Re: Solve vs. NSolve

*To*: mathgroup at smc.vnet.net*Subject*: [mg92862] Re: Solve vs. NSolve*From*: SigmundV <sigmundv at gmail.com>*Date*: Wed, 15 Oct 2008 06:01:28 -0400 (EDT)*References*: <gd1n2c$e6o$1@smc.vnet.net>

Thank you, Bill, for pointing this out. I was aware of the PowerExpand issue, but I guess I hadn't thought it fully through. On Oct 14, 10:59 am, Bill Rowe <readn... at sbcglobal.net> wrote: > On 10/13/08 at 6:21 AM, sigmu... at gmail.com (SigmundV) wrote: > > > > >Consider > >f[a_, b_, c_, k_, t_] := > >With[{\[Alpha] = a k, \[Beta] = b k}, (x - \[Alpha] Cos[t])^2/a^2 + > >(y - (\[Beta] Sin[t] + c) - c)^2/b^2 - 1 == 0]; df[a_, b_, c_, k_, > >t_] := D[f[a, b, c, k, t], t]; > >and execute > >{x, y} /. Simplify@PowerExpand@Simplify@Solve[{f[1, 2, 1/2, 4/5, t], > >df[1, 2, 1/2, 4/5, t]}, {x, y}] // Chop // N > >and > >{x, y} /. Simplify@PowerExpand@Simplify@NSolve[{f[1, 2, 1/2, 4/5, > >t], df[1, 2, 1/2, 4/5, t]}, {x, y}] // Chop > >Simplify@PowerExpand@Chop@Simplify[% /. Cos[2 t] -> (1 - 2 > >Sin[t]^2)] > >respectively. > >As you see, Solve and NSolve yield two different solutions, with the > >solution from Solve being the correct one, as can be verified by > >plugging in to the equations -- the solution from NSolve does not > >satisfy the second equation, but only the first. Can anyone explain > >this behaviour to me? > > The issue isn't NSolve vs Solve since, > > In[5]:= Solve[{f[1, 2, 1/2, 4/5, t], df[1, 2, 1/2, 4/5, t]}, {x, > y}] // FullSimplify > > Out[5]= {{x -> (4*Cos[t])/5 - Cot[t]*Sqrt[Sin[t]^2], > y -> (8*Sin[t])/5 - 2*Sqrt[Sin[t]^2] + 1}, > {x -> (4*Cos[t])/5 + Cot[t]*Sqrt[Sin[t]^2], > y -> (8*Sin[t])/5 + 2*Sqrt[Sin[t]^2] + 1}} > > and > > In[8]:= NSolve[{f[1, 2, 1/2, 4/5, t], df[1, 2, 1/2, 4/5, t]}, {x, > y}] // FullSimplify // Chop > > Out[8]= {{x -> 0.8*Cos[t] - 1.*Cot[t]*Sqrt[Sin[t]^2], > y -> 1.6*Sin[t] - 2.*Sqrt[0.49999999999999994 - > 0.49999999999999994*Cos[2*t]] + 1.}, > {x -> 0.8*Cos[t] + 1.*Cot[t]*Sqrt[Sin[t]^2], > y -> 1.6*Sin[t] + 2.*Sqrt[0.49999999999999994 - > 0.49999999999999994*Cos[2*t]] + 1.}} > > The issue is the transformations you do to these solutions. In > particular, note > > In[14]:= Sqrt[Sin[t]^2] == Sin[t] /. t -> -.5 > > Out[14]= False > > That is PowerExpand needs to be used with a lot of care since > the transformation it does is not valid for all possible values. > > =46or the case where you used Solve, PowerExpand transforms > 2*Sqrt[Sin[t]^2] to 2 Sin[t]. In the case where you used NSolve, > the mathematically equivalent expression 2.*Sqrt[0.5 - > 0.5*Cos[2*t]] is unaffected by PowerExpand. The result is two > expressions that are not equal although they started as equal.