text: #
  # test_linsolve.py
  #
  #    Test the internal implementation of linsolve.
  #
  
  from sympy.testing.pytest import raises
  
  from sympy.core.numbers import I
  from sympy.core.relational import Eq
  from sympy.core.singleton import S
  from sympy.abc import x, y, z
  
  from sympy.polys.matrices.linsolve import _linsolve
  from sympy.polys.solvers import PolyNonlinearError
  
  
  def test__linsolve():
      assert _linsolve([], [x]) == {x:x}
      assert _linsolve([S.Zero], [x]) == {x:x}
      assert _linsolve([x-1,x-2], [x]) is None
      assert _linsolve([x-1], [x]) == {x:1}
      assert _linsolve([x-1, y], [x, y]) == {x:1, y:S.Zero}
      assert _linsolve([2*I], [x]) is None
      raises(PolyNonlinearError, lambda: _linsolve([x*(1 + x)], [x]))
  
  
  def test__linsolve_float():
  
      # This should give the exact answer:
      eqs = [
          y - x,
          y - 0.0216 * x
      ]
      sol = {x:0.0, y:0.0}
      assert _linsolve(eqs, (x, y)) == sol
  
      # Other cases should be close to eps
  
      def all_close(sol1, sol2, eps=1e-15):
          close = lambda a, b: abs(a - b) < eps
          assert sol1.keys() == sol2.keys()
          return all(close(sol1[s], sol2[s]) for s in sol1)
  
      eqs = [
          0.8*x +         0.8*z + 0.2,
          0.9*x + 0.7*y + 0.2*z + 0.9,
          0.7*x + 0.2*y + 0.2*z + 0.5
      ]
      sol_exact = {x:-29/42, y:-11/21, z:37/84}
      sol_linsolve = _linsolve(eqs, [x,y,z])
      assert all_close(sol_exact, sol_linsolve)
  
      eqs = [
          0.9*x + 0.3*y + 0.4*z + 0.6,
          0.6*x + 0.9*y + 0.1*z + 0.7,
          0.4*x + 0.6*y + 0.9*z + 0.5
      ]
      sol_exact = {x:-88/175, y:-46/105, z:-1/25}
      sol_linsolve = _linsolve(eqs, [x,y,z])
      assert all_close(sol_exact, sol_linsolve)
  
      eqs = [
          0.4*x + 0.3*y + 0.6*z + 0.7,
          0.4*x + 0.3*y + 0.9*z + 0.9,
          0.7*x + 0.9*y,
      ]
      sol_exact = {x:-9/5, y:7/5, z:-2/3}
      sol_linsolve = _linsolve(eqs, [x,y,z])
      assert all_close(sol_exact, sol_linsolve)
  
      eqs = [
          x*(0.7 + 0.6*I) + y*(0.4 + 0.7*I) + z*(0.9 + 0.1*I) + 0.5,
          0.2*I*x + 0.2*I*y + z*(0.9 + 0.2*I) + 0.1,
          x*(0.9 + 0.7*I) + y*(0.9 + 0.7*I) + z*(0.9 + 0.4*I) + 0.4,
      ]
      sol_exact = {
          x:-6157/7995 - 411/5330*I,
          y:8519/15990 + 1784/7995*I,
          z:-34/533 + 107/1599*I,
      }
      sol_linsolve = _linsolve(eqs, [x,y,z])
      assert all_close(sol_exact, sol_linsolve)
  
      # XXX: This system for x and y over RR(z) is problematic.
      #
      # eqs = [
      #     x*(0.2*z + 0.9) + y*(0.5*z + 0.8) + 0.6,
      #     0.1*x*z + y*(0.1*z + 0.6) + 0.9,
      # ]
      #
      # linsolve(eqs, [x, y])
      # The solution for x comes out as
      #
      #       -3.9e-5*z**2 - 3.6e-5*z - 8.67361737988404e-20
      #  x =  ----------------------------------------------
      #           3.0e-6*z**3 - 1.3e-5*z**2 - 5.4e-5*z
      #
      # The 8e-20 in the numerator should be zero which would allow z to cancel
      # from top and bottom. It should be possible to avoid this somehow because
      # the inverse of the matrix only has a quadratic factor (the determinant)
      # in the denominator.
  
  
  def test__linsolve_deprecated():
      assert _linsolve([Eq(x**2, x**2+y)], [x, y]) == {x:x, y:S.Zero}
      assert _linsolve([(x+y)**2-x**2], [x]) == {x:-y/2}
      assert _linsolve([Eq((x+y)**2, x**2)], [x]) == {x:-y/2}
  