Re: Zero or not?

[Matthias Brennwald]

Dear Franceso and James

Thanks for you replies. I've got a few more comments:

"Francesco Potorti`" <address@hidden> wrote:
>>Ok, that's a start. If I calculate the sum of N numbers, what is the
>>maximum 'offset' of the numerical result to mathematically correct
>>result?
>
> Not a numerical expert, but the error magnitude should be bounded by
> eps*sum(abs(x[1:N])).

Are you sure this is what you intended to write? For example:

octave:5> eps
ans =  2.2204e-16
octave:6> x = [ 1E5 -1E5 ]; N = length(x);
octave:7> eps*sum(abs(x(1:N)))
ans =  4.4409e-11

I just hope (pray) that Octave can do better than this when calculating
the sum of the values in x (i.e. sum(x))...

"James Sherman Jr." <address@hidden> wrote:
> I don't know if this is even possible to quantify absolutely.  take for
> example:
>
> octave:55> 1e20+0.1-1e20
> ans = 0
>
> Because underflow happens when you add 1e20 and 0.1, the result is still
> 1e20.  So I believe the maximum offset for a set of numbers (worse case
> scenario) is something on the order  of:  (let N be the vector of numbers)
> maximumOffset = length(N)*eps*max(abs(N));
>
> Which would be like adding 0.1 to 1e20, N-1 times.  Note that this because
> eps is not an absolute value, but a relative one.

Ok, that makes sense to me. In fact, I've found the following:

octave:36> 1e20+8.197e3-1e20
ans =  16384
octave:37> 1e20+8.196e3-1e20
ans = 0

However:

octave:39> 8.196e3/1e20
ans =  8.1960e-17

This is much less than eps, which is:

octave:41> eps
ans =  2.2204e-16

So, maybe your estimate is still on the conservative side? Or am I missing
something here?

Matthias