Python numpy.outer() Examples

def _get_knmat(exog, xcov, sl):
    # Utility function, see equation 2.2 of Barber & Candes.

    nobs, nvar = exog.shape

    ash = np.linalg.inv(xcov)
    ash *= -np.outer(sl, sl)
    i, j = np.diag_indices(nvar)
    ash[i, j] += 2 * sl

    umat = np.random.normal(size=(nobs, nvar))
    u, _ = np.linalg.qr(exog)
    umat -= np.dot(u, np.dot(u.T, umat))
    umat, _ = np.linalg.qr(umat)

    ashr, xc, _ = np.linalg.svd(ash, 0)
    ashr *= np.sqrt(xc)
    ashr = ashr.T

    ex = (sl[:, None] * np.linalg.solve(xcov, exog.T)).T
    exogn = exog - ex + np.dot(umat, ashr)

    return exogn 

def test_Flip2D(par):
    """Dot-test, forward and adjoint for Flip operator on 2d signal
    """
    np.random.seed(10)
    x = {}
    x['0'] = np.outer(np.arange(par['ny']), np.ones(par['nx'])) + \
             par['imag'] * np.outer(np.arange(par['ny']), np.ones(par['nx']))
    x['1'] = np.outer(np.ones(par['ny']), np.arange(par['nx'])) + \
             par['imag'] * np.outer(np.ones(par['ny']), np.arange(par['nx']))

    for dir in [0, 1]:
        Fop = Flip(par['ny']*par['nx'], dims=(par['ny'], par['nx']),
                   dir=dir, dtype=par['dtype'])
        assert dottest(Fop, par['ny']*par['nx'], par['ny']*par['nx'])

        y = Fop * x[str(dir)].flatten()
        xadj = Fop.H * y.flatten()
        xadj = xadj.reshape(par['ny'], par['nx'])
        assert_array_equal(x[str(dir)], xadj) 

def gen_disentangler_psi_prod(psiP, psiQ):
    """generate a PurificationMPS as tensorproduct (psi_P x psi_Q).

    psiQ should have the same `sites` as psiP.
    """
    L = psiP.L
    Bs = []
    for i in range(L):
        BP = psiP.get_B(i)
        BQ = psiQ.get_B(i)
        B2 = npc.outer(BP, BQ.conj())
        B2 = B2.combine_legs([['vL', 'vL*'], ['vR', 'vR*']], qconj=[+1, -1])
        B2.ireplace_labels(['(vL.vL*)', '(vR.vR*)', 'p*'], ['vL', 'vR', 'q'])
        Bs.append(B2)
    Ss = [np.outer(S, S2).flatten() for S, S2 in zip(psiP._S, psiQ._S)]
    return purification_mps.PurificationMPS(psiP.sites, Bs, Ss) 

def test_TakeTransposeInnerOuter(self):
        # Test of take, transpose, inner, outer products
        x = arange(24)
        y = np.arange(24)
        x[5:6] = masked
        x = x.reshape(2, 3, 4)
        y = y.reshape(2, 3, 4)
        assert_equal(np.transpose(y, (2, 0, 1)), transpose(x, (2, 0, 1)))
        assert_equal(np.take(y, (2, 0, 1), 1), take(x, (2, 0, 1), 1))
        assert_equal(np.inner(filled(x, 0), filled(y, 0)),
                     inner(x, y))
        assert_equal(np.outer(filled(x, 0), filled(y, 0)),
                     outer(x, y))
        y = array(['abc', 1, 'def', 2, 3], object)
        y[2] = masked
        t = take(y, [0, 3, 4])
        assert_(t[0] == 'abc')
        assert_(t[1] == 2)
        assert_(t[2] == 3) 

def tdhf_frozen_mask(eri, kind="ov"):
    if isinstance(eri.nocc, int):
        nocc = int(eri.model.mo_occ.sum() // 2)
        mask = eri.space
    else:
        nocc = numpy.array(tuple(int(i.sum() // 2) for i in eri.model.mo_occ))
        assert numpy.all(nocc == nocc[0])
        assert numpy.all(eri.space == eri.space[0, numpy.newaxis, :])
        nocc = nocc[0]
        mask = eri.space[0]
    mask_o = mask[:nocc]
    mask_v = mask[nocc:]
    if kind == "ov":
        mask_ov = numpy.outer(mask_o, mask_v).reshape(-1)
        return numpy.tile(mask_ov, 2)
    elif kind == "1ov":
        return numpy.outer(mask_o, mask_v).reshape(-1)
    elif kind == "sov":
        mask_ov = numpy.outer(mask_o, mask_v).reshape(-1)
        nk = len(eri.model.mo_occ)
        return numpy.tile(mask_ov, 2 * nk ** 2)
    elif kind == "o,v":
        return mask_o, mask_v 

def quintic_spline(dx, dtype=np.float64):
    def inner(x):
        return 1 + x ** 3 / 12 * (-95 + 138 * x - 55 * x ** 2)

    def middle(x):
        return (x - 1) * (x - 2) / 24 * (-138 + 348 * x - 249 * x ** 2 + 55 * x ** 3)

    def outer(x):
        return (x - 2) * (x - 3) ** 2 / 24 * (-54 + 50 * x - 11 * x ** 2)

    window = np.arange(-3, 4)
    x = np.abs(dx - window)
    result = np.piecewise(
        x,
        [x <= 1, (x > 1) & (x <= 2), (x > 2) & (x <= 3)],
        [lambda x: inner(x), lambda x: middle(x), lambda x: outer(x)],
    )
    return result, window 

def map_fit(interface, state, label, inp):
    """
    Function calculates matrices ete and etde for every sample, aggregates and output them.
    """
    import numpy as np
    ete, etde = 0, 0
    out = interface.output(0)

    for row in inp:
        row = row.strip().split(state["delimiter"])  # split row
        if len(row) > 1:  # check if row is empty
            # intercept term is added to every sample
            x = np.array([(0 if v in state["missing_vals"] else float(v)) for i, v in enumerate(row) if
                          i in state["X_indices"]] + [-1])
            # map label value to 1 or -1. If label does not match set error
            y = 1 if state["y_map"][0] == row[state["y_index"]] else -1 if state["y_map"][1] == row[
                state["y_index"]] else "Error"
            ete += np.outer(x, x)
            etde += x * y
    out.add("etde", etde)
    for i, row in enumerate(ete):
        out.add(i, row) 

def test_Symmetrize2D(par):
    """Dot-test, forward and inverse for Symmetrize operator on 2d signal
    """
    np.random.seed(10)
    x = {}
    x['0'] = np.outer(np.arange(par['ny']), np.ones(par['nx'])) + \
             par['imag'] * np.outer(np.arange(par['ny']), np.ones(par['nx']))
    x['1'] = np.outer(np.ones(par['ny']), np.arange(par['nx'])) + \
             par['imag'] * np.outer(np.ones(par['ny']), np.arange(par['nx']))

    for dir in [0, 1]:
        Sop = Symmetrize(par['ny']*par['nx'],
                         dims=(par['ny'], par['nx']),
                         dir=dir, dtype=par['dtype'])
        y = Sop * x[str(dir)].flatten()
        assert dottest(Sop, y.size, par['ny']*par['nx'])

        xinv = Sop / y
        assert_array_almost_equal(x[str(dir)].ravel(), xinv, decimal=3) 

def quat2mat(quaternion):
    """
    Converts given quaternion (x, y, z, w) to matrix.

    Args:
        quaternion: vec4 float angles

    Returns:
        3x3 rotation matrix
    """
    q = np.array(quaternion, dtype=np.float32, copy=True)[[3, 0, 1, 2]]
    n = np.dot(q, q)
    if n < EPS:
        return np.identity(3)
    q *= math.sqrt(2.0 / n)
    q = np.outer(q, q)
    return np.array(
        [
            [1.0 - q[2, 2] - q[3, 3], q[1, 2] - q[3, 0], q[1, 3] + q[2, 0]],
            [q[1, 2] + q[3, 0], 1.0 - q[1, 1] - q[3, 3], q[2, 3] - q[1, 0]],
            [q[1, 3] - q[2, 0], q[2, 3] + q[1, 0], 1.0 - q[1, 1] - q[2, 2]],
        ]
    ) 

def _eigen_components(self):
        components = [(0, np.diag([1, 1, 1, 0, 1, 0, 0, 1]))]
        nontrivial_part = np.zeros((3, 3), dtype=np.complex128)
        for ij, w in zip([(1, 2), (0, 2), (0, 1)], self.weights):
            nontrivial_part[ij] = w
            nontrivial_part[ij[::-1]] = w.conjugate()
        assert np.allclose(nontrivial_part, nontrivial_part.conj().T)
        eig_vals, eig_vecs = np.linalg.eigh(nontrivial_part)
        for eig_val, eig_vec in zip(eig_vals, eig_vecs.T):
            exp_factor = -eig_val / np.pi
            proj = np.zeros((8, 8), dtype=np.complex128)
            nontrivial_indices = np.array([3, 5, 6], dtype=np.intp)
            proj[nontrivial_indices[:, np.newaxis], nontrivial_indices] = (
                np.outer(eig_vec.conjugate(), eig_vec))
            components.append((exp_factor, proj))
        return components 

def map_fit(interface, state, label, inp):
    import numpy as np
    a, b = 0, 0
    out = interface.output(0)

    for row in inp:
        row = row.strip().split(state["delimiter"])
        if len(row) > 1:
            x = np.array([1] + [(0 if v in state["missing_vals"] else float(v)) for i, v in enumerate(row) if
                                i in state["X_indices"]])
            y = float(row[state["y_index"]])
            a += np.outer(x, x)
            b += x * y
    out.add("b", b)
    for i, row in enumerate(a):
        out.add(i, row) 

def map_fit(interface, state, label, inp):
    import numpy as np
    combiner = dict([(k, [0, 0]) for k in state["samples"].keys()])
    out = interface.output(0)

    for row in inp:
        row = row.strip().split(state["delimiter"])
        if len(row) > 1:
            x = np.array([1] + [(0 if v in state["missing_vals"] else float(v)) for i, v in enumerate(row) if
                                i in state["X_indices"]])
            y = float(row[state["y_index"]])

            for test_id, x1 in state["samples"].iteritems():
                w = np.exp(np.dot(-(x1 - x).transpose(), x1 - x) / state["tau"])
                combiner[test_id][0] += w * np.outer(x, x)
                combiner[test_id][1] += w * x * y

    for k, v in combiner.iteritems():
        out.add(k + state["delimiter"] + "b", v[1])
        for i in range(len(v[0])):
            out.add(k + state["delimiter"] + "A" + state["delimiter"] + str(i), v[0][i]) 

def tdhf_frozen_mask(eri, kind="ov"):
    if isinstance(eri.nocc, int):
        nocc = int(eri.model.mo_occ.sum() // 2)
        mask = eri.space
    else:
        nocc = numpy.array(tuple(int(i.sum() // 2) for i in eri.model.mo_occ))
        assert numpy.all(nocc == nocc[0])
        assert numpy.all(eri.space == eri.space[0, numpy.newaxis, :])
        nocc = nocc[0]
        mask = eri.space[0]
    mask_o = mask[:nocc]
    mask_v = mask[nocc:]
    if kind == "ov":
        mask_ov = numpy.outer(mask_o, mask_v).reshape(-1)
        return numpy.tile(mask_ov, 2)
    elif kind == "1ov":
        return numpy.outer(mask_o, mask_v).reshape(-1)
    elif kind == "sov":
        mask_ov = numpy.outer(mask_o, mask_v).reshape(-1)
        nk = len(eri.model.mo_occ)
        return numpy.tile(mask_ov, 2 * nk ** 2)
    elif kind == "o,v":
        return mask_o, mask_v 

def tdhf_frozen_mask(eri, kind="ov"):
    if isinstance(eri.nocc, int):
        nocc = int(eri.model.mo_occ.sum() // 2)
        mask = eri.space
    else:
        nocc = numpy.array(tuple(int(i.sum() // 2) for i in eri.model.mo_occ))
        assert numpy.all(nocc == nocc[0])
        assert numpy.all(eri.space == eri.space[0, numpy.newaxis, :])
        nocc = nocc[0]
        mask = eri.space[0]
    mask_o = mask[:nocc]
    mask_v = mask[nocc:]
    if kind == "ov":
        mask_ov = numpy.outer(mask_o, mask_v).reshape(-1)
        return numpy.tile(mask_ov, 2)
    elif kind == "1ov":
        return numpy.outer(mask_o, mask_v).reshape(-1)
    elif kind == "sov":
        mask_ov = numpy.outer(mask_o, mask_v).reshape(-1)
        nk = len(eri.model.mo_occ)
        return numpy.tile(mask_ov, 2 * nk ** 2)
    elif kind == "o,v":
        return mask_o, mask_v 

def test_testTakeTransposeInnerOuter(self):
        # Test of take, transpose, inner, outer products
        x = arange(24)
        y = np.arange(24)
        x[5:6] = masked
        x = x.reshape(2, 3, 4)
        y = y.reshape(2, 3, 4)
        assert_(eq(np.transpose(y, (2, 0, 1)), transpose(x, (2, 0, 1))))
        assert_(eq(np.take(y, (2, 0, 1), 1), take(x, (2, 0, 1), 1)))
        assert_(eq(np.inner(filled(x, 0), filled(y, 0)),
                   inner(x, y)))
        assert_(eq(np.outer(filled(x, 0), filled(y, 0)),
                   outer(x, y)))
        y = array(['abc', 1, 'def', 2, 3], object)
        y[2] = masked
        t = take(y, [0, 3, 4])
        assert_(t[0] == 'abc')
        assert_(t[1] == 2)
        assert_(t[2] == 3) 

def chi2_contingency(contingency):
    """Pearson chi^2 test of independence statistic on contingency table.

    https://en.wikipedia.org/wiki/Pearson%27s_chi-squared_test#Test_of_independence
    """
    contingency = numpy.array(contingency, dtype=int, ndmin=2)
    assert contingency.ndim == 2
    n = float(numpy.sum(contingency))
    n0 = contingency.shape[0]
    n1 = contingency.shape[1]
    assert 0 < n0
    assert 0 < n1
    p0 = numpy.sum(contingency, axis=1)/n
    p1 = numpy.sum(contingency, axis=0)/n
    expected = n * numpy.outer(p0, p1)
    return numpy.sum(((contingency - expected)**2)/expected) 

def test_b2q():
    # conversion of b matrix to q
    q = np.array([1,2,3])
    s = np.sqrt(np.sum(q * q)) # vector norm
    B = np.outer(q, q)
    assert_array_almost_equal(q*s, B2q(B))
    q = np.array([1,2,3])
    # check that the sign of the vector as positive x convention
    B = np.outer(-q, -q)
    assert_array_almost_equal(q*s, B2q(B))
    q = np.array([-1, 2, 3])
    B = np.outer(q, q)
    assert_array_almost_equal(-q*s, B2q(B))
    B = np.eye(3) * -1
    assert_raises(ValueError, B2q, B)
    # no error if we up the tolerance
    q = B2q(B, tol=1) 

def ssim(img1, img2):
    C1 = (0.01 * 255)**2
    C2 = (0.03 * 255)**2

    img1 = img1.astype(np.float64)
    img2 = img2.astype(np.float64)
    kernel = cv2.getGaussianKernel(11, 1.5)
    window = np.outer(kernel, kernel.transpose())

    mu1 = cv2.filter2D(img1, -1, window)[5:-5, 5:-5]  # valid
    mu2 = cv2.filter2D(img2, -1, window)[5:-5, 5:-5]
    mu1_sq = mu1**2
    mu2_sq = mu2**2
    mu1_mu2 = mu1 * mu2
    sigma1_sq = cv2.filter2D(img1**2, -1, window)[5:-5, 5:-5] - mu1_sq
    sigma2_sq = cv2.filter2D(img2**2, -1, window)[5:-5, 5:-5] - mu2_sq
    sigma12 = cv2.filter2D(img1 * img2, -1, window)[5:-5, 5:-5] - mu1_mu2

    ssim_map = ((2 * mu1_mu2 + C1) * (2 * sigma12 + C2)) / ((mu1_sq + mu2_sq + C1) *
                                                            (sigma1_sq + sigma2_sq + C2))
    return ssim_map.mean() 

def test_4(self):
        """
        Test of take, transpose, inner, outer products.

        """
        x = self.arange(24)
        y = np.arange(24)
        x[5:6] = self.masked
        x = x.reshape(2, 3, 4)
        y = y.reshape(2, 3, 4)
        assert self.allequal(np.transpose(y, (2, 0, 1)), self.transpose(x, (2, 0, 1)))
        assert self.allequal(np.take(y, (2, 0, 1), 1), self.take(x, (2, 0, 1), 1))
        assert self.allequal(np.inner(self.filled(x, 0), self.filled(y, 0)),
                            self.inner(x, y))
        assert self.allequal(np.outer(self.filled(x, 0), self.filled(y, 0)),
                            self.outer(x, y))
        y = self.array(['abc', 1, 'def', 2, 3], object)
        y[2] = self.masked
        t = self.take(y, [0, 3, 4])
        assert t[0] == 'abc'
        assert t[1] == 2
        assert t[2] == 3 

def optimization_variables_avg_QCQP():
    l = np.array([4, 1, 1, -5, 3], dtype=float)
    A = np.array([[1, 2, 3, 4],
                  [3, 1, 2, 2],
                  [2, 3, 1, 1],
                  [1, 3, 5, 1]], dtype=float)
    A = np.vstack([-np.mean(A, axis=1), A - np.mean(A, axis=1)])
    Q = A.dot(A.T)
    A_in = np.array([[1, 5, 2, 4],
                     [5, 2, 1, 2],
                     [0, 1, 1, 1],
                     [2, 4, 3, 5]], dtype=float)
    A_in = np.vstack([-np.mean(A_in, axis=1), A_in - np.mean(A_in, axis=1)])
    Q_in = A_in.dot(A_in.T)
    Q_in += np.outer(l, l)
    return l, Q, np.ones(5), Q_in 

def test_both_limit_angle_Q_iteration(self, optimization_variables_avg_QCQP):
        l, Q, A, Q_in = optimization_variables_avg_QCQP
        # l, Q, A = optimization_variables_avg
        # Q_in = 6 * np.eye(len(l)) + np.outer(l, l)
        max_angle = 20
        m = 2e-3
        m1 = 4e-3
        f = .01
        x = optimization_methods.optimize_focality(l, Q, f,
                                                   max_el_current=m,
                                                   max_total_current=m1,
                                                   Qin=Q_in,
                                                   max_angle=max_angle)
        x_sp = optimize_focality(
            l, Q, f, max_el_current=m, max_total_current=m1,
            Qin=Q_in, max_angle=max_angle)
        assert np.linalg.norm(x, 1) <= 2 * m1 + 1e-4
        assert np.all(np.abs(x) <= m + 1e-4)
        assert np.isclose(np.sum(x), 0)
        assert np.isclose(l.dot(x), f)
        assert np.arccos(l.dot(x) / np.sqrt(x.dot(Q_in).dot(x))) <= np.deg2rad(max_angle)
        assert np.allclose(x.dot(Q.dot(x)), x_sp.dot(Q.dot(x_sp)), rtol=1e-4, atol=1e-5) 

def mutcoh(A):
    """ mutcoh calculates the mutual coherence of a matrix A, i.e. the cosine
        of the smallest angle between two columns
      
        mutual_coherence = mutcoh(A)
 
        Input:
            A ... m x n matrix
 
        Output:
            mutual_coherence """

    T = np.dot(A.conj().T,A)
    s = np.sqrt(np.diag(T))    
    S = np.diag(s)
    mutual_coherence = np.max(1.0*(T-S)/np.outer(s,s))
    
    return mutual_coherence 

def test_robust_outer_product(self):
        """
        Ensure that robust_outer_product returns the expected results.
        Unfortunately, I cannot find a good case now where using the regular
        outer product gives incorrect results. However without a compelling
        reason to remove the function, I'll trust my earlier judgement in
        creating it in the first place.
        """
        # Define a vector whose outer product we want to take
        x = np.array([1e-100, 0.01])
        outer_product = np.outer(x, x)
        robust_outer_product = cc.robust_outer_product(x, x)

        # Perform the desired tests
        self.assertIsInstance(robust_outer_product, np.ndarray)
        self.assertEqual(robust_outer_product.shape, outer_product.shape)
        npt.assert_allclose(outer_product, robust_outer_product)

        return None 

def calc_asymptotic_covariance(hessian, fisher_info_matrix):
    """
    Parameters
    ----------
    hessian : 2D ndarray.
        It should have shape `(num_vars, num_vars)`. It is the matrix of second
        derivatives of the total loss across the dataset, with respect to each
        pair of coefficients being estimated.
    fisher_info_matrix : 2D ndarray.
        It should have a shape of `(num_vars, num_vars)`.  It is the
        approximation of the negative of the expected hessian formed by taking
        the outer product of (each observation's gradient of the loss function)
        with itself, and then summing across all observations.

    Returns
    -------
    huber_white_matrix : 2D ndarray.
        Will have shape `(num_vars, num_vars)`. The entries in the returned
        matrix are calculated by the following formula:
        `hess_inverse * fisher_info_matrix * hess_inverse`.
    """
    # Calculate the inverse of the hessian
    hess_inv = scipy.linalg.inv(hessian)

    return np.dot(hess_inv, np.dot(fisher_info_matrix, hess_inv)) 

def robust_outer_product(vec_1, vec_2):
    """
    Calculates a 'robust' outer product of two vectors that may or may not
    contain very small values.

    Parameters
    ----------
    vec_1 : 1D ndarray
    vec_2 : 1D ndarray

    Returns
    -------
    outer_prod : 2D ndarray. The outer product of vec_1 and vec_2
    """
    mantissa_1, exponents_1 = np.frexp(vec_1)
    mantissa_2, exponents_2 = np.frexp(vec_2)
    new_mantissas = mantissa_1[None, :] * mantissa_2[:, None]
    new_exponents = exponents_1[None, :] + exponents_2[:, None]
    return new_mantissas * np.exp2(new_exponents) 

def reflection_matrix(point, normal):
    """Return matrix to mirror at plane defined by point and normal vector.
    >>> v0 = numpy.random.random(4) - 0.5
    >>> v0[3] = 1.0
    >>> v1 = numpy.random.random(3) - 0.5
    >>> R = reflection_matrix(v0, v1)
    >>> numpy.allclose(2., numpy.trace(R))
    True
    >>> numpy.allclose(v0, numpy.dot(R, v0))
    True
    >>> v2 = v0.copy()
    >>> v2[:3] += v1
    >>> v3 = v0.copy()
    >>> v2[:3] -= v1
    >>> numpy.allclose(v2, numpy.dot(R, v3))
    True
    """
    normal = unit_vector(normal[:3])
    M = numpy.identity(4)
    M[:3, :3] -= 2.0 * numpy.outer(normal, normal)
    M[:3, 3] = (2.0 * numpy.dot(point[:3], normal)) * normal
    return M 

def reflection_matrix(point, normal):
    """Return matrix to mirror at plane defined by point and normal vector.

    >>> v0 = numpy.random.random(4) - 0.5
    >>> v0[3] = 1.0
    >>> v1 = numpy.random.random(3) - 0.5
    >>> R = reflection_matrix(v0, v1)
    >>> numpy.allclose(2., numpy.trace(R))
    True
    >>> numpy.allclose(v0, numpy.dot(R, v0))
    True
    >>> v2 = v0.copy()
    >>> v2[:3] += v1
    >>> v3 = v0.copy()
    >>> v2[:3] -= v1
    >>> numpy.allclose(v2, numpy.dot(R, v3))
    True

    """
    normal = unit_vector(normal[:3])
    M = numpy.identity(4)
    M[:3, :3] -= 2.0 * numpy.outer(normal, normal)
    M[:3, 3] = (2.0 * numpy.dot(point[:3], normal)) * normal
    return M 

def shear_matrix(angle, direction, point, normal):
    """Return matrix to shear by angle along direction vector on shear plane.
    The shear plane is defined by a point and normal vector. The direction
    vector must be orthogonal to the plane's normal vector.
    A point P is transformed by the shear matrix into P" such that
    the vector P-P" is parallel to the direction vector and its extent is
    given by the angle of P-P'-P", where P' is the orthogonal projection
    of P onto the shear plane.
    >>> angle = (random.random() - 0.5) * 4*math.pi
    >>> direct = numpy.random.random(3) - 0.5
    >>> point = numpy.random.random(3) - 0.5
    >>> normal = numpy.cross(direct, numpy.random.random(3))
    >>> S = shear_matrix(angle, direct, point, normal)
    >>> numpy.allclose(1.0, numpy.linalg.det(S))
    True
    """
    normal = unit_vector(normal[:3])
    direction = unit_vector(direction[:3])
    if abs(numpy.dot(normal, direction)) > 1e-6:
        raise ValueError("direction and normal vectors are not orthogonal")
    angle = math.tan(angle)
    M = numpy.identity(4)
    M[:3, :3] += angle * numpy.outer(direction, normal)
    M[:3, 3] = -angle * numpy.dot(point[:3], normal) * direction
    return M 

def test_expectation_value_multisite():
    s = spin_half
    psi = mps.MPS.from_singlets(s, 6, [(0, 1), (2, 3), (4, 5)], lonely=[], bc='finite')
    SpSm = npc.outer(s.Sp.replace_labels(['p', 'p*'], ['p0', 'p0*']),
                     s.Sm.replace_labels(['p', 'p*'], ['p1', 'p1*']))
    psi1 = psi.copy()
    ev = psi.expectation_value(SpSm)
    npt.assert_almost_equal(ev, [-0.5, 0., -0.5, 0., -0.5])
    env1 = mps.MPSEnvironment(psi1, psi)
    ev = env1.expectation_value(SpSm)
    npt.assert_almost_equal(ev, [-0.5, 0., -0.5, 0., -0.5])

    psi1.apply_local_op(2, SpSm)  # multi-site operator
    ev = psi1.expectation_value(SpSm)  # normalized!
    npt.assert_almost_equal(ev, [-0.5, 0., 0.0, 0., -0.5])
    env1 = mps.MPSEnvironment(psi1, psi)
    ev = env1.expectation_value(SpSm) / psi1.overlap(psi)  # normalize
    npt.assert_almost_equal(ev, [-0.5, 0., -1., 0., -0.5]) 

def _determine_weights(self, fg1, fg2):
        # both are (arrays of) scalars
        if len(fg1.shape) == 1 and len(fg2.shape) == 1:
            weights = np.outer(fg1, fg2)
        # both are (arrays of) vectors
        elif len(fg1.shape) == 2 and len(fg2.shape) == 2:
            # TODO: tests on the second dimension...
            weights = np.dot(fg1, fg2.T)
        else:
            raise Exception("Error, shape of the observable output not handled"
                            "in RDF")
        return weights.ravel()