Updates:
Status: Started
Owner: mattpap
Cc: -mattpap
Labels: -polys Polynomial Milestone-Release0.7.0
Comment #1 on issue 2087 by mattpap: solver treats a nonlinear expression
as linear and mis-solves a true nonlinear system; fixed in polys11 but slow
http://code.google.com/p/sympy/issues/detail?id=2087
In polys11 we get the following results:
In [1]: solve([x + x*y - 3, y + x*y - 4], x, y)
Out[1]: [(-3, -2), (1, 2)]
In [2]: solve_triangulated([x + x*y - 3, y + x*y - 4], x, y)
Out[2]: [(-3, -2), (1, 2)]
In [3]: %timeit u = solve([x + x*y - 3, y + x*y - 4], x, y)
10 loops, best of 3: 24.4 ms per loop
In [4]: %timeit u = solve_triangulated([x + x*y - 3, y + x*y - 4], x, y)
100 loops, best of 3: 8.32 ms per loop
In [5]: F = [-10 + x + y**2, 20 - y + x*y + x**2]
In [6]: %time s = solve(F, x, y)
CPU times: user 0.25 s, sys: 0.00 s, total: 0.25 s
Wall time: 0.26 s
In [8]: F[0].subs(zip((x, y), s[0])).evalf()
Out[8]: 0
In [9]: F[1].subs(zip((x, y), s[0])).evalf()
Out[9]: .0e-126 - .0e-126⋅ⅈ
In [10]: F[0].subs(zip((x, y), s[1])).evalf()
Out[10]: 0
In [11]: F[1].subs(zip((x, y), s[1])).evalf()
Out[11]: .0e-126 + .0e-126⋅ⅈ
The solution 's' is very complicated because quartic root finding algorithm
has to be used. There reason for this is the Groebner basis of 'F':
In [12]: groebner(F, x, y)
Out[12]:
⎡ 2 2 3 4⎤
⎣-10 + x + y , 120 + 9⋅y - 20⋅y - y + y ⎦
which has a polynomial of degree 4 as the univariate polynomial.
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