Absolute error minimal supremum

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July 22nd 2006, 01:39 PM
OReilly

Absolute error minimal supremum

This is definition from book:

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

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

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

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

Can someone explain me on some example this smallest supremum?
July 22nd 2006, 06:02 PM
ThePerfectHacker

Quote:

Originally Posted by OReilly

This is definition from book:

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

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

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

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

Can someone explain me on some example this smallest supremum?

The book was speaking about approximations. Given a number $x$ and its approximant $x'$ the error is $\epsilon=|x-x'|$ (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 $x'$ is. Because if we did then we would know what actual value of $x$ 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 $|x-x'|<\epsilon$ (which is possible in real life). And the inequality says how close something is, for example is $\epsilon=.00002$ the approximant is 4 decimal places corret which is accurate.
July 23rd 2006, 04:29 AM
OReilly

Let me give my example if I understood correctly.

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

Highest number $x$ can be $x = 0.00159$ and smallest approximant can be $x' = 0.0015$ so we have smallest supremum $\Delta _{x'} = 0.00009$ 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?
July 23rd 2006, 04:57 AM
CaptainBlack

Quote:

Originally Posted by OReilly

This is definition from book:

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

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

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

$\Delta _{x'}$ 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.

I think we may need more information about the problem.

RonL
July 23rd 2006, 05:03 AM
OReilly

Quote:

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.

I think we may need more information about the problem.

RonL

Actually I mean only "supremum" not "smallest supremum". My mistake.