def test_rref():
e = ReductionsOnlyMatrix(0, 0, [])
assert e.rref(pivots=False) == e
e = ReductionsOnlyMatrix(1, 1, [1])
a = ReductionsOnlyMatrix(1, 1, [5])
assert e.rref(pivots=False) == a.rref(pivots=False) == e
a = ReductionsOnlyMatrix(3, 1, [1, 2, 3])
assert a.rref(pivots=False) == Matrix([[1], [0], [0]])
a = ReductionsOnlyMatrix(1, 3, [1, 2, 3])
assert a.rref(pivots=False) == Matrix([[1, 2, 3]])
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
assert a.rref(pivots=False) == Matrix([[1, 0, -1], [0, 1, 2], [0, 0, 0]])
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 1, 2, 3, 1, 2, 3])
b = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 0, 0, 0, 0, 0, 0])
c = ReductionsOnlyMatrix(3, 3, [0, 0, 0, 1, 2, 3, 0, 0, 0])
d = ReductionsOnlyMatrix(3, 3, [0, 0, 0, 0, 0, 0, 1, 2, 3])
assert a.rref(pivots=False) == \
b.rref(pivots=False) == \
c.rref(pivots=False) == \
d.rref(pivots=False) == b
e = eye_Reductions(3)
z = zeros_Reductions(3)
assert e.rref(pivots=False) == e
assert z.rref(pivots=False) == z
a = ReductionsOnlyMatrix([[0, 0, 1, 2, 2, -5, 3], [-1, 5, 2, 2, 1, -7, 5],
[0, 0, -2, -3, -3, 8, -5],
[-1, 5, 0, -1, -2, 1, 0]])
mat, pivot_offsets = a.rref()
assert mat == Matrix([[1, -5, 0, 0, 1, 1, -1], [0, 0, 1, 0, 0, -1, 1],
[0, 0, 0, 1, 1, -2, 1], [0, 0, 0, 0, 0, 0, 0]])
assert pivot_offsets == (0, 2, 3)
a = ReductionsOnlyMatrix([[S(1) / 19, S(1) / 5, 2, 3], [4, 5, 6, 7],
[8, 9, 10, 11], [12, 13, 14, 15]])
assert a.rref(pivots=False) == Matrix([[1, 0, 0, -S(76) / 157],
[0, 1, 0, -S(5) / 157],
[0, 0, 1, S(238) / 157],
[0, 0, 0, 0]])
x = Symbol('x')
a = ReductionsOnlyMatrix(2, 3, [x, 1, 1, sqrt(x), x, 1])
for i, j in zip(a.rref(pivots=False), [
1, 0,
sqrt(x) * (-x + 1) / (-x**(S(5) / 2) + x), 0, 1, 1 /
(sqrt(x) + x + 1)
]):
assert simplify(i - j).is_zero
def test_rref():
e = ReductionsOnlyMatrix(0, 0, [])
assert e.rref(pivots=False) == e
e = ReductionsOnlyMatrix(1, 1, [1])
a = ReductionsOnlyMatrix(1, 1, [5])
assert e.rref(pivots=False) == a.rref(pivots=False) == e
a = ReductionsOnlyMatrix(3, 1, [1, 2, 3])
assert a.rref(pivots=False) == Matrix([[1], [0], [0]])
a = ReductionsOnlyMatrix(1, 3, [1, 2, 3])
assert a.rref(pivots=False) == Matrix([[1, 2, 3]])
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 4, 5, 6, 7, 8, 9])
assert a.rref(pivots=False) == Matrix([
[1, 0, -1],
[0, 1, 2],
[0, 0, 0]])
a = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 1, 2, 3, 1, 2, 3])
b = ReductionsOnlyMatrix(3, 3, [1, 2, 3, 0, 0, 0, 0, 0, 0])
c = ReductionsOnlyMatrix(3, 3, [0, 0, 0, 1, 2, 3, 0, 0, 0])
d = ReductionsOnlyMatrix(3, 3, [0, 0, 0, 0, 0, 0, 1, 2, 3])
assert a.rref(pivots=False) == \
b.rref(pivots=False) == \
c.rref(pivots=False) == \
d.rref(pivots=False) == b
e = eye_Reductions(3)
z = zeros_Reductions(3)
assert e.rref(pivots=False) == e
assert z.rref(pivots=False) == z
a = ReductionsOnlyMatrix([
[ 0, 0, 1, 2, 2, -5, 3],
[-1, 5, 2, 2, 1, -7, 5],
[ 0, 0, -2, -3, -3, 8, -5],
[-1, 5, 0, -1, -2, 1, 0]])
mat, pivot_offsets = a.rref()
assert mat == Matrix([
[1, -5, 0, 0, 1, 1, -1],
[0, 0, 1, 0, 0, -1, 1],
[0, 0, 0, 1, 1, -2, 1],
[0, 0, 0, 0, 0, 0, 0]])
assert pivot_offsets == (0, 2, 3)
a = ReductionsOnlyMatrix([[S(1)/19, S(1)/5, 2, 3],
[ 4, 5, 6, 7],
[ 8, 9, 10, 11],
[ 12, 13, 14, 15]])
assert a.rref(pivots=False) == Matrix([
[1, 0, 0, -S(76)/157],
[0, 1, 0, -S(5)/157],
[0, 0, 1, S(238)/157],
[0, 0, 0, 0]])
x = Symbol('x')
a = ReductionsOnlyMatrix(2, 3, [x, 1, 1, sqrt(x), x, 1])
for i, j in zip(a.rref(pivots=False),
[1, 0, sqrt(x)*(-x + 1)/(-x**(S(5)/2) + x),
0, 1, 1/(sqrt(x) + x + 1)]):
assert simplify(i - j).is_zero