Exemplo n.º 1
0
def test_QR():
    A = Matrix([[1, 2], [2, 3]])
    Q, S = A.QRdecomposition()
    R = Rational
    assert Q == Matrix([[5**R(-1, 2), (R(2) / 5) * (R(1) / 5)**R(-1, 2)],
                        [2 * 5**R(-1, 2), (-R(1) / 5) * (R(1) / 5)**R(-1, 2)]])
    assert S == Matrix([[5**R(1, 2), 8 * 5**R(-1, 2)],
                        [0, (R(1) / 5)**R(1, 2)]])
    assert Q * S == A
    assert Q.T * Q == eye(2)

    A = Matrix([[1, 1, 1], [1, 1, 3], [2, 3, 4]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[12, 0, -51], [6, 0, 167], [-4, 0, 24]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    x = Symbol('x')
    A = Matrix([x])
    Q, R = A.QRdecomposition()
    assert Q == Matrix([x / Abs(x)])
    assert R == Matrix([Abs(x)])

    A = Matrix([[x, 0], [0, x]])
    Q, R = A.QRdecomposition()
    assert Q == x / Abs(x) * Matrix([[1, 0], [0, 1]])
    assert R == Abs(x) * Matrix([[1, 0], [0, 1]])
Exemplo n.º 2
0
def test_QR_float():
    A = Matrix([[1, 1], [1, 1.01]])
    Q, R = A.QRdecomposition()
    assert allclose(Q * R, A)
    assert allclose(Q * Q.T, Matrix.eye(2))
    assert allclose(Q.T * Q, Matrix.eye(2))

    A = Matrix([[1, 1], [1, 1.001]])
    Q, R = A.QRdecomposition()
    assert allclose(Q * R, A)
    assert allclose(Q * Q.T, Matrix.eye(2))
    assert allclose(Q.T * Q, Matrix.eye(2))
Exemplo n.º 3
0
def test_QR():
    A = Matrix([[1, 2], [2, 3]])
    Q, S = A.QRdecomposition()
    R = Rational
    assert Q == Matrix([[5**R(-1, 2), (R(2) / 5) * (R(1) / 5)**R(-1, 2)],
                        [2 * 5**R(-1, 2), (-R(1) / 5) * (R(1) / 5)**R(-1, 2)]])
    assert S == Matrix([[5**R(1, 2), 8 * 5**R(-1, 2)],
                        [0, (R(1) / 5)**R(1, 2)]])
    assert Q * S == A
    assert Q.T * Q == eye(2)

    A = Matrix([[1, 1, 1], [1, 1, 3], [2, 3, 4]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R
Exemplo n.º 4
0
def test_QR_non_square():
    # Narrow (cols < rows) matrices
    A = Matrix([[9, 0, 26], [12, 0, -7], [0, 4, 4], [0, -3, -3]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[1, -1, 4], [1, 4, -2], [1, 4, 2], [1, -1, 0]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix(2, 1, [1, 2])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    # Wide (cols > rows) matrices
    A = Matrix([[1, 2, 3], [4, 5, 6]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[1, 2, 3, 4], [1, 4, 9, 16], [1, 8, 27, 64]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix(1, 2, [1, 2])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R
Exemplo n.º 5
0
def test_QR_trivial():
    # Rank deficient matrices
    A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[1, 1, 1], [2, 2, 2], [3, 3, 3], [4, 4, 4]]).T
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    # Zero rank matrices
    A = Matrix([[0, 0, 0]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[0, 0, 0]]).T
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[0, 0, 0], [0, 0, 0]])
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[0, 0, 0], [0, 0, 0]]).T
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    # Rank deficient matrices with zero norm from beginning columns
    A = Matrix([[0, 0, 0], [1, 2, 3]]).T
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0]]).T
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[0, 0, 0, 0], [1, 2, 3, 4], [0, 0, 0, 0], [2, 4, 6, 8]]).T
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R

    A = Matrix([[0, 0, 0], [0, 0, 0], [0, 0, 0], [1, 2, 3]]).T
    Q, R = A.QRdecomposition()
    assert Q.T * Q == eye(Q.cols)
    assert R.is_upper
    assert A == Q * R