475 lines
14 KiB
Python
475 lines
14 KiB
Python
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from sympy.core.function import expand_mul
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from sympy.core.numbers import I, Rational
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from sympy.core.singleton import S
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from sympy.core.symbol import Symbol
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from sympy.functions.elementary.miscellaneous import sqrt
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from sympy.functions.elementary.complexes import Abs
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from sympy.simplify.simplify import simplify
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from sympy.matrices.matrices import NonSquareMatrixError
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from sympy.matrices import Matrix, zeros, eye, SparseMatrix
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from sympy.abc import x, y, z
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from sympy.testing.pytest import raises, slow
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from sympy.testing.matrices import allclose
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def test_LUdecomp():
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testmat = Matrix([[0, 2, 5, 3],
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[3, 3, 7, 4],
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[8, 4, 0, 2],
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[-2, 6, 3, 4]])
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L, U, p = testmat.LUdecomposition()
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assert L.is_lower
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assert U.is_upper
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assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)
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testmat = Matrix([[6, -2, 7, 4],
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[0, 3, 6, 7],
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[1, -2, 7, 4],
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[-9, 2, 6, 3]])
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L, U, p = testmat.LUdecomposition()
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assert L.is_lower
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assert U.is_upper
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assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4)
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# non-square
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testmat = Matrix([[1, 2, 3],
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[4, 5, 6],
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[7, 8, 9],
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[10, 11, 12]])
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L, U, p = testmat.LUdecomposition(rankcheck=False)
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assert L.is_lower
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assert U.is_upper
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assert (L*U).permute_rows(p, 'backward') - testmat == zeros(4, 3)
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# square and singular
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testmat = Matrix([[1, 2, 3],
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[2, 4, 6],
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[4, 5, 6]])
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L, U, p = testmat.LUdecomposition(rankcheck=False)
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assert L.is_lower
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assert U.is_upper
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assert (L*U).permute_rows(p, 'backward') - testmat == zeros(3)
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M = Matrix(((1, x, 1), (2, y, 0), (y, 0, z)))
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L, U, p = M.LUdecomposition()
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assert L.is_lower
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assert U.is_upper
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assert (L*U).permute_rows(p, 'backward') - M == zeros(3)
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mL = Matrix((
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(1, 0, 0),
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(2, 3, 0),
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))
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assert mL.is_lower is True
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assert mL.is_upper is False
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mU = Matrix((
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(1, 2, 3),
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(0, 4, 5),
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))
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assert mU.is_lower is False
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assert mU.is_upper is True
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# test FF LUdecomp
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M = Matrix([[1, 3, 3],
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[3, 2, 6],
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[3, 2, 2]])
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P, L, Dee, U = M.LUdecompositionFF()
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assert P*M == L*Dee.inv()*U
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M = Matrix([[1, 2, 3, 4],
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[3, -1, 2, 3],
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[3, 1, 3, -2],
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[6, -1, 0, 2]])
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P, L, Dee, U = M.LUdecompositionFF()
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assert P*M == L*Dee.inv()*U
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M = Matrix([[0, 0, 1],
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[2, 3, 0],
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[3, 1, 4]])
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P, L, Dee, U = M.LUdecompositionFF()
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assert P*M == L*Dee.inv()*U
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# issue 15794
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M = Matrix(
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[[1, 2, 3],
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[4, 5, 6],
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[7, 8, 9]]
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)
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raises(ValueError, lambda : M.LUdecomposition_Simple(rankcheck=True))
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def test_singular_value_decompositionD():
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A = Matrix([[1, 2], [2, 1]])
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U, S, V = A.singular_value_decomposition()
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assert U * S * V.T == A
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assert U.T * U == eye(U.cols)
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assert V.T * V == eye(V.cols)
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B = Matrix([[1, 2]])
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U, S, V = B.singular_value_decomposition()
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assert U * S * V.T == B
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assert U.T * U == eye(U.cols)
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assert V.T * V == eye(V.cols)
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C = Matrix([
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[1, 0, 0, 0, 2],
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[0, 0, 3, 0, 0],
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[0, 0, 0, 0, 0],
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[0, 2, 0, 0, 0],
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])
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U, S, V = C.singular_value_decomposition()
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assert U * S * V.T == C
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assert U.T * U == eye(U.cols)
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assert V.T * V == eye(V.cols)
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D = Matrix([[Rational(1, 3), sqrt(2)], [0, Rational(1, 4)]])
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U, S, V = D.singular_value_decomposition()
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assert simplify(U.T * U) == eye(U.cols)
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assert simplify(V.T * V) == eye(V.cols)
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assert simplify(U * S * V.T) == D
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def test_QR():
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A = Matrix([[1, 2], [2, 3]])
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Q, S = A.QRdecomposition()
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R = Rational
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assert Q == Matrix([
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[ 5**R(-1, 2), (R(2)/5)*(R(1)/5)**R(-1, 2)],
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[2*5**R(-1, 2), (-R(1)/5)*(R(1)/5)**R(-1, 2)]])
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assert S == Matrix([[5**R(1, 2), 8*5**R(-1, 2)], [0, (R(1)/5)**R(1, 2)]])
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assert Q*S == A
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assert Q.T * Q == eye(2)
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A = Matrix([[1, 1, 1], [1, 1, 3], [2, 3, 4]])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[12, 0, -51], [6, 0, 167], [-4, 0, 24]])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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x = Symbol('x')
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A = Matrix([x])
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Q, R = A.QRdecomposition()
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assert Q == Matrix([x / Abs(x)])
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assert R == Matrix([Abs(x)])
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A = Matrix([[x, 0], [0, x]])
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Q, R = A.QRdecomposition()
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assert Q == x / Abs(x) * Matrix([[1, 0], [0, 1]])
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assert R == Abs(x) * Matrix([[1, 0], [0, 1]])
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def test_QR_non_square():
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# Narrow (cols < rows) matrices
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A = Matrix([[9, 0, 26], [12, 0, -7], [0, 4, 4], [0, -3, -3]])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[1, -1, 4], [1, 4, -2], [1, 4, 2], [1, -1, 0]])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix(2, 1, [1, 2])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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# Wide (cols > rows) matrices
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A = Matrix([[1, 2, 3], [4, 5, 6]])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[1, 2, 3, 4], [1, 4, 9, 16], [1, 8, 27, 64]])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix(1, 2, [1, 2])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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def test_QR_trivial():
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# Rank deficient matrices
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A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]]).T
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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# Zero rank matrices
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A = Matrix([[0, 0, 0]])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[0, 0, 0]]).T
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[0, 0, 0], [0, 0, 0]])
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[0, 0, 0], [0, 0, 0]]).T
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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# Rank deficient matrices with zero norm from beginning columns
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A = Matrix([[0, 0, 0], [1, 2, 3]]).T
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0]]).T
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0], [2, 4, 6, 8]]).T
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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A = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 2, 3]]).T
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Q, R = A.QRdecomposition()
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assert Q.T * Q == eye(Q.cols)
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assert R.is_upper
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assert A == Q*R
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def test_QR_float():
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A = Matrix([[1, 1], [1, 1.01]])
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Q, R = A.QRdecomposition()
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assert allclose(Q * R, A)
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assert allclose(Q * Q.T, Matrix.eye(2))
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assert allclose(Q.T * Q, Matrix.eye(2))
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A = Matrix([[1, 1], [1, 1.001]])
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Q, R = A.QRdecomposition()
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assert allclose(Q * R, A)
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assert allclose(Q * Q.T, Matrix.eye(2))
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assert allclose(Q.T * Q, Matrix.eye(2))
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def test_LUdecomposition_Simple_iszerofunc():
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# Test if callable passed to matrices.LUdecomposition_Simple() as iszerofunc keyword argument is used inside
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# matrices.LUdecomposition_Simple()
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magic_string = "I got passed in!"
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def goofyiszero(value):
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raise ValueError(magic_string)
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try:
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lu, p = Matrix([[1, 0], [0, 1]]).LUdecomposition_Simple(iszerofunc=goofyiszero)
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except ValueError as err:
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assert magic_string == err.args[0]
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return
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assert False
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def test_LUdecomposition_iszerofunc():
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# Test if callable passed to matrices.LUdecomposition() as iszerofunc keyword argument is used inside
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# matrices.LUdecomposition_Simple()
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magic_string = "I got passed in!"
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def goofyiszero(value):
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raise ValueError(magic_string)
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try:
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l, u, p = Matrix([[1, 0], [0, 1]]).LUdecomposition(iszerofunc=goofyiszero)
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except ValueError as err:
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assert magic_string == err.args[0]
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return
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assert False
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def test_LDLdecomposition():
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raises(NonSquareMatrixError, lambda: Matrix((1, 2)).LDLdecomposition())
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raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition())
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raises(ValueError, lambda: Matrix(((5 + I, 0), (0, 1))).LDLdecomposition())
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raises(ValueError, lambda: Matrix(((1, 5), (5, 1))).LDLdecomposition())
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raises(ValueError, lambda: Matrix(((1, 2), (3, 4))).LDLdecomposition(hermitian=False))
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A = Matrix(((1, 5), (5, 1)))
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L, D = A.LDLdecomposition(hermitian=False)
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assert L * D * L.T == A
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A = Matrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
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L, D = A.LDLdecomposition()
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assert L * D * L.T == A
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assert L.is_lower
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assert L == Matrix([[1, 0, 0], [ Rational(3, 5), 1, 0], [Rational(-1, 5), Rational(1, 3), 1]])
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assert D.is_diagonal()
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assert D == Matrix([[25, 0, 0], [0, 9, 0], [0, 0, 9]])
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A = Matrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
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L, D = A.LDLdecomposition()
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assert expand_mul(L * D * L.H) == A
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assert L.expand() == Matrix([[1, 0, 0], [I/2, 1, 0], [S.Half - I/2, 0, 1]])
|
||
|
|
assert D.expand() == Matrix(((4, 0, 0), (0, 1, 0), (0, 0, 9)))
|
||
|
|
|
||
|
|
raises(NonSquareMatrixError, lambda: SparseMatrix((1, 2)).LDLdecomposition())
|
||
|
|
raises(ValueError, lambda: SparseMatrix(((1, 2), (3, 4))).LDLdecomposition())
|
||
|
|
raises(ValueError, lambda: SparseMatrix(((5 + I, 0), (0, 1))).LDLdecomposition())
|
||
|
|
raises(ValueError, lambda: SparseMatrix(((1, 5), (5, 1))).LDLdecomposition())
|
||
|
|
raises(ValueError, lambda: SparseMatrix(((1, 2), (3, 4))).LDLdecomposition(hermitian=False))
|
||
|
|
A = SparseMatrix(((1, 5), (5, 1)))
|
||
|
|
L, D = A.LDLdecomposition(hermitian=False)
|
||
|
|
assert L * D * L.T == A
|
||
|
|
A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
|
||
|
|
L, D = A.LDLdecomposition()
|
||
|
|
assert L * D * L.T == A
|
||
|
|
assert L.is_lower
|
||
|
|
assert L == Matrix([[1, 0, 0], [ Rational(3, 5), 1, 0], [Rational(-1, 5), Rational(1, 3), 1]])
|
||
|
|
assert D.is_diagonal()
|
||
|
|
assert D == Matrix([[25, 0, 0], [0, 9, 0], [0, 0, 9]])
|
||
|
|
A = SparseMatrix(((4, -2*I, 2 + 2*I), (2*I, 2, -1 + I), (2 - 2*I, -1 - I, 11)))
|
||
|
|
L, D = A.LDLdecomposition()
|
||
|
|
assert expand_mul(L * D * L.H) == A
|
||
|
|
assert L == Matrix(((1, 0, 0), (I/2, 1, 0), (S.Half - I/2, 0, 1)))
|
||
|
|
assert D == Matrix(((4, 0, 0), (0, 1, 0), (0, 0, 9)))
|
||
|
|
|
||
|
|
def test_pinv_succeeds_with_rank_decomposition_method():
|
||
|
|
# Test rank decomposition method of pseudoinverse succeeding
|
||
|
|
As = [Matrix([
|
||
|
|
[61, 89, 55, 20, 71, 0],
|
||
|
|
[62, 96, 85, 85, 16, 0],
|
||
|
|
[69, 56, 17, 4, 54, 0],
|
||
|
|
[10, 54, 91, 41, 71, 0],
|
||
|
|
[ 7, 30, 10, 48, 90, 0],
|
||
|
|
[0,0,0,0,0,0]])]
|
||
|
|
for A in As:
|
||
|
|
A_pinv = A.pinv(method="RD")
|
||
|
|
AAp = A * A_pinv
|
||
|
|
ApA = A_pinv * A
|
||
|
|
assert simplify(AAp * A) == A
|
||
|
|
assert simplify(ApA * A_pinv) == A_pinv
|
||
|
|
assert AAp.H == AAp
|
||
|
|
assert ApA.H == ApA
|
||
|
|
|
||
|
|
def test_rank_decomposition():
|
||
|
|
a = Matrix(0, 0, [])
|
||
|
|
c, f = a.rank_decomposition()
|
||
|
|
assert f.is_echelon
|
||
|
|
assert c.cols == f.rows == a.rank()
|
||
|
|
assert c * f == a
|
||
|
|
|
||
|
|
a = Matrix(1, 1, [5])
|
||
|
|
c, f = a.rank_decomposition()
|
||
|
|
assert f.is_echelon
|
||
|
|
assert c.cols == f.rows == a.rank()
|
||
|
|
assert c * f == a
|
||
|
|
|
||
|
|
a = Matrix(3, 3, [1, 2, 3, 1, 2, 3, 1, 2, 3])
|
||
|
|
c, f = a.rank_decomposition()
|
||
|
|
assert f.is_echelon
|
||
|
|
assert c.cols == f.rows == a.rank()
|
||
|
|
assert c * f == a
|
||
|
|
|
||
|
|
a = Matrix([
|
||
|
|
[0, 0, 1, 2, 2, -5, 3],
|
||
|
|
[-1, 5, 2, 2, 1, -7, 5],
|
||
|
|
[0, 0, -2, -3, -3, 8, -5],
|
||
|
|
[-1, 5, 0, -1, -2, 1, 0]])
|
||
|
|
c, f = a.rank_decomposition()
|
||
|
|
assert f.is_echelon
|
||
|
|
assert c.cols == f.rows == a.rank()
|
||
|
|
assert c * f == a
|
||
|
|
|
||
|
|
|
||
|
|
@slow
|
||
|
|
def test_upper_hessenberg_decomposition():
|
||
|
|
A = Matrix([
|
||
|
|
[1, 0, sqrt(3)],
|
||
|
|
[sqrt(2), Rational(1, 2), 2],
|
||
|
|
[1, Rational(1, 4), 3],
|
||
|
|
])
|
||
|
|
H, P = A.upper_hessenberg_decomposition()
|
||
|
|
assert simplify(P * P.H) == eye(P.cols)
|
||
|
|
assert simplify(P.H * P) == eye(P.cols)
|
||
|
|
assert H.is_upper_hessenberg
|
||
|
|
assert (simplify(P * H * P.H)) == A
|
||
|
|
|
||
|
|
|
||
|
|
B = Matrix([
|
||
|
|
[1, 2, 10],
|
||
|
|
[8, 2, 5],
|
||
|
|
[3, 12, 34],
|
||
|
|
])
|
||
|
|
H, P = B.upper_hessenberg_decomposition()
|
||
|
|
assert simplify(P * P.H) == eye(P.cols)
|
||
|
|
assert simplify(P.H * P) == eye(P.cols)
|
||
|
|
assert H.is_upper_hessenberg
|
||
|
|
assert simplify(P * H * P.H) == B
|
||
|
|
|
||
|
|
C = Matrix([
|
||
|
|
[1, sqrt(2), 2, 3],
|
||
|
|
[0, 5, 3, 4],
|
||
|
|
[1, 1, 4, sqrt(5)],
|
||
|
|
[0, 2, 2, 3]
|
||
|
|
])
|
||
|
|
|
||
|
|
H, P = C.upper_hessenberg_decomposition()
|
||
|
|
assert simplify(P * P.H) == eye(P.cols)
|
||
|
|
assert simplify(P.H * P) == eye(P.cols)
|
||
|
|
assert H.is_upper_hessenberg
|
||
|
|
assert simplify(P * H * P.H) == C
|
||
|
|
|
||
|
|
D = Matrix([
|
||
|
|
[1, 2, 3],
|
||
|
|
[-3, 5, 6],
|
||
|
|
[4, -8, 9],
|
||
|
|
])
|
||
|
|
H, P = D.upper_hessenberg_decomposition()
|
||
|
|
assert simplify(P * P.H) == eye(P.cols)
|
||
|
|
assert simplify(P.H * P) == eye(P.cols)
|
||
|
|
assert H.is_upper_hessenberg
|
||
|
|
assert simplify(P * H * P.H) == D
|
||
|
|
|
||
|
|
E = Matrix([
|
||
|
|
[1, 0, 0, 0],
|
||
|
|
[0, 1, 0, 0],
|
||
|
|
[1, 1, 0, 1],
|
||
|
|
[1, 1, 1, 0]
|
||
|
|
])
|
||
|
|
|
||
|
|
H, P = E.upper_hessenberg_decomposition()
|
||
|
|
assert simplify(P * P.H) == eye(P.cols)
|
||
|
|
assert simplify(P.H * P) == eye(P.cols)
|
||
|
|
assert H.is_upper_hessenberg
|
||
|
|
assert simplify(P * H * P.H) == E
|