Comment #4 on issue 2405 by matt...@gmail.com: solve and solve_poly_system
can give ambiguous results
http://code.google.com/p/sympy/issues/detail?id=2405
There will be always some ambiguity as long as solvers aren't returning
mappings. However, it's not solve_poly_system() where the discussed
ambiguity comes from. This is because if you construct and later unify
polynomials, there are rules for this.
In the first case polys assume that x > y (x comes before y), so:
In [1]: parallel_poly_from_expr([x-y-3,x+y+5])
Out[1]:
([Poly(x - y - 3, x, y, domain='ZZ'), Poly(x + y + 5, x, y, domain='ZZ')],
{domain: ℤ, gens: (x, y), polys: False})
But you can change this (without affecting generators and domain
computations):
In [2]: parallel_poly_from_expr([x-y-3,x+y+5], wrt=y)
Out[2]:
([Poly(-y + x - 3, y, x, domain='ZZ'), Poly(y + x + 5, y, x, domain='ZZ')],
{domain: ℤ, gens: (y, x), polys: False, wrt: [y]})
Or using another syntax:
In [3]: parallel_poly_from_expr([x-y-3,x+y+5], sort='y > x')
Out[3]:
([Poly(-y + x - 3, y, x, domain='ZZ'), Poly(y + x + 5, y, x, domain='ZZ')],
{domain: ℤ, gens: (y, x), polys: False, sort: [y, x]})
In all other cases the polynomials are already constructed, so unification
rules have to be taken into account. If you have two polynomials with the
same generators up to a permutation, then poly tries to preserve the order
of generators as much as possible, giving the first polynomial precedence,
e.g.:
In [14]: from sympy.polys.polyutils import _unify_gens
In [15]: _unify_gens([x, y], [x, y])
Out[15]: (x, y)
In [16]: _unify_gens([x, y], [y, x])
Out[16]: (x, y)
In [17]: _unify_gens([y, x], [y, x])
Out[17]: (y, x)
In [18]: _unify_gens([y, x], [x, y])
Out[18]: (y, x)
This behavior may be seen very complex (maybe it's even too complex), but
there is no randomness here. You will always get the same results, unless
you tell polynomial manipulation functions to behave differently.
Of course for a user it may be a little too much to know all those rules,
but this is his choice to let SymPy figure out what problem he wants to
solve. It would help a lot here if solve variables were present in the
output.
If you are looking for a source of confusion then lines 159-165 in
solvers.py are good candidates (ordering by hash where order is meaningful).
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