def test_solve_biquadratic():
x0, y0, x1, y1, r = symbols('x0 y0 x1 y1 r')
f_1 = (x - 1)**2 + (y - 1)**2 - r**2
f_2 = (x - 2)**2 + (y - 2)**2 - r**2
assert solve_poly_system([f_1, f_2], x, y) == \
[(Rational(3, 2) - sqrt(-1 + 2*r**2)/2, Rational(3, 2) + sqrt(-1 + 2*r**2)/2),
(Rational(3, 2) + sqrt(-1 + 2*r**2)/2, Rational(3, 2) - sqrt(-1 + 2*r**2)/2)]
f_1 = (x - 1)**2 + (y - 2)**2 - r**2
f_2 = (x - 1)**2 + (y - 1)**2 - r**2
assert solve_poly_system([f_1, f_2], x, y) == \
[(1 - sqrt(((2*r - 1)*(2*r + 1)))/2, Rational(3, 2)),
(1 + sqrt(((2*r - 1)*(2*r + 1)))/2, Rational(3, 2))]
def query(expr):
return expr.is_Pow and expr.exp is S.Half
f_1 = (x - 1 )**2 + (y - 2)**2 - r**2
f_2 = (x - x1)**2 + (y - 1)**2 - r**2
result = solve_poly_system([f_1, f_2], x, y)
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(r.count(query) == 1 for r in flatten(result))
f_1 = (x - x0)**2 + (y - y0)**2 - r**2
f_2 = (x - x1)**2 + (y - y1)**2 - r**2
result = solve_poly_system([f_1, f_2], x, y)
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(len(r.find(query)) == 1 for r in flatten(result))
def test_solve_biquadratic():
x0, y0, x1, y1, r = symbols('x0 y0 x1 y1 r')
f_1 = (x - 1)**2 + (y - 1)**2 - r**2
f_2 = (x - 2)**2 + (y - 2)**2 - r**2
assert (solve_poly_system([f_1, f_2], x, y) ==
[{x: Rational(3, 2) - sqrt(-1 + 2*r**2)/2,
y: Rational(3, 2) + sqrt(-1 + 2*r**2)/2},
{x: Rational(3, 2) + sqrt(-1 + 2*r**2)/2,
y: Rational(3, 2) - sqrt(-1 + 2*r**2)/2}])
f_1 = (x - 1)**2 + (y - 2)**2 - r**2
f_2 = (x - 1)**2 + (y - 1)**2 - r**2
assert (solve_poly_system([f_1, f_2], x, y) ==
[{x: 1 - sqrt(((2*r - 1)*(2*r + 1)))/2, y: Rational(3, 2)},
{x: 1 + sqrt(((2*r - 1)*(2*r + 1)))/2, y: Rational(3, 2)}])
def query(expr):
return expr.is_Pow and expr.exp is S.Half
f_1 = (x - 1 )**2 + (y - 2)**2 - r**2
f_2 = (x - x1)**2 + (y - 1)**2 - r**2
result = [tuple(r.values()) for r in solve_poly_system([f_1, f_2], x, y)]
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(r.count(query) == 1 for r in flatten(result))
f_1 = (x - x0)**2 + (y - y0)**2 - r**2
f_2 = (x - x1)**2 + (y - y1)**2 - r**2
result = [tuple(r.values()) for r in solve_poly_system([f_1, f_2], x, y)]
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(len(r.find(query)) == 1 for r in flatten(result))
eqs = [y**2 - 4 + x, y*2 + 3*x - 7]
assert solve_poly_system(eqs, x, y) == [{x: Rational(11, 9),
y: Rational(5, 3)},
{x: 3, y: -1}]
eqs = [y + x**2 - 3, -y + x - 4]
assert solve_poly_system(eqs, x, y) == [{x: -S.Half + sqrt(29)/2,
y: Rational(-9, 2) + sqrt(29)/2},
{x: -sqrt(29)/2 - S.Half,
y: Rational(-9, 2) - sqrt(29)/2}]
def _new(cls, *args, **kwargs):
shape, flat_list = cls._handle_ndarray_creation_inputs(*args, **kwargs)
flat_list = flatten(flat_list)
self = object.__new__(cls)
self._shape = shape
self._array = list(flat_list)
self._rank = len(shape)
self._loop_size = functools.reduce(lambda x, y: x * y,
shape) if shape else 0
return self
def test_flatten():
assert flatten((1, (1,))) == [1, 1]
assert flatten((x, (x,))) == [x, x]
ls = [[(-2, -1), (1, 2)], [(0, 0)]]
assert flatten(ls, levels=0) == ls
assert flatten(ls, levels=1) == [(-2, -1), (1, 2), (0, 0)]
assert flatten(ls, levels=2) == [-2, -1, 1, 2, 0, 0]
assert flatten(ls, levels=3) == [-2, -1, 1, 2, 0, 0]
pytest.raises(ValueError, lambda: flatten(ls, levels=-1))
class MyOp(Basic):
pass
assert flatten([MyOp(x, y), z]) == [MyOp(x, y), z]
assert flatten([MyOp(x, y), z], cls=MyOp) == [x, y, z]
assert flatten({1, 11, 2}) == list({1, 11, 2})
def _new(cls, *args, **kwargs):
shape, flat_list = cls._handle_ndarray_creation_inputs(*args, **kwargs)
shape = Tuple(*map(_sympify, shape))
flat_list = flatten(flat_list)
flat_list = Tuple(*flat_list)
self = Basic.__new__(cls, flat_list, shape, **kwargs)
self._shape = shape
self._array = tuple(flat_list)
self._rank = len(shape)
self._loop_size = functools.reduce(lambda x, y: x * y,
shape) if shape else 0
return self
def __new__(cls, *args, **kwargs):
shape, flat_list = cls._handle_ndarray_creation_inputs(*args, **kwargs)
self = object.__new__(cls)
self._shape = shape
self._rank = len(shape)
self._loop_size = functools.reduce(lambda x, y: x * y,
shape) if shape else 0
# Sparse array:
if isinstance(flat_list, (dict, Dict)):
self._sparse_array = dict(flat_list)
return self
self._sparse_array = {}
for i, el in enumerate(flatten(flat_list)):
if el != 0:
self._sparse_array[i] = _sympify(el)
return self
def __new__(cls, *args, **kwargs):
shape, flat_list = cls._handle_ndarray_creation_inputs(*args, **kwargs)
shape = Tuple(*map(_sympify, shape))
loop_size = functools.reduce(lambda x, y: x * y, shape) if shape else 0
# Sparse array:
if isinstance(flat_list, (dict, Dict)):
sparse_array = Dict(flat_list)
else:
sparse_array = {}
for i, el in enumerate(flatten(flat_list)):
if el != 0:
sparse_array[i] = _sympify(el)
sparse_array = Dict(sparse_array)
self = Expr.__new__(cls, sparse_array, shape, **kwargs)
self._shape = shape
self._rank = len(shape)
self._loop_size = loop_size
self._sparse_array = sparse_array
return self
def test_solve_biquadratic():
x0, y0, x1, y1, r = symbols('x0 y0 x1 y1 r')
f_1 = (x - 1)**2 + (y - 1)**2 - r**2
f_2 = (x - 2)**2 + (y - 2)**2 - r**2
assert (solve_poly_system([f_1, f_2], x, y) == [{
x:
Rational(3, 2) - sqrt(-1 + 2 * r**2) / 2,
y:
Rational(3, 2) + sqrt(-1 + 2 * r**2) / 2
}, {
x:
Rational(3, 2) + sqrt(-1 + 2 * r**2) / 2,
y:
Rational(3, 2) - sqrt(-1 + 2 * r**2) / 2
}])
f_1 = (x - 1)**2 + (y - 2)**2 - r**2
f_2 = (x - 1)**2 + (y - 1)**2 - r**2
assert (solve_poly_system([f_1, f_2], x, y) == [{
x:
1 - sqrt(((2 * r - 1) * (2 * r + 1))) / 2,
y:
Rational(3, 2)
}, {
x:
1 + sqrt(((2 * r - 1) * (2 * r + 1))) / 2,
y:
Rational(3, 2)
}])
def query(expr):
return expr.is_Pow and expr.exp is Rational(1, 2)
f_1 = (x - 1)**2 + (y - 2)**2 - r**2
f_2 = (x - x1)**2 + (y - 1)**2 - r**2
result = [tuple(r.values()) for r in solve_poly_system([f_1, f_2], x, y)]
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(r.count(query) == 1 for r in flatten(result))
f_1 = (x - x0)**2 + (y - y0)**2 - r**2
f_2 = (x - x1)**2 + (y - y1)**2 - r**2
result = [tuple(r.values()) for r in solve_poly_system([f_1, f_2], x, y)]
assert len(result) == 2 and all(len(r) == 2 for r in result)
assert all(len(r.find(query)) == 1 for r in flatten(result))
eqs = [y**2 - 4 + x, y * 2 + 3 * x - 7]
assert solve_poly_system(eqs, x, y) == [{
x: Rational(11, 9),
y: Rational(5, 3)
}, {
x: 3,
y: -1
}]
eqs = [y + x**2 - 3, -y + x - 4]
assert solve_poly_system(eqs, x, y) == [{
x: Rational(-1, 2) + sqrt(29) / 2,
y: Rational(-9, 2) + sqrt(29) / 2
}, {
x: -sqrt(29) / 2 - Rational(1, 2),
y: Rational(-9, 2) - sqrt(29) / 2
}]
# issue sympy/sympy#6785
roots = solve_poly_system([((x - 5)**2 / 250000 +
(y - Rational(5, 10))**2 / 250000) - 1, x], x,
y)
assert roots == [{
x: 0,
y: Rational(1, 2) + 15 * sqrt(1111)
}, {
x: 0,
y: Rational(1, 2) - 15 * sqrt(1111)
}]
roots = solve_poly_system([((x - 5)**2 / 250000 +
(y - 5.0 / 10)**2 / 250000) - 1, x], x, y)
assert len(roots) == 2
assert roots[0][x] == roots[1][x] == 0
assert roots[0][y].epsilon_eq(-499.474999374969, 1e12)
assert roots[1][y].epsilon_eq(+500.474999374969, 1e12)