Re: floating point comparison

[Email_To...Hans.Olsson@dna.lth.se (Hans Olsson)]

In article <dewar.871925372@merv>, Robert Dewar <dewar@merv.cs.nyu.edu> wrote:
>Andrew says
>
><<        Strongly disagree. There ARE roundoff errors even in the IEEE 754
>arithemtic model. Moreover, the standard clearly specifies rounding models.>>
>
>You completely miss the point I am making.
>
>There are no *errors*, the discrepancies between IEEE arithmetic and
>real arithmetic are not errors, they are simply differences that come from
>two different arithmetic models.
>
>When we have integer arithmetic and we divide 10 by 3 to get 3, we do
>not say this is an error. The result is different from the mathematical
>value of 10.0/3.0, but there is no error here, just a different arithmetic 
>model.
>
>I know perfectly well that the phrase "rounding error" is well established,
>but my point is that calling it an error leads people into the niave trap
>of thinking of floating-point arithmetic as being real arithmetic.
>
>In fact I received quite a few email messages, from some quite interesting
>people :-) saying that they agreed that it was a pity that the term
>rounding error had ever got into the literature, but of course
>it is much too entrnched to get rid of.
>
>But your repsonse tends to make me think that you are indeed fallling
>into the trap of thinking of these xdiscepancies as errors. It's a mistake!

No, it's a way of seeing things. By seeing computer arithmetic as
real arithmetic combined with rounding errors obeying some simple rules,
some algorithms can easily be analyzed.

That model of arithmetic is not appropiate for all purposes, but it's 
in general appropiate for the numerical analysis I'm interested in.
IEEE makes the rounding predictable, which can help in some cases, and 
in _those_cases_ the term rounding error can be misleading.

Note that calling a well-defined and predictable discrepancy for error
instead of the result of different arithmetic/formula/model is very common in
numerical analysis, because it often gives a simple insight.
In some cases the errors are further analyzed, but are still called errors.

Consider discretizations of ODE/DAE/PDE. One could make an equally good
case that the error of a Runge-Kutta method is not an error, but the 
time-discretization is a completely different problem and should
be analyzed as such. Seeing the Runge-Kutta discretization as a new problem
is appropiate in some cases, but I'm still happy with the term "global error".

BTW:
Does there even exist error at all in your view of numerical analysis?
(Excluding programming errors).
If so, can you give any examples?
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