Finding symbolic solutions to nonlinear system of algebraic equations using Sympy

Question

Here are a system of two nonlinear equations :

In an answer to a previous question it has been shown that this system of equations doesn't possess explicit solutions(here). But it is also noted that this system of equations is already an implicit representation of solutions to the system.

My questions are:

Can anyone explain what does implicit representation of the solutions to the system mean ?

Can I exploit this feature to obtain analytical expressions for x and y using sympy ?

Answer 1

You can iterate to an approximation of the solution if you have values for the constants a,b,c,d,g,h. Numerical solvers do not need an explicit representation for x and y, they just need two expressions that define the desired relationships. That's what you have. They implicitly define what x and y must be in terms of those relationships.

By graphing or guessing or some other means you get an initial guess for the solution and then let the numerical solver take that guess and refine it to a desired precision. This is done whenever someone wants a solution to one or more equations that cannot be solved explicitly.

In your case it might look like this:

>>> from sympy import *
>>> from sympy.abc import x, y
>>> a = 0.12
>>> b = -2.031529100521498e-30
>>> c = b
>>> d = 1
>>> g = 11
>>> h = 0
>>> def eqs():
...   F = (y + a/2)*x*(2*x**2 - 2*y*a + 2*d**2 + b + c) - (x*y*g**2)
...   G = (x**2 - y*a + b + d**2)*(x**2 - y*a + c+d**2) - 4*((y+a/2)**2)*x**2 - (h**2 + (x**2)*(g**2))
...   return F, G
>>> nsolve(eqs(),(x,y),(1,1))
Matrix([
[  0.091656069634048],
[0.00101689322921863]])

(Oscar already showed how to get numerical solutions for all roots.) If you now change one of the parameters by a small amount, you might find another solution near this solution. e.g. if you used g = 12 instead of g = 11 then you could do:

>>> g = 12
>>> nsolve(eqs(),(x,y),(0.09,.001))  # using previous solution 
Matrix([
[  0.0839072097131664],
[0.000851016936303022]])