On Sat, 26 Jan 2008 15:34:54 +0100, Axel Vogt <&noreply@[EMAIL PROTECTED]
>
wrote:
>quasi wrote:
>> On Sat, 26 Jan 2008 06:39:09 -0500, quasi <quasi@[EMAIL PROTECTED]
> wrote:
>>
>>> On Sat, 26 Jan 2008 03:07:45 -0800 (PST), Vladimir Bondarenko
>>> <vb@[EMAIL PROTECTED]
> wrote:
>>>
>>>> ----------------------------------------------------------------
>>>>
>>>> An exact 1-D integration challenge - 48 -
>>>> (go and give a kick to all stupid CASs!)
>>>>
>>>>
http://groups.google.com/group/sci.math.symbolic/browse_thread/thread/28416525f0de8f90/1d8cd83e96cb7f63?#1d8cd83e96cb7f63
>>>>
>>>> ----------------------------------------------------------------
>>>>
>>>> N[Integrate[
>>>>
>>>> Sqrt[Sqrt[2] + Sqrt[z] + Sqrt[2 + 2 Sqrt[2] Sqrt[z] + 2 z]],
>>>>
>>>> {z, 0, 1}]]
>>>>
>>>> ----------------------------------------------------------------
>>>> VERSION OUTPUT RESOLUTION
>>>> ----------------------------------------------------------------
>>>>
>>>> Mathematica 6.0 2.67602 <-------------------------------- BUG
>>>>
>>>> Mathematica 5.2 3.66533 <-------------------------------- BUG
>>> Let f(z) = sqrt(sqrt(2) + sqrt(z) + sqrt(2 + 2*sqrt(2)*sqrt(z) + 2*z))
>>>
>>> Then it's completely obvious that f has domain [0,infinity) and that
>>> is strictly increasing.
>>>
>>> Hence the integral of f on the interval [0,1] must be strictly between
>>> f(0) and f(1).
>>>
>>> Approximately, we have
>>>
>>> f(0) = 1.681792831
>>>
>>> and
>>>
>>> f(1) = 2.242172940
>>>
>>> which makes it clear that Mathematica's outputs are ridiculous.
>>>
>>> Maple 9.5 gives a result of 2.062759840, which at least seems
>>> reasonable. Whether it's correct or not, I'm not sure.
>>>
>>> quasi
>>
>> A CAS should never return a wrong answer. Returning no answer is
>> better than a wrong one.
>>
>> For this example in particular, Mathematica really has no excuse for
>> getting it wrong. It's not like the function has wild variations which
>> might throw off a numerical integration. This is a positive,
>> increasing function, easily evaluated. For a CAS, the numerical
>> integration of such a function should be effortless.
>>
>> quasi
>
>Finding a possible (numerical) error is only half the job
>of a tester.
I disagree -- a tester needs to find a reproducible bug -- period. An
instance is sufficient, the simpler, the better.
The fixers (on the Mathematica side) are the ones who need to
investigate further.
>I would expect him to locate it closer by deforming the situation,
>easiest would be the bounds.
I would expect that Mathematica should sense a possible problem and
deform the function and/or the bounds automatically, if needed.
>Then one could say something whether MMA is disturbed
>by the vertical tangent in 0.
If a vertical (or near vertical) tangent is an im****tant consideration
for the internal algorithm, then the CAS implementation should
incor****ate a way to automatically detect and deal with that
contingency.
>Changing z ---> t^2 should do it.
If the user gets an actual answer, why should the user have to check
to see if it's correct? Worse, if the integral is used as part of a
user-written program, the user doesn't even get to look at the answer
-- it just gets fed to the next piece of the program. Shouldn't a
supposed world-class CAS at least offer an option to automatically
verify/certify numerical calculations to within a specified accuracy?
quasi
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