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VerifySolutions setting

• To: mathgroup at smc.vnet.net
• Subject: [mg71363] VerifySolutions setting
• From: "dimitris" <dimmechan at yahoo.com>
• Date: Thu, 16 Nov 2006 00:52:37 -0500 (EST)

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

• Follow-Ups:
  • Re: VerifySolutions setting
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>