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equation solving - NSolve finds real-valued results in version 9, but not in version 10



UPDATE Reals also was used in the code.


Actually, I tried the following simple piece of code.


P[T_, V_] := -(1/V^2) + T/V + (2 T)/(-1 + V)^3 + (4 T)/(-1 + V)^2;
NSolve[
  {
   D[P[T, V], {V, 1}] == 0,
   D[P[T, V], {V, 2}] == 0
   },
  {T, V},
  Reals

  ] // TableForm

And in ver. 9, the output was,


{
 {T -> -9.67712, V -> -2.35529},
 {T -> -5.12191, V -> -0.778707},
 {T -> 0.0943287, V -> 7.66613}
}

and in case of ver. 10.3.1, obtained,



{}

That is, these two versions apparently return different outputs.


Whats is the cause? Is there any version dependency in Mathematica?



Answer



Not a solution, more of an extended comment.


Clearly there are real solutions, the curves below do cross


ContourPlot[
 Evaluate[{D[P[T, V], {V, 1}] == 0, 
   D[P[T, V], {V, 2}] == 0}], {V, -10, 10}, {T, -10, 10}, 

 PlotPoints -> 40]

enter image description here


You can get the real-valued solutions version 9 gave via


Solve[{N@D[P[T, V], {V, 1}] == 0, 
  N@D[P[T, V], {V, 2}] == 0}, {T, V}, Reals]


During evaluation of Solve::ratnz: Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result. >>




(* {{T -> -9.67712, V -> -2.35529}, {T -> -5.12191, 
  V -> -0.778707}, {T -> 0.0943287, V -> 7.66613}} *)

You can also get these answers with NSolve (with a little more precision) by using the Method option


NSolve[
   {
     D[P[T, V], {V, 1}] == 0,
     D[P[T, V], {V, 2}] == 0
     },
   {T, V}

   , Reals, Method -> "UseSlicingHyperplanes"] 
(* {{T -> 0.0943287, V -> 7.66613}, {T -> -9.67712, 
  V -> -2.35529}, {T -> -5.12191, V -> -0.778707}} *)

{D[P[T, V], {V, 1}] , D[P[T, V], {V, 2}]} /. %
(* {{-1.12757*10^-17, 
  3.25261*10^-18}, {1.11022*10^-15, -1.9984*10^-15}, {3.55271*10^-15, 
  1.59872*10^-14}} *)

Why is this happening? I don't know (hence the disclaimer at the top). I know that the functions NSolve and the like are constantly undergoing development, and some of those very developers post here. They would be very interested in hearing about this I think.



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