[Cuis-dev] Re : Re: testing Float results

[H. Fernandes]

Would be nice to have within DrGeo, especially when computing locus 

Dr. Geo -- http://gnu.org/s/dr-geo

----- ken.dickey--- via Cuis-dev <cuis-dev at lists.cuis.st> a écrit :
> On 2024-06-03 16:36, Mark Volkmann via Cuis-dev wrote:
> 
> > Is there a function that tests whether two Float values are "close" 
> > (within some delta)?
> 
> Really, it depends on what you expect to use numbers for.
> 
> Numerical methods using Floats are frequently unstable and "the wrong 
> answer fast".
> 
> https://people.cs.pitt.edu/~cho/cs1541/current/handouts/goldberg.pdf
> 
> Various strategies have been devised to make numerical calculations more 
> robust.
> 
> One simple idea, used in several systems such as Mathematica, is to keep 
> the highest and lowest possible values a function computes and carry the 
> calculation of values throughout a calculation. You then expect the 
> "exact" value to be within this interval.
> 
> https://en.wikipedia.org/wiki/Interval_arithmetic
> 
> If the interval is small, you may have a high confidence in a close 
> result. If the "answer" is a humongous interval, you better do the 
> error analysis.
> 
> An interesting variant of this is Ball Arithmetic, where the answer is 
> not an interval but lives within a (potentially multidimensional) 
> hypersphere.
> 
> https://www.texmacs.org/joris/ball/ball.html
> 
> NB: I am not mathematician enough to evaluate Ball Arithmetic.
> 
> I have used Interval Arithmetic for some cases. Interestingly, there 
> was an Apple function grapher which used Interval Arithmetic to 
> automatically calculate function values to the required precision. This 
> meant that you could look at f(x) = x * sin(x) near zero, and keep 
> "inzooming" to smaller and smaller ranges and still see an accurate 
> graph of this "squiggle function".
> 
> There are some very interesting discussions about real numbers -- and 
> holes between them -- in Lakoff & Nunez's _Where Mathematics Comes From_
> 
> HTH,
> -KenD
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