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| from sympy.functions.elementary.miscellaneous import sqrt | |
| from sympy.simplify.powsimp import powsimp | |
| from sympy.testing.pytest import raises | |
| from sympy.core.expr import unchanged | |
| from sympy.core import symbols, S | |
| from sympy.matrices import Identity, MatrixSymbol, ImmutableMatrix, ZeroMatrix, OneMatrix, Matrix | |
| from sympy.matrices.exceptions import NonSquareMatrixError | |
| from sympy.matrices.expressions import MatPow, MatAdd, MatMul | |
| from sympy.matrices.expressions.inverse import Inverse | |
| from sympy.matrices.expressions.matexpr import MatrixElement | |
| n, m, l, k = symbols('n m l k', integer=True) | |
| A = MatrixSymbol('A', n, m) | |
| B = MatrixSymbol('B', m, l) | |
| C = MatrixSymbol('C', n, n) | |
| D = MatrixSymbol('D', n, n) | |
| E = MatrixSymbol('E', m, n) | |
| def test_entry_matrix(): | |
| X = ImmutableMatrix([[1, 2], [3, 4]]) | |
| assert MatPow(X, 0)[0, 0] == 1 | |
| assert MatPow(X, 0)[0, 1] == 0 | |
| assert MatPow(X, 1)[0, 0] == 1 | |
| assert MatPow(X, 1)[0, 1] == 2 | |
| assert MatPow(X, 2)[0, 0] == 7 | |
| def test_entry_symbol(): | |
| from sympy.concrete import Sum | |
| assert MatPow(C, 0)[0, 0] == 1 | |
| assert MatPow(C, 0)[0, 1] == 0 | |
| assert MatPow(C, 1)[0, 0] == C[0, 0] | |
| assert isinstance(MatPow(C, 2)[0, 0], Sum) | |
| assert isinstance(MatPow(C, n)[0, 0], MatrixElement) | |
| def test_as_explicit_symbol(): | |
| X = MatrixSymbol('X', 2, 2) | |
| assert MatPow(X, 0).as_explicit() == ImmutableMatrix(Identity(2)) | |
| assert MatPow(X, 1).as_explicit() == X.as_explicit() | |
| assert MatPow(X, 2).as_explicit() == (X.as_explicit())**2 | |
| assert MatPow(X, n).as_explicit() == ImmutableMatrix([ | |
| [(X ** n)[0, 0], (X ** n)[0, 1]], | |
| [(X ** n)[1, 0], (X ** n)[1, 1]], | |
| ]) | |
| a = MatrixSymbol("a", 3, 1) | |
| b = MatrixSymbol("b", 3, 1) | |
| c = MatrixSymbol("c", 3, 1) | |
| expr = (a.T*b)**S.Half | |
| assert expr.as_explicit() == Matrix([[sqrt(a[0, 0]*b[0, 0] + a[1, 0]*b[1, 0] + a[2, 0]*b[2, 0])]]) | |
| expr = c*(a.T*b)**S.Half | |
| m = sqrt(a[0, 0]*b[0, 0] + a[1, 0]*b[1, 0] + a[2, 0]*b[2, 0]) | |
| assert expr.as_explicit() == Matrix([[c[0, 0]*m], [c[1, 0]*m], [c[2, 0]*m]]) | |
| expr = (a*b.T)**S.Half | |
| denom = sqrt(a[0, 0]*b[0, 0] + a[1, 0]*b[1, 0] + a[2, 0]*b[2, 0]) | |
| expected = (a*b.T).as_explicit()/denom | |
| assert expr.as_explicit() == expected | |
| expr = X**-1 | |
| det = X[0, 0]*X[1, 1] - X[1, 0]*X[0, 1] | |
| expected = Matrix([[X[1, 1], -X[0, 1]], [-X[1, 0], X[0, 0]]])/det | |
| assert expr.as_explicit() == expected | |
| expr = X**m | |
| assert expr.as_explicit() == X.as_explicit()**m | |
| def test_as_explicit_matrix(): | |
| A = ImmutableMatrix([[1, 2], [3, 4]]) | |
| assert MatPow(A, 0).as_explicit() == ImmutableMatrix(Identity(2)) | |
| assert MatPow(A, 1).as_explicit() == A | |
| assert MatPow(A, 2).as_explicit() == A**2 | |
| assert MatPow(A, -1).as_explicit() == A.inv() | |
| assert MatPow(A, -2).as_explicit() == (A.inv())**2 | |
| # less expensive than testing on a 2x2 | |
| A = ImmutableMatrix([4]) | |
| assert MatPow(A, S.Half).as_explicit() == A**S.Half | |
| def test_doit_symbol(): | |
| assert MatPow(C, 0).doit() == Identity(n) | |
| assert MatPow(C, 1).doit() == C | |
| assert MatPow(C, -1).doit() == C.I | |
| for r in [2, S.Half, S.Pi, n]: | |
| assert MatPow(C, r).doit() == MatPow(C, r) | |
| def test_doit_matrix(): | |
| X = ImmutableMatrix([[1, 2], [3, 4]]) | |
| assert MatPow(X, 0).doit() == ImmutableMatrix(Identity(2)) | |
| assert MatPow(X, 1).doit() == X | |
| assert MatPow(X, 2).doit() == X**2 | |
| assert MatPow(X, -1).doit() == X.inv() | |
| assert MatPow(X, -2).doit() == (X.inv())**2 | |
| # less expensive than testing on a 2x2 | |
| assert MatPow(ImmutableMatrix([4]), S.Half).doit() == ImmutableMatrix([2]) | |
| X = ImmutableMatrix([[0, 2], [0, 4]]) # det() == 0 | |
| raises(ValueError, lambda: MatPow(X,-1).doit()) | |
| raises(ValueError, lambda: MatPow(X,-2).doit()) | |
| def test_nonsquare(): | |
| A = MatrixSymbol('A', 2, 3) | |
| B = ImmutableMatrix([[1, 2, 3], [4, 5, 6]]) | |
| for r in [-1, 0, 1, 2, S.Half, S.Pi, n]: | |
| raises(NonSquareMatrixError, lambda: MatPow(A, r)) | |
| raises(NonSquareMatrixError, lambda: MatPow(B, r)) | |
| def test_doit_equals_pow(): #17179 | |
| X = ImmutableMatrix ([[1,0],[0,1]]) | |
| assert MatPow(X, n).doit() == X**n == X | |
| def test_doit_nested_MatrixExpr(): | |
| X = ImmutableMatrix([[1, 2], [3, 4]]) | |
| Y = ImmutableMatrix([[2, 3], [4, 5]]) | |
| assert MatPow(MatMul(X, Y), 2).doit() == (X*Y)**2 | |
| assert MatPow(MatAdd(X, Y), 2).doit() == (X + Y)**2 | |
| def test_identity_power(): | |
| k = Identity(n) | |
| assert MatPow(k, 4).doit() == k | |
| assert MatPow(k, n).doit() == k | |
| assert MatPow(k, -3).doit() == k | |
| assert MatPow(k, 0).doit() == k | |
| l = Identity(3) | |
| assert MatPow(l, n).doit() == l | |
| assert MatPow(l, -1).doit() == l | |
| assert MatPow(l, 0).doit() == l | |
| def test_zero_power(): | |
| z1 = ZeroMatrix(n, n) | |
| assert MatPow(z1, 3).doit() == z1 | |
| raises(ValueError, lambda:MatPow(z1, -1).doit()) | |
| assert MatPow(z1, 0).doit() == Identity(n) | |
| assert MatPow(z1, n).doit() == z1 | |
| raises(ValueError, lambda:MatPow(z1, -2).doit()) | |
| z2 = ZeroMatrix(4, 4) | |
| assert MatPow(z2, n).doit() == z2 | |
| raises(ValueError, lambda:MatPow(z2, -3).doit()) | |
| assert MatPow(z2, 2).doit() == z2 | |
| assert MatPow(z2, 0).doit() == Identity(4) | |
| raises(ValueError, lambda:MatPow(z2, -1).doit()) | |
| def test_OneMatrix_power(): | |
| o = OneMatrix(3, 3) | |
| assert o ** 0 == Identity(3) | |
| assert o ** 1 == o | |
| assert o * o == o ** 2 == 3 * o | |
| assert o * o * o == o ** 3 == 9 * o | |
| o = OneMatrix(n, n) | |
| assert o * o == o ** 2 == n * o | |
| # powsimp necessary as n ** (n - 2) * n does not produce n ** (n - 1) | |
| assert powsimp(o ** (n - 1) * o) == o ** n == n ** (n - 1) * o | |
| def test_transpose_power(): | |
| from sympy.matrices.expressions.transpose import Transpose as TP | |
| assert (C*D).T**5 == ((C*D)**5).T == (D.T * C.T)**5 | |
| assert ((C*D).T**5).T == (C*D)**5 | |
| assert (C.T.I.T)**7 == C**-7 | |
| assert (C.T**l).T**k == C**(l*k) | |
| assert ((E.T * A.T)**5).T == (A*E)**5 | |
| assert ((A*E).T**5).T**7 == (A*E)**35 | |
| assert TP(TP(C**2 * D**3)**5).doit() == (C**2 * D**3)**5 | |
| assert ((D*C)**-5).T**-5 == ((D*C)**25).T | |
| assert (((D*C)**l).T**k).T == (D*C)**(l*k) | |
| def test_Inverse(): | |
| assert Inverse(MatPow(C, 0)).doit() == Identity(n) | |
| assert Inverse(MatPow(C, 1)).doit() == Inverse(C) | |
| assert Inverse(MatPow(C, 2)).doit() == MatPow(C, -2) | |
| assert Inverse(MatPow(C, -1)).doit() == C | |
| assert MatPow(Inverse(C), 0).doit() == Identity(n) | |
| assert MatPow(Inverse(C), 1).doit() == Inverse(C) | |
| assert MatPow(Inverse(C), 2).doit() == MatPow(C, -2) | |
| assert MatPow(Inverse(C), -1).doit() == C | |
| def test_combine_powers(): | |
| assert (C ** 1) ** 1 == C | |
| assert (C ** 2) ** 3 == MatPow(C, 6) | |
| assert (C ** -2) ** -3 == MatPow(C, 6) | |
| assert (C ** -1) ** -1 == C | |
| assert (((C ** 2) ** 3) ** 4) ** 5 == MatPow(C, 120) | |
| assert (C ** n) ** n == C ** (n ** 2) | |
| def test_unchanged(): | |
| assert unchanged(MatPow, C, 0) | |
| assert unchanged(MatPow, C, 1) | |
| assert unchanged(MatPow, Inverse(C), -1) | |
| assert unchanged(Inverse, MatPow(C, -1), -1) | |
| assert unchanged(MatPow, MatPow(C, -1), -1) | |
| assert unchanged(MatPow, MatPow(C, 1), 1) | |
| def test_no_exponentiation(): | |
| # if this passes, Pow.as_numer_denom should recognize | |
| # MatAdd as exponent | |
| raises(NotImplementedError, lambda: 3**(-2*C)) | |