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

**References**:**VerifySolutions setting***From:*"dimitris" <dimmechan@yahoo.com>