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Math Help - Absolute error minimal supremum

  1. #1
    Senior Member OReilly's Avatar
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    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?
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  2. #2
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    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.
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  3. #3
    Senior Member OReilly's Avatar
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    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?
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  4. #4
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    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
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  5. #5
    Senior Member OReilly's Avatar
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    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.
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