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System of 2 nonlinear equations: no result

max ma
Posted 10 years ago

Why I can not get the answers of x, y, what's wrong with it? Pls guide me, Thanks in advance.

enter image description here
POSTED BY: max ma
18 Replies
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Cheating or not MMA has problems with this equation. I hope that development team fix this issue, will improve in a future release.

Thanks for Reply.
POSTED BY: Mariusz Iwaniuk

With a bit of cheating we can get candidate solutions for x^2 and y^2 as below. The caveat is that they might be incorrect due to branch cut issues.

 exprs = {1/2 \[Lambda] y^(1/2)
      x^(-(1/2)) (p - 1/2 x^2 - 1/2 y^2) + (a - 
       k p + \[Lambda] (x y)^(1/2)) (-x), 
   1/2 \[Lambda] x^(1/2)
      y^(-(1/2)) (p - 1/2 x^2 - 1/2 y^2) + (a - 
       k p + \[Lambda] (x y)^(1/2)) (-y)};
e2 = Numerator[Together[exprs]];
e3 = PowerExpand[e2 /. {x -> x^2, y -> y^2}]

(* Out[11]= {-4 a x^3 + 4 k p x^3 + 2 p y \[Lambda] - 5 x^4 y \[Lambda] -
   y^5 \[Lambda], -4 a y^3 + 4 k p y^3 + 2 p x \[Lambda] - 
  x^5 \[Lambda] - 5 x y^4 \[Lambda]} *)

Solving this modified system is not hard.

Timing[solnsb = Solve[e3 == 0, {x, y}];]

Out[21]= {1.356464, Null}

There are 25 solutions. I show the first three are shown below. One would need to take square roots though, since we solved (modulo branch cut issues) for (x^2, y^2) pairs rather than (x,y).

{{x -> 0, 
  y -> 0}, {x -> (
   1/(-2 a p \[Lambda] + 
    2 k p^2 \[Lambda]))((
     4 a^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
        2 k^2 p^2)/\[Lambda]^2 - (
        2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
            3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) - (
     8 a k p (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
        2 k^2 p^2)/\[Lambda]^2 - (
        2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
            3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) + (
     4 k^2 p^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
        2 k^2 p^2)/\[Lambda]^2 - (
        2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
            3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) - (
     p \[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
        4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - (
        2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
            3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/Sqrt[
     3] + (\[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
        4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - ({{x -> 0, 
          y -> 0}, {x -> (
           1/(-2 a p \[Lambda] + 
            2 k p^2 \[Lambda]))((
             4 a^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
                2 k^2 p^2)/\[Lambda]^2 - (
                2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                    3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) - (
             8 a k p (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
                2 k^2 p^2)/\[Lambda]^2 - (
                2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                    3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) + (
             4 k^2 p^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
                2 k^2 p^2)/\[Lambda]^2 - (
                2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                    3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) - (
             p \[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
                4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - (
                2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                    3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/Sqrt[
             3] + (\[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
                4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - (
                2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                    3 p \[Lambda]^2)])/\[Lambda]^2)^(7/4))/(3 Sqrt[3])), 
          y -> -((3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
              2 k^2 p^2)/\[Lambda]^2 - (
              2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
                  3 p \[Lambda]^2)])/\[Lambda]^2)^(1/4)/Sqrt[3])},
{x -> (1/(-2 a p \[Lambda] + 
   2 k p^2 \[Lambda]))(-((
     4 I a^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
        2 k^2 p^2)/\[Lambda]^2 - (
        2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
            3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3])) + (
    8 I a k p (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
       2 k^2 p^2)/\[Lambda]^2 - (
       2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
           3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) - (
    4 I k^2 p^2 (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
       2 k^2 p^2)/\[Lambda]^2 - (
       2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
           3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/(3 Sqrt[3]) + (
    I p \[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
       4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - (
       2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
           3 p \[Lambda]^2)])/\[Lambda]^2)^(3/4))/Sqrt[3] - (
    I \[Lambda]^2 (3 p - (2 a^2)/\[Lambda]^2 + (
       4 a k p)/\[Lambda]^2 - (2 k^2 p^2)/\[Lambda]^2 - (
       2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
           3 p \[Lambda]^2)])/\[Lambda]^2)^(7/4))/(3 Sqrt[3])), 
 y -> -((I (3 p - (2 a^2)/\[Lambda]^2 + (4 a k p)/\[Lambda]^2 - (
      2 k^2 p^2)/\[Lambda]^2 - (
      2 Sqrt[(a - k p)^2 (a^2 - 2 a k p + k^2 p^2 - 
          3 p \[Lambda]^2)])/\[Lambda]^2)^(1/4))/Sqrt[3])}
..
POSTED BY: Daniel Lichtblau

Mr. Frank Kampas you made ??a mistake.

We want to solve these equations:

eq1 = FullSimplify[1/2 ? y^(1/2) x^(-(1/2))(p - 1/2 x^2 - 1/2 y^2) + (a - k p + ? (x y)^(1/2)) (-x) == 0]
eq2 = FullSimplify[1/2 ? x^(1/2) y^(-(1/2))(p - 1/2 x^2 - 1/2 y^2) + (a - k p + ? (x y)^(1/2)) (-y) == 0]

not those:

eq1 = FullSimplify[(? ? + ? ? y) (p - ? x^2 - (1 - ?) y^2 - b - d) + (a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 ? x) == 0]
eq2 = FullSimplify[(? (1 - ?) + ? ? x) (p - ? x^2 - (1 - ?) y^2 - b - d) + (a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 (1 - ?) y) == 0]

My Laptop has 4 GBytes of RAM.With these 2 equations Kernel level of consumption of RAM is only 55MB and did not increase for the next hours calculating.

See file attachments:

Attachments:
POSTED BY: Mariusz Iwaniuk
max ma
max ma
Posted 10 years ago

Not sure, looks like it is, it just say "calculating.....", and waited several hours, still no result. Thanks.
POSTED BY: max ma
max ma
max ma
Posted 10 years ago

Yes, you are right, transferring Windows version is not helpful. Let me try Maples. Thanks a lot.
POSTED BY: max ma 
In[1]:= eq1 = 
  FullSimplify[(\[Lambda] \[Alpha] + \[Lambda] \[Mu] y) (p - \[Theta] \
x^2 - (1 - \[Theta]) y^2 - b - d) + (a - 
        k p + \[Lambda] (\[Alpha] x + (1 - \[Alpha]) y + \[Lambda] \
\[Mu] y)) (-2 \[Theta] x) == 0];
eq2 = FullSimplify[(\[Lambda] (1 - \[Alpha]) + \[Lambda] \[Mu] x) (p \
- \[Theta] x^2 - (1 - \[Theta]) y^2 - b - d) + (a - 
        k p + \[Lambda] (\[Alpha] x + (1 - \[Alpha]) y + \[Lambda] \
\[Mu] y)) (-2 (1 - \[Theta]) y) == 0];

In[3]:= AbsoluteTiming[Solve[Join[eq1, eq2], {x, y}];]

Out[3]= {0.627934, Null}

Are you running out of memory? My PC has 16 GBytes of RAM.
POSTED BY: Frank Kampas

Try with this ?

eq1 = FullSimplify[1/2 ? y^(1/2) x^(-(1/2))(p - 1/2 x^2 - 1/2 y^2) + (a - k p + ? (x y)^(1/2)) (-x) == 0]
eq2 = FullSimplify[1/2 ? x^(1/2) y^(-(1/2))(p - 1/2 x^2 - 1/2 y^2) + (a - k p + ? (x y)^(1/2)) (-y) == 0]
Solve[Join[eq1, eq2], {x, y}]

I am waiting and nothing.

Maples solutions after few seconds. enter image description here
POSTED BY: Mariusz Iwaniuk

Mac or Windows, both versions are the same.MMA has difficulty with this equations.After several hours of computation I'm not saw any result.I do not know why it has such problems. Only numericall is a quick solution with MMA. Maple 2015 solves these equations very quickly.

Attachments:
POSTED BY: Mariusz Iwaniuk
max ma
max ma
Posted 10 years ago

Really appreciate your help. My MMA version is 10.2 too, but it is Mac version, not Windows version. I reduced the symbolic coefficients as much as possible, and still not work, should I transfer Windows version also?

Attachments:
POSTED BY: max ma

Maybe you can start MMA with a fresh kernel.I'm using Mathematica 10.2 and works fine.

Attachments:
POSTED BY: Mariusz Iwaniuk
max ma
max ma
Posted 10 years ago

Thanks for your help. I tried this method, but still no result, could you guide me your .nb code? Thanks a lot.
POSTED BY: max ma
max ma
max ma
Posted 10 years ago

Thanks for your help. I tried this method, but still no result, could you guide me your code? Thanks a lot.
POSTED BY: max ma
max ma
max ma
Posted 10 years ago

Thanks for your help. Trying to reduce the variables numbers, but can not reduce all...but know the reason why it can not run. Thanks.
POSTED BY: max ma
POSTED BY: Frank Kampas 
Clear["Global`*"]
eq1 = FullSimplify[(? ? + ? ? y) (p - ? x^2 - (1 - ?) y^2 - b - d) + 
(a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 ? x) == 0]
eq2 = FullSimplify[(? (1 - ?) + ? ? x) (p - ? x^2 - (1 - ?) y^2 - b - d) + 
(a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 (1 - ?) y) == 0]

for Y:

Y = Solve[Eliminate[{eq1, eq2}, x], y]

for X:

X = Solve[Eliminate[{eq1, eq2}, y], x]
POSTED BY: Mariusz Iwaniuk

Your system is of high degree with symbolic coefficients. A full symbolic solution may be too complicated. If you lower your expectations, you get quick explicit solutions for specific numeric values of the parameters, for example

With[{\[Lambda] = 1, \[Alpha] = -1, \[Mu] = 1, p = 1, \[Theta] = 1, 
  b = 1, d = 1, a = 1, k = 1},
 Solve[{(\[Lambda] \[Alpha] + \[Lambda] \[Mu] y) (p - \[Theta] x^2 - \
(1 - \[Theta]) y^2 - b - d) + (a - 
        k p + \[Lambda] (\[Alpha] x + (1 - \[Alpha]) y + \[Lambda] \
\[Mu] y)) (-2 \[Theta] x) == 
    0, (\[Lambda] (1 - \[Alpha]) + \[Lambda] \[Mu] x) (p - \[Theta] \
x^2 - (1 - \[Theta]) y^2 - b - d) + (a - 
        k p + \[Lambda] (\[Alpha] x + (1 - \[Alpha]) y + \[Lambda] \
\[Mu] y)) (-2 (1 - \[Theta]) y) == 0}, {x, y}]]

You can go interactive too:

Manipulate[
 ContourPlot[{(\[Lambda] \[Alpha] + \[Lambda] \[Mu] y) (p - \[Theta] \
x^2 - (1 - \[Theta]) y^2 - b - d) + (a - 
        k p + \[Lambda] (\[Alpha] x + (1 - \[Alpha]) y + \[Lambda] \
\[Mu] y)) (-2 \[Theta] x) == 
    0, (\[Lambda] (1 - \[Alpha]) + \[Lambda] \[Mu] x) (p - \[Theta] \
x^2 - (1 - \[Theta]) y^2 - b - d) + (a - 
        k p + \[Lambda] (\[Alpha] x + (1 - \[Alpha]) y + \[Lambda] \
\[Mu] y)) (-2 (1 - \[Theta]) y) == 0}, {x, -1, 1}, {y, -1, 1}],
 {\[Lambda], -1, 1}, {\[Alpha], -1, 1}, {\[Mu], -1, 1}, {p, -1, 
  1}, {\[Theta], -1, 1}, {b, -1, 1}, {d, -1, 1}, {a, -1, 1}, {k, -1, 
  1}]
POSTED BY: Gianluca Gorni
max ma
max ma
Posted 10 years ago

Added code and the .nb file. Thanks.

Solve[{(? ? + ? ? y) (p - ? x^2 - (1 - ?) y^2 - b - d) + 
(a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 ? x) == 0, 
(? (1 - ?) + ? ? x) (p - ? x^2 - (1 - ?) y^2 - b - d) + 
(a - k p + ? (? x + (1 - ?) y + ? ? y)) (-2 (1 - ?) y) == 0}, {x, y}]
Attachments:
POSTED BY: max ma

Please post CODE not the image or attach the notebook.

POSTED BY: EDITORIAL BOARD
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