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rl/Lib/site-packages/sympy/polys/matrices/tests/test_ddm.py

@@ -0,0 +1,558 @@
+from sympy.testing.pytest import raises
+from sympy.external.gmpy import GROUND_TYPES
+
+from sympy.polys import ZZ, QQ
+
+from sympy.polys.matrices.ddm import DDM
+from sympy.polys.matrices.exceptions import (
+    DMShapeError, DMNonInvertibleMatrixError, DMDomainError,
+    DMBadInputError)
+
+
+def test_DDM_init():
+    items = [[ZZ(0), ZZ(1), ZZ(2)], [ZZ(3), ZZ(4), ZZ(5)]]
+    shape = (2, 3)
+    ddm = DDM(items, shape, ZZ)
+    assert ddm.shape == shape
+    assert ddm.rows == 2
+    assert ddm.cols == 3
+    assert ddm.domain == ZZ
+
+    raises(DMBadInputError, lambda: DDM([[ZZ(2), ZZ(3)]], (2, 2), ZZ))
+    raises(DMBadInputError, lambda: DDM([[ZZ(1)], [ZZ(2), ZZ(3)]], (2, 2), ZZ))
+
+
+def test_DDM_getsetitem():
+    ddm = DDM([[ZZ(2), ZZ(3)], [ZZ(4), ZZ(5)]], (2, 2), ZZ)
+
+    assert ddm[0][0] == ZZ(2)
+    assert ddm[0][1] == ZZ(3)
+    assert ddm[1][0] == ZZ(4)
+    assert ddm[1][1] == ZZ(5)
+
+    raises(IndexError, lambda: ddm[2][0])
+    raises(IndexError, lambda: ddm[0][2])
+
+    ddm[0][0] = ZZ(-1)
+    assert ddm[0][0] == ZZ(-1)
+
+
+def test_DDM_str():
+    ddm = DDM([[ZZ(0), ZZ(1)], [ZZ(2), ZZ(3)]], (2, 2), ZZ)
+    if GROUND_TYPES == 'gmpy': # pragma: no cover
+        assert str(ddm) == '[[0, 1], [2, 3]]'
+        assert repr(ddm) == 'DDM([[mpz(0), mpz(1)], [mpz(2), mpz(3)]], (2, 2), ZZ)'
+    else:        # pragma: no cover
+        assert repr(ddm) == 'DDM([[0, 1], [2, 3]], (2, 2), ZZ)'
+        assert str(ddm) == '[[0, 1], [2, 3]]'
+
+
+def test_DDM_eq():
+    items = [[ZZ(0), ZZ(1)], [ZZ(2), ZZ(3)]]
+    ddm1 = DDM(items, (2, 2), ZZ)
+    ddm2 = DDM(items, (2, 2), ZZ)
+
+    assert (ddm1 == ddm1) is True
+    assert (ddm1 == items) is False
+    assert (items == ddm1) is False
+    assert (ddm1 == ddm2) is True
+    assert (ddm2 == ddm1) is True
+
+    assert (ddm1 != ddm1) is False
+    assert (ddm1 != items) is True
+    assert (items != ddm1) is True
+    assert (ddm1 != ddm2) is False
+    assert (ddm2 != ddm1) is False
+
+    ddm3 = DDM([[ZZ(0), ZZ(1)], [ZZ(3), ZZ(3)]], (2, 2), ZZ)
+    ddm3 = DDM(items, (2, 2), QQ)
+
+    assert (ddm1 == ddm3) is False
+    assert (ddm3 == ddm1) is False
+    assert (ddm1 != ddm3) is True
+    assert (ddm3 != ddm1) is True
+
+
+def test_DDM_convert_to():
+    ddm = DDM([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
+    assert ddm.convert_to(ZZ) == ddm
+    ddmq = ddm.convert_to(QQ)
+    assert ddmq.domain == QQ
+
+
+def test_DDM_zeros():
+    ddmz = DDM.zeros((3, 4), QQ)
+    assert list(ddmz) == [[QQ(0)] * 4] * 3
+    assert ddmz.shape == (3, 4)
+    assert ddmz.domain == QQ
+
+def test_DDM_ones():
+    ddmone = DDM.ones((2, 3), QQ)
+    assert list(ddmone) == [[QQ(1)] * 3] * 2
+    assert ddmone.shape == (2, 3)
+    assert ddmone.domain == QQ
+
+def test_DDM_eye():
+    ddmz = DDM.eye(3, QQ)
+    f = lambda i, j: QQ(1) if i == j else QQ(0)
+    assert list(ddmz) == [[f(i, j) for i in range(3)] for j in range(3)]
+    assert ddmz.shape == (3, 3)
+    assert ddmz.domain == QQ
+
+
+def test_DDM_copy():
+    ddm1 = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
+    ddm2 = ddm1.copy()
+    assert (ddm1 == ddm2) is True
+    ddm1[0][0] = QQ(-1)
+    assert (ddm1 == ddm2) is False
+    ddm2[0][0] = QQ(-1)
+    assert (ddm1 == ddm2) is True
+
+
+def test_DDM_transpose():
+    ddm = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
+    ddmT = DDM([[QQ(1), QQ(2)]], (1, 2), QQ)
+    assert ddm.transpose() == ddmT
+    ddm02 = DDM([], (0, 2), QQ)
+    ddm02T = DDM([[], []], (2, 0), QQ)
+    assert ddm02.transpose() == ddm02T
+    assert ddm02T.transpose() == ddm02
+    ddm0 = DDM([], (0, 0), QQ)
+    assert ddm0.transpose() == ddm0
+
+
+def test_DDM_add():
+    A = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
+    B = DDM([[ZZ(3)], [ZZ(4)]], (2, 1), ZZ)
+    C = DDM([[ZZ(4)], [ZZ(6)]], (2, 1), ZZ)
+    AQ = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
+    assert A + B == A.add(B) == C
+
+    raises(DMShapeError, lambda: A + DDM([[ZZ(5)]], (1, 1), ZZ))
+    raises(TypeError, lambda: A + ZZ(1))
+    raises(TypeError, lambda: ZZ(1) + A)
+    raises(DMDomainError, lambda: A + AQ)
+    raises(DMDomainError, lambda: AQ + A)
+
+
+def test_DDM_sub():
+    A = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
+    B = DDM([[ZZ(3)], [ZZ(4)]], (2, 1), ZZ)
+    C = DDM([[ZZ(-2)], [ZZ(-2)]], (2, 1), ZZ)
+    AQ = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
+    D = DDM([[ZZ(5)]], (1, 1), ZZ)
+    assert A - B == A.sub(B) == C
+
+    raises(TypeError, lambda: A - ZZ(1))
+    raises(TypeError, lambda: ZZ(1) - A)
+    raises(DMShapeError, lambda: A - D)
+    raises(DMShapeError, lambda: D - A)
+    raises(DMShapeError, lambda: A.sub(D))
+    raises(DMShapeError, lambda: D.sub(A))
+    raises(DMDomainError, lambda: A - AQ)
+    raises(DMDomainError, lambda: AQ - A)
+    raises(DMDomainError, lambda: A.sub(AQ))
+    raises(DMDomainError, lambda: AQ.sub(A))
+
+
+def test_DDM_neg():
+    A = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
+    An = DDM([[ZZ(-1)], [ZZ(-2)]], (2, 1), ZZ)
+    assert -A == A.neg() == An
+    assert -An == An.neg() == A
+
+
+def test_DDM_mul():
+    A = DDM([[ZZ(1)]], (1, 1), ZZ)
+    A2 = DDM([[ZZ(2)]], (1, 1), ZZ)
+    assert A * ZZ(2) == A2
+    assert ZZ(2) * A == A2
+    raises(TypeError, lambda: [[1]] * A)
+    raises(TypeError, lambda: A * [[1]])
+
+
+def test_DDM_matmul():
+    A = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
+    B = DDM([[ZZ(3), ZZ(4)]], (1, 2), ZZ)
+    AB = DDM([[ZZ(3), ZZ(4)], [ZZ(6), ZZ(8)]], (2, 2), ZZ)
+    BA = DDM([[ZZ(11)]], (1, 1), ZZ)
+
+    assert A @ B == A.matmul(B) == AB
+    assert B @ A == B.matmul(A) == BA
+
+    raises(TypeError, lambda: A @ 1)
+    raises(TypeError, lambda: A @ [[3, 4]])
+
+    Bq = DDM([[QQ(3), QQ(4)]], (1, 2), QQ)
+
+    raises(DMDomainError, lambda: A @ Bq)
+    raises(DMDomainError, lambda: Bq @ A)
+
+    C = DDM([[ZZ(1)]], (1, 1), ZZ)
+
+    assert A @ C == A.matmul(C) == A
+
+    raises(DMShapeError, lambda: C @ A)
+    raises(DMShapeError, lambda: C.matmul(A))
+
+    Z04 = DDM([], (0, 4), ZZ)
+    Z40 = DDM([[]]*4, (4, 0), ZZ)
+    Z50 = DDM([[]]*5, (5, 0), ZZ)
+    Z05 = DDM([], (0, 5), ZZ)
+    Z45 = DDM([[0] * 5] * 4, (4, 5), ZZ)
+    Z54 = DDM([[0] * 4] * 5, (5, 4), ZZ)
+    Z00 = DDM([], (0, 0), ZZ)
+
+    assert Z04 @ Z45 == Z04.matmul(Z45) == Z05
+    assert Z45 @ Z50 == Z45.matmul(Z50) == Z40
+    assert Z00 @ Z04 == Z00.matmul(Z04) == Z04
+    assert Z50 @ Z00 == Z50.matmul(Z00) == Z50
+    assert Z00 @ Z00 == Z00.matmul(Z00) == Z00
+    assert Z50 @ Z04 == Z50.matmul(Z04) == Z54
+
+    raises(DMShapeError, lambda: Z05 @ Z40)
+    raises(DMShapeError, lambda: Z05.matmul(Z40))
+
+
+def test_DDM_hstack():
+    A = DDM([[ZZ(1), ZZ(2), ZZ(3)]], (1, 3), ZZ)
+    B = DDM([[ZZ(4), ZZ(5)]], (1, 2), ZZ)
+    C = DDM([[ZZ(6)]], (1, 1), ZZ)
+
+    Ah = A.hstack(B)
+    assert Ah.shape == (1, 5)
+    assert Ah.domain == ZZ
+    assert Ah == DDM([[ZZ(1), ZZ(2), ZZ(3), ZZ(4), ZZ(5)]], (1, 5), ZZ)
+
+    Ah = A.hstack(B, C)
+    assert Ah.shape == (1, 6)
+    assert Ah.domain == ZZ
+    assert Ah == DDM([[ZZ(1), ZZ(2), ZZ(3), ZZ(4), ZZ(5), ZZ(6)]], (1, 6), ZZ)
+
+
+def test_DDM_vstack():
+    A = DDM([[ZZ(1)], [ZZ(2)], [ZZ(3)]], (3, 1), ZZ)
+    B = DDM([[ZZ(4)], [ZZ(5)]], (2, 1), ZZ)
+    C = DDM([[ZZ(6)]], (1, 1), ZZ)
+
+    Ah = A.vstack(B)
+    assert Ah.shape == (5, 1)
+    assert Ah.domain == ZZ
+    assert Ah == DDM([[ZZ(1)], [ZZ(2)], [ZZ(3)], [ZZ(4)], [ZZ(5)]], (5, 1), ZZ)
+
+    Ah = A.vstack(B, C)
+    assert Ah.shape == (6, 1)
+    assert Ah.domain == ZZ
+    assert Ah == DDM([[ZZ(1)], [ZZ(2)], [ZZ(3)], [ZZ(4)], [ZZ(5)], [ZZ(6)]], (6, 1), ZZ)
+
+
+def test_DDM_applyfunc():
+    A = DDM([[ZZ(1), ZZ(2), ZZ(3)]], (1, 3), ZZ)
+    B = DDM([[ZZ(2), ZZ(4), ZZ(6)]], (1, 3), ZZ)
+    assert A.applyfunc(lambda x: 2*x, ZZ) == B
+
+def test_DDM_rref():
+
+    A = DDM([], (0, 4), QQ)
+    assert A.rref() == (A, [])
+
+    A = DDM([[QQ(0), QQ(1)], [QQ(1), QQ(1)]], (2, 2), QQ)
+    Ar = DDM([[QQ(1), QQ(0)], [QQ(0), QQ(1)]], (2, 2), QQ)
+    pivots = [0, 1]
+    assert A.rref() == (Ar, pivots)
+
+    A = DDM([[QQ(1), QQ(2), QQ(1)], [QQ(3), QQ(4), QQ(1)]], (2, 3), QQ)
+    Ar = DDM([[QQ(1), QQ(0), QQ(-1)], [QQ(0), QQ(1), QQ(1)]], (2, 3), QQ)
+    pivots = [0, 1]
+    assert A.rref() == (Ar, pivots)
+
+    A = DDM([[QQ(3), QQ(4), QQ(1)], [QQ(1), QQ(2), QQ(1)]], (2, 3), QQ)
+    Ar = DDM([[QQ(1), QQ(0), QQ(-1)], [QQ(0), QQ(1), QQ(1)]], (2, 3), QQ)
+    pivots = [0, 1]
+    assert A.rref() == (Ar, pivots)
+
+    A = DDM([[QQ(1), QQ(0)], [QQ(1), QQ(3)], [QQ(0), QQ(1)]], (3, 2), QQ)
+    Ar = DDM([[QQ(1), QQ(0)], [QQ(0), QQ(1)], [QQ(0), QQ(0)]], (3, 2), QQ)
+    pivots = [0, 1]
+    assert A.rref() == (Ar, pivots)
+
+    A = DDM([[QQ(1), QQ(0), QQ(1)], [QQ(3), QQ(0), QQ(1)]], (2, 3), QQ)
+    Ar = DDM([[QQ(1), QQ(0), QQ(0)], [QQ(0), QQ(0), QQ(1)]], (2, 3), QQ)
+    pivots = [0, 2]
+    assert A.rref() == (Ar, pivots)
+
+
+def test_DDM_nullspace():
+    # more tests are in test_nullspace.py
+    A = DDM([[QQ(1), QQ(1)], [QQ(1), QQ(1)]], (2, 2), QQ)
+    Anull = DDM([[QQ(-1), QQ(1)]], (1, 2), QQ)
+    nonpivots = [1]
+    assert A.nullspace() == (Anull, nonpivots)
+
+
+def test_DDM_particular():
+    A = DDM([[QQ(1), QQ(0)]], (1, 2), QQ)
+    assert A.particular() == DDM.zeros((1, 1), QQ)
+
+
+def test_DDM_det():
+    # 0x0 case
+    A = DDM([], (0, 0), ZZ)
+    assert A.det() == ZZ(1)
+
+    # 1x1 case
+    A = DDM([[ZZ(2)]], (1, 1), ZZ)
+    assert A.det() == ZZ(2)
+
+    # 2x2 case
+    A = DDM([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
+    assert A.det() == ZZ(-2)
+
+    # 3x3 with swap
+    A = DDM([[ZZ(1), ZZ(2), ZZ(3)], [ZZ(1), ZZ(2), ZZ(4)], [ZZ(1), ZZ(2), ZZ(5)]], (3, 3), ZZ)
+    assert A.det() == ZZ(0)
+
+    # 2x2 QQ case
+    A = DDM([[QQ(1, 2), QQ(1, 2)], [QQ(1, 3), QQ(1, 4)]], (2, 2), QQ)
+    assert A.det() == QQ(-1, 24)
+
+    # Nonsquare error
+    A = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
+    raises(DMShapeError, lambda: A.det())
+
+    # Nonsquare error with empty matrix
+    A = DDM([], (0, 1), ZZ)
+    raises(DMShapeError, lambda: A.det())
+
+
+def test_DDM_inv():
+    A = DDM([[QQ(1, 1), QQ(2, 1)], [QQ(3, 1), QQ(4, 1)]], (2, 2), QQ)
+    Ainv = DDM([[QQ(-2, 1), QQ(1, 1)], [QQ(3, 2), QQ(-1, 2)]], (2, 2), QQ)
+    assert A.inv() == Ainv
+
+    A = DDM([[QQ(1), QQ(2)]], (1, 2), QQ)
+    raises(DMShapeError, lambda: A.inv())
+
+    A = DDM([[ZZ(2)]], (1, 1), ZZ)
+    raises(DMDomainError, lambda: A.inv())
+
+    A = DDM([], (0, 0), QQ)
+    assert A.inv() == A
+
+    A = DDM([[QQ(1), QQ(2)], [QQ(2), QQ(4)]], (2, 2), QQ)
+    raises(DMNonInvertibleMatrixError, lambda: A.inv())
+
+
+def test_DDM_lu():
+    A = DDM([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
+    L, U, swaps = A.lu()
+    assert L == DDM([[QQ(1), QQ(0)], [QQ(3), QQ(1)]], (2, 2), QQ)
+    assert U == DDM([[QQ(1), QQ(2)], [QQ(0), QQ(-2)]], (2, 2), QQ)
+    assert swaps == []
+
+    A = [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 1, 2]]
+    Lexp = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [0, 0, 1, 1]]
+    Uexp = [[1, 0, 0, 0], [0, 0, 0, 0], [0, 0, 1, 1], [0, 0, 0, 1]]
+    to_dom = lambda rows, dom: [[dom(e) for e in row] for row in rows]
+    A = DDM(to_dom(A, QQ), (4, 4), QQ)
+    Lexp = DDM(to_dom(Lexp, QQ), (4, 4), QQ)
+    Uexp = DDM(to_dom(Uexp, QQ), (4, 4), QQ)
+    L, U, swaps = A.lu()
+    assert L == Lexp
+    assert U == Uexp
+    assert swaps == []
+
+
+def test_DDM_lu_solve():
+    # Basic example
+    A = DDM([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
+    b = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
+    x = DDM([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
+    assert A.lu_solve(b) == x
+
+    # Example with swaps
+    A = DDM([[QQ(0), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
+    assert A.lu_solve(b) == x
+
+    # Overdetermined, consistent
+    A = DDM([[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]], (3, 2), QQ)
+    b = DDM([[QQ(1)], [QQ(2)], [QQ(3)]], (3, 1), QQ)
+    assert A.lu_solve(b) == x
+
+    # Overdetermined, inconsistent
+    b = DDM([[QQ(1)], [QQ(2)], [QQ(4)]], (3, 1), QQ)
+    raises(DMNonInvertibleMatrixError, lambda: A.lu_solve(b))
+
+    # Square, noninvertible
+    A = DDM([[QQ(1), QQ(2)], [QQ(1), QQ(2)]], (2, 2), QQ)
+    b = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
+    raises(DMNonInvertibleMatrixError, lambda: A.lu_solve(b))
+
+    # Underdetermined
+    A = DDM([[QQ(1), QQ(2)]], (1, 2), QQ)
+    b = DDM([[QQ(3)]], (1, 1), QQ)
+    raises(NotImplementedError, lambda: A.lu_solve(b))
+
+    # Domain mismatch
+    bz = DDM([[ZZ(1)], [ZZ(2)]], (2, 1), ZZ)
+    raises(DMDomainError, lambda: A.lu_solve(bz))
+
+    # Shape mismatch
+    b3 = DDM([[QQ(1)], [QQ(2)], [QQ(3)]], (3, 1), QQ)
+    raises(DMShapeError, lambda: A.lu_solve(b3))
+
+
+def test_DDM_charpoly():
+    A = DDM([], (0, 0), ZZ)
+    assert A.charpoly() == [ZZ(1)]
+
+    A = DDM([
+        [ZZ(1), ZZ(2), ZZ(3)],
+        [ZZ(4), ZZ(5), ZZ(6)],
+        [ZZ(7), ZZ(8), ZZ(9)]], (3, 3), ZZ)
+    Avec = [ZZ(1), ZZ(-15), ZZ(-18), ZZ(0)]
+    assert A.charpoly() == Avec
+
+    A = DDM([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
+    raises(DMShapeError, lambda: A.charpoly())
+
+
+def test_DDM_getitem():
+    dm = DDM([
+        [ZZ(1), ZZ(2), ZZ(3)],
+        [ZZ(4), ZZ(5), ZZ(6)],
+        [ZZ(7), ZZ(8), ZZ(9)]], (3, 3), ZZ)
+
+    assert dm.getitem(1, 1) == ZZ(5)
+    assert dm.getitem(1, -2) == ZZ(5)
+    assert dm.getitem(-1, -3) == ZZ(7)
+
+    raises(IndexError, lambda: dm.getitem(3, 3))
+
+
+def test_DDM_setitem():
+    dm = DDM.zeros((3, 3), ZZ)
+    dm.setitem(0, 0, 1)
+    dm.setitem(1, -2, 1)
+    dm.setitem(-1, -1, 1)
+    assert dm == DDM.eye(3, ZZ)
+
+    raises(IndexError, lambda: dm.setitem(3, 3, 0))
+
+
+def test_DDM_extract_slice():
+    dm = DDM([
+        [ZZ(1), ZZ(2), ZZ(3)],
+        [ZZ(4), ZZ(5), ZZ(6)],
+        [ZZ(7), ZZ(8), ZZ(9)]], (3, 3), ZZ)
+
+    assert dm.extract_slice(slice(0, 3), slice(0, 3)) == dm
+    assert dm.extract_slice(slice(1, 3), slice(-2)) == DDM([[4], [7]], (2, 1), ZZ)
+    assert dm.extract_slice(slice(1, 3), slice(-2)) == DDM([[4], [7]], (2, 1), ZZ)
+    assert dm.extract_slice(slice(2, 3), slice(-2)) == DDM([[ZZ(7)]], (1, 1), ZZ)
+    assert dm.extract_slice(slice(0, 2), slice(-2)) == DDM([[1], [4]], (2, 1), ZZ)
+    assert dm.extract_slice(slice(-1), slice(-1)) == DDM([[1, 2], [4, 5]], (2, 2), ZZ)
+
+    assert dm.extract_slice(slice(2), slice(3, 4)) == DDM([[], []], (2, 0), ZZ)
+    assert dm.extract_slice(slice(3, 4), slice(2)) == DDM([], (0, 2), ZZ)
+    assert dm.extract_slice(slice(3, 4), slice(3, 4)) == DDM([], (0, 0), ZZ)
+
+
+def test_DDM_extract():
+    dm1 = DDM([
+        [ZZ(1), ZZ(2), ZZ(3)],
+        [ZZ(4), ZZ(5), ZZ(6)],
+        [ZZ(7), ZZ(8), ZZ(9)]], (3, 3), ZZ)
+    dm2 = DDM([
+        [ZZ(6), ZZ(4)],
+        [ZZ(3), ZZ(1)]], (2, 2), ZZ)
+    assert dm1.extract([1, 0], [2, 0]) == dm2
+    assert dm1.extract([-2, 0], [-1, 0]) == dm2
+
+    assert dm1.extract([], []) == DDM.zeros((0, 0), ZZ)
+    assert dm1.extract([1], []) == DDM.zeros((1, 0), ZZ)
+    assert dm1.extract([], [1]) == DDM.zeros((0, 1), ZZ)
+
+    raises(IndexError, lambda: dm2.extract([2], [0]))
+    raises(IndexError, lambda: dm2.extract([0], [2]))
+    raises(IndexError, lambda: dm2.extract([-3], [0]))
+    raises(IndexError, lambda: dm2.extract([0], [-3]))
+
+
+def test_DDM_flat():
+    dm = DDM([
+        [ZZ(6), ZZ(4)],
+        [ZZ(3), ZZ(1)]], (2, 2), ZZ)
+    assert dm.flat() == [ZZ(6), ZZ(4), ZZ(3), ZZ(1)]
+
+
+def test_DDM_is_zero_matrix():
+    A = DDM([[QQ(1), QQ(0)], [QQ(0), QQ(0)]], (2, 2), QQ)
+    Azero = DDM.zeros((1, 2), QQ)
+    assert A.is_zero_matrix() is False
+    assert Azero.is_zero_matrix() is True
+
+
+def test_DDM_is_upper():
+    # Wide matrices:
+    A = DDM([
+        [QQ(1), QQ(2), QQ(3), QQ(4)],
+        [QQ(0), QQ(5), QQ(6), QQ(7)],
+        [QQ(0), QQ(0), QQ(8), QQ(9)]
+    ], (3, 4), QQ)
+    B = DDM([
+        [QQ(1), QQ(2), QQ(3), QQ(4)],
+        [QQ(0), QQ(5), QQ(6), QQ(7)],
+        [QQ(0), QQ(7), QQ(8), QQ(9)]
+    ], (3, 4), QQ)
+    assert A.is_upper() is True
+    assert B.is_upper() is False
+
+    # Tall matrices:
+    A = DDM([
+        [QQ(1), QQ(2), QQ(3)],
+        [QQ(0), QQ(5), QQ(6)],
+        [QQ(0), QQ(0), QQ(8)],
+        [QQ(0), QQ(0), QQ(0)]
+    ], (4, 3), QQ)
+    B = DDM([
+        [QQ(1), QQ(2), QQ(3)],
+        [QQ(0), QQ(5), QQ(6)],
+        [QQ(0), QQ(0), QQ(8)],
+        [QQ(0), QQ(0), QQ(10)]
+    ], (4, 3), QQ)
+    assert A.is_upper() is True
+    assert B.is_upper() is False
+
+
+def test_DDM_is_lower():
+    # Tall matrices:
+    A = DDM([
+        [QQ(1), QQ(2), QQ(3), QQ(4)],
+        [QQ(0), QQ(5), QQ(6), QQ(7)],
+        [QQ(0), QQ(0), QQ(8), QQ(9)]
+    ], (3, 4), QQ).transpose()
+    B = DDM([
+        [QQ(1), QQ(2), QQ(3), QQ(4)],
+        [QQ(0), QQ(5), QQ(6), QQ(7)],
+        [QQ(0), QQ(7), QQ(8), QQ(9)]
+    ], (3, 4), QQ).transpose()
+    assert A.is_lower() is True
+    assert B.is_lower() is False
+
+    # Wide matrices:
+    A = DDM([
+        [QQ(1), QQ(2), QQ(3)],
+        [QQ(0), QQ(5), QQ(6)],
+        [QQ(0), QQ(0), QQ(8)],
+        [QQ(0), QQ(0), QQ(0)]
+    ], (4, 3), QQ).transpose()
+    B = DDM([
+        [QQ(1), QQ(2), QQ(3)],
+        [QQ(0), QQ(5), QQ(6)],
+        [QQ(0), QQ(0), QQ(8)],
+        [QQ(0), QQ(0), QQ(10)]
+    ], (4, 3), QQ).transpose()
+    assert A.is_lower() is True
+    assert B.is_lower() is False

rl/Lib/site-packages/sympy/polys/matrices/tests/test_dense.py

@@ -0,0 +1,350 @@
+from sympy.testing.pytest import raises
+
+from sympy.polys import ZZ, QQ
+
+from sympy.polys.matrices.ddm import DDM
+from sympy.polys.matrices.dense import (
+        ddm_transpose,
+        ddm_iadd, ddm_isub, ddm_ineg, ddm_imatmul, ddm_imul, ddm_irref,
+        ddm_idet, ddm_iinv, ddm_ilu, ddm_ilu_split, ddm_ilu_solve, ddm_berk)
+
+from sympy.polys.matrices.exceptions import (
+    DMDomainError,
+    DMNonInvertibleMatrixError,
+    DMNonSquareMatrixError,
+    DMShapeError,
+)
+
+
+def test_ddm_transpose():
+    a = [[1, 2], [3, 4]]
+    assert ddm_transpose(a) == [[1, 3], [2, 4]]
+
+
+def test_ddm_iadd():
+    a = [[1, 2], [3, 4]]
+    b = [[5, 6], [7, 8]]
+    ddm_iadd(a, b)
+    assert a == [[6, 8], [10, 12]]
+
+
+def test_ddm_isub():
+    a = [[1, 2], [3, 4]]
+    b = [[5, 6], [7, 8]]
+    ddm_isub(a, b)
+    assert a == [[-4, -4], [-4, -4]]
+
+
+def test_ddm_ineg():
+    a = [[1, 2], [3, 4]]
+    ddm_ineg(a)
+    assert a == [[-1, -2], [-3, -4]]
+
+
+def test_ddm_matmul():
+    a = [[1, 2], [3, 4]]
+    ddm_imul(a, 2)
+    assert a == [[2, 4], [6, 8]]
+
+    a = [[1, 2], [3, 4]]
+    ddm_imul(a, 0)
+    assert a == [[0, 0], [0, 0]]
+
+
+def test_ddm_imatmul():
+    a = [[1, 2, 3], [4, 5, 6]]
+    b = [[1, 2], [3, 4], [5, 6]]
+
+    c1 = [[0, 0], [0, 0]]
+    ddm_imatmul(c1, a, b)
+    assert c1 == [[22, 28], [49, 64]]
+
+    c2 = [[0, 0, 0], [0, 0, 0], [0, 0, 0]]
+    ddm_imatmul(c2, b, a)
+    assert c2 == [[9, 12, 15], [19, 26, 33], [29, 40, 51]]
+
+    b3 = [[1], [2], [3]]
+    c3 = [[0], [0]]
+    ddm_imatmul(c3, a, b3)
+    assert c3 == [[14], [32]]
+
+
+def test_ddm_irref():
+    # Empty matrix
+    A = []
+    Ar = []
+    pivots = []
+    assert ddm_irref(A) == pivots
+    assert A == Ar
+
+    # Standard square case
+    A = [[QQ(0), QQ(1)], [QQ(1), QQ(1)]]
+    Ar = [[QQ(1), QQ(0)], [QQ(0), QQ(1)]]
+    pivots = [0, 1]
+    assert ddm_irref(A) == pivots
+    assert A == Ar
+
+    # m < n  case
+    A = [[QQ(1), QQ(2), QQ(1)], [QQ(3), QQ(4), QQ(1)]]
+    Ar = [[QQ(1), QQ(0), QQ(-1)], [QQ(0), QQ(1), QQ(1)]]
+    pivots = [0, 1]
+    assert ddm_irref(A) == pivots
+    assert A == Ar
+
+    # same m < n  but reversed
+    A = [[QQ(3), QQ(4), QQ(1)], [QQ(1), QQ(2), QQ(1)]]
+    Ar = [[QQ(1), QQ(0), QQ(-1)], [QQ(0), QQ(1), QQ(1)]]
+    pivots = [0, 1]
+    assert ddm_irref(A) == pivots
+    assert A == Ar
+
+    # m > n case
+    A = [[QQ(1), QQ(0)], [QQ(1), QQ(3)], [QQ(0), QQ(1)]]
+    Ar = [[QQ(1), QQ(0)], [QQ(0), QQ(1)], [QQ(0), QQ(0)]]
+    pivots = [0, 1]
+    assert ddm_irref(A) == pivots
+    assert A == Ar
+
+    # Example with missing pivot
+    A = [[QQ(1), QQ(0), QQ(1)], [QQ(3), QQ(0), QQ(1)]]
+    Ar = [[QQ(1), QQ(0), QQ(0)], [QQ(0), QQ(0), QQ(1)]]
+    pivots = [0, 2]
+    assert ddm_irref(A) == pivots
+    assert A == Ar
+
+    # Example with missing pivot and no replacement
+    A = [[QQ(0), QQ(1)], [QQ(0), QQ(2)], [QQ(1), QQ(0)]]
+    Ar = [[QQ(1), QQ(0)], [QQ(0), QQ(1)], [QQ(0), QQ(0)]]
+    pivots = [0, 1]
+    assert ddm_irref(A) == pivots
+    assert A == Ar
+
+
+def test_ddm_idet():
+    A = []
+    assert ddm_idet(A, ZZ) == ZZ(1)
+
+    A = [[ZZ(2)]]
+    assert ddm_idet(A, ZZ) == ZZ(2)
+
+    A = [[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]]
+    assert ddm_idet(A, ZZ) == ZZ(-2)
+
+    A = [[ZZ(1), ZZ(2), ZZ(3)], [ZZ(1), ZZ(2), ZZ(4)], [ZZ(1), ZZ(3), ZZ(5)]]
+    assert ddm_idet(A, ZZ) == ZZ(-1)
+
+    A = [[ZZ(1), ZZ(2), ZZ(3)], [ZZ(1), ZZ(2), ZZ(4)], [ZZ(1), ZZ(2), ZZ(5)]]
+    assert ddm_idet(A, ZZ) == ZZ(0)
+
+    A = [[QQ(1, 2), QQ(1, 2)], [QQ(1, 3), QQ(1, 4)]]
+    assert ddm_idet(A, QQ) == QQ(-1, 24)
+
+
+def test_ddm_inv():
+    A = []
+    Ainv = []
+    ddm_iinv(Ainv, A, QQ)
+    assert Ainv == A
+
+    A = []
+    Ainv = []
+    raises(DMDomainError, lambda: ddm_iinv(Ainv, A, ZZ))
+
+    A = [[QQ(1), QQ(2)]]
+    Ainv = [[QQ(0), QQ(0)]]
+    raises(DMNonSquareMatrixError, lambda: ddm_iinv(Ainv, A, QQ))
+
+    A = [[QQ(1, 1), QQ(2, 1)], [QQ(3, 1), QQ(4, 1)]]
+    Ainv = [[QQ(0), QQ(0)], [QQ(0), QQ(0)]]
+    Ainv_expected = [[QQ(-2, 1), QQ(1, 1)], [QQ(3, 2), QQ(-1, 2)]]
+    ddm_iinv(Ainv, A, QQ)
+    assert Ainv == Ainv_expected
+
+    A = [[QQ(1, 1), QQ(2, 1)], [QQ(2, 1), QQ(4, 1)]]
+    Ainv = [[QQ(0), QQ(0)], [QQ(0), QQ(0)]]
+    raises(DMNonInvertibleMatrixError, lambda: ddm_iinv(Ainv, A, QQ))
+
+
+def test_ddm_ilu():
+    A = []
+    Alu = []
+    swaps = ddm_ilu(A)
+    assert A == Alu
+    assert swaps == []
+
+    A = [[]]
+    Alu = [[]]
+    swaps = ddm_ilu(A)
+    assert A == Alu
+    assert swaps == []
+
+    A = [[QQ(1), QQ(2)], [QQ(3), QQ(4)]]
+    Alu = [[QQ(1), QQ(2)], [QQ(3), QQ(-2)]]
+    swaps = ddm_ilu(A)
+    assert A == Alu
+    assert swaps == []
+
+    A = [[QQ(0), QQ(2)], [QQ(3), QQ(4)]]
+    Alu = [[QQ(3), QQ(4)], [QQ(0), QQ(2)]]
+    swaps = ddm_ilu(A)
+    assert A == Alu
+    assert swaps == [(0, 1)]
+
+    A = [[QQ(1), QQ(2), QQ(3)], [QQ(4), QQ(5), QQ(6)], [QQ(7), QQ(8), QQ(9)]]
+    Alu = [[QQ(1), QQ(2), QQ(3)], [QQ(4), QQ(-3), QQ(-6)], [QQ(7), QQ(2), QQ(0)]]
+    swaps = ddm_ilu(A)
+    assert A == Alu
+    assert swaps == []
+
+    A = [[QQ(0), QQ(1), QQ(2)], [QQ(0), QQ(1), QQ(3)], [QQ(1), QQ(1), QQ(2)]]
+    Alu = [[QQ(1), QQ(1), QQ(2)], [QQ(0), QQ(1), QQ(3)], [QQ(0), QQ(1), QQ(-1)]]
+    swaps = ddm_ilu(A)
+    assert A == Alu
+    assert swaps == [(0, 2)]
+
+    A = [[QQ(1), QQ(2), QQ(3)], [QQ(4), QQ(5), QQ(6)]]
+    Alu = [[QQ(1), QQ(2), QQ(3)], [QQ(4), QQ(-3), QQ(-6)]]
+    swaps = ddm_ilu(A)
+    assert A == Alu
+    assert swaps == []
+
+    A = [[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]]
+    Alu = [[QQ(1), QQ(2)], [QQ(3), QQ(-2)], [QQ(5), QQ(2)]]
+    swaps = ddm_ilu(A)
+    assert A == Alu
+    assert swaps == []
+
+
+def test_ddm_ilu_split():
+    U = []
+    L = []
+    Uexp = []
+    Lexp = []
+    swaps = ddm_ilu_split(L, U, QQ)
+    assert U == Uexp
+    assert L == Lexp
+    assert swaps == []
+
+    U = [[]]
+    L = [[QQ(1)]]
+    Uexp = [[]]
+    Lexp = [[QQ(1)]]
+    swaps = ddm_ilu_split(L, U, QQ)
+    assert U == Uexp
+    assert L == Lexp
+    assert swaps == []
+
+    U = [[QQ(1), QQ(2)], [QQ(3), QQ(4)]]
+    L = [[QQ(1), QQ(0)], [QQ(0), QQ(1)]]
+    Uexp = [[QQ(1), QQ(2)], [QQ(0), QQ(-2)]]
+    Lexp = [[QQ(1), QQ(0)], [QQ(3), QQ(1)]]
+    swaps = ddm_ilu_split(L, U, QQ)
+    assert U == Uexp
+    assert L == Lexp
+    assert swaps == []
+
+    U = [[QQ(1), QQ(2), QQ(3)], [QQ(4), QQ(5), QQ(6)]]
+    L = [[QQ(1), QQ(0)], [QQ(0), QQ(1)]]
+    Uexp = [[QQ(1), QQ(2), QQ(3)], [QQ(0), QQ(-3), QQ(-6)]]
+    Lexp = [[QQ(1), QQ(0)], [QQ(4), QQ(1)]]
+    swaps = ddm_ilu_split(L, U, QQ)
+    assert U == Uexp
+    assert L == Lexp
+    assert swaps == []
+
+    U = [[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]]
+    L = [[QQ(1), QQ(0), QQ(0)], [QQ(0), QQ(1), QQ(0)], [QQ(0), QQ(0), QQ(1)]]
+    Uexp = [[QQ(1), QQ(2)], [QQ(0), QQ(-2)], [QQ(0), QQ(0)]]
+    Lexp = [[QQ(1), QQ(0), QQ(0)], [QQ(3), QQ(1), QQ(0)], [QQ(5), QQ(2), QQ(1)]]
+    swaps = ddm_ilu_split(L, U, QQ)
+    assert U == Uexp
+    assert L == Lexp
+    assert swaps == []
+
+
+def test_ddm_ilu_solve():
+    # Basic example
+    # A = [[QQ(1), QQ(2)], [QQ(3), QQ(4)]]
+    U = [[QQ(1), QQ(2)], [QQ(0), QQ(-2)]]
+    L = [[QQ(1), QQ(0)], [QQ(3), QQ(1)]]
+    swaps = []
+    b = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
+    x = DDM([[QQ(0)], [QQ(0)]], (2, 1), QQ)
+    xexp = DDM([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
+    ddm_ilu_solve(x, L, U, swaps, b)
+    assert x == xexp
+
+    # Example with swaps
+    # A = [[QQ(0), QQ(2)], [QQ(3), QQ(4)]]
+    U = [[QQ(3), QQ(4)], [QQ(0), QQ(2)]]
+    L = [[QQ(1), QQ(0)], [QQ(0), QQ(1)]]
+    swaps = [(0, 1)]
+    b = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
+    x = DDM([[QQ(0)], [QQ(0)]], (2, 1), QQ)
+    xexp = DDM([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
+    ddm_ilu_solve(x, L, U, swaps, b)
+    assert x == xexp
+
+    # Overdetermined, consistent
+    # A = DDM([[QQ(1), QQ(2)], [QQ(3), QQ(4)], [QQ(5), QQ(6)]], (3, 2), QQ)
+    U = [[QQ(1), QQ(2)], [QQ(0), QQ(-2)], [QQ(0), QQ(0)]]
+    L = [[QQ(1), QQ(0), QQ(0)], [QQ(3), QQ(1), QQ(0)], [QQ(5), QQ(2), QQ(1)]]
+    swaps = []
+    b = DDM([[QQ(1)], [QQ(2)], [QQ(3)]], (3, 1), QQ)
+    x = DDM([[QQ(0)], [QQ(0)]], (2, 1), QQ)
+    xexp = DDM([[QQ(0)], [QQ(1, 2)]], (2, 1), QQ)
+    ddm_ilu_solve(x, L, U, swaps, b)
+    assert x == xexp
+
+    # Overdetermined, inconsistent
+    b = DDM([[QQ(1)], [QQ(2)], [QQ(4)]], (3, 1), QQ)
+    raises(DMNonInvertibleMatrixError, lambda: ddm_ilu_solve(x, L, U, swaps, b))
+
+    # Square, noninvertible
+    # A = DDM([[QQ(1), QQ(2)], [QQ(1), QQ(2)]], (2, 2), QQ)
+    U = [[QQ(1), QQ(2)], [QQ(0), QQ(0)]]
+    L = [[QQ(1), QQ(0)], [QQ(1), QQ(1)]]
+    swaps = []
+    b = DDM([[QQ(1)], [QQ(2)]], (2, 1), QQ)
+    raises(DMNonInvertibleMatrixError, lambda: ddm_ilu_solve(x, L, U, swaps, b))
+
+    # Underdetermined
+    # A = DDM([[QQ(1), QQ(2)]], (1, 2), QQ)
+    U = [[QQ(1), QQ(2)]]
+    L = [[QQ(1)]]
+    swaps = []
+    b = DDM([[QQ(3)]], (1, 1), QQ)
+    raises(NotImplementedError, lambda: ddm_ilu_solve(x, L, U, swaps, b))
+
+    # Shape mismatch
+    b3 = DDM([[QQ(1)], [QQ(2)], [QQ(3)]], (3, 1), QQ)
+    raises(DMShapeError, lambda: ddm_ilu_solve(x, L, U, swaps, b3))
+
+    # Empty shape mismatch
+    U = [[QQ(1)]]
+    L = [[QQ(1)]]
+    swaps = []
+    x = [[QQ(1)]]
+    b = []
+    raises(DMShapeError, lambda: ddm_ilu_solve(x, L, U, swaps, b))
+
+    # Empty system
+    U = []
+    L = []
+    swaps = []
+    b = []
+    x = []
+    ddm_ilu_solve(x, L, U, swaps, b)
+    assert x == []
+
+
+def test_ddm_charpoly():
+    A = []
+    assert ddm_berk(A, ZZ) == [[ZZ(1)]]
+
+    A = [[ZZ(1), ZZ(2), ZZ(3)], [ZZ(4), ZZ(5), ZZ(6)], [ZZ(7), ZZ(8), ZZ(9)]]
+    Avec = [[ZZ(1)], [ZZ(-15)], [ZZ(-18)], [ZZ(0)]]
+    assert ddm_berk(A, ZZ) == Avec
+
+    A = DDM([[ZZ(1), ZZ(2)]], (1, 2), ZZ)
+    raises(DMShapeError, lambda: ddm_berk(A, ZZ))

rl/Lib/site-packages/sympy/polys/matrices/tests/test_domainscalar.py

@@ -0,0 +1,153 @@
+from sympy.testing.pytest import raises
+
+from sympy.core.symbol import S
+from sympy.polys import ZZ, QQ
+from sympy.polys.matrices.domainscalar import DomainScalar
+from sympy.polys.matrices.domainmatrix import DomainMatrix
+
+
+def test_DomainScalar___new__():
+    raises(TypeError, lambda: DomainScalar(ZZ(1), QQ))
+    raises(TypeError, lambda: DomainScalar(ZZ(1), 1))
+
+
+def test_DomainScalar_new():
+    A = DomainScalar(ZZ(1), ZZ)
+    B = A.new(ZZ(4), ZZ)
+    assert B == DomainScalar(ZZ(4), ZZ)
+
+
+def test_DomainScalar_repr():
+    A = DomainScalar(ZZ(1), ZZ)
+    assert repr(A) in {'1', 'mpz(1)'}
+
+
+def test_DomainScalar_from_sympy():
+    expr = S(1)
+    B = DomainScalar.from_sympy(expr)
+    assert B == DomainScalar(ZZ(1), ZZ)
+
+
+def test_DomainScalar_to_sympy():
+    B = DomainScalar(ZZ(1), ZZ)
+    expr = B.to_sympy()
+    assert expr.is_Integer and expr == 1
+
+
+def test_DomainScalar_to_domain():
+    A = DomainScalar(ZZ(1), ZZ)
+    B = A.to_domain(QQ)
+    assert B == DomainScalar(QQ(1), QQ)
+
+
+def test_DomainScalar_convert_to():
+    A = DomainScalar(ZZ(1), ZZ)
+    B = A.convert_to(QQ)
+    assert B == DomainScalar(QQ(1), QQ)
+
+
+def test_DomainScalar_unify():
+    A = DomainScalar(ZZ(1), ZZ)
+    B = DomainScalar(QQ(2), QQ)
+    A, B = A.unify(B)
+    assert A.domain == B.domain == QQ
+
+
+def test_DomainScalar_add():
+    A = DomainScalar(ZZ(1), ZZ)
+    B = DomainScalar(QQ(2), QQ)
+    assert A + B == DomainScalar(QQ(3), QQ)
+
+    raises(TypeError, lambda: A + 1.5)
+
+def test_DomainScalar_sub():
+    A = DomainScalar(ZZ(1), ZZ)
+    B = DomainScalar(QQ(2), QQ)
+    assert A - B == DomainScalar(QQ(-1), QQ)
+
+    raises(TypeError, lambda: A - 1.5)
+
+def test_DomainScalar_mul():
+    A = DomainScalar(ZZ(1), ZZ)
+    B = DomainScalar(QQ(2), QQ)
+    dm = DomainMatrix([[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]], (2, 2), ZZ)
+    assert A * B == DomainScalar(QQ(2), QQ)
+    assert A * dm == dm
+    assert B * 2 == DomainScalar(QQ(4), QQ)
+
+    raises(TypeError, lambda: A * 1.5)
+
+
+def test_DomainScalar_floordiv():
+    A = DomainScalar(ZZ(-5), ZZ)
+    B = DomainScalar(QQ(2), QQ)
+    assert A // B == DomainScalar(QQ(-5, 2), QQ)
+    C = DomainScalar(ZZ(2), ZZ)
+    assert A // C == DomainScalar(ZZ(-3), ZZ)
+
+    raises(TypeError, lambda: A // 1.5)
+
+
+def test_DomainScalar_mod():
+    A = DomainScalar(ZZ(5), ZZ)
+    B = DomainScalar(QQ(2), QQ)
+    assert A % B == DomainScalar(QQ(0), QQ)
+    C = DomainScalar(ZZ(2), ZZ)
+    assert A % C == DomainScalar(ZZ(1), ZZ)
+
+    raises(TypeError, lambda: A % 1.5)
+
+
+def test_DomainScalar_divmod():
+    A = DomainScalar(ZZ(5), ZZ)
+    B = DomainScalar(QQ(2), QQ)
+    assert divmod(A, B) == (DomainScalar(QQ(5, 2), QQ), DomainScalar(QQ(0), QQ))
+    C = DomainScalar(ZZ(2), ZZ)
+    assert divmod(A, C) == (DomainScalar(ZZ(2), ZZ), DomainScalar(ZZ(1), ZZ))
+
+    raises(TypeError, lambda: divmod(A, 1.5))
+
+
+def test_DomainScalar_pow():
+    A = DomainScalar(ZZ(-5), ZZ)
+    B = A**(2)
+    assert B == DomainScalar(ZZ(25), ZZ)
+
+    raises(TypeError, lambda: A**(1.5))
+
+
+def test_DomainScalar_pos():
+    A = DomainScalar(QQ(2), QQ)
+    B = DomainScalar(QQ(2), QQ)
+    assert  +A == B
+
+
+def test_DomainScalar_neg():
+    A = DomainScalar(QQ(2), QQ)
+    B = DomainScalar(QQ(-2), QQ)
+    assert  -A == B
+
+
+def test_DomainScalar_eq():
+    A = DomainScalar(QQ(2), QQ)
+    assert A == A
+    B = DomainScalar(ZZ(-5), ZZ)
+    assert A != B
+    C = DomainScalar(ZZ(2), ZZ)
+    assert A != C
+    D = [1]
+    assert A != D
+
+
+def test_DomainScalar_isZero():
+    A = DomainScalar(ZZ(0), ZZ)
+    assert A.is_zero() == True
+    B = DomainScalar(ZZ(1), ZZ)
+    assert B.is_zero() == False
+
+
+def test_DomainScalar_isOne():
+    A = DomainScalar(ZZ(1), ZZ)
+    assert A.is_one() == True
+    B = DomainScalar(ZZ(0), ZZ)
+    assert B.is_one() == False

rl/Lib/site-packages/sympy/polys/matrices/tests/test_eigen.py

@@ -0,0 +1,90 @@
+"""
+Tests for the sympy.polys.matrices.eigen module
+"""
+
+from sympy.core.singleton import S
+from sympy.functions.elementary.miscellaneous import sqrt
+from sympy.matrices.dense import Matrix
+
+from sympy.polys.agca.extensions import FiniteExtension
+from sympy.polys.domains import QQ
+from sympy.polys.polytools import Poly
+from sympy.polys.rootoftools import CRootOf
+from sympy.polys.matrices.domainmatrix import DomainMatrix
+
+from sympy.polys.matrices.eigen import dom_eigenvects, dom_eigenvects_to_sympy
+
+
+def test_dom_eigenvects_rational():
+    # Rational eigenvalues
+    A = DomainMatrix([[QQ(1), QQ(2)], [QQ(1), QQ(2)]], (2, 2), QQ)
+    rational_eigenvects = [
+        (QQ, QQ(3), 1, DomainMatrix([[QQ(1), QQ(1)]], (1, 2), QQ)),
+        (QQ, QQ(0), 1, DomainMatrix([[QQ(-2), QQ(1)]], (1, 2), QQ)),
+    ]
+    assert dom_eigenvects(A) == (rational_eigenvects, [])
+
+    # Test converting to Expr:
+    sympy_eigenvects = [
+        (S(3), 1, [Matrix([1, 1])]),
+        (S(0), 1, [Matrix([-2, 1])]),
+    ]
+    assert dom_eigenvects_to_sympy(rational_eigenvects, [], Matrix) == sympy_eigenvects
+
+
+def test_dom_eigenvects_algebraic():
+    # Algebraic eigenvalues
+    A = DomainMatrix([[QQ(1), QQ(2)], [QQ(3), QQ(4)]], (2, 2), QQ)
+    Avects = dom_eigenvects(A)
+
+    # Extract the dummy to build the expected result:
+    lamda = Avects[1][0][1].gens[0]
+    irreducible = Poly(lamda**2 - 5*lamda - 2, lamda, domain=QQ)
+    K = FiniteExtension(irreducible)
+    KK = K.from_sympy
+    algebraic_eigenvects = [
+        (K, irreducible, 1, DomainMatrix([[KK((lamda-4)/3), KK(1)]], (1, 2), K)),
+    ]
+    assert Avects == ([], algebraic_eigenvects)
+
+    # Test converting to Expr:
+    sympy_eigenvects = [
+        (S(5)/2 - sqrt(33)/2, 1, [Matrix([[-sqrt(33)/6 - S(1)/2], [1]])]),
+        (S(5)/2 + sqrt(33)/2, 1, [Matrix([[-S(1)/2 + sqrt(33)/6], [1]])]),
+    ]
+    assert dom_eigenvects_to_sympy([], algebraic_eigenvects, Matrix) == sympy_eigenvects
+
+
+def test_dom_eigenvects_rootof():
+    # Algebraic eigenvalues
+    A = DomainMatrix([
+        [0, 0, 0, 0, -1],
+        [1, 0, 0, 0, 1],
+        [0, 1, 0, 0, 0],
+        [0, 0, 1, 0, 0],
+        [0, 0, 0, 1, 0]], (5, 5), QQ)
+    Avects = dom_eigenvects(A)
+
+    # Extract the dummy to build the expected result:
+    lamda = Avects[1][0][1].gens[0]
+    irreducible = Poly(lamda**5 - lamda + 1, lamda, domain=QQ)
+    K = FiniteExtension(irreducible)
+    KK = K.from_sympy
+    algebraic_eigenvects = [
+        (K, irreducible, 1,
+            DomainMatrix([
+                [KK(lamda**4-1), KK(lamda**3), KK(lamda**2), KK(lamda), KK(1)]
+            ], (1, 5), K)),
+    ]
+    assert Avects == ([], algebraic_eigenvects)
+
+    # Test converting to Expr (slow):
+    l0, l1, l2, l3, l4 = [CRootOf(lamda**5 - lamda + 1, i) for i in range(5)]
+    sympy_eigenvects = [
+        (l0, 1, [Matrix([-1 + l0**4, l0**3, l0**2, l0, 1])]),
+        (l1, 1, [Matrix([-1 + l1**4, l1**3, l1**2, l1, 1])]),
+        (l2, 1, [Matrix([-1 + l2**4, l2**3, l2**2, l2, 1])]),
+        (l3, 1, [Matrix([-1 + l3**4, l3**3, l3**2, l3, 1])]),
+        (l4, 1, [Matrix([-1 + l4**4, l4**3, l4**2, l4, 1])]),
+    ]
+    assert dom_eigenvects_to_sympy([], algebraic_eigenvects, Matrix) == sympy_eigenvects

rl/Lib/site-packages/sympy/polys/matrices/tests/test_inverse.py

@@ -0,0 +1,193 @@
+from sympy import ZZ, Matrix
+from sympy.polys.matrices import DM, DomainMatrix
+from sympy.polys.matrices.dense import ddm_iinv
+from sympy.polys.matrices.exceptions import DMNonInvertibleMatrixError
+from sympy.matrices.exceptions import NonInvertibleMatrixError
+
+import pytest
+from sympy.testing.pytest import raises
+from sympy.core.numbers import all_close
+
+from sympy.abc import x
+
+
+# Examples are given as adjugate matrix and determinant adj_det should match
+# these exactly but inv_den only matches after cancel_denom.
+
+
+INVERSE_EXAMPLES = [
+
+    (
+        'zz_1',
+        DomainMatrix([], (0, 0), ZZ),
+        DomainMatrix([], (0, 0), ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_2',
+        DM([[2]], ZZ),
+        DM([[1]], ZZ),
+        ZZ(2),
+    ),
+
+    (
+        'zz_3',
+        DM([[2, 0],
+            [0, 2]], ZZ),
+        DM([[2, 0],
+            [0, 2]], ZZ),
+        ZZ(4),
+    ),
+
+    (
+        'zz_4',
+        DM([[1, 2],
+            [3, 4]], ZZ),
+        DM([[ 4, -2],
+            [-3,  1]], ZZ),
+        ZZ(-2),
+    ),
+
+    (
+        'zz_5',
+        DM([[2, 2, 0],
+            [0, 2, 2],
+            [0, 0, 2]], ZZ),
+        DM([[4, -4, 4],
+            [0, 4, -4],
+            [0, 0,  4]], ZZ),
+        ZZ(8),
+    ),
+
+    (
+        'zz_6',
+        DM([[1, 2, 3],
+            [4, 5, 6],
+            [7, 8, 9]], ZZ),
+        DM([[-3,   6, -3],
+            [ 6, -12,  6],
+            [-3,   6, -3]], ZZ),
+        ZZ(0),
+    ),
+]
+
+
+@pytest.mark.parametrize('name, A, A_inv, den', INVERSE_EXAMPLES)
+def test_Matrix_inv(name, A, A_inv, den):
+
+    def _check(**kwargs):
+        if den != 0:
+            assert A.inv(**kwargs) == A_inv
+        else:
+            raises(NonInvertibleMatrixError, lambda: A.inv(**kwargs))
+
+    K = A.domain
+    A = A.to_Matrix()
+    A_inv = A_inv.to_Matrix() / K.to_sympy(den)
+    _check()
+    for method in ['GE', 'LU', 'ADJ', 'CH', 'LDL', 'QR']:
+        _check(method=method)
+
+
+@pytest.mark.parametrize('name, A, A_inv, den', INVERSE_EXAMPLES)
+def test_dm_inv_den(name, A, A_inv, den):
+    if den != 0:
+        A_inv_f, den_f = A.inv_den()
+        assert A_inv_f.cancel_denom(den_f) == A_inv.cancel_denom(den)
+    else:
+        raises(DMNonInvertibleMatrixError, lambda: A.inv_den())
+
+
+@pytest.mark.parametrize('name, A, A_inv, den', INVERSE_EXAMPLES)
+def test_dm_inv(name, A, A_inv, den):
+    A = A.to_field()
+    if den != 0:
+        A_inv = A_inv.to_field() / den
+        assert A.inv() == A_inv
+    else:
+        raises(DMNonInvertibleMatrixError, lambda: A.inv())
+
+
+@pytest.mark.parametrize('name, A, A_inv, den', INVERSE_EXAMPLES)
+def test_ddm_inv(name, A, A_inv, den):
+    A = A.to_field().to_ddm()
+    if den != 0:
+        A_inv = (A_inv.to_field() / den).to_ddm()
+        assert A.inv() == A_inv
+    else:
+        raises(DMNonInvertibleMatrixError, lambda: A.inv())
+
+
+@pytest.mark.parametrize('name, A, A_inv, den', INVERSE_EXAMPLES)
+def test_sdm_inv(name, A, A_inv, den):
+    A = A.to_field().to_sdm()
+    if den != 0:
+        A_inv = (A_inv.to_field() / den).to_sdm()
+        assert A.inv() == A_inv
+    else:
+        raises(DMNonInvertibleMatrixError, lambda: A.inv())
+
+
+@pytest.mark.parametrize('name, A, A_inv, den', INVERSE_EXAMPLES)
+def test_dense_ddm_iinv(name, A, A_inv, den):
+    A = A.to_field().to_ddm().copy()
+    K = A.domain
+    A_result = A.copy()
+    if den != 0:
+        A_inv = (A_inv.to_field() / den).to_ddm()
+        ddm_iinv(A_result, A, K)
+        assert A_result == A_inv
+    else:
+        raises(DMNonInvertibleMatrixError, lambda: ddm_iinv(A_result, A, K))
+
+
+@pytest.mark.parametrize('name, A, A_inv, den', INVERSE_EXAMPLES)
+def test_Matrix_adjugate(name, A, A_inv, den):
+    A = A.to_Matrix()
+    A_inv = A_inv.to_Matrix()
+    assert A.adjugate() == A_inv
+    for method in ["bareiss", "berkowitz", "bird", "laplace", "lu"]:
+        assert A.adjugate(method=method) == A_inv
+
+
+@pytest.mark.parametrize('name, A, A_inv, den', INVERSE_EXAMPLES)
+def test_dm_adj_det(name, A, A_inv, den):
+    assert A.adj_det() == (A_inv, den)
+
+
+def test_inverse_inexact():
+
+    M = Matrix([[x-0.3, -0.06, -0.22],
+                [-0.46, x-0.48, -0.41],
+                [-0.14, -0.39, x-0.64]])
+
+    Mn = Matrix([[1.0*x**2 - 1.12*x + 0.1473, 0.06*x + 0.0474, 0.22*x - 0.081],
+                 [0.46*x - 0.237, 1.0*x**2 - 0.94*x + 0.1612, 0.41*x - 0.0218],
+                 [0.14*x + 0.1122, 0.39*x - 0.1086, 1.0*x**2 - 0.78*x + 0.1164]])
+
+    d = 1.0*x**3 - 1.42*x**2 + 0.4249*x - 0.0546540000000002
+
+    Mi = Mn / d
+
+    M_dm = M.to_DM()
+    M_dmd = M_dm.to_dense()
+    M_dm_num, M_dm_den = M_dm.inv_den()
+    M_dmd_num, M_dmd_den = M_dmd.inv_den()
+
+    # XXX: We don't check M_dm().to_field().inv() which currently uses division
+    # and produces a more complicate result from gcd cancellation failing.
+    # DomainMatrix.inv() over RR(x) should be changed to clear denominators and
+    # use DomainMatrix.inv_den().
+
+    Minvs = [
+        M.inv(),
+        (M_dm_num.to_field() / M_dm_den).to_Matrix(),
+        (M_dmd_num.to_field() / M_dmd_den).to_Matrix(),
+        M_dm_num.to_Matrix() / M_dm_den.as_expr(),
+        M_dmd_num.to_Matrix() / M_dmd_den.as_expr(),
+    ]
+
+    for Minv in Minvs:
+        for Mi1, Mi2 in zip(Minv.flat(), Mi.flat()):
+            assert all_close(Mi2, Mi1)

rl/Lib/site-packages/sympy/polys/matrices/tests/test_linsolve.py

@@ -0,0 +1,111 @@
+#
+# 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():
+    raises(PolyNonlinearError, lambda:
+        _linsolve([Eq(x**2, x**2 + y)], [x, y]))
+    raises(PolyNonlinearError, lambda:
+        _linsolve([(x + y)**2 - x**2], [x]))
+    raises(PolyNonlinearError, lambda:
+        _linsolve([Eq((x + y)**2, x**2)], [x]))

rl/Lib/site-packages/sympy/polys/matrices/tests/test_lll.py

@@ -0,0 +1,145 @@
+from sympy.polys.domains import ZZ, QQ
+from sympy.polys.matrices import DM
+from sympy.polys.matrices.domainmatrix import DomainMatrix
+from sympy.polys.matrices.exceptions import DMRankError, DMValueError, DMShapeError, DMDomainError
+from sympy.polys.matrices.lll import _ddm_lll, ddm_lll, ddm_lll_transform
+from sympy.testing.pytest import raises
+
+
+def test_lll():
+    normal_test_data = [
+        (
+            DM([[1, 0, 0, 0, -20160],
+                [0, 1, 0, 0, 33768],
+                [0, 0, 1, 0, 39578],
+                [0, 0, 0, 1, 47757]], ZZ),
+            DM([[10, -3, -2, 8, -4],
+                [3, -9, 8, 1, -11],
+                [-3, 13, -9, -3, -9],
+                [-12, -7, -11, 9, -1]], ZZ)
+        ),
+        (
+            DM([[20, 52, 3456],
+                [14, 31, -1],
+                [34, -442, 0]], ZZ),
+            DM([[14, 31, -1],
+                [188, -101, -11],
+                [236, 13, 3443]], ZZ)
+        ),
+        (
+            DM([[34, -1, -86, 12],
+                [-54, 34, 55, 678],
+                [23, 3498, 234, 6783],
+                [87, 49, 665, 11]], ZZ),
+            DM([[34, -1, -86, 12],
+                [291, 43, 149, 83],
+                [-54, 34, 55, 678],
+                [-189, 3077, -184, -223]], ZZ)
+        )
+    ]
+    delta = QQ(5, 6)
+    for basis_dm, reduced_dm in normal_test_data:
+        reduced = _ddm_lll(basis_dm.rep.to_ddm(), delta=delta)[0]
+        assert reduced == reduced_dm.rep.to_ddm()
+
+        reduced = ddm_lll(basis_dm.rep.to_ddm(), delta=delta)
+        assert reduced == reduced_dm.rep.to_ddm()
+
+        reduced, transform = _ddm_lll(basis_dm.rep.to_ddm(), delta=delta, return_transform=True)
+        assert reduced == reduced_dm.rep.to_ddm()
+        assert transform.matmul(basis_dm.rep.to_ddm()) == reduced_dm.rep.to_ddm()
+
+        reduced, transform = ddm_lll_transform(basis_dm.rep.to_ddm(), delta=delta)
+        assert reduced == reduced_dm.rep.to_ddm()
+        assert transform.matmul(basis_dm.rep.to_ddm()) == reduced_dm.rep.to_ddm()
+
+        reduced = basis_dm.rep.lll(delta=delta)
+        assert reduced == reduced_dm.rep
+
+        reduced, transform = basis_dm.rep.lll_transform(delta=delta)
+        assert reduced == reduced_dm.rep
+        assert transform.matmul(basis_dm.rep) == reduced_dm.rep
+
+        reduced = basis_dm.rep.to_sdm().lll(delta=delta)
+        assert reduced == reduced_dm.rep.to_sdm()
+
+        reduced, transform = basis_dm.rep.to_sdm().lll_transform(delta=delta)
+        assert reduced == reduced_dm.rep.to_sdm()
+        assert transform.matmul(basis_dm.rep.to_sdm()) == reduced_dm.rep.to_sdm()
+
+        reduced = basis_dm.lll(delta=delta)
+        assert reduced == reduced_dm
+
+        reduced, transform = basis_dm.lll_transform(delta=delta)
+        assert reduced == reduced_dm
+        assert transform.matmul(basis_dm) == reduced_dm
+
+
+def test_lll_linear_dependent():
+    linear_dependent_test_data = [
+        DM([[0, -1, -2, -3],
+            [1, 0, -1, -2],
+            [2, 1, 0, -1],
+            [3, 2, 1, 0]], ZZ),
+        DM([[1, 0, 0, 1],
+            [0, 1, 0, 1],
+            [0, 0, 1, 1],
+            [1, 2, 3, 6]], ZZ),
+        DM([[3, -5, 1],
+            [4, 6, 0],
+            [10, -4, 2]], ZZ)
+    ]
+    for not_basis in linear_dependent_test_data:
+        raises(DMRankError, lambda: _ddm_lll(not_basis.rep.to_ddm()))
+        raises(DMRankError, lambda: ddm_lll(not_basis.rep.to_ddm()))
+        raises(DMRankError, lambda: not_basis.rep.lll())
+        raises(DMRankError, lambda: not_basis.rep.to_sdm().lll())
+        raises(DMRankError, lambda: not_basis.lll())
+        raises(DMRankError, lambda: _ddm_lll(not_basis.rep.to_ddm(), return_transform=True))
+        raises(DMRankError, lambda: ddm_lll_transform(not_basis.rep.to_ddm()))
+        raises(DMRankError, lambda: not_basis.rep.lll_transform())
+        raises(DMRankError, lambda: not_basis.rep.to_sdm().lll_transform())
+        raises(DMRankError, lambda: not_basis.lll_transform())
+
+
+def test_lll_wrong_delta():
+    dummy_matrix = DomainMatrix.ones((3, 3), ZZ)
+    for wrong_delta in [QQ(-1, 4), QQ(0, 1), QQ(1, 4), QQ(1, 1), QQ(100, 1)]:
+        raises(DMValueError, lambda: _ddm_lll(dummy_matrix.rep, delta=wrong_delta))
+        raises(DMValueError, lambda: ddm_lll(dummy_matrix.rep, delta=wrong_delta))
+        raises(DMValueError, lambda: dummy_matrix.rep.lll(delta=wrong_delta))
+        raises(DMValueError, lambda: dummy_matrix.rep.to_sdm().lll(delta=wrong_delta))
+        raises(DMValueError, lambda: dummy_matrix.lll(delta=wrong_delta))
+        raises(DMValueError, lambda: _ddm_lll(dummy_matrix.rep, delta=wrong_delta, return_transform=True))
+        raises(DMValueError, lambda: ddm_lll_transform(dummy_matrix.rep, delta=wrong_delta))
+        raises(DMValueError, lambda: dummy_matrix.rep.lll_transform(delta=wrong_delta))
+        raises(DMValueError, lambda: dummy_matrix.rep.to_sdm().lll_transform(delta=wrong_delta))
+        raises(DMValueError, lambda: dummy_matrix.lll_transform(delta=wrong_delta))
+
+
+def test_lll_wrong_shape():
+    wrong_shape_matrix = DomainMatrix.ones((4, 3), ZZ)
+    raises(DMShapeError, lambda: _ddm_lll(wrong_shape_matrix.rep))
+    raises(DMShapeError, lambda: ddm_lll(wrong_shape_matrix.rep))
+    raises(DMShapeError, lambda: wrong_shape_matrix.rep.lll())
+    raises(DMShapeError, lambda: wrong_shape_matrix.rep.to_sdm().lll())
+    raises(DMShapeError, lambda: wrong_shape_matrix.lll())
+    raises(DMShapeError, lambda: _ddm_lll(wrong_shape_matrix.rep, return_transform=True))
+    raises(DMShapeError, lambda: ddm_lll_transform(wrong_shape_matrix.rep))
+    raises(DMShapeError, lambda: wrong_shape_matrix.rep.lll_transform())
+    raises(DMShapeError, lambda: wrong_shape_matrix.rep.to_sdm().lll_transform())
+    raises(DMShapeError, lambda: wrong_shape_matrix.lll_transform())
+
+
+def test_lll_wrong_domain():
+    wrong_domain_matrix = DomainMatrix.ones((3, 3), QQ)
+    raises(DMDomainError, lambda: _ddm_lll(wrong_domain_matrix.rep))
+    raises(DMDomainError, lambda: ddm_lll(wrong_domain_matrix.rep))
+    raises(DMDomainError, lambda: wrong_domain_matrix.rep.lll())
+    raises(DMDomainError, lambda: wrong_domain_matrix.rep.to_sdm().lll())
+    raises(DMDomainError, lambda: wrong_domain_matrix.lll())
+    raises(DMDomainError, lambda: _ddm_lll(wrong_domain_matrix.rep, return_transform=True))
+    raises(DMDomainError, lambda: ddm_lll_transform(wrong_domain_matrix.rep))
+    raises(DMDomainError, lambda: wrong_domain_matrix.rep.lll_transform())
+    raises(DMDomainError, lambda: wrong_domain_matrix.rep.to_sdm().lll_transform())
+    raises(DMDomainError, lambda: wrong_domain_matrix.lll_transform())

rl/Lib/site-packages/sympy/polys/matrices/tests/test_normalforms.py

@@ -0,0 +1,75 @@
+from sympy.testing.pytest import raises
+
+from sympy.core.symbol import Symbol
+from sympy.polys.matrices.normalforms import (
+    invariant_factors, smith_normal_form,
+    hermite_normal_form, _hermite_normal_form, _hermite_normal_form_modulo_D)
+from sympy.polys.domains import ZZ, QQ
+from sympy.polys.matrices import DomainMatrix, DM
+from sympy.polys.matrices.exceptions import DMDomainError, DMShapeError
+
+
+def test_smith_normal():
+
+    m = DM([[12, 6, 4, 8], [3, 9, 6, 12], [2, 16, 14, 28], [20, 10, 10, 20]], ZZ)
+    smf = DM([[1, 0, 0, 0], [0, 10, 0, 0], [0, 0, -30, 0], [0, 0, 0, 0]], ZZ)
+    assert smith_normal_form(m).to_dense() == smf
+
+    x = Symbol('x')
+    m = DM([[x-1,  1, -1],
+            [  0,  x, -1],
+            [  0, -1,  x]], QQ[x])
+    dx = m.domain.gens[0]
+    assert invariant_factors(m) == (1, dx-1, dx**2-1)
+
+    zr = DomainMatrix([], (0, 2), ZZ)
+    zc = DomainMatrix([[], []], (2, 0), ZZ)
+    assert smith_normal_form(zr).to_dense() == zr
+    assert smith_normal_form(zc).to_dense() == zc
+
+    assert smith_normal_form(DM([[2, 4]], ZZ)).to_dense() == DM([[2, 0]], ZZ)
+    assert smith_normal_form(DM([[0, -2]], ZZ)).to_dense() == DM([[-2, 0]], ZZ)
+    assert smith_normal_form(DM([[0], [-2]], ZZ)).to_dense() == DM([[-2], [0]], ZZ)
+
+    m =   DM([[3, 0, 0, 0], [0, 0, 0, 0], [0, 0, 2, 0]], ZZ)
+    snf = DM([[1, 0, 0, 0], [0, 6, 0, 0], [0, 0, 0, 0]], ZZ)
+    assert smith_normal_form(m).to_dense() == snf
+
+    raises(ValueError, lambda: smith_normal_form(DM([[1]], ZZ[x])))
+
+
+def test_hermite_normal():
+    m = DM([[2, 7, 17, 29, 41], [3, 11, 19, 31, 43], [5, 13, 23, 37, 47]], ZZ)
+    hnf = DM([[1, 0, 0], [0, 2, 1], [0, 0, 1]], ZZ)
+    assert hermite_normal_form(m) == hnf
+    assert hermite_normal_form(m, D=ZZ(2)) == hnf
+    assert hermite_normal_form(m, D=ZZ(2), check_rank=True) == hnf
+
+    m = m.transpose()
+    hnf = DM([[37, 0, 19], [222, -6, 113], [48, 0, 25], [0, 2, 1], [0, 0, 1]], ZZ)
+    assert hermite_normal_form(m) == hnf
+    raises(DMShapeError, lambda: _hermite_normal_form_modulo_D(m, ZZ(96)))
+    raises(DMDomainError, lambda: _hermite_normal_form_modulo_D(m, QQ(96)))
+
+    m = DM([[8, 28, 68, 116, 164], [3, 11, 19, 31, 43], [5, 13, 23, 37, 47]], ZZ)
+    hnf = DM([[4, 0, 0], [0, 2, 1], [0, 0, 1]], ZZ)
+    assert hermite_normal_form(m) == hnf
+    assert hermite_normal_form(m, D=ZZ(8)) == hnf
+    assert hermite_normal_form(m, D=ZZ(8), check_rank=True) == hnf
+
+    m = DM([[10, 8, 6, 30, 2], [45, 36, 27, 18, 9], [5, 4, 3, 2, 1]], ZZ)
+    hnf = DM([[26, 2], [0, 9], [0, 1]], ZZ)
+    assert hermite_normal_form(m) == hnf
+
+    m = DM([[2, 7], [0, 0], [0, 0]], ZZ)
+    hnf = DM([[1], [0], [0]], ZZ)
+    assert hermite_normal_form(m) == hnf
+
+    m = DM([[-2, 1], [0, 1]], ZZ)
+    hnf = DM([[2, 1], [0, 1]], ZZ)
+    assert hermite_normal_form(m) == hnf
+
+    m = DomainMatrix([[QQ(1)]], (1, 1), QQ)
+    raises(DMDomainError, lambda: hermite_normal_form(m))
+    raises(DMDomainError, lambda: _hermite_normal_form(m))
+    raises(DMDomainError, lambda: _hermite_normal_form_modulo_D(m, ZZ(1)))

rl/Lib/site-packages/sympy/polys/matrices/tests/test_nullspace.py

@@ -0,0 +1,209 @@
+from sympy import ZZ, Matrix
+from sympy.polys.matrices import DM, DomainMatrix
+from sympy.polys.matrices.ddm import DDM
+from sympy.polys.matrices.sdm import SDM
+
+import pytest
+
+zeros = lambda shape, K: DomainMatrix.zeros(shape, K).to_dense()
+eye = lambda n, K: DomainMatrix.eye(n, K).to_dense()
+
+
+#
+# DomainMatrix.nullspace can have a divided answer or can return an undivided
+# uncanonical answer. The uncanonical answer is not unique but we can make it
+# unique by making it primitive (remove gcd). The tests here all show the
+# primitive form. We test two things:
+#
+#   A.nullspace().primitive()[1] == answer.
+#   A.nullspace(divide_last=True) == _divide_last(answer).
+#
+# The nullspace as returned by DomainMatrix and related classes is the
+# transpose of the nullspace as returned by Matrix. Matrix returns a list of
+# of column vectors whereas DomainMatrix returns a matrix whose rows are the
+# nullspace vectors.
+#
+
+
+NULLSPACE_EXAMPLES = [
+
+    (
+        'zz_1',
+         DM([[ 1, 2, 3]], ZZ),
+         DM([[-2, 1, 0],
+             [-3, 0, 1]], ZZ),
+    ),
+
+    (
+        'zz_2',
+         zeros((0, 0), ZZ),
+         zeros((0, 0), ZZ),
+    ),
+
+    (
+        'zz_3',
+        zeros((2, 0), ZZ),
+        zeros((0, 0), ZZ),
+    ),
+
+    (
+        'zz_4',
+        zeros((0, 2), ZZ),
+        eye(2, ZZ),
+    ),
+
+    (
+        'zz_5',
+        zeros((2, 2), ZZ),
+        eye(2, ZZ),
+    ),
+
+    (
+        'zz_6',
+        DM([[1, 2],
+            [3, 4]], ZZ),
+        zeros((0, 2), ZZ),
+    ),
+
+    (
+        'zz_7',
+        DM([[1, 1],
+            [1, 1]], ZZ),
+        DM([[-1, 1]], ZZ),
+    ),
+
+    (
+        'zz_8',
+        DM([[1],
+            [1]], ZZ),
+        zeros((0, 1), ZZ),
+    ),
+
+    (
+        'zz_9',
+        DM([[1, 1]], ZZ),
+        DM([[-1, 1]], ZZ),
+    ),
+
+    (
+        'zz_10',
+        DM([[0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
+            [1, 0, 0, 0, 0, 0, 1, 0, 0, 0],
+            [0, 1, 0, 0, 0, 0, 0, 1, 0, 0],
+            [0, 0, 0, 1, 0, 0, 0, 0, 1, 0],
+            [0, 0, 0, 0, 1, 0, 0, 0, 0, 1]], ZZ),
+        DM([[ 0,  0, 1,  0,  0, 0, 0, 0, 0, 0],
+            [-1,  0, 0,  0,  0, 0, 1, 0, 0, 0],
+            [ 0, -1, 0,  0,  0, 0, 0, 1, 0, 0],
+            [ 0,  0, 0, -1,  0, 0, 0, 0, 1, 0],
+            [ 0,  0, 0,  0, -1, 0, 0, 0, 0, 1]], ZZ),
+    ),
+
+]
+
+
+def _to_DM(A, ans):
+    """Convert the answer to DomainMatrix."""
+    if isinstance(A, DomainMatrix):
+        return A.to_dense()
+    elif isinstance(A, DDM):
+        return DomainMatrix(list(A), A.shape, A.domain).to_dense()
+    elif isinstance(A, SDM):
+        return DomainMatrix(dict(A), A.shape, A.domain).to_dense()
+    else:
+        assert False # pragma: no cover
+
+
+def _divide_last(null):
+    """Normalize the nullspace by the rightmost non-zero entry."""
+    null = null.to_field()
+
+    if null.is_zero_matrix:
+        return null
+
+    rows = []
+    for i in range(null.shape[0]):
+        for j in reversed(range(null.shape[1])):
+            if null[i, j]:
+                rows.append(null[i, :] / null[i, j])
+                break
+        else:
+            assert False # pragma: no cover
+
+    return DomainMatrix.vstack(*rows)
+
+
+def _check_primitive(null, null_ans):
+    """Check that the primitive of the answer matches."""
+    null = _to_DM(null, null_ans)
+    cont, null_prim = null.primitive()
+    assert null_prim == null_ans
+
+
+def _check_divided(null, null_ans):
+    """Check the divided answer."""
+    null = _to_DM(null, null_ans)
+    null_ans_norm = _divide_last(null_ans)
+    assert null == null_ans_norm
+
+
+@pytest.mark.parametrize('name, A, A_null', NULLSPACE_EXAMPLES)
+def test_Matrix_nullspace(name, A, A_null):
+    A = A.to_Matrix()
+
+    A_null_cols = A.nullspace()
+
+    # We have to patch up the case where the nullspace is empty
+    if A_null_cols:
+        A_null_found = Matrix.hstack(*A_null_cols)
+    else:
+        A_null_found = Matrix.zeros(A.cols, 0)
+
+    A_null_found = A_null_found.to_DM().to_field().to_dense()
+
+    # The Matrix result is the transpose of DomainMatrix result.
+    A_null_found = A_null_found.transpose()
+
+    _check_divided(A_null_found, A_null)
+
+
+@pytest.mark.parametrize('name, A, A_null', NULLSPACE_EXAMPLES)
+def test_dm_dense_nullspace(name, A, A_null):
+    A = A.to_field().to_dense()
+    A_null_found = A.nullspace(divide_last=True)
+    _check_divided(A_null_found, A_null)
+
+
+@pytest.mark.parametrize('name, A, A_null', NULLSPACE_EXAMPLES)
+def test_dm_sparse_nullspace(name, A, A_null):
+    A = A.to_field().to_sparse()
+    A_null_found = A.nullspace(divide_last=True)
+    _check_divided(A_null_found, A_null)
+
+
+@pytest.mark.parametrize('name, A, A_null', NULLSPACE_EXAMPLES)
+def test_ddm_nullspace(name, A, A_null):
+    A = A.to_field().to_ddm()
+    A_null_found, _ = A.nullspace()
+    _check_divided(A_null_found, A_null)
+
+
+@pytest.mark.parametrize('name, A, A_null', NULLSPACE_EXAMPLES)
+def test_sdm_nullspace(name, A, A_null):
+    A = A.to_field().to_sdm()
+    A_null_found, _ = A.nullspace()
+    _check_divided(A_null_found, A_null)
+
+
+@pytest.mark.parametrize('name, A, A_null', NULLSPACE_EXAMPLES)
+def test_dm_dense_nullspace_fracfree(name, A, A_null):
+    A = A.to_dense()
+    A_null_found = A.nullspace()
+    _check_primitive(A_null_found, A_null)
+
+
+@pytest.mark.parametrize('name, A, A_null', NULLSPACE_EXAMPLES)
+def test_dm_sparse_nullspace_fracfree(name, A, A_null):
+    A = A.to_sparse()
+    A_null_found = A.nullspace()
+    _check_primitive(A_null_found, A_null)

rl/Lib/site-packages/sympy/polys/matrices/tests/test_rref.py

@@ -0,0 +1,737 @@
+from sympy import ZZ, QQ, ZZ_I, EX, Matrix, eye, zeros, symbols
+from sympy.polys.matrices import DM, DomainMatrix
+from sympy.polys.matrices.dense import ddm_irref_den, ddm_irref
+from sympy.polys.matrices.ddm import DDM
+from sympy.polys.matrices.sdm import SDM, sdm_irref, sdm_rref_den
+
+import pytest
+
+
+#
+# The dense and sparse implementations of rref_den are ddm_irref_den and
+# sdm_irref_den. These can give results that differ by some factor and also
+# give different results if the order of the rows is changed. The tests below
+# show all results on lowest terms as should be returned by cancel_denom.
+#
+# The EX domain is also a case where the dense and sparse implementations
+# can give results in different forms: the results should be equivalent but
+# are not canonical because EX does not have a canonical form.
+#
+
+
+a, b, c, d = symbols('a, b, c, d')
+
+
+qq_large_1 = DM([
+[  (1,2),  (1,3),  (1,5),  (1,7), (1,11), (1,13), (1,17), (1,19), (1,23), (1,29), (1,31)],
+[ (1,37), (1,41), (1,43), (1,47), (1,53), (1,59), (1,61), (1,67), (1,71), (1,73), (1,79)],
+[ (1,83), (1,89), (1,97),(1,101),(1,103),(1,107),(1,109),(1,113),(1,127),(1,131),(1,137)],
+[(1,139),(1,149),(1,151),(1,157),(1,163),(1,167),(1,173),(1,179),(1,181),(1,191),(1,193)],
+[(1,197),(1,199),(1,211),(1,223),(1,227),(1,229),(1,233),(1,239),(1,241),(1,251),(1,257)],
+[(1,263),(1,269),(1,271),(1,277),(1,281),(1,283),(1,293),(1,307),(1,311),(1,313),(1,317)],
+[(1,331),(1,337),(1,347),(1,349),(1,353),(1,359),(1,367),(1,373),(1,379),(1,383),(1,389)],
+[(1,397),(1,401),(1,409),(1,419),(1,421),(1,431),(1,433),(1,439),(1,443),(1,449),(1,457)],
+[(1,461),(1,463),(1,467),(1,479),(1,487),(1,491),(1,499),(1,503),(1,509),(1,521),(1,523)],
+[(1,541),(1,547),(1,557),(1,563),(1,569),(1,571),(1,577),(1,587),(1,593),(1,599),(1,601)],
+[(1,607),(1,613),(1,617),(1,619),(1,631),(1,641),(1,643),(1,647),(1,653),(1,659),(1,661)]],
+        QQ)
+
+qq_large_2 = qq_large_1 + 10**100 * DomainMatrix.eye(11, QQ)
+
+
+RREF_EXAMPLES = [
+    (
+        'zz_1',
+         DM([[1, 2, 3]], ZZ),
+         DM([[1, 2, 3]], ZZ),
+         ZZ(1),
+    ),
+
+    (
+        'zz_2',
+         DomainMatrix([], (0, 0), ZZ),
+         DomainMatrix([], (0, 0), ZZ),
+         ZZ(1),
+    ),
+
+    (
+        'zz_3',
+        DM([[1, 2],
+            [3, 4]], ZZ),
+        DM([[1, 0],
+            [0, 1]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_4',
+        DM([[1, 0],
+            [3, 4]], ZZ),
+        DM([[1, 0],
+            [0, 1]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_5',
+        DM([[0, 2],
+            [3, 4]], ZZ),
+        DM([[1, 0],
+            [0, 1]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_6',
+        DM([[1, 2, 3],
+            [4, 5, 6],
+            [7, 8, 9]], ZZ),
+        DM([[1, 0, -1],
+            [0, 1,  2],
+            [0, 0,  0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_7',
+        DM([[0, 0, 0],
+            [0, 0, 0],
+            [1, 0, 0]], ZZ),
+        DM([[1, 0, 0],
+            [0, 0, 0],
+            [0, 0, 0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_8',
+        DM([[0, 0, 0],
+            [0, 0, 0],
+            [0, 0, 0]], ZZ),
+        DM([[0, 0, 0],
+            [0, 0, 0],
+            [0, 0, 0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_9',
+        DM([[1, 1, 0],
+            [0, 0, 2],
+            [0, 0, 0]], ZZ),
+        DM([[1, 1, 0],
+            [0, 0, 1],
+            [0, 0, 0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_10',
+        DM([[2, 2, 0],
+            [0, 0, 2],
+            [0, 0, 0]], ZZ),
+        DM([[1, 1, 0],
+            [0, 0, 1],
+            [0, 0, 0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_11',
+        DM([[2, 2, 0],
+            [0, 2, 2],
+            [0, 0, 2]], ZZ),
+        DM([[1, 0, 0],
+            [0, 1, 0],
+            [0, 0, 1]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_12',
+        DM([[ 1,  2,  3],
+            [ 4,  5,  6],
+            [ 7,  8,  9],
+            [10, 11, 12]], ZZ),
+        DM([[1,  0, -1],
+            [0,  1,  2],
+            [0,  0,  0],
+            [0,  0,  0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_13',
+        DM([[ 1,  2,  3],
+            [ 4,  5,  6],
+            [ 7,  8,  9],
+            [10, 11, 13]], ZZ),
+        DM([[ 1,  0,  0],
+            [ 0,  1,  0],
+            [ 0,  0,  1],
+            [ 0,  0,  0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_14',
+        DM([[1, 2,  4, 3],
+            [4, 5, 10, 6],
+            [7, 8, 16, 9]], ZZ),
+        DM([[1, 0, 0, -1],
+            [0, 1, 2,  2],
+            [0, 0, 0,  0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_15',
+        DM([[1, 2,  4, 3],
+            [4, 5, 10, 6],
+            [7, 8, 17, 9]], ZZ),
+        DM([[1, 0, 0, -1],
+            [0, 1, 0,  2],
+            [0, 0, 1,  0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_16',
+        DM([[1, 2, 0, 1],
+            [1, 1, 9, 0]], ZZ),
+        DM([[1, 0, 18, -1],
+            [0, 1, -9,  1]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_17',
+        DM([[1, 1, 1],
+            [1, 2, 2]], ZZ),
+        DM([[1, 0, 0],
+            [0, 1, 1]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        # Here the sparse implementation and dense implementation give very
+        # different denominators: 4061232 and -1765176.
+        'zz_18',
+        DM([[94, 24,  0, 27, 0],
+            [79,  0,  0,  0, 0],
+            [85, 16, 71, 81, 0],
+            [ 0,  0, 72, 77, 0],
+            [21,  0, 34,  0, 0]], ZZ),
+        DM([[ 1,  0,  0,  0, 0],
+            [ 0,  1,  0,  0, 0],
+            [ 0,  0,  1,  0, 0],
+            [ 0,  0,  0,  1, 0],
+            [ 0,  0,  0,  0, 0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        # Let's have a denominator that cannot be cancelled.
+        'zz_19',
+        DM([[1, 2, 4],
+            [4, 5, 6]], ZZ),
+        DM([[3, 0, -8],
+            [0, 3, 10]], ZZ),
+        ZZ(3),
+    ),
+
+    (
+        'zz_20',
+        DM([[0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 4]], ZZ),
+        DM([[0, 0, 0, 0, 1],
+            [0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_21',
+        DM([[0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
+            [1, 0, 0, 0, 0, 0, 1, 0, 0, 0],
+            [0, 1, 0, 0, 0, 0, 0, 1, 0, 0],
+            [0, 0, 0, 1, 0, 0, 0, 0, 1, 0],
+            [0, 0, 0, 0, 1, 0, 0, 0, 0, 1]], ZZ),
+        DM([[1, 0, 0, 0, 0, 0, 1, 0, 0, 0],
+            [0, 1, 0, 0, 0, 0, 0, 1, 0, 0],
+            [0, 0, 0, 1, 0, 0, 0, 0, 1, 0],
+            [0, 0, 0, 0, 1, 0, 0, 0, 0, 1],
+            [0, 0, 0, 0, 0, 1, 0, 0, 0, 0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_22',
+        DM([[1, 1, 1, 0, 1],
+            [1, 1, 0, 1, 0],
+            [1, 0, 1, 0, 1],
+            [1, 1, 0, 1, 0],
+            [1, 0, 0, 0, 0]], ZZ),
+        DM([[1, 0, 0, 0, 0],
+            [0, 1, 0, 0, 0],
+            [0, 0, 1, 0, 1],
+            [0, 0, 0, 1, 0],
+            [0, 0, 0, 0, 0]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_large_1',
+        DM([
+[ 0,  0,  0, 81,  0,  0, 75,  0,  0,  0,  0,  0,  0, 27,  0,  0,  0,  0,  0,  0],
+[ 0,  0,  0,  0,  0, 86,  0, 92, 79, 54,  0,  7,  0,  0,  0,  0, 79,  0,  0,  0],
+[89, 54, 81,  0,  0, 20,  0,  0,  0,  0,  0,  0, 51,  0, 94,  0,  0, 77,  0,  0],
+[ 0,  0,  0, 96,  0,  0,  0,  0,  0,  0,  0,  0, 48, 29,  0,  0,  5,  0, 32,  0],
+[ 0, 70,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 60,  0,  0,  0, 11],
+[ 0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 37,  0, 43,  0,  0],
+[ 0,  0,  0,  0,  0, 38, 91,  0,  0,  0,  0, 38,  0,  0,  0,  0,  0, 26,  0,  0],
+[69,  0,  0,  0,  0,  0, 94,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 55],
+[ 0, 13, 18, 49, 49, 88,  0,  0, 35, 54,  0,  0, 51,  0,  0,  0,  0,  0,  0, 87],
+[ 0,  0,  0,  0, 31,  0, 40,  0,  0,  0,  0,  0,  0, 50,  0,  0,  0,  0, 88,  0],
+[ 0,  0,  0,  0,  0,  0,  0,  0, 98,  0,  0,  0, 15, 53,  0, 92,  0,  0,  0,  0],
+[ 0,  0,  0, 95,  0,  0,  0, 36,  0,  0,  0,  0,  0, 72,  0,  0,  0,  0, 73, 19],
+[ 0, 65, 14, 96,  0,  0,  0,  0,  0,  0,  0,  0,  0, 90,  0,  0,  0, 34,  0,  0],
+[ 0,  0,  0, 16, 39, 44,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 51,  0,  0],
+[ 0, 17,  0,  0,  0, 99, 84, 13, 50, 84,  0,  0,  0,  0, 95,  0, 43, 33, 20,  0],
+[79,  0, 17, 52, 99, 12, 69,  0, 98,  0, 68,  0,  0,  0,  0,  0,  0,  0,  0,  0],
+[ 0,  0,  0, 82,  0, 44,  0,  0,  0, 97,  0,  0,  0,  0,  0, 10,  0,  0, 31,  0],
+[ 0,  0, 21,  0, 67,  0,  0,  0,  0,  0,  4,  0, 50,  0,  0,  0, 33,  0,  0,  0],
+[ 0,  0,  0,  0,  9, 42,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  8],
+[ 0, 77,  0,  0,  0,  0,  0,  0,  0,  0, 34, 93,  0,  0,  0,  0, 47,  0,  0,  0]],
+           ZZ),
+        DM([[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
+            [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]], ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_large_2',
+        DM([
+[ 0,  0,  0,  0, 50,  0,  6, 81,  0,  1, 86,  0,  0, 98, 82, 94,  4,  0,  0, 29],
+[ 0, 44, 43,  0, 62,  0,  0,  0, 60,  0,  0,  0,  0, 71,  9,  0, 57, 41,  0, 93],
+[ 0,  0, 28,  0, 74, 89, 42,  0, 28,  0,  6,  0,  0,  0, 44,  0,  0,  0, 77, 19],
+[ 0, 21, 82,  0, 30, 88,  0, 89, 68,  0,  0,  0, 79, 41,  0,  0, 99,  0,  0,  0],
+[31,  0,  0,  0, 19, 64,  0,  0, 79,  0,  5,  0, 72, 10, 60, 32, 64, 59,  0, 24],
+[ 0,  0,  0,  0,  0, 57,  0, 94,  0, 83, 20,  0,  0,  9, 31,  0, 49, 26, 58,  0],
+[ 0, 65, 56, 31, 64,  0,  0,  0,  0,  0,  0, 52, 85,  0,  0,  0,  0, 51,  0,  0],
+[ 0, 35,  0,  0,  0, 69,  0,  0, 64,  0,  0,  0,  0, 70,  0,  0, 90,  0, 75, 76],
+[69,  7,  0, 90,  0,  0, 84,  0, 47, 69, 19, 20, 42,  0,  0, 32, 71, 35,  0,  0],
+[39,  0, 90,  0,  0,  4, 85,  0,  0, 55,  0,  0,  0, 35, 67, 40,  0, 40,  0, 77],
+[98, 63,  0, 71,  0, 50,  0,  2, 61,  0, 38,  0,  0,  0,  0, 75,  0, 40, 33, 56],
+[ 0, 73,  0, 64,  0, 38,  0, 35, 61,  0,  0, 52,  0,  7,  0, 51,  0,  0,  0, 34],
+[ 0,  0, 28,  0, 34,  5, 63, 45, 14, 42, 60, 16, 76, 54, 99,  0, 28, 30,  0,  0],
+[58, 37, 14,  0,  0,  0, 94,  0,  0, 90,  0,  0,  0,  0,  0,  0,  0,  8, 90, 53],
+[86, 74, 94,  0, 49, 10, 60,  0, 40, 18,  0,  0,  0, 31, 60, 24,  0,  1,  0, 29],
+[53,  0,  0, 97,  0,  0, 58,  0,  0, 39, 44, 47,  0,  0,  0, 12, 50,  0,  0, 11],
+[ 4,  0, 92, 10, 28,  0,  0, 89,  0,  0, 18, 54, 23, 39,  0,  2,  0, 48,  0, 92],
+[ 0,  0, 90, 77, 95, 33,  0,  0, 49, 22, 39,  0,  0,  0,  0,  0,  0, 40,  0,  0],
+[96,  0,  0,  0,  0, 38, 86,  0, 22, 76,  0,  0,  0,  0, 83, 88, 95, 65, 72,  0],
+[81, 65,  0,  4, 60,  0, 19,  0,  0, 68,  0,  0, 89,  0, 67, 22,  0,  0, 55, 33]],
+           ZZ),
+        DM([
+[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0],
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]],
+           ZZ),
+        ZZ(1),
+    ),
+
+    (
+        'zz_large_3',
+        DM([
+[62,35,89,58,22,47,30,28,52,72,17,56,80,26,64,21,10,35,24,42,96,32,23,50,92,37,76,94,63,66],
+[20,47,96,34,10,98,19,6,29,2,19,92,61,94,38,41,32,9,5,94,31,58,27,41,72,85,61,62,40,46],
+[69,26,35,68,25,52,94,13,38,65,81,10,29,15,5,4,13,99,85,0,80,51,60,60,26,77,85,2,87,25],
+[99,58,69,15,52,12,18,7,27,56,12,54,21,92,38,95,33,83,28,1,44,8,29,84,92,12,2,25,46,46],
+[93,13,55,48,35,87,24,40,23,35,25,32,0,19,0,85,4,79,26,11,46,75,7,96,76,11,7,57,99,75],
+[128,85,26,51,161,173,77,78,85,103,123,58,91,147,38,91,161,36,123,81,102,25,75,59,17,150,112,65,77,143],
+[15,59,61,82,12,83,34,8,94,71,66,7,91,21,48,69,26,12,64,38,97,87,38,15,51,33,93,43,66,89],
+[74,74,53,39,69,90,41,80,32,66,40,83,87,87,61,38,12,80,24,49,37,90,19,33,56,0,46,57,56,60],
+[82,11,0,25,56,58,39,49,92,93,80,38,19,62,33,85,19,61,14,30,45,91,97,34,97,53,92,28,33,43],
+[83,79,41,16,95,35,53,45,26,4,71,76,61,69,69,72,87,92,59,72,54,11,22,83,8,57,77,55,19,22],
+[49,34,13,31,72,77,52,70,46,41,37,6,42,66,35,6,75,33,62,57,30,14,26,31,9,95,89,13,12,90],
+[29,3,49,30,51,32,77,41,38,50,16,1,87,81,93,88,58,91,83,0,38,67,29,64,60,84,5,60,23,28],
+[79,51,13,20,89,96,25,8,39,62,86,52,49,81,3,85,86,3,61,24,72,11,49,28,8,55,23,52,65,53],
+[96,86,73,20,41,20,37,18,10,61,85,24,40,83,69,41,4,92,23,99,64,33,18,36,32,56,60,98,39,24],
+[32,62,47,80,51,66,17,1,9,30,65,75,75,88,99,92,64,53,53,86,38,51,41,14,35,18,39,25,26,32],
+[39,21,8,16,33,6,35,85,75,62,43,34,18,68,71,28,32,18,12,0,81,53,1,99,3,5,45,99,35,33],
+[19,95,89,45,75,94,92,5,84,93,34,17,50,56,79,98,68,82,65,81,51,90,5,95,33,71,46,61,14,7],
+[53,92,8,49,67,84,21,79,49,95,66,48,36,14,62,97,26,45,58,31,83,48,11,89,67,72,91,34,56,89],
+[56,76,99,92,40,8,0,16,15,48,35,72,91,46,81,14,86,60,51,7,33,12,53,78,48,21,3,89,15,79],
+[81,43,33,49,6,49,36,32,57,74,87,91,17,37,31,17,67,1,40,38,69,8,3,48,59,37,64,97,11,3],
+[98,48,77,16,2,48,57,38,63,59,79,35,16,71,60,86,71,41,14,76,80,97,77,69,4,58,22,55,26,73],
+[80,47,78,44,31,48,47,29,29,62,19,21,17,24,19,3,53,93,97,57,13,54,12,10,77,66,60,75,32,21],
+[86,63,2,13,71,38,86,23,18,15,91,65,77,65,9,92,50,0,17,42,99,80,99,27,10,99,92,9,87,84],
+[66,27,72,13,13,15,72,75,39,3,14,71,15,68,10,19,49,54,11,29,47,20,63,13,97,47,24,62,16,96],
+[42,63,83,60,49,68,9,53,75,87,40,25,12,63,0,12,0,95,46,46,55,25,89,1,51,1,1,96,80,52],
+[35,9,97,13,86,39,66,48,41,57,23,38,11,9,35,72,88,13,41,60,10,64,71,23,1,5,23,57,6,19],
+[70,61,5,50,72,60,77,13,41,94,1,45,52,22,99,47,27,18,99,42,16,48,26,9,88,77,10,94,11,92],
+[55,68,58,2,72,56,81,52,79,37,1,40,21,46,27,60,37,13,97,42,85,98,69,60,76,44,42,46,29,73],
+[73,0,43,17,89,97,45,2,68,14,55,60,95,2,74,85,88,68,93,76,38,76,2,51,45,76,50,79,56,18],
+[72,58,41,39,24,80,23,79,44,7,98,75,30,6,85,60,20,58,77,71,90,51,38,80,30,15,33,10,82,8]],
+            ZZ),
+        Matrix([
+    [eye(29) * 2028539767964472550625641331179545072876560857886207583101,
+     Matrix([ 4260575808093245475167216057435155595594339172099000182569,
+              169148395880755256182802335904188369274227936894862744452,
+              4915975976683942569102447281579134986891620721539038348914,
+              6113916866367364958834844982578214901958429746875633283248,
+              5585689617819894460378537031623265659753379011388162534838,
+              359776822829880747716695359574308645968094838905181892423,
+              -2800926112141776386671436511182421432449325232461665113305,
+              941642292388230001722444876624818265766384442910688463158,
+              3648811843256146649321864698600908938933015862008642023935,
+              -4104526163246702252932955226754097174212129127510547462419,
+              -704814955438106792441896903238080197619233342348191408078,
+              1640882266829725529929398131287244562048075707575030019335,
+              -4068330845192910563212155694231438198040299927120544468520,
+              136589038308366497790495711534532612862715724187671166593,
+              2544937011460702462290799932536905731142196510605191645593,
+              755591839174293940486133926192300657264122907519174116472,
+              -3683838489869297144348089243628436188645897133242795965021,
+              -522207137101161299969706310062775465103537953077871128403,
+              -2260451796032703984456606059649402832441331339246756656334,
+              -6476809325293587953616004856993300606040336446656916663680,
+              3521944238996782387785653800944972787867472610035040989081,
+              2270762115788407950241944504104975551914297395787473242379,
+              -3259947194628712441902262570532921252128444706733549251156,
+              -5624569821491886970999097239695637132075823246850431083557,
+              -3262698255682055804320585332902837076064075936601504555698,
+              5786719943788937667411185880136324396357603606944869545501,
+              -955257841973865996077323863289453200904051299086000660036,
+              -1294235552446355326174641248209752679127075717918392702116,
+              -3718353510747301598130831152458342785269166356215331448279,
+             ]),],
+        [zeros(1, 29), zeros(1, 1)],
+        ]).to_DM().to_dense(),
+        ZZ(2028539767964472550625641331179545072876560857886207583101),
+    ),
+
+
+    (
+        'qq_1',
+        DM([[(1,2), 0], [0, 2]], QQ),
+        DM([[1, 0], [0, 1]], QQ),
+        QQ(1),
+    ),
+
+    (
+        # Standard square case
+        'qq_2',
+        DM([[0, 1],
+            [1, 1]], QQ),
+        DM([[1, 0],
+            [0, 1]], QQ),
+        QQ(1),
+    ),
+
+    (
+        # m < n  case
+        'qq_3',
+        DM([[1, 2, 1],
+            [3, 4, 1]], QQ),
+        DM([[1, 0, -1],
+            [0, 1,  1]], QQ),
+        QQ(1),
+    ),
+
+    (
+        # same m < n  but reversed
+        'qq_4',
+        DM([[3, 4, 1],
+            [1, 2, 1]], QQ),
+        DM([[1, 0, -1],
+            [0, 1,  1]], QQ),
+        QQ(1),
+    ),
+
+    (
+        # m > n case
+        'qq_5',
+        DM([[1, 0],
+            [1, 3],
+            [0, 1]], QQ),
+        DM([[1, 0],
+            [0, 1],
+            [0, 0]], QQ),
+        QQ(1),
+    ),
+
+    (
+        # Example with missing pivot
+        'qq_6',
+        DM([[1, 0, 1],
+            [3, 0, 1]], QQ),
+        DM([[1, 0, 0],
+            [0, 0, 1]], QQ),
+        QQ(1),
+    ),
+
+    (
+        # This is intended to trigger the threshold where we give up on
+        # clearing denominators.
+        'qq_large_1',
+        qq_large_1,
+        DomainMatrix.eye(11, QQ).to_dense(),
+        QQ(1),
+    ),
+
+    (
+        # This is intended to trigger the threshold where we use rref_den over
+        # QQ.
+        'qq_large_2',
+        qq_large_2,
+        DomainMatrix.eye(11, QQ).to_dense(),
+        QQ(1),
+    ),
+
+    (
+        # Example with missing pivot and no replacement
+
+        # This example is just enough to show a different result from the dense
+        # and sparse versions of the algorithm:
+        #
+        #   >>> A = Matrix([[0, 1], [0, 2], [1, 0]])
+        #   >>> A.to_DM().to_sparse().rref_den()[0].to_Matrix()
+        #   Matrix([
+        #   [1, 0],
+        #   [0, 1],
+        #   [0, 0]])
+        #   >>> A.to_DM().to_dense().rref_den()[0].to_Matrix()
+        #   Matrix([
+        #   [2, 0],
+        #   [0, 2],
+        #   [0, 0]])
+        #
+        'qq_7',
+        DM([[0, 1],
+            [0, 2],
+            [1, 0]], QQ),
+        DM([[1, 0],
+            [0, 1],
+            [0, 0]], QQ),
+        QQ(1),
+    ),
+
+    (
+        # Gaussian integers
+        'zz_i_1',
+        DM([[(0,1), 1, 1],
+            [    1, 1, 1]], ZZ_I),
+        DM([[1, 0, 0],
+            [0, 1, 1]], ZZ_I),
+        ZZ_I(1),
+    ),
+
+    (
+        # EX: test_issue_23718
+        'EX_1',
+        DM([
+        [a, b, 1],
+        [c, d, 1]], EX),
+        DM([[a*d - b*c,         0, -b + d],
+            [        0, a*d - b*c,  a - c]], EX),
+        EX(a*d - b*c),
+    ),
+
+]
+
+
+def _to_DM(A, ans):
+    """Convert the answer to DomainMatrix."""
+    if isinstance(A, DomainMatrix):
+        return A.to_dense()
+    elif isinstance(A, Matrix):
+        return A.to_DM(ans.domain).to_dense()
+
+    if not (hasattr(A, 'shape') and hasattr(A, 'domain')):
+        shape, domain = ans.shape, ans.domain
+    else:
+        shape, domain = A.shape, A.domain
+
+    if isinstance(A, (DDM, list)):
+        return DomainMatrix(list(A), shape, domain).to_dense()
+    elif isinstance(A, (SDM, dict)):
+        return DomainMatrix(dict(A), shape, domain).to_dense()
+    else:
+        assert False # pragma: no cover
+
+
+def _pivots(A_rref):
+    """Return the pivots from the rref of A."""
+    return tuple(sorted(map(min, A_rref.to_sdm().values())))
+
+
+def _check_cancel(result, rref_ans, den_ans):
+    """Check the cancelled result."""
+    rref, den, pivots = result
+    if isinstance(rref, (DDM, SDM, list, dict)):
+        assert type(pivots) is list
+        pivots = tuple(pivots)
+    rref = _to_DM(rref, rref_ans)
+    rref2, den2 = rref.cancel_denom(den)
+    assert rref2 == rref_ans
+    assert den2 == den_ans
+    assert pivots == _pivots(rref)
+
+
+def _check_divide(result, rref_ans, den_ans):
+    """Check the divided result."""
+    rref, pivots = result
+    if isinstance(rref, (DDM, SDM, list, dict)):
+        assert type(pivots) is list
+        pivots = tuple(pivots)
+    rref_ans = rref_ans.to_field() / den_ans
+    rref = _to_DM(rref, rref_ans)
+    assert rref == rref_ans
+    assert _pivots(rref) == pivots
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_Matrix_rref(name, A, A_rref, den):
+    K = A.domain
+    A = A.to_Matrix()
+    A_rref_found, pivots = A.rref()
+    if K.is_EX:
+        A_rref_found = A_rref_found.expand()
+    _check_divide((A_rref_found, pivots), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_dm_dense_rref(name, A, A_rref, den):
+    A = A.to_field()
+    _check_divide(A.rref(), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_dm_dense_rref_den(name, A, A_rref, den):
+    _check_cancel(A.rref_den(), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_dm_sparse_rref(name, A, A_rref, den):
+    A = A.to_field().to_sparse()
+    _check_divide(A.rref(), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_dm_sparse_rref_den(name, A, A_rref, den):
+    A = A.to_sparse()
+    _check_cancel(A.rref_den(), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_dm_sparse_rref_den_keep_domain(name, A, A_rref, den):
+    A = A.to_sparse()
+    A_rref_f, den_f, pivots_f = A.rref_den(keep_domain=False)
+    A_rref_f = A_rref_f.to_field() / den_f
+    _check_divide((A_rref_f, pivots_f), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_dm_sparse_rref_den_keep_domain_CD(name, A, A_rref, den):
+    A = A.to_sparse()
+    A_rref_f, den_f, pivots_f = A.rref_den(keep_domain=False, method='CD')
+    A_rref_f = A_rref_f.to_field() / den_f
+    _check_divide((A_rref_f, pivots_f), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_dm_sparse_rref_den_keep_domain_GJ(name, A, A_rref, den):
+    A = A.to_sparse()
+    A_rref_f, den_f, pivots_f = A.rref_den(keep_domain=False, method='GJ')
+    A_rref_f = A_rref_f.to_field() / den_f
+    _check_divide((A_rref_f, pivots_f), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_ddm_rref_den(name, A, A_rref, den):
+    A = A.to_ddm()
+    _check_cancel(A.rref_den(), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_sdm_rref_den(name, A, A_rref, den):
+    A = A.to_sdm()
+    _check_cancel(A.rref_den(), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_ddm_rref(name, A, A_rref, den):
+    A = A.to_field().to_ddm()
+    _check_divide(A.rref(), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_sdm_rref(name, A, A_rref, den):
+    A = A.to_field().to_sdm()
+    _check_divide(A.rref(), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_ddm_irref(name, A, A_rref, den):
+    A = A.to_field().to_ddm().copy()
+    pivots_found = ddm_irref(A)
+    _check_divide((A, pivots_found), A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_ddm_irref_den(name, A, A_rref, den):
+    A = A.to_ddm().copy()
+    (den_found, pivots_found) = ddm_irref_den(A, A.domain)
+    result = (A, den_found, pivots_found)
+    _check_cancel(result, A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_sparse_sdm_rref(name, A, A_rref, den):
+    A = A.to_field().to_sdm()
+    _check_divide(sdm_irref(A)[:2], A_rref, den)
+
+
+@pytest.mark.parametrize('name, A, A_rref, den', RREF_EXAMPLES)
+def test_sparse_sdm_rref_den(name, A, A_rref, den):
+    A = A.to_sdm().copy()
+    K = A.domain
+    _check_cancel(sdm_rref_den(A, K), A_rref, den)

rl/Lib/site-packages/sympy/polys/matrices/tests/test_sdm.py

@@ -0,0 +1,428 @@
+"""
+Tests for the basic functionality of the SDM class.
+"""
+
+from itertools import product
+
+from sympy.core.singleton import S
+from sympy.external.gmpy import GROUND_TYPES
+from sympy.testing.pytest import raises
+
+from sympy.polys.domains import QQ, ZZ, EXRAW
+from sympy.polys.matrices.sdm import SDM
+from sympy.polys.matrices.ddm import DDM
+from sympy.polys.matrices.exceptions import (DMBadInputError, DMDomainError,
+                                             DMShapeError)
+
+
+def test_SDM():
+    A = SDM({0:{0:ZZ(1)}}, (2, 2), ZZ)
+    assert A.domain == ZZ
+    assert A.shape == (2, 2)
+    assert dict(A) == {0:{0:ZZ(1)}}
+
+    raises(DMBadInputError, lambda: SDM({5:{1:ZZ(0)}}, (2, 2), ZZ))
+    raises(DMBadInputError, lambda: SDM({0:{5:ZZ(0)}}, (2, 2), ZZ))
+
+
+def test_DDM_str():
+    sdm = SDM({0:{0:ZZ(1)}, 1:{1:ZZ(1)}}, (2, 2), ZZ)
+    assert str(sdm) == '{0: {0: 1}, 1: {1: 1}}'
+    if GROUND_TYPES == 'gmpy': # pragma: no cover
+        assert repr(sdm) == 'SDM({0: {0: mpz(1)}, 1: {1: mpz(1)}}, (2, 2), ZZ)'
+    else:        # pragma: no cover
+        assert repr(sdm) == 'SDM({0: {0: 1}, 1: {1: 1}}, (2, 2), ZZ)'
+
+
+def test_SDM_new():
+    A = SDM({0:{0:ZZ(1)}}, (2, 2), ZZ)
+    B = A.new({}, (2, 2), ZZ)
+    assert B == SDM({}, (2, 2), ZZ)
+
+
+def test_SDM_copy():
+    A = SDM({0:{0:ZZ(1)}}, (2, 2), ZZ)
+    B = A.copy()
+    assert A == B
+    A[0][0] = ZZ(2)
+    assert A != B
+
+
+def test_SDM_from_list():
+    A = SDM.from_list([[ZZ(0), ZZ(1)], [ZZ(1), ZZ(0)]], (2, 2), ZZ)
+    assert A == SDM({0:{1:ZZ(1)}, 1:{0:ZZ(1)}}, (2, 2), ZZ)
+
+    raises(DMBadInputError, lambda: SDM.from_list([[ZZ(0)], [ZZ(0), ZZ(1)]], (2, 2), ZZ))
+    raises(DMBadInputError, lambda: SDM.from_list([[ZZ(0), ZZ(1)]], (2, 2), ZZ))
+
+
+def test_SDM_to_list():
+    A = SDM({0:{1: ZZ(1)}}, (2, 2), ZZ)
+    assert A.to_list() == [[ZZ(0), ZZ(1)], [ZZ(0), ZZ(0)]]
+
+    A = SDM({}, (0, 2), ZZ)
+    assert A.to_list() == []
+
+    A = SDM({}, (2, 0), ZZ)
+    assert A.to_list() == [[], []]
+
+
+def test_SDM_to_list_flat():
+    A = SDM({0:{1: ZZ(1)}}, (2, 2), ZZ)
+    assert A.to_list_flat() == [ZZ(0), ZZ(1), ZZ(0), ZZ(0)]
+
+
+def test_SDM_to_dok():
+    A = SDM({0:{1: ZZ(1)}}, (2, 2), ZZ)
+    assert A.to_dok() == {(0, 1): ZZ(1)}
+
+
+def test_SDM_from_ddm():
+    A = DDM([[ZZ(1), ZZ(0)], [ZZ(1), ZZ(0)]], (2, 2), ZZ)
+    B = SDM.from_ddm(A)
+    assert B.domain == ZZ
+    assert B.shape == (2, 2)
+    assert dict(B) == {0:{0:ZZ(1)}, 1:{0:ZZ(1)}}
+
+
+def test_SDM_to_ddm():
+    A = SDM({0:{1: ZZ(1)}}, (2, 2), ZZ)
+    B = DDM([[ZZ(0), ZZ(1)], [ZZ(0), ZZ(0)]], (2, 2), ZZ)
+    assert A.to_ddm() == B
+
+
+def test_SDM_to_sdm():
+    A = SDM({0:{1: ZZ(1)}}, (2, 2), ZZ)
+    assert A.to_sdm() == A
+
+
+def test_SDM_getitem():
+    A = SDM({0:{1:ZZ(1)}}, (2, 2), ZZ)
+    assert A.getitem(0, 0) == ZZ.zero
+    assert A.getitem(0, 1) == ZZ.one
+    assert A.getitem(1, 0) == ZZ.zero
+    assert A.getitem(-2, -2) == ZZ.zero
+    assert A.getitem(-2, -1) == ZZ.one
+    assert A.getitem(-1, -2) == ZZ.zero
+    raises(IndexError, lambda: A.getitem(2, 0))
+    raises(IndexError, lambda: A.getitem(0, 2))
+
+
+def test_SDM_setitem():
+    A = SDM({0:{1:ZZ(1)}}, (2, 2), ZZ)
+    A.setitem(0, 0, ZZ(1))
+    assert A == SDM({0:{0:ZZ(1), 1:ZZ(1)}}, (2, 2), ZZ)
+    A.setitem(1, 0, ZZ(1))
+    assert A == SDM({0:{0:ZZ(1), 1:ZZ(1)}, 1:{0:ZZ(1)}}, (2, 2), ZZ)
+    A.setitem(1, 0, ZZ(0))
+    assert A == SDM({0:{0:ZZ(1), 1:ZZ(1)}}, (2, 2), ZZ)
+    # Repeat the above test so that this time the row is empty
+    A.setitem(1, 0, ZZ(0))
+    assert A == SDM({0:{0:ZZ(1), 1:ZZ(1)}}, (2, 2), ZZ)
+    A.setitem(0, 0, ZZ(0))
+    assert A == SDM({0:{1:ZZ(1)}}, (2, 2), ZZ)
+    # This time the row is there but column is empty
+    A.setitem(0, 0, ZZ(0))
+    assert A == SDM({0:{1:ZZ(1)}}, (2, 2), ZZ)
+    raises(IndexError, lambda: A.setitem(2, 0, ZZ(1)))
+    raises(IndexError, lambda: A.setitem(0, 2, ZZ(1)))
+
+
+def test_SDM_extract_slice():
+    A = SDM({0:{0:ZZ(1), 1:ZZ(2)}, 1:{0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
+    B = A.extract_slice(slice(1, 2), slice(1, 2))
+    assert B == SDM({0:{0:ZZ(4)}}, (1, 1), ZZ)
+
+
+def test_SDM_extract():
+    A = SDM({0:{0:ZZ(1), 1:ZZ(2)}, 1:{0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
+    B = A.extract([1], [1])
+    assert B == SDM({0:{0:ZZ(4)}}, (1, 1), ZZ)
+    B = A.extract([1, 0], [1, 0])
+    assert B == SDM({0:{0:ZZ(4), 1:ZZ(3)}, 1:{0:ZZ(2), 1:ZZ(1)}}, (2, 2), ZZ)
+    B = A.extract([1, 1], [1, 1])
+    assert B == SDM({0:{0:ZZ(4), 1:ZZ(4)}, 1:{0:ZZ(4), 1:ZZ(4)}}, (2, 2), ZZ)
+    B = A.extract([-1], [-1])
+    assert B == SDM({0:{0:ZZ(4)}}, (1, 1), ZZ)
+
+    A = SDM({}, (2, 2), ZZ)
+    B = A.extract([0, 1, 0], [0, 0])
+    assert B == SDM({}, (3, 2), ZZ)
+
+    A = SDM({0:{0:ZZ(1), 1:ZZ(2)}, 1:{0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
+    assert A.extract([], []) == SDM.zeros((0, 0), ZZ)
+    assert A.extract([1], []) == SDM.zeros((1, 0), ZZ)
+    assert A.extract([], [1]) == SDM.zeros((0, 1), ZZ)
+
+    raises(IndexError, lambda: A.extract([2], [0]))
+    raises(IndexError, lambda: A.extract([0], [2]))
+    raises(IndexError, lambda: A.extract([-3], [0]))
+    raises(IndexError, lambda: A.extract([0], [-3]))
+
+
+def test_SDM_zeros():
+    A = SDM.zeros((2, 2), ZZ)
+    assert A.domain == ZZ
+    assert A.shape == (2, 2)
+    assert dict(A) == {}
+
+def test_SDM_ones():
+    A = SDM.ones((1, 2), QQ)
+    assert A.domain == QQ
+    assert A.shape == (1, 2)
+    assert dict(A) == {0:{0:QQ(1), 1:QQ(1)}}
+
+def test_SDM_eye():
+    A = SDM.eye((2, 2), ZZ)
+    assert A.domain == ZZ
+    assert A.shape == (2, 2)
+    assert dict(A) == {0:{0:ZZ(1)}, 1:{1:ZZ(1)}}
+
+
+def test_SDM_diag():
+    A = SDM.diag([ZZ(1), ZZ(2)], ZZ, (2, 3))
+    assert A == SDM({0:{0:ZZ(1)}, 1:{1:ZZ(2)}}, (2, 3), ZZ)
+
+
+def test_SDM_transpose():
+    A = SDM({0:{0:ZZ(1), 1:ZZ(2)}, 1:{0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
+    B = SDM({0:{0:ZZ(1), 1:ZZ(3)}, 1:{0:ZZ(2), 1:ZZ(4)}}, (2, 2), ZZ)
+    assert A.transpose() == B
+
+    A = SDM({0:{1:ZZ(2)}}, (2, 2), ZZ)
+    B = SDM({1:{0:ZZ(2)}}, (2, 2), ZZ)
+    assert A.transpose() == B
+
+    A = SDM({0:{1:ZZ(2)}}, (1, 2), ZZ)
+    B = SDM({1:{0:ZZ(2)}}, (2, 1), ZZ)
+    assert A.transpose() == B
+
+
+def test_SDM_mul():
+    A = SDM({0:{0:ZZ(2)}}, (2, 2), ZZ)
+    B = SDM({0:{0:ZZ(4)}}, (2, 2), ZZ)
+    assert A*ZZ(2) == B
+    assert ZZ(2)*A == B
+
+    raises(TypeError, lambda: A*QQ(1, 2))
+    raises(TypeError, lambda: QQ(1, 2)*A)
+
+
+def test_SDM_mul_elementwise():
+    A = SDM({0:{0:ZZ(2), 1:ZZ(2)}}, (2, 2), ZZ)
+    B = SDM({0:{0:ZZ(4)}, 1:{0:ZZ(3)}}, (2, 2), ZZ)
+    C = SDM({0:{0:ZZ(8)}}, (2, 2), ZZ)
+    assert A.mul_elementwise(B) == C
+    assert B.mul_elementwise(A) == C
+
+    Aq = A.convert_to(QQ)
+    A1 = SDM({0:{0:ZZ(1)}}, (1, 1), ZZ)
+
+    raises(DMDomainError, lambda: Aq.mul_elementwise(B))
+    raises(DMShapeError, lambda: A1.mul_elementwise(B))
+
+
+def test_SDM_matmul():
+    A = SDM({0:{0:ZZ(2)}}, (2, 2), ZZ)
+    B = SDM({0:{0:ZZ(4)}}, (2, 2), ZZ)
+    assert A.matmul(A) == A*A == B
+
+    C = SDM({0:{0:ZZ(2)}}, (2, 2), QQ)
+    raises(DMDomainError, lambda: A.matmul(C))
+
+    A = SDM({0:{0:ZZ(1), 1:ZZ(2)}, 1:{0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
+    B = SDM({0:{0:ZZ(7), 1:ZZ(10)}, 1:{0:ZZ(15), 1:ZZ(22)}}, (2, 2), ZZ)
+    assert A.matmul(A) == A*A == B
+
+    A22 = SDM({0:{0:ZZ(4)}}, (2, 2), ZZ)
+    A32 = SDM({0:{0:ZZ(2)}}, (3, 2), ZZ)
+    A23 = SDM({0:{0:ZZ(4)}}, (2, 3), ZZ)
+    A33 = SDM({0:{0:ZZ(8)}}, (3, 3), ZZ)
+    A22 = SDM({0:{0:ZZ(8)}}, (2, 2), ZZ)
+    assert A32.matmul(A23) == A33
+    assert A23.matmul(A32) == A22
+    # XXX: @ not supported by SDM...
+    #assert A32.matmul(A23) == A32 @ A23 == A33
+    #assert A23.matmul(A32) == A23 @ A32 == A22
+    #raises(DMShapeError, lambda: A23 @ A22)
+    raises(DMShapeError, lambda: A23.matmul(A22))
+
+    A = SDM({0: {0: ZZ(-1), 1: ZZ(1)}}, (1, 2), ZZ)
+    B = SDM({0: {0: ZZ(-1)}, 1: {0: ZZ(-1)}}, (2, 1), ZZ)
+    assert A.matmul(B) == A*B == SDM({}, (1, 1), ZZ)
+
+
+def test_matmul_exraw():
+
+    def dm(d):
+        result = {}
+        for i, row in d.items():
+            row = {j:val for j, val in row.items() if val}
+            if row:
+                result[i] = row
+        return SDM(result, (2, 2), EXRAW)
+
+    values = [S.NegativeInfinity, S.NegativeOne, S.Zero, S.One, S.Infinity]
+    for a, b, c, d in product(*[values]*4):
+        Ad = dm({0: {0:a, 1:b}, 1: {0:c, 1:d}})
+        Ad2 = dm({0: {0:a*a + b*c, 1:a*b + b*d}, 1:{0:c*a + d*c, 1: c*b + d*d}})
+        assert Ad * Ad == Ad2
+
+
+def test_SDM_add():
+    A = SDM({0:{1:ZZ(1)}, 1:{0:ZZ(2), 1:ZZ(3)}}, (2, 2), ZZ)
+    B = SDM({0:{0:ZZ(1)}, 1:{0:ZZ(-2), 1:ZZ(3)}}, (2, 2), ZZ)
+    C = SDM({0:{0:ZZ(1), 1:ZZ(1)}, 1:{1:ZZ(6)}}, (2, 2), ZZ)
+    assert A.add(B) == B.add(A) == A + B == B + A == C
+
+    A = SDM({0:{1:ZZ(1)}}, (2, 2), ZZ)
+    B = SDM({0:{0:ZZ(1)}, 1:{0:ZZ(-2), 1:ZZ(3)}}, (2, 2), ZZ)
+    C = SDM({0:{0:ZZ(1), 1:ZZ(1)}, 1:{0:ZZ(-2), 1:ZZ(3)}}, (2, 2), ZZ)
+    assert A.add(B) == B.add(A) == A + B == B + A == C
+
+    raises(TypeError, lambda: A + [])
+
+
+def test_SDM_sub():
+    A = SDM({0:{1:ZZ(1)}, 1:{0:ZZ(2), 1:ZZ(3)}}, (2, 2), ZZ)
+    B = SDM({0:{0:ZZ(1)}, 1:{0:ZZ(-2), 1:ZZ(3)}}, (2, 2), ZZ)
+    C = SDM({0:{0:ZZ(-1), 1:ZZ(1)}, 1:{0:ZZ(4)}}, (2, 2), ZZ)
+    assert A.sub(B) == A - B == C
+
+    raises(TypeError, lambda: A - [])
+
+
+def test_SDM_neg():
+    A = SDM({0:{1:ZZ(1)}, 1:{0:ZZ(2), 1:ZZ(3)}}, (2, 2), ZZ)
+    B = SDM({0:{1:ZZ(-1)}, 1:{0:ZZ(-2), 1:ZZ(-3)}}, (2, 2), ZZ)
+    assert A.neg() == -A == B
+
+
+def test_SDM_convert_to():
+    A = SDM({0:{1:ZZ(1)}, 1:{0:ZZ(2), 1:ZZ(3)}}, (2, 2), ZZ)
+    B = SDM({0:{1:QQ(1)}, 1:{0:QQ(2), 1:QQ(3)}}, (2, 2), QQ)
+    C = A.convert_to(QQ)
+    assert C == B
+    assert C.domain == QQ
+
+    D = A.convert_to(ZZ)
+    assert D == A
+    assert D.domain == ZZ
+
+
+def test_SDM_hstack():
+    A = SDM({0:{1:ZZ(1)}}, (2, 2), ZZ)
+    B = SDM({1:{1:ZZ(1)}}, (2, 2), ZZ)
+    AA = SDM({0:{1:ZZ(1), 3:ZZ(1)}}, (2, 4), ZZ)
+    AB = SDM({0:{1:ZZ(1)}, 1:{3:ZZ(1)}}, (2, 4), ZZ)
+    assert SDM.hstack(A) == A
+    assert SDM.hstack(A, A) == AA
+    assert SDM.hstack(A, B) == AB
+
+
+def test_SDM_vstack():
+    A = SDM({0:{1:ZZ(1)}}, (2, 2), ZZ)
+    B = SDM({1:{1:ZZ(1)}}, (2, 2), ZZ)
+    AA = SDM({0:{1:ZZ(1)}, 2:{1:ZZ(1)}}, (4, 2), ZZ)
+    AB = SDM({0:{1:ZZ(1)}, 3:{1:ZZ(1)}}, (4, 2), ZZ)
+    assert SDM.vstack(A) == A
+    assert SDM.vstack(A, A) == AA
+    assert SDM.vstack(A, B) == AB
+
+
+def test_SDM_applyfunc():
+    A = SDM({0:{1:ZZ(1)}}, (2, 2), ZZ)
+    B = SDM({0:{1:ZZ(2)}}, (2, 2), ZZ)
+    assert A.applyfunc(lambda x: 2*x, ZZ) == B
+
+
+def test_SDM_inv():
+    A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
+    B = SDM({0:{0:QQ(-2), 1:QQ(1)}, 1:{0:QQ(3, 2), 1:QQ(-1, 2)}}, (2, 2), QQ)
+    assert A.inv() == B
+
+
+def test_SDM_det():
+    A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
+    assert A.det() == QQ(-2)
+
+
+def test_SDM_lu():
+    A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
+    L = SDM({0:{0:QQ(1)}, 1:{0:QQ(3), 1:QQ(1)}}, (2, 2), QQ)
+    #U = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(-2)}}, (2, 2), QQ)
+    #swaps = []
+    # This doesn't quite work. U has some nonzero elements in the lower part.
+    #assert A.lu() == (L, U, swaps)
+    assert A.lu()[0] == L
+
+
+def test_SDM_lu_solve():
+    A = SDM({0:{0:QQ(1), 1:QQ(2)}, 1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
+    b = SDM({0:{0:QQ(1)}, 1:{0:QQ(2)}}, (2, 1), QQ)
+    x = SDM({1:{0:QQ(1, 2)}}, (2, 1), QQ)
+    assert A.matmul(x) == b
+    assert A.lu_solve(b) == x
+
+
+def test_SDM_charpoly():
+    A = SDM({0:{0:ZZ(1), 1:ZZ(2)}, 1:{0:ZZ(3), 1:ZZ(4)}}, (2, 2), ZZ)
+    assert A.charpoly() == [ZZ(1), ZZ(-5), ZZ(-2)]
+
+
+def test_SDM_nullspace():
+    # More tests are in test_nullspace.py
+    A = SDM({0:{0:QQ(1), 1:QQ(1)}}, (2, 2), QQ)
+    assert A.nullspace()[0] == SDM({0:{0:QQ(-1), 1:QQ(1)}}, (1, 2), QQ)
+
+
+def test_SDM_rref():
+    # More tests are in test_rref.py
+
+    A = SDM({0:{0:QQ(1), 1:QQ(2)},
+             1:{0:QQ(3), 1:QQ(4)}}, (2, 2), QQ)
+    A_rref = SDM({0:{0:QQ(1)}, 1:{1:QQ(1)}}, (2, 2), QQ)
+    assert A.rref() == (A_rref, [0, 1])
+
+    A = SDM({0: {0: QQ(1), 1: QQ(2), 2: QQ(2)},
+             1: {0: QQ(3),           2: QQ(4)}}, (2, 3), ZZ)
+    A_rref = SDM({0: {0: QQ(1,1), 2: QQ(4,3)},
+                  1: {1: QQ(1,1), 2: QQ(1,3)}}, (2, 3), QQ)
+    assert A.rref() == (A_rref, [0, 1])
+
+
+def test_SDM_particular():
+    A = SDM({0:{0:QQ(1)}}, (2, 2), QQ)
+    Apart = SDM.zeros((1, 2), QQ)
+    assert A.particular() == Apart
+
+
+def test_SDM_is_zero_matrix():
+    A = SDM({0: {0: QQ(1)}}, (2, 2), QQ)
+    Azero = SDM.zeros((1, 2), QQ)
+    assert A.is_zero_matrix() is False
+    assert Azero.is_zero_matrix() is True
+
+
+def test_SDM_is_upper():
+    A = SDM({0: {0: QQ(1), 1: QQ(2), 2: QQ(3), 3: QQ(4)},
+                       1: {1: QQ(5), 2: QQ(6), 3: QQ(7)},
+                                 2: {2: QQ(8), 3: QQ(9)}}, (3, 4), QQ)
+    B = SDM({0: {0: QQ(1), 1: QQ(2), 2: QQ(3), 3: QQ(4)},
+                       1: {1: QQ(5), 2: QQ(6), 3: QQ(7)},
+                       2: {1: QQ(7), 2: QQ(8), 3: QQ(9)}}, (3, 4), QQ)
+    assert A.is_upper() is True
+    assert B.is_upper() is False
+
+
+def test_SDM_is_lower():
+    A = SDM({0: {0: QQ(1), 1: QQ(2), 2: QQ(3), 3: QQ(4)},
+                       1: {1: QQ(5), 2: QQ(6), 3: QQ(7)},
+                                 2: {2: QQ(8), 3: QQ(9)}}, (3, 4), QQ
+            ).transpose()
+    B = SDM({0: {0: QQ(1), 1: QQ(2), 2: QQ(3), 3: QQ(4)},
+                       1: {1: QQ(5), 2: QQ(6), 3: QQ(7)},
+                       2: {1: QQ(7), 2: QQ(8), 3: QQ(9)}}, (3, 4), QQ
+            ).transpose()
+    assert A.is_lower() is True
+    assert B.is_lower() is False

rl/Lib/site-packages/sympy/polys/matrices/tests/test_xxm.py

@@ -0,0 +1,864 @@
+#
+# Test basic features of DDM, SDM and DFM.
+#
+# These three types are supposed to be interchangeable, so we should use the
+# same tests for all of them for the most part.
+#
+# The tests here cover the basic part of the inerface that the three types
+# should expose and that DomainMatrix should mostly rely on.
+#
+# More in-depth tests of the heavier algorithms like rref etc should go in
+# their own test files.
+#
+# Any new methods added to the DDM, SDM or DFM classes should be tested here
+# and added to all classes.
+#
+
+from sympy.external.gmpy import GROUND_TYPES
+
+from sympy import ZZ, QQ, GF, ZZ_I, symbols
+
+from sympy.polys.matrices.exceptions import (
+    DMBadInputError,
+    DMDomainError,
+    DMNonSquareMatrixError,
+    DMNonInvertibleMatrixError,
+    DMShapeError,
+)
+
+from sympy.polys.matrices.domainmatrix import DM, DomainMatrix, DDM, SDM, DFM
+
+from sympy.testing.pytest import raises, skip
+import pytest
+
+
+def test_XXM_constructors():
+    """Test the DDM, etc constructors."""
+
+    lol = [
+        [ZZ(1), ZZ(2)],
+        [ZZ(3), ZZ(4)],
+        [ZZ(5), ZZ(6)],
+    ]
+    dod = {
+        0: {0: ZZ(1), 1: ZZ(2)},
+        1: {0: ZZ(3), 1: ZZ(4)},
+        2: {0: ZZ(5), 1: ZZ(6)},
+    }
+
+    lol_0x0 = []
+    lol_0x2 = []
+    lol_2x0 = [[], []]
+    dod_0x0 = {}
+    dod_0x2 = {}
+    dod_2x0 = {}
+
+    lol_bad = [
+        [ZZ(1), ZZ(2)],
+        [ZZ(3), ZZ(4)],
+        [ZZ(5), ZZ(6), ZZ(7)],
+    ]
+    dod_bad = {
+        0: {0: ZZ(1), 1: ZZ(2)},
+        1: {0: ZZ(3), 1: ZZ(4)},
+        2: {0: ZZ(5), 1: ZZ(6), 2: ZZ(7)},
+    }
+
+    XDM_dense = [DDM]
+    XDM_sparse = [SDM]
+
+    if GROUND_TYPES == 'flint':
+        XDM_dense.append(DFM)
+
+    for XDM in XDM_dense:
+
+        A = XDM(lol, (3, 2), ZZ)
+        assert A.rows == 3
+        assert A.cols == 2
+        assert A.domain == ZZ
+        assert A.shape == (3, 2)
+        if XDM is not DFM:
+            assert ZZ.of_type(A[0][0]) is True
+        else:
+            assert ZZ.of_type(A.rep[0, 0]) is True
+
+        Adm = DomainMatrix(lol, (3, 2), ZZ)
+        if XDM is DFM:
+            assert Adm.rep == A
+            assert Adm.rep.to_ddm() != A
+        elif GROUND_TYPES == 'flint':
+            assert Adm.rep.to_ddm() == A
+            assert Adm.rep != A
+        else:
+            assert Adm.rep == A
+            assert Adm.rep.to_ddm() == A
+
+        assert XDM(lol_0x0, (0, 0), ZZ).shape == (0, 0)
+        assert XDM(lol_0x2, (0, 2), ZZ).shape == (0, 2)
+        assert XDM(lol_2x0, (2, 0), ZZ).shape == (2, 0)
+        raises(DMBadInputError, lambda: XDM(lol, (2, 3), ZZ))
+        raises(DMBadInputError, lambda: XDM(lol_bad, (3, 2), ZZ))
+        raises(DMBadInputError, lambda: XDM(dod, (3, 2), ZZ))
+
+    for XDM in XDM_sparse:
+
+        A = XDM(dod, (3, 2), ZZ)
+        assert A.rows == 3
+        assert A.cols == 2
+        assert A.domain == ZZ
+        assert A.shape == (3, 2)
+        assert ZZ.of_type(A[0][0]) is True
+
+        assert DomainMatrix(dod, (3, 2), ZZ).rep == A
+
+        assert XDM(dod_0x0, (0, 0), ZZ).shape == (0, 0)
+        assert XDM(dod_0x2, (0, 2), ZZ).shape == (0, 2)
+        assert XDM(dod_2x0, (2, 0), ZZ).shape == (2, 0)
+        raises(DMBadInputError, lambda: XDM(dod, (2, 3), ZZ))
+        raises(DMBadInputError, lambda: XDM(lol, (3, 2), ZZ))
+        raises(DMBadInputError, lambda: XDM(dod_bad, (3, 2), ZZ))
+
+    raises(DMBadInputError, lambda: DomainMatrix(lol, (2, 3), ZZ))
+    raises(DMBadInputError, lambda: DomainMatrix(lol_bad, (3, 2), ZZ))
+    raises(DMBadInputError, lambda: DomainMatrix(dod_bad, (3, 2), ZZ))
+
+
+def test_XXM_eq():
+    """Test equality for DDM, SDM, DFM and DomainMatrix."""
+
+    lol1 = [[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]]
+    dod1 = {0: {0: ZZ(1), 1: ZZ(2)}, 1: {0: ZZ(3), 1: ZZ(4)}}
+
+    lol2 = [[ZZ(1), ZZ(2)], [ZZ(3), ZZ(5)]]
+    dod2 = {0: {0: ZZ(1), 1: ZZ(2)}, 1: {0: ZZ(3), 1: ZZ(5)}}
+
+    A1_ddm = DDM(lol1, (2, 2), ZZ)
+    A1_sdm = SDM(dod1, (2, 2), ZZ)
+    A1_dm_d = DomainMatrix(lol1, (2, 2), ZZ)
+    A1_dm_s = DomainMatrix(dod1, (2, 2), ZZ)
+
+    A2_ddm = DDM(lol2, (2, 2), ZZ)
+    A2_sdm = SDM(dod2, (2, 2), ZZ)
+    A2_dm_d = DomainMatrix(lol2, (2, 2), ZZ)
+    A2_dm_s = DomainMatrix(dod2, (2, 2), ZZ)
+
+    A1_all = [A1_ddm, A1_sdm, A1_dm_d, A1_dm_s]
+    A2_all = [A2_ddm, A2_sdm, A2_dm_d, A2_dm_s]
+
+    if GROUND_TYPES == 'flint':
+
+        A1_dfm = DFM([[1, 2], [3, 4]], (2, 2), ZZ)
+        A2_dfm = DFM([[1, 2], [3, 5]], (2, 2), ZZ)
+
+        A1_all.append(A1_dfm)
+        A2_all.append(A2_dfm)
+
+    for n, An in enumerate(A1_all):
+        for m, Am in enumerate(A1_all):
+            if n == m:
+                assert (An == Am) is True
+                assert (An != Am) is False
+            else:
+                assert (An == Am) is False
+                assert (An != Am) is True
+
+    for n, An in enumerate(A2_all):
+        for m, Am in enumerate(A2_all):
+            if n == m:
+                assert (An == Am) is True
+                assert (An != Am) is False
+            else:
+                assert (An == Am) is False
+                assert (An != Am) is True
+
+    for n, A1 in enumerate(A1_all):
+        for m, A2 in enumerate(A2_all):
+            assert (A1 == A2) is False
+            assert (A1 != A2) is True
+
+
+def test_to_XXM():
+    """Test to_ddm etc. for DDM, SDM, DFM and DomainMatrix."""
+
+    lol = [[ZZ(1), ZZ(2)], [ZZ(3), ZZ(4)]]
+    dod = {0: {0: ZZ(1), 1: ZZ(2)}, 1: {0: ZZ(3), 1: ZZ(4)}}
+
+    A_ddm = DDM(lol, (2, 2), ZZ)
+    A_sdm = SDM(dod, (2, 2), ZZ)
+    A_dm_d = DomainMatrix(lol, (2, 2), ZZ)
+    A_dm_s = DomainMatrix(dod, (2, 2), ZZ)
+
+    A_all = [A_ddm, A_sdm, A_dm_d, A_dm_s]
+
+    if GROUND_TYPES == 'flint':
+        A_dfm = DFM(lol, (2, 2), ZZ)
+        A_all.append(A_dfm)
+
+    for A in A_all:
+        assert A.to_ddm() == A_ddm
+        assert A.to_sdm() == A_sdm
+        if GROUND_TYPES != 'flint':
+            raises(NotImplementedError, lambda: A.to_dfm())
+            assert A.to_dfm_or_ddm() == A_ddm
+
+        # Add e.g. DDM.to_DM()?
+        # assert A.to_DM() == A_dm
+
+    if GROUND_TYPES == 'flint':
+        for A in A_all:
+            assert A.to_dfm() == A_dfm
+            for K in [ZZ, QQ, GF(5), ZZ_I]:
+                if isinstance(A, DFM) and not DFM._supports_domain(K):
+                    raises(NotImplementedError, lambda: A.convert_to(K))
+                else:
+                    A_K = A.convert_to(K)
+                    if DFM._supports_domain(K):
+                        A_dfm_K = A_dfm.convert_to(K)
+                        assert A_K.to_dfm() == A_dfm_K
+                        assert A_K.to_dfm_or_ddm() == A_dfm_K
+                    else:
+                        raises(NotImplementedError, lambda: A_K.to_dfm())
+                        assert A_K.to_dfm_or_ddm() == A_ddm.convert_to(K)
+
+
+def test_DFM_domains():
+    """Test which domains are supported by DFM."""
+
+    x, y = symbols('x, y')
+
+    if GROUND_TYPES in ('python', 'gmpy'):
+
+        supported = []
+        flint_funcs = {}
+        not_supported = [ZZ, QQ, GF(5), QQ[x], QQ[x,y]]
+
+    elif GROUND_TYPES == 'flint':
+
+        import flint
+        supported = [ZZ, QQ]
+        flint_funcs = {
+            ZZ: flint.fmpz_mat,
+            QQ: flint.fmpq_mat,
+        }
+        not_supported = [
+            # This could be supported but not yet implemented in SymPy:
+            GF(5),
+            # Other domains could be supported but not implemented as matrices
+            # in python-flint:
+            QQ[x],
+            QQ[x,y],
+            QQ.frac_field(x,y),
+            # Others would potentially never be supported by python-flint:
+            ZZ_I,
+        ]
+
+    else:
+        assert False, "Unknown GROUND_TYPES: %s" % GROUND_TYPES
+
+    for domain in supported:
+        assert DFM._supports_domain(domain) is True
+        assert DFM._get_flint_func(domain) == flint_funcs[domain]
+    for domain in not_supported:
+        assert DFM._supports_domain(domain) is False
+        raises(NotImplementedError, lambda: DFM._get_flint_func(domain))
+
+
+def _DM(lol, typ, K):
+    """Make a DM of type typ over K from lol."""
+    A = DM(lol, K)
+
+    if typ == 'DDM':
+        return A.to_ddm()
+    elif typ == 'SDM':
+        return A.to_sdm()
+    elif typ == 'DFM':
+        if GROUND_TYPES != 'flint':
+            skip("DFM not supported in this ground type")
+        return A.to_dfm()
+    else:
+        assert False, "Unknown type %s" % typ
+
+
+def _DMZ(lol, typ):
+    """Make a DM of type typ over ZZ from lol."""
+    return _DM(lol, typ, ZZ)
+
+
+def _DMQ(lol, typ):
+    """Make a DM of type typ over QQ from lol."""
+    return _DM(lol, typ, QQ)
+
+
+def DM_ddm(lol, K):
+    """Make a DDM over K from lol."""
+    return _DM(lol, 'DDM', K)
+
+
+def DM_sdm(lol, K):
+    """Make a SDM over K from lol."""
+    return _DM(lol, 'SDM', K)
+
+
+def DM_dfm(lol, K):
+    """Make a DFM over K from lol."""
+    return _DM(lol, 'DFM', K)
+
+
+def DMZ_ddm(lol):
+    """Make a DDM from lol."""
+    return _DMZ(lol, 'DDM')
+
+
+def DMZ_sdm(lol):
+    """Make a SDM from lol."""
+    return _DMZ(lol, 'SDM')
+
+
+def DMZ_dfm(lol):
+    """Make a DFM from lol."""
+    return _DMZ(lol, 'DFM')
+
+
+def DMQ_ddm(lol):
+    """Make a DDM from lol."""
+    return _DMQ(lol, 'DDM')
+
+
+def DMQ_sdm(lol):
+    """Make a SDM from lol."""
+    return _DMQ(lol, 'SDM')
+
+
+def DMQ_dfm(lol):
+    """Make a DFM from lol."""
+    return _DMQ(lol, 'DFM')
+
+
+DM_all = [DM_ddm, DM_sdm, DM_dfm]
+DMZ_all = [DMZ_ddm, DMZ_sdm, DMZ_dfm]
+DMQ_all = [DMQ_ddm, DMQ_sdm, DMQ_dfm]
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XDM_getitem(DM):
+    """Test getitem for DDM, etc."""
+
+    lol = [[0, 1], [2, 0]]
+    A = DM(lol)
+    m, n = A.shape
+
+    indices = [-3, -2, -1, 0, 1, 2]
+
+    for i in indices:
+        for j in indices:
+            if -2 <= i < m and -2 <= j < n:
+                assert A.getitem(i, j) == ZZ(lol[i][j])
+            else:
+                raises(IndexError, lambda: A.getitem(i, j))
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XDM_setitem(DM):
+    """Test setitem for DDM, etc."""
+
+    A = DM([[0, 1, 2], [3, 4, 5]])
+
+    A.setitem(0, 0, ZZ(6))
+    assert A == DM([[6, 1, 2], [3, 4, 5]])
+
+    A.setitem(0, 1, ZZ(7))
+    assert A == DM([[6, 7, 2], [3, 4, 5]])
+
+    A.setitem(0, 2, ZZ(8))
+    assert A == DM([[6, 7, 8], [3, 4, 5]])
+
+    A.setitem(0, -1, ZZ(9))
+    assert A == DM([[6, 7, 9], [3, 4, 5]])
+
+    A.setitem(0, -2, ZZ(10))
+    assert A == DM([[6, 10, 9], [3, 4, 5]])
+
+    A.setitem(0, -3, ZZ(11))
+    assert A == DM([[11, 10, 9], [3, 4, 5]])
+
+    raises(IndexError, lambda: A.setitem(0, 3, ZZ(12)))
+    raises(IndexError, lambda: A.setitem(0, -4, ZZ(13)))
+
+    A.setitem(1, 0, ZZ(14))
+    assert A == DM([[11, 10, 9], [14, 4, 5]])
+
+    A.setitem(1, 1, ZZ(15))
+    assert A == DM([[11, 10, 9], [14, 15, 5]])
+
+    A.setitem(-1, 1, ZZ(16))
+    assert A == DM([[11, 10, 9], [14, 16, 5]])
+
+    A.setitem(-2, 1, ZZ(17))
+    assert A == DM([[11, 17, 9], [14, 16, 5]])
+
+    raises(IndexError, lambda: A.setitem(2, 0, ZZ(18)))
+    raises(IndexError, lambda: A.setitem(-3, 0, ZZ(19)))
+
+    A.setitem(1, 2, ZZ(0))
+    assert A == DM([[11, 17, 9], [14, 16, 0]])
+
+    A.setitem(1, -2, ZZ(0))
+    assert A == DM([[11, 17, 9], [14, 0, 0]])
+
+    A.setitem(1, -3, ZZ(0))
+    assert A == DM([[11, 17, 9], [0, 0, 0]])
+
+    A.setitem(0, 0, ZZ(0))
+    assert A == DM([[0, 17, 9], [0, 0, 0]])
+
+    A.setitem(0, -1, ZZ(0))
+    assert A == DM([[0, 17, 0], [0, 0, 0]])
+
+    A.setitem(0, 0, ZZ(0))
+    assert A == DM([[0, 17, 0], [0, 0, 0]])
+
+    A.setitem(0, -2, ZZ(0))
+    assert A == DM([[0, 0, 0], [0, 0, 0]])
+
+    A.setitem(0, -3, ZZ(1))
+    assert A == DM([[1, 0, 0], [0, 0, 0]])
+
+
+class _Sliced:
+    def __getitem__(self, item):
+        return item
+
+
+_slice = _Sliced()
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_extract_slice(DM):
+    A = DM([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    assert A.extract_slice(*_slice[:,:]) == A
+    assert A.extract_slice(*_slice[1:,:]) == DM([[4, 5, 6], [7, 8, 9]])
+    assert A.extract_slice(*_slice[1:,1:]) == DM([[5, 6], [8, 9]])
+    assert A.extract_slice(*_slice[1:,:-1]) == DM([[4, 5], [7, 8]])
+    assert A.extract_slice(*_slice[1:,:-1:2]) == DM([[4], [7]])
+    assert A.extract_slice(*_slice[:,::2]) == DM([[1, 3], [4, 6], [7, 9]])
+    assert A.extract_slice(*_slice[::2,:]) == DM([[1, 2, 3], [7, 8, 9]])
+    assert A.extract_slice(*_slice[::2,::2]) == DM([[1, 3], [7, 9]])
+    assert A.extract_slice(*_slice[::2,::-2]) == DM([[3, 1], [9, 7]])
+    assert A.extract_slice(*_slice[::-2,::2]) == DM([[7, 9], [1, 3]])
+    assert A.extract_slice(*_slice[::-2,::-2]) == DM([[9, 7], [3, 1]])
+    assert A.extract_slice(*_slice[:,::-1]) == DM([[3, 2, 1], [6, 5, 4], [9, 8, 7]])
+    assert A.extract_slice(*_slice[::-1,:]) == DM([[7, 8, 9], [4, 5, 6], [1, 2, 3]])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_extract(DM):
+
+    A = DM([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+
+    assert A.extract([0, 1, 2], [0, 1, 2]) == A
+    assert A.extract([1, 2], [1, 2]) == DM([[5, 6], [8, 9]])
+    assert A.extract([1, 2], [0, 1]) == DM([[4, 5], [7, 8]])
+    assert A.extract([1, 2], [0, 2]) == DM([[4, 6], [7, 9]])
+    assert A.extract([1, 2], [0]) == DM([[4], [7]])
+    assert A.extract([1, 2], []) == DM([[1]]).zeros((2, 0), ZZ)
+    assert A.extract([], [0, 1, 2]) == DM([[1]]).zeros((0, 3), ZZ)
+
+    raises(IndexError, lambda: A.extract([1, 2], [0, 3]))
+    raises(IndexError, lambda: A.extract([1, 2], [0, -4]))
+    raises(IndexError, lambda: A.extract([3, 1], [0, 1]))
+    raises(IndexError, lambda: A.extract([-4, 2], [3, 1]))
+
+    B = DM([[0, 0, 0], [0, 0, 0], [0, 0, 0]])
+    assert B.extract([1, 2], [1, 2]) == DM([[0, 0], [0, 0]])
+
+
+def test_XXM_str():
+
+    A = DomainMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]], (3, 3), ZZ)
+
+    assert str(A) == \
+        'DomainMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]], (3, 3), ZZ)'
+    assert str(A.to_ddm()) == \
+        '[[1, 2, 3], [4, 5, 6], [7, 8, 9]]'
+    assert str(A.to_sdm()) == \
+        '{0: {0: 1, 1: 2, 2: 3}, 1: {0: 4, 1: 5, 2: 6}, 2: {0: 7, 1: 8, 2: 9}}'
+
+    assert repr(A) == \
+        'DomainMatrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]], (3, 3), ZZ)'
+    assert repr(A.to_ddm()) == \
+        'DDM([[1, 2, 3], [4, 5, 6], [7, 8, 9]], (3, 3), ZZ)'
+    assert repr(A.to_sdm()) == \
+        'SDM({0: {0: 1, 1: 2, 2: 3}, 1: {0: 4, 1: 5, 2: 6}, 2: {0: 7, 1: 8, 2: 9}}, (3, 3), ZZ)'
+
+    B = DomainMatrix({0: {0: ZZ(1), 1: ZZ(2)}, 1: {0: ZZ(3)}}, (2, 2), ZZ)
+
+    assert str(B) == \
+        'DomainMatrix({0: {0: 1, 1: 2}, 1: {0: 3}}, (2, 2), ZZ)'
+    assert str(B.to_ddm()) == \
+        '[[1, 2], [3, 0]]'
+    assert str(B.to_sdm()) == \
+        '{0: {0: 1, 1: 2}, 1: {0: 3}}'
+
+    assert repr(B) == \
+        'DomainMatrix({0: {0: 1, 1: 2}, 1: {0: 3}}, (2, 2), ZZ)'
+
+    if GROUND_TYPES != 'gmpy':
+        assert repr(B.to_ddm()) == \
+            'DDM([[1, 2], [3, 0]], (2, 2), ZZ)'
+        assert repr(B.to_sdm()) == \
+            'SDM({0: {0: 1, 1: 2}, 1: {0: 3}}, (2, 2), ZZ)'
+    else:
+        assert repr(B.to_ddm()) == \
+            'DDM([[mpz(1), mpz(2)], [mpz(3), mpz(0)]], (2, 2), ZZ)'
+        assert repr(B.to_sdm()) == \
+            'SDM({0: {0: mpz(1), 1: mpz(2)}, 1: {0: mpz(3)}}, (2, 2), ZZ)'
+
+    if GROUND_TYPES == 'flint':
+
+        assert str(A.to_dfm()) == \
+            '[[1, 2, 3], [4, 5, 6], [7, 8, 9]]'
+        assert str(B.to_dfm()) == \
+            '[[1, 2], [3, 0]]'
+
+        assert repr(A.to_dfm()) == \
+            'DFM([[1, 2, 3], [4, 5, 6], [7, 8, 9]], (3, 3), ZZ)'
+        assert repr(B.to_dfm()) == \
+            'DFM([[1, 2], [3, 0]], (2, 2), ZZ)'
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_from_list(DM):
+    T = type(DM([[0]]))
+
+    lol = [[1, 2, 4], [4, 5, 6]]
+    lol_ZZ = [[ZZ(1), ZZ(2), ZZ(4)], [ZZ(4), ZZ(5), ZZ(6)]]
+    lol_ZZ_bad = [[ZZ(1), ZZ(2), ZZ(4)], [ZZ(4), ZZ(5), ZZ(6), ZZ(7)]]
+
+    assert T.from_list(lol_ZZ, (2, 3), ZZ) == DM(lol)
+    raises(DMBadInputError, lambda: T.from_list(lol_ZZ_bad, (3, 2), ZZ))
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_to_list(DM):
+    lol = [[1, 2, 4], [4, 5, 6]]
+    assert DM(lol).to_list() == [[ZZ(1), ZZ(2), ZZ(4)], [ZZ(4), ZZ(5), ZZ(6)]]
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_to_list_flat(DM):
+    lol = [[1, 2, 4], [4, 5, 6]]
+    assert DM(lol).to_list_flat() == [ZZ(1), ZZ(2), ZZ(4), ZZ(4), ZZ(5), ZZ(6)]
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_from_list_flat(DM):
+    T = type(DM([[0]]))
+    flat = [ZZ(1), ZZ(2), ZZ(4), ZZ(4), ZZ(5), ZZ(6)]
+    assert T.from_list_flat(flat, (2, 3), ZZ) == DM([[1, 2, 4], [4, 5, 6]])
+    raises(DMBadInputError, lambda: T.from_list_flat(flat, (3, 3), ZZ))
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_to_flat_nz(DM):
+    M = DM([[1, 2, 0], [0, 0, 0], [0, 0, 3]])
+    elements = [ZZ(1), ZZ(2), ZZ(3)]
+    indices = ((0, 0), (0, 1), (2, 2))
+    assert M.to_flat_nz() == (elements, (indices, M.shape))
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_from_flat_nz(DM):
+    T = type(DM([[0]]))
+    elements = [ZZ(1), ZZ(2), ZZ(3)]
+    indices = ((0, 0), (0, 1), (2, 2))
+    data = (indices, (3, 3))
+    result = DM([[1, 2, 0], [0, 0, 0], [0, 0, 3]])
+    assert T.from_flat_nz(elements, data, ZZ) == result
+    raises(DMBadInputError, lambda: T.from_flat_nz(elements, (indices, (2, 3)), ZZ))
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_to_dod(DM):
+    dod = {0: {0: ZZ(1), 2: ZZ(4)}, 1: {0: ZZ(4), 1: ZZ(5), 2: ZZ(6)}}
+    assert DM([[1, 0, 4], [4, 5, 6]]).to_dod() == dod
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_from_dod(DM):
+    T = type(DM([[0]]))
+    dod = {0: {0: ZZ(1), 2: ZZ(4)}, 1: {0: ZZ(4), 1: ZZ(5), 2: ZZ(6)}}
+    assert T.from_dod(dod, (2, 3), ZZ) == DM([[1, 0, 4], [4, 5, 6]])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_to_dok(DM):
+    dod = {(0, 0): ZZ(1), (0, 2): ZZ(4),
+           (1, 0): ZZ(4), (1, 1): ZZ(5), (1, 2): ZZ(6)}
+    assert DM([[1, 0, 4], [4, 5, 6]]).to_dok() == dod
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_from_dok(DM):
+    T = type(DM([[0]]))
+    dod = {(0, 0): ZZ(1), (0, 2): ZZ(4),
+           (1, 0): ZZ(4), (1, 1): ZZ(5), (1, 2): ZZ(6)}
+    assert T.from_dok(dod, (2, 3), ZZ) == DM([[1, 0, 4], [4, 5, 6]])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_iter_values(DM):
+    values = [ZZ(1), ZZ(4), ZZ(4), ZZ(5), ZZ(6)]
+    assert sorted(DM([[1, 0, 4], [4, 5, 6]]).iter_values()) == values
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_iter_items(DM):
+    items = [((0, 0), ZZ(1)), ((0, 2), ZZ(4)),
+             ((1, 0), ZZ(4)), ((1, 1), ZZ(5)), ((1, 2), ZZ(6))]
+    assert sorted(DM([[1, 0, 4], [4, 5, 6]]).iter_items()) == items
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_from_ddm(DM):
+    T = type(DM([[0]]))
+    ddm = DDM([[1, 2, 4], [4, 5, 6]], (2, 3), ZZ)
+    assert T.from_ddm(ddm) == DM([[1, 2, 4], [4, 5, 6]])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_zeros(DM):
+    T = type(DM([[0]]))
+    assert T.zeros((2, 3), ZZ) == DM([[0, 0, 0], [0, 0, 0]])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_ones(DM):
+    T = type(DM([[0]]))
+    assert T.ones((2, 3), ZZ) == DM([[1, 1, 1], [1, 1, 1]])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_eye(DM):
+    T = type(DM([[0]]))
+    assert T.eye(3, ZZ) == DM([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
+    assert T.eye((3, 2), ZZ) == DM([[1, 0], [0, 1], [0, 0]])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_diag(DM):
+    T = type(DM([[0]]))
+    assert T.diag([1, 2, 3], ZZ) == DM([[1, 0, 0], [0, 2, 0], [0, 0, 3]])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_transpose(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]])
+    assert A.transpose() == DM([[1, 4], [2, 5], [3, 6]])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_add(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]])
+    B = DM([[1, 2, 3], [4, 5, 6]])
+    C = DM([[2, 4, 6], [8, 10, 12]])
+    assert A.add(B) == C
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_sub(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]])
+    B = DM([[1, 2, 3], [4, 5, 6]])
+    C = DM([[0, 0, 0], [0, 0, 0]])
+    assert A.sub(B) == C
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_mul(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]])
+    b = ZZ(2)
+    assert A.mul(b) == DM([[2, 4, 6], [8, 10, 12]])
+    assert A.rmul(b) == DM([[2, 4, 6], [8, 10, 12]])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_matmul(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]])
+    B = DM([[1, 2], [3, 4], [5, 6]])
+    C = DM([[22, 28], [49, 64]])
+    assert A.matmul(B) == C
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_mul_elementwise(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]])
+    B = DM([[1, 2, 3], [4, 5, 6]])
+    C = DM([[1, 4, 9], [16, 25, 36]])
+    assert A.mul_elementwise(B) == C
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_neg(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]])
+    C = DM([[-1, -2, -3], [-4, -5, -6]])
+    assert A.neg() == C
+
+
+@pytest.mark.parametrize('DM', DM_all)
+def test_XXM_convert_to(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]], ZZ)
+    B = DM([[1, 2, 3], [4, 5, 6]], QQ)
+    assert A.convert_to(QQ) == B
+    assert B.convert_to(ZZ) == A
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_scc(DM):
+    A = DM([
+        [0, 1, 0, 0, 0, 0],
+        [1, 0, 0, 0, 0, 0],
+        [0, 0, 1, 0, 0, 0],
+        [0, 0, 0, 1, 0, 1],
+        [0, 0, 0, 0, 1, 0],
+        [0, 0, 0, 1, 0, 1]])
+    assert A.scc() == [[0, 1], [2], [3, 5], [4]]
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_hstack(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]])
+    B = DM([[7, 8], [9, 10]])
+    C = DM([[1, 2, 3, 7, 8], [4, 5, 6, 9, 10]])
+    ABC = DM([[1, 2, 3, 7, 8, 1, 2, 3, 7, 8],
+              [4, 5, 6, 9, 10, 4, 5, 6, 9, 10]])
+    assert A.hstack(B) == C
+    assert A.hstack(B, C) == ABC
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_vstack(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]])
+    B = DM([[7, 8, 9]])
+    C = DM([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    ABC = DM([[1, 2, 3], [4, 5, 6], [7, 8, 9], [1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    assert A.vstack(B) == C
+    assert A.vstack(B, C) == ABC
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_applyfunc(DM):
+    A = DM([[1, 2, 3], [4, 5, 6]])
+    B = DM([[2, 4, 6], [8, 10, 12]])
+    assert A.applyfunc(lambda x: 2*x, ZZ) == B
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_is_upper(DM):
+    assert DM([[1, 2, 3], [0, 5, 6]]).is_upper() is True
+    assert DM([[1, 2, 3], [4, 5, 6]]).is_upper() is False
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_is_lower(DM):
+    assert DM([[1, 0, 0], [4, 5, 0]]).is_lower() is True
+    assert DM([[1, 2, 3], [4, 5, 6]]).is_lower() is False
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_is_diagonal(DM):
+    assert DM([[1, 0, 0], [0, 5, 0]]).is_diagonal() is True
+    assert DM([[1, 2, 3], [4, 5, 6]]).is_diagonal() is False
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_diagonal(DM):
+    assert DM([[1, 0, 0], [0, 5, 0]]).diagonal() == [1, 5]
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_is_zero_matrix(DM):
+    assert DM([[0, 0, 0], [0, 0, 0]]).is_zero_matrix() is True
+    assert DM([[1, 0, 0], [0, 0, 0]]).is_zero_matrix() is False
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_det_ZZ(DM):
+    assert DM([[1, 2, 3], [4, 5, 6], [7, 8, 9]]).det() == 0
+    assert DM([[1, 2, 3], [4, 5, 6], [7, 8, 10]]).det() == -3
+
+
+@pytest.mark.parametrize('DM', DMQ_all)
+def test_XXM_det_QQ(DM):
+    dM1 = DM([[(1,2), (2,3)], [(3,4), (4,5)]])
+    assert dM1.det() == QQ(-1,10)
+
+
+@pytest.mark.parametrize('DM', DMQ_all)
+def test_XXM_inv_QQ(DM):
+    dM1 = DM([[(1,2), (2,3)], [(3,4), (4,5)]])
+    dM2 = DM([[(-8,1), (20,3)], [(15,2), (-5,1)]])
+    assert dM1.inv() == dM2
+    assert dM1.matmul(dM2) == DM([[1, 0], [0, 1]])
+
+    dM3 = DM([[(1,2), (2,3)], [(1,4), (1,3)]])
+    raises(DMNonInvertibleMatrixError, lambda: dM3.inv())
+
+    dM4 = DM([[(1,2), (2,3), (3,4)], [(1,4), (1,3), (1,2)]])
+    raises(DMNonSquareMatrixError, lambda: dM4.inv())
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_inv_ZZ(DM):
+    dM1 = DM([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
+    # XXX: Maybe this should return a DM over QQ instead?
+    # XXX: Handle unimodular matrices?
+    raises(DMDomainError, lambda: dM1.inv())
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_charpoly_ZZ(DM):
+    dM1 = DM([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
+    assert dM1.charpoly() == [1, -16, -12, 3]
+
+
+@pytest.mark.parametrize('DM', DMQ_all)
+def test_XXM_charpoly_QQ(DM):
+    dM1 = DM([[(1,2), (2,3)], [(3,4), (4,5)]])
+    assert dM1.charpoly() == [QQ(1,1), QQ(-13,10), QQ(-1,10)]
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_lu_solve_ZZ(DM):
+    dM1 = DM([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
+    dM2 = DM([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
+    raises(DMDomainError, lambda: dM1.lu_solve(dM2))
+
+
+@pytest.mark.parametrize('DM', DMQ_all)
+def test_XXM_lu_solve_QQ(DM):
+    dM1 = DM([[1, 2, 3], [4, 5, 6], [7, 8, 10]])
+    dM2 = DM([[1, 0, 0], [0, 1, 0], [0, 0, 1]])
+    dM3 = DM([[(-2,3),(-4,3),(1,1)],[(-2,3),(11,3),(-2,1)],[(1,1),(-2,1),(1,1)]])
+    assert dM1.lu_solve(dM2) == dM3 == dM1.inv()
+
+    dM4 = DM([[1, 2, 3], [4, 5, 6]])
+    dM5 = DM([[1, 0], [0, 1], [0, 0]])
+    raises(DMShapeError, lambda: dM4.lu_solve(dM5))
+
+
+@pytest.mark.parametrize('DM', DMQ_all)
+def test_XXM_nullspace_QQ(DM):
+    dM1 = DM([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
+    # XXX: Change the signature to just return the nullspace. Possibly
+    # returning the rank or nullity makes sense but the list of nonpivots is
+    # not useful.
+    assert dM1.nullspace() == (DM([[1, -2, 1]]), [2])
+
+
+@pytest.mark.parametrize('DM', DMZ_all)
+def test_XXM_lll(DM):
+    M = DM([[1, 2, 3], [4, 5, 20]])
+    M_lll = DM([[1, 2, 3], [-1, -5, 5]])
+    T = DM([[1, 0], [-5, 1]])
+    assert M.lll() == M_lll
+    assert M.lll_transform() == (M_lll, T)
+    assert T.matmul(M) == M_lll