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Thread: Error Propagation - Multiplication of xy

  1. #1
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    Post Error Propagation - Multiplication of xy

    What is the effect of computing xy on the size of existing relative errors in the stored values of x and y?

    This is what I have done:

    We have x̃=(1+ε_x)x and ỹ=(1+ε_y)y so that
    x̃ỹ =(1 + ε_x + ε_y + ε_x*ε_y) xy
    => (x̃ỹ/xy) - 1 = ε_x + ε_y + ε_x*ε_y
    => ε_xy = ε_x + ε_y + ε_x*ε_y

    Do I now get rid of ε_x*ε_y because that error is too small to include? (Is this the correct reason or is it something else?)

    So that I'm left with
    ε_xy = ε_x + ε_y? If this is correct, what does it mean?
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  2. #2
    Grand Panjandrum
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    Re: Error Propagation - Multiplication of xy

    Quote Originally Posted by CourtneyMoon View Post
    What is the effect of computing xy on the size of existing relative errors in the stored values of x and y?

    This is what I have done:

    We have x̃=(1+ε_x)x and ỹ=(1+ε_y)y so that
    x̃ỹ =(1 + ε_x + ε_y + ε_x*ε_y) xy
    So:

    \widetilde{xy}=(1+\varepsilon_x+\varepsilon_y+ \varepsilon_x \varepsilon_y)xy

    Now as both \varepsilon_x and \varepsilon_y are small we may ignore their product here to get:

    \widetilde{xy}\approx(1+\varepsilon_x+\varepsilon_  y)xy

    so:

    \varepsilon_{xy} \approx ...

    Of course you do not need to make the small errors assumption and approximation and just take the relative error straight from the first equation above.

    CB
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