Solve failing to solve system of two polynomial equations

Question

I have two equations:

$$P_a=P_a(x,y)$$

$$ P_b=1-P_a(x,y) $$ $P_a$, $P_b$, $x$ and $y$ are probabilities so their values are between 0 and 1.

If I know the values of $x$ and $y$ I can calculate easily $P_a$ and $P_b$, the problems is I have $P_a$ and $P_b$, and I need to calculate $x$ and $y$. I had some ideas but any of them worked:

• The polynomials are high order and I tried to simplify the expressions, but I still get errors.
• I tried reduce also with assumptions, I still get the errors.
• I thought maybe the Inverse function could help but I have two equations and two variables and failed to use it in this case.
• I generated a list of points, then used Interpolation and the expression I got in the 3D Plot is similar to the original polynomial BUT I don't have any explicit expression to work with so it was not useful. Apparently there is an internal calculation inside Mathematica but you don't get an expression. I tried InterpolatingPolynomial but got an error.
• I tried a Taylor expansion to get a simpler expression but only worked around $x$,$y\approx 0.5$. I saw in the plot that for values close to 1 there were some strange things.

So folks, does anybody know how could I get $x$ and $y$ as function of $P_a$ and $P_b$?

Or at least, does anybody know how to calculate $x$ and $y$ if I know $P_a$ and $P_b$?

I have lost 6 days with this problem and I am desperate.

This is the expression for $P_a$:

Pa = 1/(-wb + wa (-1 + 2 wb))*
  wa (-1 + wb) (-(-1 + wb)^5 (1 + 5 wb) + 
    15 wa (-1 + wb)^4 wb (1 + 5 wb) - 
    5 wa^2 (-1 + wb)^3 wb (-11 + wb + 70 wb^2) + 
    5 wa^3 (-1 + wb)^2 wb (17 - 69 wb - 28 wb^2 + 140 wb^3) - 
    5 wa^6 wb (1 - 14 wb + 56 wb^2 - 84 wb^3 + 42 wb^4) + 
    3 wa^4 wb (-23 + 202 wb - 473 wb^2 + 252 wb^3 + 252 wb^4 - 
       210 wb^5) + 
    wa^5 wb (29 - 331 wb + 1064 wb^2 - 1176 wb^3 + 210 wb^4 + 
       210 wb^5))

I have to mention that I have the same problem with an expression that is like a 100 times larger than this one. Hopefully what I learn from this calculation could be applied in the more complicated polynomial.

I have to mention that sometimes when I push Enter to start a calculation Mathematica doesn't respond. Should I increase the memory dedicated to calculations to avoid this problem?

It says "running" but I can wait hours and I don't get an answer.

This is for example the code I used to try to Solve this polynomial:

NSolve[0.5 == 
   1/(-wb + wa (-1 + 2 wb)) *
     wa (-1 + wb) (-(-1 + wb)^5 (1 + 5 wb) + 
      15 wa (-1 + wb)^4 wb (1 + 5 wb) - 
      5 wa^2 (-1 + wb)^3 wb (-11 + wb + 70 wb^2) + 
      5 wa^3 (-1 + wb)^2 wb (17 - 69 wb - 28 wb^2 + 140 wb^3) - 
      5 wa^6 wb (1 - 14 wb + 56 wb^2 - 84 wb^3 + 42 wb^4) + 
      3 wa^4 wb (-23 + 202 wb - 473 wb^2 + 252 wb^3 + 252 wb^4 - 
         210 wb^5) + 
      wa^5 wb (29 - 331 wb + 1064 wb^2 - 1176 wb^3 + 210 wb^4 + 
         210 wb^5)) && 
  0.5 == 1 - (1/(-wb + wa (-1 + 2 wb)) *
       wa (-1 + wb) (-(-1 + wb)^5 (1 + 5 wb) + 
        15 wa (-1 + wb)^4 wb (1 + 5 wb) - 
        5 wa^2 (-1 + wb)^3 wb (-11 + wb + 70 wb^2) + 
        5 wa^3 (-1 + wb)^2 wb (17 - 69 wb - 28 wb^2 + 140 wb^3) - 
        5 wa^6 wb (1 - 14 wb + 56 wb^2 - 84 wb^3 + 42 wb^4) + 
        3 wa^4 wb (-23 + 202 wb - 473 wb^2 + 252 wb^3 + 252 wb^4 - 
           210 wb^5) + 
        wa^5 wb (29 - 331 wb + 1064 wb^2 - 1176 wb^3 + 210 wb^4 + 
           210 wb^5))) && 0 < wa < 1 && 0 < wb < 1, {wa, wb}]

These are the errors I get:

NSolve::svars: Equations may not give solutions for all "solve" variables. >>
NSolve::ratnz: NSolve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result. >>

And this is the result:

(*{{wb -> ConditionalExpression[wa, 0 < wa < 1.]}}*)

I know the polynomials contain a lot of terms and could be high order, but the range is between 0 and 1, maybe there is a way to take advantage of that to get the solution.

Answer 1

This is more of an extended comment.

Knowing $P_b=1-P_a$ doesn't add another equation to the party (as @Daniel Lichtblau pointed out). In other words, knowing $P_a$ and $P_b$ doesn't give you two equations for the two unknowns $x$ and $y$. However, if you know $P_a$ and $x$, then you could solve (numerically) for $y$. (Below I've used your notation of wa and wb interchangeably for $x$ and $y$.)

First show the contour plot:

prob[wa_, wb_] := 1/(-wb + wa (-1 + 2 wb)) wa (-1 + wb) (-(-1 + wb)^5 (1 + 5 wb) +
        15 wa (-1 + wb)^4 wb (1 + 5 wb) - 
        5 wa^2 (-1 + wb)^3 wb (-11 + wb + 70 wb^2) + 
        5 wa^3 (-1 + wb)^2 wb (17 - 69 wb - 28 wb^2 + 140 wb^3) -
        5 wa^6 wb (1 - 14 wb + 56 wb^2 - 84 wb^3 + 42 wb^4) + 
        3 wa^4 wb (-23 + 202 wb - 473 wb^2 + 252 wb^3 + 252 wb^4 - 210 wb^5) + 
        wa^5 wb (29 - 331 wb + 1064 wb^2 - 1176 wb^3 + 210 wb^4 + 210 wb^5))

ContourPlot[prob[wa, wb], {wa, 0, 1}, {wb, 0, 1}, ContourLabels -> True,
  Contours -> {0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95},
  Frame -> True, FrameLabel -> {Style["wa", Large, Bold, Black],
    Style["wb", Large, Bold, Black]}]

One way to solve for wb given values of wa and pa is to use NSolve:

wa = 0.8;
pa = 0.1;
wb /. NSolve[pa == prob[wa, wb] && 0 < wb < 1, wb][[1]]
(* 0.9625101428796428 *)

An alternative but less exact approach is to interpolate:

y = Interpolation[
  Flatten[Table[{{wa, prob[wa, wb]}, wb}, {wa, 0.005, 0.995, 0.005},
     {wb, 0.005, 0.995, 0.005}], 1],
  InterpolationOrder -> 1]

x = 0.8;
pa = 0.1;
y[x, pa]
(* 0.9624580040494005 *)

(* Check to see if we get back close to the original value of pa *)
prob[x, y[x, pa]]
(* 0.1001401896838602 *)

The interpolation method is less exact but likely much, much quicker if you have lots of evaluations to make.