[R] Logical inconsistency
Wacek Kusnierczyk
Waclaw.Marcin.Kusnierczyk at idi.ntnu.no
Sat Dec 6 11:02:32 CET 2008
Stavros Macrakis wrote:
> On Fri, Dec 5, 2008 at 8:18 AM, Wacek Kusnierczyk <
> Waclaw.Marcin.Kusnierczyk at idi.ntnu.no> wrote:
>
>
>> well, this answer the question only partially. this explains why a
>> system with finite precision arithmetic, such as r, will fail to be
>> logically correct in certain cases. it does not explain why r, a
>> language said to isolate a user from the underlying implementational
>> choices, would have to fail this way.
>>
>>
>
> Assuming you are not trolling....
>
you never know these days. point taken.
> There are systems which provide arbitrary (but finite) precision floating
> point, exact rational arithmetic (to the limits of machine memory), and even
> symbolic calculation on constant expressions like sin(2+pi/23)^2 as well as
> expressions with variables such as e^-x/(x^2+1) -- for example, the Maxima
> free symbolic algebra system (maxima.sourceforge.net).
>
> The bc program is far more primitive and limited than that: it is performing
> decimal *fixed*-point arithmetic (though the precision is settable), so it
> may get 8.8-7.8 = 1, but (2/3)*3 => 1.99999999999999999998 and 9^-30 => 0
> (with the default scale=20). So much for "logical consistency".
>
yepp, though (2/3)*3 not evaluating to 2 is again not a must, is it.
you know scheme would say (* (/ 2 3) 3) is 2, and similarly for (* (/
2.0 3.0) 3.0), though it would say no to (= (- 8.8 7.8) 1.0).
it's all about representations, but representations are typically below
the interface. i did not complain, or troll, about r not implementing
arbitrary precision, just remarked that the example and its explanation
expose a user to the internals.
> Exact symbolic calculation can be very useful, but Maxima (and Mathematica
> and Maple etc.) has different applications and different limitations
> (implementational and theoretical) from a system like R. Though Maxima can
> factor (for example) x^7+x^5+x^4+x^3-x^2-x+2 into (x+1)*(x^2-x+2)*(x^4-x+1),
> and give the exact roots, it is rarely useful to know that one root is
> -3^(3/4)*(2^(1/6)*%i*sqrt(((sqrt(229)*%i+3*sqrt(3))^(2/3)+4*2^(2/3))*sqrt(2^(1/3)*(sqrt(229)*%i+3*sqrt(3))^(2/3)+8)+2*2^(2/3)*3^(3/4)*sqrt(sqrt(229)*%i+3*sqrt(3)))+(2^(1/3)*(sqrt(229)*%i+3*sqrt(3))^(2/3)+8)^(3/4))/(6*2^(1/3)*(sqrt(229)*%i+3*sqrt(3))^(1/6)*(2^(1/3)*(sqrt(229)*%i+3*sqrt(3))^(2/3)+8)^(1/4));
> the approximation -0.93409928946053*%i-0.7271360844912 (which Maxima can
> also calculate -- to 100000 digits if you like) is more useful.
>
sure. but the examples you give are somewhat extreme if compared to the
original question about 8.8-7.8 not being 1. the result is finely
explained by reference to the underlying representation, but this does
break the assumption of isolation from underlying details, claimed here
once upon a time.
vQ
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