Error propagation with two functions, two unknowns.

[Hepth]

If I have two independent variables x,y, and two measurements, m1, m2 with errors. And the dependence is thus:
$$m_1 \pm \delta m_1 = f[x,y]$$
$$m_2 \pm \delta m_2 = g[x,y]$$

Now in my case, f and g are complicated expressions of x and y with no simple solution. (Actually I think i can solve one for x, but not for y).

Now if the equations were easy, I could solve for x and y:
$$x \pm \delta_x = F[m_1, m_2,...]$$
$$y \pm \delta_y = G[m_1, m_2,...]$$

And from there add the errors in quadrature to get the x and y errors.

BUT if I can't solve for x and y independently, and I must use numerical solutions to get the results ( I can, its easy). How can I go about getting the ERRORS? Is there another way I can solve for the errors and numerically solve for them, or a different method?

I have Mathematica if that helps.

 

[Stephen Tashi]

You should explain what you mean by "error". Are you doing numerical approximations that involve an error in that sense? Are you doing a stochastic simulation where randomness causes an "error"? Or are you teking physical measurements with equipment that has a specified precision?

 

[haruspex]

If I have two independent variables x,y, and two measurements, m1, m2 with errors. And the dependence is thus:
$$m_1 \pm \delta m_1 = f[x,y]$$
$$m_2 \pm \delta m_2 = g[x,y]$$

Now in my case, f and g are complicated expressions of x and y with no simple solution. (Actually I think i can solve one for x, but not for y).

Now if the equations were easy, I could solve for x and y:
$$x \pm \delta_x = F[m_1, m_2,...]$$
$$y \pm \delta_y = G[m_1, m_2,...]$$

Not sure if I've understood completely, but see if this helps.
Are f, g differentiable? Can you evaluate the derivatives at (m₁, m₂)? If so, can write $$\delta m_1 = \delta f = f_x \delta x + f_y \delta y; \delta m_2 = \delta g = g_x \delta x + g_y \delta y$$
Evaluating f_x etc. at (m₁, m₂), solve to find δx, δy.
Will need to check that the second order terms are not important.