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MathGroup Archive 2006

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Re: VerifySolutions setting

  • To: mathgroup at smc.vnet.net
  • Subject: [mg71416] Re: [mg71363] VerifySolutions setting
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Fri, 17 Nov 2006 04:30:50 -0500 (EST)
  • References: <200611160552.AAA08198@smc.vnet.net>

On 16 Nov 2006, at 14:52, dimitris wrote:

>
> Information["VerifySolutions", LongForm -> True]
>
> "VerifySolutions is an option to Solve and related functions. With
> VerifySolutions -> True, solutions are verified and extraneous
> solutions are eliminated. With VerifySolutions -> False, they are not.
> Note that only those numerical solutions that are obviously extraneous
> are eliminated."
> Attributes[VerifySolutions] = {Protected}
>
> Consider the following equation
>
> eq=x^(1/3) + x^(1/2) == a;
>
> With[{a = -3}, Solve[x^(1/3) + x^(1/2) == a, x, VerifySolutions ->
> Automatic]] (*default*)
> {}
>
> With[{a = -3}, Solve[x^(1/3) + x^(1/2) == a, x, VerifySolutions ->
> True]]          (*admissable 1*)
> {}
>
> With[{a = -3}, Solve[x^(1/3) + x^(1/2) == a, x, VerifySolutions ->
> False]]         (*admissable 2*)
> {{x -> 46/3 + (1/3)*(136109/2 - (6561*Sqrt[85])/2)^(1/3) +
> (1/3)*((1/2)*(136109 + 6561*Sqrt[85]))^(1/3)}, {x -> 46/3 - (1/6)*(1 +
> I*Sqrt[3])*(136109/2 - (6561*Sqrt[85])/2)^(1/3) -
>      (1/6)*(1 - I*Sqrt[3])*((1/2)*(136109 + 6561*Sqrt[85]))^(1/3)},
>   {x -> 46/3 - (1/6)*(1 - I*Sqrt[3])*(136109/2 -
> (6561*Sqrt[85])/2)^(1/3) -
>      (1/6)*(1 + I*Sqrt[3])*((1/2)*(136109 + 6561*Sqrt[85]))^(1/3)}}
>
> Are there cases witch the default setting fails and VerifySolutions ->
> True must be used in order
> to eliminate extreneous roots?
>
> Thanks a lot,
> Dimitris
>


Plenty. For example:


Solve[ 1 + x*1 + Sqrt[ 1 + 1^2]* Sqrt[ 1 + x^2] == 0, x,  
VerifySolutions ->
False]
{{x -> 1}}

Solve[1 + (x*1 + Sqrt[1 + 1^2]*Sqrt[1 + x^2]) == 0, x]
{}


Andrzej Kozlowski
Tokyo, Japan


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