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| """ | |
| 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 | |