# Relative errors product

• Jul 25th 2006, 02:58 AM
OReilly
Relative errors product
Can someone show me the proof that relative error of product of approximant numbers is equal to sum of relative errors of each factors?

I need to show that $\delta _{x'y'} = \delta _{x'} + \delta _{y'}$

I know that
$\left| {xy - x'y'} \right| = \left| {x'} \right|\Delta _{y'} + \left| {y'} \right|\Delta _{x'} + \Delta _{x'} \Delta _{y'}$

If we divide both sides with $\left| {x'y'} \right|$ we get that
$\frac{{\left| {xy - x'y'} \right|}}{{\left| {x'y'} \right|}} = \frac{{\left| {x'} \right|\Delta _{y'} }}{{\left| {x'y'} \right|}} + \frac{{\left| {y'} \right|\Delta _{x'} }}{{\left| {x'y'} \right|}} + \frac{{\Delta _{x'} \Delta _{y'} }}{{\left| {x'y'} \right|}}$
$\frac{{\Delta _{x'y'} }}{{\left| {x'y'} \right|}} = \frac{{\Delta _{y'} }}{{\left| {y'} \right|}} + \frac{{\Delta _{x'} }}{{\left| {x'} \right|}} + \frac{{\Delta _{x'} \Delta _{y'} }}{{\left| {x'y'} \right|}}$

$\delta _{x'y'} = \delta _{x'} + \delta _{x'} + \frac{{\Delta _{x'} \Delta _{y'} }}{{\left| {x'y'} \right|}}$

That's how I tried to prove but I don't know what to do with $\frac{{\Delta _{x'} \Delta _{y'} }}{{\left| {x'y'} \right|}}$
• Jul 25th 2006, 06:25 AM
CaptainBlack
Quote:

Originally Posted by OReilly
Can someone show me the proof that relative error of product of approximant numbers is equal to sum of relative errors of each factors?

I need to show that $\delta _{x'y'} = \delta _{x'} + \delta _{y'}$

I know that
$\left| {xy - x'y'} \right| = \left| {x'} \right|\Delta _{y'} + \left| {y'} \right|\Delta _{x'} + \Delta _{x'} \Delta _{y'}$

If we divide both sides with $\left| {x'y'} \right|$ we get that
$\frac{{\left| {xy - x'y'} \right|}}{{\left| {x'y'} \right|}} = \frac{{\left| {x'} \right|\Delta _{y'} }}{{\left| {x'y'} \right|}} + \frac{{\left| {y'} \right|\Delta _{x'} }}{{\left| {x'y'} \right|}} + \frac{{\Delta _{x'} \Delta _{y'} }}{{\left| {x'y'} \right|}}$
$\frac{{\Delta _{x'y'} }}{{\left| {x'y'} \right|}} = \frac{{\Delta _{y'} }}{{\left| {y'} \right|}} + \frac{{\Delta _{x'} }}{{\left| {x'} \right|}} + \frac{{\Delta _{x'} \Delta _{y'} }}{{\left| {x'y'} \right|}}$

$\delta _{x'y'} = \delta _{x'} + \delta _{x'} + \frac{{\Delta _{x'} \Delta _{y'} }}{{\left| {x'y'} \right|}}$

That's how I tried to prove but I don't know what to do with $\frac{{\Delta _{x'} \Delta _{y'} }}{{\left| {x'y'} \right|}}$

You throw it out as being of second degree in relative error, and so
if the relative errors are relativly small this is negligable in comparison.

RonL