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Error propagation with two functions, two unknowns.

  1. Aug 6, 2012 #1

    Hepth

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    Gold Member

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

    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:
    [tex]
    x \pm \delta_x = F[m_1, m_2,...]
    [/tex]
    [tex]
    y \pm \delta_y = G[m_1, m_2,...]
    [/tex]

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

    BUT if I cant 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.
     
  2. jcsd
  3. Aug 7, 2012 #2

    Stephen Tashi

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    Science Advisor

    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?
     
  4. Aug 8, 2012 #3

    haruspex

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    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 [tex]\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[/tex]
    Evaluating fx etc. at (m1, m2), solve to find δx, δy.
    Will need to check that the second order terms are not important.
     
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