# Thread: Error Propagation - Multiplication of xy

1. ## 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?

2. ## Re: Error Propagation - Multiplication of xy

Originally Posted by CourtneyMoon
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:

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

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

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

so:

$\displaystyle \varepsilon_{xy} \approx ...$

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