variables x and y and I want to know the variance of the solution variables. Basically the same type of equations, difference lies in the coefficients of the functions. $a,b, x0, d_{ij}, c, e, f$ are all known.

$f1(x,y_1) = a + b (x-x_0) - \ln{y_1} - \ln\dfrac{\sqrt{\dfrac{2dx}{c}+1}-1}{d} - 2*\dfrac{\sqrt{\dfrac{2dx}{c}+1}-1}{d} * \sum_{j=1}^2 y_jd_{ij}= 0$

$f2(x,y_1) = e + f (x-x_0) - \ln{(1-y_1)} - \ln\dfrac{\sqrt{\dfrac{2dx}{c}+1}-1}{d} - 2*\dfrac{\sqrt{\dfrac{2dx}{c}+1}-1}{d} * \sum_{j=1}^2 y_jd_{ij}= 0$

Two more functional relationships between: $y_2 = 1-y_1 (1.)$ $d = \sum_{i=1}^2 \sum_{j=1}^2 y_i y_j d_{ij} (2.) $$Solving the system of these equations using a numerical procedure I get the values of x and y, the solution variables. Because the variables a,b,e,f have their statistical uncertainties, their uncertainties will propagate into the solution also. a and b are fixed values and not dependent on x, but x and the partial derivatives of x are dependent of d, and d is dependent on y. In case of one nonlinear equations with one unknown variable x (y_1 is then = 1), I found the variance using implicit differentation. I wanted to expand this method on 2-nonlinear equations and I'll show how I wanted to do it: Because x is defined implicitly in both equations, I used the implicit differentiation theorem to find the partial derivatives of x in terms of the error affected variables for both functions: \frac{\partial x}{\partial a} = -\frac{\displaystyle\frac{\partial f1}{\partial a}}{\displaystyle\frac{\partial f1}{\partial p}}, \frac{\partial x}{\partial b} = -\frac{\displaystyle\frac{\partial f1}{\partial b}}{\displaystyle\frac{\partial f1}{\partial p}}, (\frac{\partial x}{\partial y})_1 = -\frac{\displaystyle\frac{\partial f1}{\partial y}}{\displaystyle\frac{\partial f1}{\partial p}}. \frac{\partial x}{\partial e} = -\frac{\displaystyle\frac{\partial f2}{\partial e}}{\displaystyle\frac{\partial f2}{\partial p}}, \frac{\partial x}{\partial f} = -\frac{\displaystyle\frac{\partial f2}{\partial f}}{\displaystyle\frac{\partial f2}{\partial p}}, (\frac{\partial x}{\partial y})_2 = -\frac{\displaystyle\frac{\partial f2}{\partial y}}{\displaystyle\frac{\partial f2}{\partial p}} All partial derivatives were approximated with the difference quotient and are known. (dx/dy) was written with subscripts 1 and 2 to mark the difference. Now, using the formula for error propagation (neglecting covariances) for both equations:$$\sigma^2_x = \left(\frac{\partial x}{\partial a}\right)^2 \sigma^2_{a} + \left(\frac{\partial x}{\partial b}\right)^2 \sigma^2_{b} + \left(\frac{\partial x}{\partial y}\right)^2_{1} \sigma^2_{y}\sigma^2_x = \left(\frac{\partial x}{\partial e}\right)^2 \sigma^2_{e} + \left(\frac{\partial x}{\partial f}\right)^2 \sigma^2_{f} + \left(\frac{\partial x}{\partial y}\right)^2_{2} \sigma^2_{y}$$Everything except the variances$$\sigma^2_x, \sigma^2_y$$is known in the last two equations. My idea was to bring them on one side and solve numerically:$$\sigma^2_x - \left(\frac{\partial x}{\partial y}\right)^2_{1} \sigma^2_{y} = \left(\frac{\partial x}{\partial a}\right)^2 \sigma^2_{a} + \left(\frac{\partial x}{\partial b}\right)^2 \sigma^2_{b} \sigma^2_x - \left(\frac{\partial x}{\partial y}\right)^2_{2} \sigma^2_{y}= \left(\frac{\partial x}{\partial e}\right)^2 \sigma^2_{e} + \left(\frac{\partial x}{\partial f}\right)^2 \sigma^2_{f}$\$

I tried this, but it didnt yield me the expected results.

I got expected results in the case of one nonlinear equations with one variable, but I assume it happened so, because the variable x wasn't dependent on any other variable. In the case of two equation, the derivatives of x become dependent on the value of the solution y, and I don't know if it is possible to use the error propagation law without considering this dependence. Is there any other way to find the variance or I made a mistake in my try to do it with implicit differentiaon?