Michael Orlitzky wrote:
> Jason Grout wrote:
>> I think there are several points here:
>>
>> 1. The moment Sage sees a decimal, it starts using approximate, floating
>> point arithmetic. If you don't want that, then don't use decimals; use
>> fractions. This is consistent with most mathematical software (though
>> other software may have better algorithms that minimize the effect of
>> the round-off errors).
>>
>> 2. Sage currently does not deal very well with round-off error in linear
>> algebra. This needs to be fixed. Furthermore, Sage doesn't tell you
>> (like matlab *sometimes* does) when round-off error might be a big
>> issue. For your matrix, Sage probably should have warned you that
>> round-off error might lead to an incorrect response.
>>
>> I've opened up http://trac.sagemath.org/sage_trac/ticket/7392 and
>> http://trac.sagemath.org/sage_trac/ticket/7391 to address specific
>> issues that came up in this thread. Implementing numerically stable
>> arbitrary precision linear algebra is a much bigger project, so I'm
>> holding off on opening up a ticket until someone starts making it happen.
>
> Thanks. If there's a warning, this is a lot less dangerous.
>
>
>> Don't use decimals; use fractions. Then Sage does everything exactly,
>> with no round-off error.
>
> But how would you go about that? You'd have to convert them all by hand,
> which sort of defeats the purpose. It would be nice if we could, say,
> map (lambda x: x.exact_rational()) the matrix, but once the decimal
> number is stored as floating point, its representation is already
> incorrect. For example,
Apparently this tries harder to make a "nice" fraction:
sage: QQ(0.3)
3/10
So you could do this like Simon says in his post, or like this:
sage: n = matrix(QQ, [ [-0.3, 0.2, 0.1],
[0.2, -0.4, 0.4],
[0.1, 0.2, -0.5] ])
or
sage: n.apply_map(lambda x: QQ(x))
[-3/10 1/5 1/10]
[ 1/5 -2/5 2/5]
[ 1/10 1/5 -1/2]
>
> I admit to not searching very hard, but is there even any mention in the
> docs that this is going to happen as soon as you introduce a decimal point?
That's a good question. If you find a place in the docs that is very
natural to talk about floating point numbers and computers (which is a
very general subject), that would probably be the place for mentioning
this. Feel free to send in a patch, or even just the text that should
be added!
Thanks,
Jason
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