Re: Solve : missing elims in the Mathematica 8 version

*To*: mathgroup at smc.vnet.net*Subject*: [mg115054] Re: Solve : missing elims in the Mathematica 8 version*From*: Adam Strzebonski <adams at wolfram.com>*Date*: Wed, 29 Dec 2010 06:00:00 -0500 (EST)

Andrzej Kozlowski wrote: > On 28 Dec 2010, at 12:49, W. Deinhard wrote: > >> ?Hi, >> in Mathematica 7 Solve allowed to specify the variables I wanted to eliminate. >> How can I do that in Mathtematica 8 ? >> >> The syntax is no longer >> >> Solve[eqns,vars,elims] >> >> but >> >> Solve[expr,vars,dom] >> >> Thanks , bye Walter. >> > > First, the old syntax still works but now it is undocumented (?): > > In[33]:= Solve[{1 == x^5 + y^5, a - b == x + y, b + a == x*y}, {x, > y}, {a, b}] > > Solve::svars:Equations may not give solutions for all "solve" variables. >> > > {{y -> (1 - x^5)^(1/ > 5)}, {y -> (-(-1)^(1/5))*(1 - x^5)^(1/5)}, > {y -> (-1)^(2/5)*(1 - x^5)^(1/5)}, > {y -> (-(-1)^(3/5))*(1 - x^5)^(1/5)}, > {y -> (-1)^(4/5)*(1 - x^5)^(1/5)}} > > This, in fact, is the same answer as the one you get if you explicitly use Eliminate: > > Solve[ > Eliminate[{1 == x^5 + y^5, a - b == x + y, b + a == x*y}, {a, > b}], {x, y}] > > Solve::svars:Equations may not give solutions for all "solve" variables. >> > > {{y -> (1 - x^5)^(1/ > 5)}, {y -> (-(-1)^(1/5))*(1 - x^5)^(1/5)}, > {y -> (-1)^(2/5)*(1 - x^5)^(1/5)}, > {y -> (-(-1)^(3/5))*(1 - x^5)^(1/5)}, > {y -> (-1)^(4/5)*(1 - x^5)^(1/5)}} > > I am not sure if the lack of documentation for the former usage is an oversight or it is now deprecated. > > Andrzej Kozlowski The new Solve syntax allows use of arbitrary combinations of quantifiers. "Elimination variables" are a special case, namely Solve[eqns, vars, elims] is equivalent to Solve[Exists[elims, eqns], vars] In[1]:= Solve[Exists[{a, b}, 1 == x^5 + y^5 && a - b == x + y && b + a == x*y], y] 5 1/5 1/5 5 1/5 Out[1]= {{y -> (1 - x ) }, {y -> -((-1) (1 - x ) )}, 2/5 5 1/5 3/5 5 1/5 > {y -> (-1) (1 - x ) }, {y -> -((-1) (1 - x ) )}, 4/5 5 1/5 > {y -> (-1) (1 - x ) }} Note, that the last argument of a quantifier needs to be a Boolean formula, so one should use eq1 && eq2 && ... rather than {eq1, eq2, ...} The old syntax still works, but is now deprecated because of a possible confusion with the domain argument. In[2]:= Solve[x^4==1, x, Reals] Out[2]= {{x -> -1}, {x -> 1}} In[3]:= Solve[x^4==1, x, Real] Solve::bdomv: Warning: Real is not a valid domain specification. Mathematica is assuming it is a variable to eliminate. Out[3]= {{x -> -1}, {x -> -I}, {x -> I}, {x -> 1}} Best regards, Adam Strzebonski Wolfram Research