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Re: Re: Re: Reduce/Solve
>> In this type of equations the appearance of "phantom" solutions is a
consequence of the method used for solving them. It is not a feature of
Mathematica or of Solve.
What does that mean?
1) WRI programmers didn't choose to apply that method?
2) Solve gets credit for right answers, but can't be blamed for wrong ones?
I don't think so.
EVERYTHING that happens in Mathematica is a feature, bug, or limitation of Mathematica; that certainly includes any errors that result from the methods used.
Bobby
On Sat, 14 Aug 2004 01:50:31 -0400 (EDT), Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>
> On 13 Aug 2004, at 11:56, Andy Kowar wrote:
>
>>
>> I have a problem that the experts on Solve/Reduce participating in
>> this thread might help me with.
>>
>> I define an ellipse in parametric form:
>> r[t_]:={a*Cos[t],b*Sin[t]}
>> and the tangent
>> dr[t_]:=Evaluate[D[r[t], t]]
>> I define a vector
>> q={k,l};
>> I want to find points on the ellipse such that the tangent is parallel
>> to a given vector:
>> q={k,l};
>> I expect two two solutions.
>>
>> I define the equation:
>> eq = dr[t] == lambda q;
>> The following command
>> sol = Simplify[Solve[eq, {t, lambda}], {a > b > 0, k > 0,
>> l\[Element]Reals}]
>> produces the following warning
>> "Solve::ifun: Inverse functions are being used by Solve,
>> so some solutions may not be found; use Reduce for complete
>> solution information"
>> and the following four solutions (here in the InputForm)
>> {{lambda -> -((a*b)/Sqrt[b^2*k^2 + a^2*l^2]),
>> t -> -ArcCos[-((a*l)/Sqrt[b^2*k^2 + a^2*l^2])]},
>> {{lambda -> -((a*b)/Sqrt[b^2*k^2 + a^2*l^2]),
>> t -> -ArcCos[-((a*l)/Sqrt[b^2*k^2 + a^2*l^2])]},
>> {lambda -> -((a*b)/Sqrt[b^2*k^2 + a^2*l^2]),
>> t -> ArcCos[-((a*l)/Sqrt[b^2*k^2 + a^2*l^2])]},
>> {lambda -> (a*b)/Sqrt[b^2*k^2 + a^2*l^2],
>> t -> -ArcCos[(a*l)/Sqrt[b^2*k^2 + a^2*l^2]]},
>> {lambda -> (a*b)/Sqrt[b^2*k^2 + a^2*l^2],
>> t -> ArcCos[(a*l)/Sqrt[b^2*k^2 + a^2*l^2]]}}
>>
>> The warning suggests that some solutions might be missing. In fact,
>> Solve produces two extra expressions that are not solutions.
>> eq /. sol // Simplify[#, {a > b > 0, k > 0, l\[Element]Reals}] &
>> returns
>> {False,True,True,False}
>>
>> My questions are:
>>
>> 1. Is that a bug or feature that Solve produces expressions that are
>> not solutions for the original equations?
>>
>> 2. How to make Solve to return only 'true' solutions?
>>
>> On a side note, I tried Reduce only once because Mathematica froze my
>> PC.
>>
>> AK
>>
>>
> In this type of equations the appearance of "phantom" solutions is a
> consequence of the method used for solving them. It is not a feature of
> Mathematica or of Solve. The only way to eliminate these phantoms is to
> perform the sort of verification you have done. Mathematica does not do
> it automatically because in most cases of equations with non-numerical
> parameters such a verification is impossible and trying to do so would
> simply mean that Mathematica would appear to hang. In your case
> verification can be accomplished only with the additional information
> about the parameters included in Simplify but Solve cannot use it.
> Reduce, however, can:
>
>
>
> FullSimplify[Reduce[Flatten[{Thread[eq],
> {a > b > 0, k > 0, l â?? Reals}}], {t, lambda}],
> {a > b > 0, k > 0, l â?? Reals}]
>
>
> C[1] â?? Integers &&
> (
> t == 2*(ArcCot[(b*k)/(a*l - Sqrt[b^2*k^2 +
> a^2*l^2])] + Pi*C[1]) &&
> lambda == (a*b)/Sqrt[b^2*k^2 + a^2*l^2] ||
> t == 2*(ArcCot[(b*k)/(a*l + Sqrt[b^2*k^2 +
> a^2*l^2])] + Pi*C[1]) &&
> (a*b)/Sqrt[b^2*k^2 + a^2*l^2] + lambda == 0)
>
> (Mathematica 5.0 under Mac OS X 10.3.5)
>
> Andrzej Kozlowski
> Chiba, Japan
> http://www.mimuw.edu.pl/~akoz/
>
>
>
--
DrBob at bigfoot.com
www.eclecticdreams.net
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