284 lines
12 KiB
Python
284 lines
12 KiB
Python
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from sympy.assumptions.ask import (Q, ask)
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from sympy.core.symbol import Symbol
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from sympy.matrices.expressions.diagonal import (DiagMatrix, DiagonalMatrix)
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from sympy.matrices.dense import Matrix
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from sympy.matrices.expressions import (MatrixSymbol, Identity, ZeroMatrix,
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OneMatrix, Trace, MatrixSlice, Determinant, BlockMatrix, BlockDiagMatrix)
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from sympy.matrices.expressions.factorizations import LofLU
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from sympy.testing.pytest import XFAIL
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X = MatrixSymbol('X', 2, 2)
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Y = MatrixSymbol('Y', 2, 3)
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Z = MatrixSymbol('Z', 2, 2)
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A1x1 = MatrixSymbol('A1x1', 1, 1)
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B1x1 = MatrixSymbol('B1x1', 1, 1)
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C0x0 = MatrixSymbol('C0x0', 0, 0)
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V1 = MatrixSymbol('V1', 2, 1)
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V2 = MatrixSymbol('V2', 2, 1)
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def test_square():
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assert ask(Q.square(X))
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assert not ask(Q.square(Y))
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assert ask(Q.square(Y*Y.T))
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def test_invertible():
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assert ask(Q.invertible(X), Q.invertible(X))
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assert ask(Q.invertible(Y)) is False
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assert ask(Q.invertible(X*Y), Q.invertible(X)) is False
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assert ask(Q.invertible(X*Z), Q.invertible(X)) is None
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assert ask(Q.invertible(X*Z), Q.invertible(X) & Q.invertible(Z)) is True
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assert ask(Q.invertible(X.T)) is None
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assert ask(Q.invertible(X.T), Q.invertible(X)) is True
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assert ask(Q.invertible(X.I)) is True
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assert ask(Q.invertible(Identity(3))) is True
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assert ask(Q.invertible(ZeroMatrix(3, 3))) is False
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assert ask(Q.invertible(OneMatrix(1, 1))) is True
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assert ask(Q.invertible(OneMatrix(3, 3))) is False
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assert ask(Q.invertible(X), Q.fullrank(X) & Q.square(X))
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def test_singular():
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assert ask(Q.singular(X)) is None
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assert ask(Q.singular(X), Q.invertible(X)) is False
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assert ask(Q.singular(X), ~Q.invertible(X)) is True
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@XFAIL
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def test_invertible_fullrank():
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assert ask(Q.invertible(X), Q.fullrank(X)) is True
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def test_invertible_BlockMatrix():
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assert ask(Q.invertible(BlockMatrix([Identity(3)]))) == True
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assert ask(Q.invertible(BlockMatrix([ZeroMatrix(3, 3)]))) == False
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X = Matrix([[1, 2, 3], [3, 5, 4]])
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Y = Matrix([[4, 2, 7], [2, 3, 5]])
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# non-invertible A block
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assert ask(Q.invertible(BlockMatrix([
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[Matrix.ones(3, 3), Y.T],
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[X, Matrix.eye(2)],
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]))) == True
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# non-invertible B block
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assert ask(Q.invertible(BlockMatrix([
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[Y.T, Matrix.ones(3, 3)],
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[Matrix.eye(2), X],
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]))) == True
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# non-invertible C block
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assert ask(Q.invertible(BlockMatrix([
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[X, Matrix.eye(2)],
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[Matrix.ones(3, 3), Y.T],
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]))) == True
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# non-invertible D block
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assert ask(Q.invertible(BlockMatrix([
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[Matrix.eye(2), X],
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[Y.T, Matrix.ones(3, 3)],
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]))) == True
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def test_invertible_BlockDiagMatrix():
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assert ask(Q.invertible(BlockDiagMatrix(Identity(3), Identity(5)))) == True
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assert ask(Q.invertible(BlockDiagMatrix(ZeroMatrix(3, 3), Identity(5)))) == False
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assert ask(Q.invertible(BlockDiagMatrix(Identity(3), OneMatrix(5, 5)))) == False
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def test_symmetric():
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assert ask(Q.symmetric(X), Q.symmetric(X))
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assert ask(Q.symmetric(X*Z), Q.symmetric(X)) is None
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assert ask(Q.symmetric(X*Z), Q.symmetric(X) & Q.symmetric(Z)) is True
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assert ask(Q.symmetric(X + Z), Q.symmetric(X) & Q.symmetric(Z)) is True
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assert ask(Q.symmetric(Y)) is False
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assert ask(Q.symmetric(Y*Y.T)) is True
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assert ask(Q.symmetric(Y.T*X*Y)) is None
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assert ask(Q.symmetric(Y.T*X*Y), Q.symmetric(X)) is True
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assert ask(Q.symmetric(X**10), Q.symmetric(X)) is True
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assert ask(Q.symmetric(A1x1)) is True
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assert ask(Q.symmetric(A1x1 + B1x1)) is True
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assert ask(Q.symmetric(A1x1 * B1x1)) is True
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assert ask(Q.symmetric(V1.T*V1)) is True
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assert ask(Q.symmetric(V1.T*(V1 + V2))) is True
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assert ask(Q.symmetric(V1.T*(V1 + V2) + A1x1)) is True
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assert ask(Q.symmetric(MatrixSlice(Y, (0, 1), (1, 2)))) is True
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assert ask(Q.symmetric(Identity(3))) is True
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assert ask(Q.symmetric(ZeroMatrix(3, 3))) is True
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assert ask(Q.symmetric(OneMatrix(3, 3))) is True
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def _test_orthogonal_unitary(predicate):
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assert ask(predicate(X), predicate(X))
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assert ask(predicate(X.T), predicate(X)) is True
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assert ask(predicate(X.I), predicate(X)) is True
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assert ask(predicate(X**2), predicate(X))
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assert ask(predicate(Y)) is False
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assert ask(predicate(X)) is None
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assert ask(predicate(X), ~Q.invertible(X)) is False
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assert ask(predicate(X*Z*X), predicate(X) & predicate(Z)) is True
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assert ask(predicate(Identity(3))) is True
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assert ask(predicate(ZeroMatrix(3, 3))) is False
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assert ask(Q.invertible(X), predicate(X))
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assert not ask(predicate(X + Z), predicate(X) & predicate(Z))
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def test_orthogonal():
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_test_orthogonal_unitary(Q.orthogonal)
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def test_unitary():
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_test_orthogonal_unitary(Q.unitary)
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assert ask(Q.unitary(X), Q.orthogonal(X))
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def test_fullrank():
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assert ask(Q.fullrank(X), Q.fullrank(X))
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assert ask(Q.fullrank(X**2), Q.fullrank(X))
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assert ask(Q.fullrank(X.T), Q.fullrank(X)) is True
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assert ask(Q.fullrank(X)) is None
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assert ask(Q.fullrank(Y)) is None
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assert ask(Q.fullrank(X*Z), Q.fullrank(X) & Q.fullrank(Z)) is True
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assert ask(Q.fullrank(Identity(3))) is True
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assert ask(Q.fullrank(ZeroMatrix(3, 3))) is False
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assert ask(Q.fullrank(OneMatrix(1, 1))) is True
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assert ask(Q.fullrank(OneMatrix(3, 3))) is False
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assert ask(Q.invertible(X), ~Q.fullrank(X)) == False
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def test_positive_definite():
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assert ask(Q.positive_definite(X), Q.positive_definite(X))
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assert ask(Q.positive_definite(X.T), Q.positive_definite(X)) is True
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assert ask(Q.positive_definite(X.I), Q.positive_definite(X)) is True
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assert ask(Q.positive_definite(Y)) is False
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assert ask(Q.positive_definite(X)) is None
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assert ask(Q.positive_definite(X**3), Q.positive_definite(X))
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assert ask(Q.positive_definite(X*Z*X),
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Q.positive_definite(X) & Q.positive_definite(Z)) is True
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assert ask(Q.positive_definite(X), Q.orthogonal(X))
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assert ask(Q.positive_definite(Y.T*X*Y),
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Q.positive_definite(X) & Q.fullrank(Y)) is True
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assert not ask(Q.positive_definite(Y.T*X*Y), Q.positive_definite(X))
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assert ask(Q.positive_definite(Identity(3))) is True
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assert ask(Q.positive_definite(ZeroMatrix(3, 3))) is False
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assert ask(Q.positive_definite(OneMatrix(1, 1))) is True
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assert ask(Q.positive_definite(OneMatrix(3, 3))) is False
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assert ask(Q.positive_definite(X + Z), Q.positive_definite(X) &
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Q.positive_definite(Z)) is True
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assert not ask(Q.positive_definite(-X), Q.positive_definite(X))
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assert ask(Q.positive(X[1, 1]), Q.positive_definite(X))
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def test_triangular():
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assert ask(Q.upper_triangular(X + Z.T + Identity(2)), Q.upper_triangular(X) &
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Q.lower_triangular(Z)) is True
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assert ask(Q.upper_triangular(X*Z.T), Q.upper_triangular(X) &
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Q.lower_triangular(Z)) is True
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assert ask(Q.lower_triangular(Identity(3))) is True
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assert ask(Q.lower_triangular(ZeroMatrix(3, 3))) is True
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assert ask(Q.upper_triangular(ZeroMatrix(3, 3))) is True
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assert ask(Q.lower_triangular(OneMatrix(1, 1))) is True
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assert ask(Q.upper_triangular(OneMatrix(1, 1))) is True
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assert ask(Q.lower_triangular(OneMatrix(3, 3))) is False
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assert ask(Q.upper_triangular(OneMatrix(3, 3))) is False
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assert ask(Q.triangular(X), Q.unit_triangular(X))
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assert ask(Q.upper_triangular(X**3), Q.upper_triangular(X))
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assert ask(Q.lower_triangular(X**3), Q.lower_triangular(X))
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def test_diagonal():
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assert ask(Q.diagonal(X + Z.T + Identity(2)), Q.diagonal(X) &
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Q.diagonal(Z)) is True
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assert ask(Q.diagonal(ZeroMatrix(3, 3)))
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assert ask(Q.diagonal(OneMatrix(1, 1))) is True
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assert ask(Q.diagonal(OneMatrix(3, 3))) is False
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assert ask(Q.lower_triangular(X) & Q.upper_triangular(X), Q.diagonal(X))
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assert ask(Q.diagonal(X), Q.lower_triangular(X) & Q.upper_triangular(X))
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assert ask(Q.symmetric(X), Q.diagonal(X))
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assert ask(Q.triangular(X), Q.diagonal(X))
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assert ask(Q.diagonal(C0x0))
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assert ask(Q.diagonal(A1x1))
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assert ask(Q.diagonal(A1x1 + B1x1))
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assert ask(Q.diagonal(A1x1*B1x1))
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assert ask(Q.diagonal(V1.T*V2))
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assert ask(Q.diagonal(V1.T*(X + Z)*V1))
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assert ask(Q.diagonal(MatrixSlice(Y, (0, 1), (1, 2)))) is True
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assert ask(Q.diagonal(V1.T*(V1 + V2))) is True
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assert ask(Q.diagonal(X**3), Q.diagonal(X))
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assert ask(Q.diagonal(Identity(3)))
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assert ask(Q.diagonal(DiagMatrix(V1)))
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assert ask(Q.diagonal(DiagonalMatrix(X)))
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def test_non_atoms():
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assert ask(Q.real(Trace(X)), Q.positive(Trace(X)))
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@XFAIL
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def test_non_trivial_implies():
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X = MatrixSymbol('X', 3, 3)
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Y = MatrixSymbol('Y', 3, 3)
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assert ask(Q.lower_triangular(X+Y), Q.lower_triangular(X) &
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Q.lower_triangular(Y)) is True
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assert ask(Q.triangular(X), Q.lower_triangular(X)) is True
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assert ask(Q.triangular(X+Y), Q.lower_triangular(X) &
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Q.lower_triangular(Y)) is True
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def test_MatrixSlice():
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X = MatrixSymbol('X', 4, 4)
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B = MatrixSlice(X, (1, 3), (1, 3))
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C = MatrixSlice(X, (0, 3), (1, 3))
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assert ask(Q.symmetric(B), Q.symmetric(X))
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assert ask(Q.invertible(B), Q.invertible(X))
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assert ask(Q.diagonal(B), Q.diagonal(X))
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assert ask(Q.orthogonal(B), Q.orthogonal(X))
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assert ask(Q.upper_triangular(B), Q.upper_triangular(X))
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assert not ask(Q.symmetric(C), Q.symmetric(X))
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assert not ask(Q.invertible(C), Q.invertible(X))
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assert not ask(Q.diagonal(C), Q.diagonal(X))
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assert not ask(Q.orthogonal(C), Q.orthogonal(X))
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assert not ask(Q.upper_triangular(C), Q.upper_triangular(X))
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def test_det_trace_positive():
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X = MatrixSymbol('X', 4, 4)
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assert ask(Q.positive(Trace(X)), Q.positive_definite(X))
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assert ask(Q.positive(Determinant(X)), Q.positive_definite(X))
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def test_field_assumptions():
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X = MatrixSymbol('X', 4, 4)
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Y = MatrixSymbol('Y', 4, 4)
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assert ask(Q.real_elements(X), Q.real_elements(X))
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assert not ask(Q.integer_elements(X), Q.real_elements(X))
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assert ask(Q.complex_elements(X), Q.real_elements(X))
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assert ask(Q.complex_elements(X**2), Q.real_elements(X))
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assert ask(Q.real_elements(X**2), Q.integer_elements(X))
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assert ask(Q.real_elements(X+Y), Q.real_elements(X)) is None
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assert ask(Q.real_elements(X+Y), Q.real_elements(X) & Q.real_elements(Y))
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from sympy.matrices.expressions.hadamard import HadamardProduct
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assert ask(Q.real_elements(HadamardProduct(X, Y)),
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Q.real_elements(X) & Q.real_elements(Y))
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assert ask(Q.complex_elements(X+Y), Q.real_elements(X) & Q.complex_elements(Y))
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assert ask(Q.real_elements(X.T), Q.real_elements(X))
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assert ask(Q.real_elements(X.I), Q.real_elements(X) & Q.invertible(X))
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assert ask(Q.real_elements(Trace(X)), Q.real_elements(X))
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assert ask(Q.integer_elements(Determinant(X)), Q.integer_elements(X))
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assert not ask(Q.integer_elements(X.I), Q.integer_elements(X))
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alpha = Symbol('alpha')
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assert ask(Q.real_elements(alpha*X), Q.real_elements(X) & Q.real(alpha))
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assert ask(Q.real_elements(LofLU(X)), Q.real_elements(X))
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e = Symbol('e', integer=True, negative=True)
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assert ask(Q.real_elements(X**e), Q.real_elements(X) & Q.invertible(X))
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assert ask(Q.real_elements(X**e), Q.real_elements(X)) is None
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def test_matrix_element_sets():
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X = MatrixSymbol('X', 4, 4)
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assert ask(Q.real(X[1, 2]), Q.real_elements(X))
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assert ask(Q.integer(X[1, 2]), Q.integer_elements(X))
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assert ask(Q.complex(X[1, 2]), Q.complex_elements(X))
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assert ask(Q.integer_elements(Identity(3)))
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assert ask(Q.integer_elements(ZeroMatrix(3, 3)))
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assert ask(Q.integer_elements(OneMatrix(3, 3)))
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from sympy.matrices.expressions.fourier import DFT
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assert ask(Q.complex_elements(DFT(3)))
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def test_matrix_element_sets_slices_blocks():
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X = MatrixSymbol('X', 4, 4)
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assert ask(Q.integer_elements(X[:, 3]), Q.integer_elements(X))
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assert ask(Q.integer_elements(BlockMatrix([[X], [X]])),
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Q.integer_elements(X))
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def test_matrix_element_sets_determinant_trace():
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assert ask(Q.integer(Determinant(X)), Q.integer_elements(X))
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assert ask(Q.integer(Trace(X)), Q.integer_elements(X))
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