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Topic: An independent integration test suite
Replies: 42   Last Post: Jul 25, 2013 6:09 PM

 Messages: [ Previous | Next ]
 Nasser Abbasi Posts: 6,654 Registered: 2/7/05
Re: An independent integration test suite
Posted: Jul 24, 2013 3:14 AM

On 7/24/2013 1:25 AM, daly@axiom-developer.org wrote:
> On Wednesday, July 24, 2013 1:21:51 AM UTC-4, Albert Rich wrote:
>> On Tuesday, July 23, 2013 2:32:20 PM UTC-10, da...@axiom-developer.org wrote:
>>
>>
>>

>>> We differ on some results because Rubi (or whatever program you are using
>>
>>> to create the optimal results) assumes that square roots have only a single
>>
>>> positive value. Axiom does not want to make this simplification so given
>>
>>> sqrt(3)*sqrt(7)*sqrt(21) - 21
>>
>>> Axiom will not simplify this to zero but the Rubi test suite does.
>>
>>
>>
>> Surly there must be a way to tell Axiom to choose the principal branch so sqrt(4)-2 will simplify to zero?
>>
>>
>>
>> Albert

>
> Actually, as you can guess, the real results are more complicated.
> Given
>
> (sqrt(b^2*x^2-4)-sqrt(b*x-2)*sqrt(b*x+2))/
> (sqrt(b*x-2)*sqrt(b*x+2)*sqrt(b^2*x^2-4))
>
> under what conditions is this identically zero?
>
> If I extract the numerator and ask if it zero?
> Axiom replies false. Is there an algorithm to show
> that this is identically zero? I'd be happy to
> implement it. I reread a paper on the subject and
> now I'm more confused than when I started.
>
> Tim
>

hello Tim;

This is in M 9.01: For domain Reals, Mathematica
gives conditions for zero

--------------------------
Clear[x, b];
num = (Sqrt[b^2*x^2 - 4] - Sqrt[b*x - 2]*Sqrt[b*x + 2]);
den = Sqrt[b*x - 2]*Sqrt[b*x + 2]*Sqrt[b^2*x^2 - 4];
Reduce[(num/den) == 0, {x, b}, Reals]
------------------------------

(x < 0 && b < 2/x) || (x > 0 && b > 2/x)

But in Complex domain, it gives warnings due to branch cuts,
so not possible to decide on conditions for sure in this
case:

-------------------------------
Clear[x, b];
num = (Sqrt[b^2*x^2 - 4] - Sqrt[b*x - 2]*Sqrt[b*x + 2]);
den = Sqrt[b*x - 2]*Sqrt[b*x + 2]*Sqrt[b^2*x^2 - 4];
Reduce[(num/den) == 0, {x, b}, Complexes]
-------------------------------

----------------------
During evaluation of In[29]:= Reduce::useq: The answer found by
Reduce contains unsolved equation(s) {0==(-Sqrt[-2+Times[<<2>>]]
Sqrt[2+Times[<<2>>]]+Sqrt[-4+Power[<<2>>] Power[<<2>>]])/Sqrt[2+b x]}.
A likely reason for this is that the solution set depends on branch
cuts of Mathematica functions. >>

Out[32]= Sqrt[2 + b x] != 0 &&
0 == (-Sqrt[-2 + b x] Sqrt[2 + b x] +
Sqrt[-4 + b^2 x^2])/Sqrt[2 + b x] && -2 + b x != 0
------------------

But if one wants one instance which makes num/den = 0, then
there is a command for that:

FindInstance[(num/den) == 0, {x, b}, Complexes]
{{x -> 0, b -> 0}}

--Nasser