# Thread: Absolute error minimal supremum

1. ## Absolute error minimal supremum

This is definition from book:

1. Number $\displaystyle \Delta (x') = \left| {x - x'} \right|$ represents absolute error of number $\displaystyle x'$

Then it says that in real life is often impossible to find out value of absolute error $\displaystyle \Delta (x')$ but it is possible to find out smallest supremum of $\displaystyle \Delta (x')$.

So $\displaystyle \Delta (x') = \left| {x - x'} \right| \le \Delta _{x'}$

$\displaystyle \Delta _{x'}$ is smallest supremum.

Can someone explain me on some example this smallest supremum?

2. Originally Posted by OReilly
This is definition from book:

1. Number $\displaystyle \Delta (x') = \left| {x - x'} \right|$ represents absolute error of number $\displaystyle x'$

Then it says that in real life is often impossible to find out value of absolute error $\displaystyle \Delta (x')$ but it is possible to find out smallest supremum of $\displaystyle \Delta (x')$.

So $\displaystyle \Delta (x') = \left| {x - x'} \right| \le \Delta _{x'}$

$\displaystyle \Delta _{x'}$ is smallest supremum.

Can someone explain me on some example this smallest supremum?
The book was speaking about approximations. Given a number $\displaystyle x$ and its approximant $\displaystyle x'$ the error is $\displaystyle \epsilon=|x-x'|$ (because you need to know which one is larger which is same thing as using absolute). But in real life we usally do not know how exactly close an approximant $\displaystyle x'$ is. Because if we did then we would know what actual value of $\displaystyle x$ is, (which we do not know and thus we approximate).
However, when we want to determine how close something is to something else, we use an inequality $\displaystyle |x-x'|<\epsilon$ (which is possible in real life). And the inequality says how close something is, for example is $\displaystyle \epsilon=.00002$ the approximant is 4 decimal places corret which is accurate.

3. Let me give my example if I understood correctly.

If we have some number $\displaystyle x$ so that $\displaystyle \frac{{15}}{{10^4 }} < x < \frac{{16}}{{10^4 }}$ (x is with accuracy of $\displaystyle 10^{ - 5}$) and we need to find approximate number $\displaystyle x'$ with accuracy of $\displaystyle 10^{ - 4}$ then smallest supremum of absolute error would be $\displaystyle \Delta _{x'} = 0.00009$ because $\displaystyle \frac{{15}}{{10^4 }} \le x' \le \frac{{16}}{{10^4 }}$.

Highest number $\displaystyle x$ can be $\displaystyle x = 0.00159$ and smallest approximant can be $\displaystyle x' = 0.0015$ so we have smallest supremum $\displaystyle \Delta _{x'} = 0.00009$ which means that is the highest absolute error that can be.

I am little confused about term smallest supremum and I would like explanation of that.

Am I right?

4. Originally Posted by OReilly
This is definition from book:

1. Number $\displaystyle \Delta (x') = \left| {x - x'} \right|$ represents absolute error of number $\displaystyle x'$

Then it says that in real life is often impossible to find out value of absolute error $\displaystyle \Delta (x')$ but it is possible to find out smallest supremum of $\displaystyle \Delta (x')$.

So $\displaystyle \Delta (x') = \left| {x - x'} \right| \le \Delta _{x'}$

$\displaystyle \Delta _{x'}$ is smallest supremum.

Can someone explain me on some example this smallest supremum?
I am petrubed by the use of the term "smallest supremum". The supremum
is a least upper bound. The term "smallest supremum" seems to be usually
associated with a family of numerical methods, of other entities, where the
"best" is the one which has the smallest supremum of an appropriate measure of the error.

RonL

5. Originally Posted by CaptainBlack
I am petrubed by the use of the term "smallest supremum". The supremum
is a least upper bound. The term "smallest supremum" seems to be usually
associated with a family of numerical methods, of other entities, where the
"best" is the one which has the smallest supremum of an appropriate measure of the error.