def ifftshift(x,axes=None):
    """
    Inverse of fftshift.

    Parameters
    ----------
    x : array_like
        Input array.
    axes : int or shape tuple, optional
        Axes over which to calculate.  Defaults to None which is over all axes.

    See Also
    --------
    fftshift

    """
    tmp = asarray(x)
    ndim = len(tmp.shape)
    if axes is None:
        axes = range(ndim)
    y = tmp
    for k in axes:
        n = tmp.shape[k]
        p2 = n-(n+1)/2
        mylist = concatenate((arange(p2,n),arange(p2)))
        y = take(y,mylist,k)
    return y

def fftshift(x,axes=None):
    """
    Shift zero-frequency component to center of spectrum.

    This function swaps half-spaces for all axes listed (defaults to all).
    If len(x) is even then the Nyquist component is y[0].

    Parameters
    ----------
    x : array_like
        Input array.
    axes : int or shape tuple, optional
        Axes over which to shift.  Default is None which shifts all axes.

    See Also
    --------
    ifftshift

    """
    tmp = asarray(x)
    ndim = len(tmp.shape)
    if axes is None:
        axes = range(ndim)
    y = tmp
    for k in axes:
        n = tmp.shape[k]
        p2 = (n+1)/2
        mylist = concatenate((arange(p2,n),arange(p2)))
        y = take(y,mylist,k)
    return y

def fftshift(x, axes=None):
    """
    Shift the zero-frequency component to the center of the spectrum.

    This function swaps half-spaces for all axes listed (defaults to all).
    Note that ``y[0]`` is the Nyquist component only if ``len(x)`` is even.

    Parameters
    ----------
    x : array_like
        Input array.
    axes : int or shape tuple, optional
        Axes over which to shift.  Default is None, which shifts all axes.

    Returns
    -------
    y : ndarray
        The shifted array.

    See Also
    --------
    ifftshift : The inverse of `fftshift`.

    Examples
    --------
    >>> freqs = np.fft.fftfreq(10, 0.1)
    >>> freqs
    array([ 0.,  1.,  2.,  3.,  4., -5., -4., -3., -2., -1.])
    >>> np.fft.fftshift(freqs)
    array([-5., -4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.])

    Shift the zero-frequency component only along the second axis:

    >>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
    >>> freqs
    array([[ 0.,  1.,  2.],
           [ 3.,  4., -4.],
           [-3., -2., -1.]])
    >>> np.fft.fftshift(freqs, axes=(1,))
    array([[ 2.,  0.,  1.],
           [-4.,  3.,  4.],
           [-1., -3., -2.]])

    """
    tmp = asarray(x)
    ndim = len(tmp.shape)
    if axes is None:
        axes = list(range(ndim))
    elif isinstance(axes, (int, nt.integer)):
        axes = (axes,)
    y = tmp
    for k in axes:
        n = tmp.shape[k]
        p2 = (n+1)//2
        mylist = concatenate((arange(p2,n),arange(p2)))
        y = take(y,mylist,k)
    return y

def fftfreq(n,d=1.0):
    """ fftfreq(n, d=1.0) -> f

    DFT sample frequencies

    The returned float array contains the frequency bins in
    cycles/unit (with zero at the start) given a window length n and a
    sample spacing d:

      f = [0,1,...,n/2-1,-n/2,...,-1]/(d*n)         if n is even
      f = [0,1,...,(n-1)/2,-(n-1)/2,...,-1]/(d*n)   if n is odd
    """
    assert isinstance(n,types.IntType) or isinstance(n, integer)
    return hstack((arange(0,(n-1)/2 + 1), arange(-(n/2),0))) / (n*d)

 def original_fftshift(x, axes=None):
     """ How fftshift was implemented in v1.14"""
     tmp = asarray(x)
     ndim = tmp.ndim
     if axes is None:
         axes = list(range(ndim))
     elif isinstance(axes, integer_types):
         axes = (axes,)
     y = tmp
     for k in axes:
         n = tmp.shape[k]
         p2 = (n + 1) // 2
         mylist = concatenate((arange(p2, n), arange(p2)))
         y = take(y, mylist, k)
     return y

def det(a):
    """Compute the determinant of a matrix

    Parameters
    ----------
    a : array-like, shape (M, M)

    Returns
    -------
    det : float or complex
        Determinant of a

    Notes
    -----
    The determinant is computed via LU factorization, LAPACK routine z/dgetrf.
    """
    a = asarray(a)
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    a = _fastCopyAndTranspose(t, a)
    n = a.shape[0]
    if isComplexType(t):
        lapack_routine = lapack_lite.zgetrf
    else:
        lapack_routine = lapack_lite.dgetrf
    pivots = zeros((n,), fortran_int)
    results = lapack_routine(n, n, a, n, pivots, 0)
    info = results['info']
    if (info < 0):
        raise TypeError, "Illegal input to Fortran routine"
    elif (info > 0):
        return 0.0
    sign = add.reduce(pivots != arange(1, n+1)) % 2
    return (1.-2.*sign)*multiply.reduce(diagonal(a), axis=-1)

def ifftshift(x,axes=None):
    """ ifftshift(x,axes=None) - > y

    Inverse of fftshift.
    """
    tmp = asarray(x)
    ndim = len(tmp.shape)
    if axes is None:
        axes = range(ndim)
    y = tmp
    for k in axes:
        n = tmp.shape[k]
        p2 = n-(n+1)/2
        mylist = concatenate((arange(p2,n),arange(p2)))
        y = take(y,mylist,k)
    return y

 def test_strided(self):
     a = arange(12)
     b = a[::2]
     low, high = utils.byte_bounds(b)
     # the largest pointer address is lost (even numbers only in the
     # stride), and compensate addresses for striding by 2
     assert_equal(high - low, b.size * 2 * b.itemsize - b.itemsize)

def ifftshift(x, axes=None):
    """
    The inverse of `fftshift`. Although identical for even-length `x`, the
    functions differ by one sample for odd-length `x`.

    Parameters
    ----------
    x : array_like
        Input array.
    axes : int or shape tuple, optional
        Axes over which to calculate.  Defaults to None, which shifts all axes.

    Returns
    -------
    y : ndarray
        The shifted array.

    See Also
    --------
    fftshift : Shift zero-frequency component to the center of the spectrum.

    Examples
    --------
    >>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
    >>> freqs
    array([[ 0.,  1.,  2.],
           [ 3.,  4., -4.],
           [-3., -2., -1.]])
    >>> np.fft.ifftshift(np.fft.fftshift(freqs))
    array([[ 0.,  1.,  2.],
           [ 3.,  4., -4.],
           [-3., -2., -1.]])

    """
    tmp = asarray(x)
    ndim = len(tmp.shape)
    if axes is None:
        axes = list(range(ndim))
    elif isinstance(axes, integer_types):
        axes = (axes,)
    y = tmp
    for k in axes:
        n = tmp.shape[k]
        p2 = n-(n+1)//2
        mylist = concatenate((arange(p2, n), arange(p2)))
        y = take(y, mylist, k)
    return y

def fftfreq(n, d=1.0):
    """
    Return the Discrete Fourier Transform sample frequencies.

    The returned float array `f` contains the frequency bin centers in cycles 
    per unit of the sample spacing (with zero at the start).  For instance, if 
    the sample spacing is in seconds, then the frequency unit is cycles/second.

    Given a window length `n` and a sample spacing `d`::

      f = [0, 1, ...,   n/2-1,     -n/2, ..., -1] / (d*n)   if n is even
      f = [0, 1, ..., (n-1)/2, -(n-1)/2, ..., -1] / (d*n)   if n is odd

    Parameters
    ----------
    n : int
        Window length.
    d : scalar, optional
        Sample spacing (inverse of the sampling rate). Defaults to 1.
        
    Returns
    -------
    f : ndarray
        Array of length `n` containing the sample frequencies.

    Examples
    --------
    >>> signal = np.array([-2, 8, 6, 4, 1, 0, 3, 5], dtype=float)
    >>> fourier = np.fft.fft(signal)
    >>> n = signal.size
    >>> timestep = 0.1
    >>> freq = np.fft.fftfreq(n, d=timestep)
    >>> freq
    array([ 0.  ,  1.25,  2.5 ,  3.75, -5.  , -3.75, -2.5 , -1.25])

    """
    if not (isinstance(n,types.IntType) or isinstance(n, integer)):
        raise ValueError("n should be an integer")
    val = 1.0 / (n * d)
    results = empty(n, int)
    N = (n-1)//2 + 1
    p1 = arange(0, N, dtype=int)
    results[:N] = p1
    p2 = arange(-(n//2), 0, dtype=int)
    results[N:] = p2
    return results * val

def fftfreq(n,d=1.0):
    """
    Discrete Fourier Transform sample frequencies.

    The returned float array contains the frequency bins in
    cycles/unit (with zero at the start) given a window length `n` and a
    sample spacing `d`.
    ::

      f = [0,1,...,n/2-1,-n/2,...,-1]/(d*n)         if n is even
      f = [0,1,...,(n-1)/2,-(n-1)/2,...,-1]/(d*n)   if n is odd

    Parameters
    ----------
    n : int
        Window length.
    d : scalar
        Sample spacing.

    Returns
    -------
    out : ndarray, shape(`n`,)
        Sample frequencies.

    Examples
    --------
    >>> signal = np.array([-2.,  8., -6.,  4.,  1., 0.,  3.,  5.])
    >>> fourier = np.fft.fft(signal)
    >>> n = len(signal)
    >>> timestep = 0.1
    >>> freq = np.fft.fftfreq(n, d=timestep)
    >>> freq
    array([ 0.  ,  1.25,  2.5 ,  3.75, -5.  , -3.75, -2.5 , -1.25])

    """
    assert isinstance(n,types.IntType) or isinstance(n, integer)
    val = 1.0/(n*d)
    results = empty(n, int)
    N = (n-1)//2 + 1
    p1 = arange(0,N,dtype=int)
    results[:N] = p1
    p2 = arange(-(n//2),0,dtype=int)
    results[N:] = p2
    return results * val

def ifftshift(x, axes=None):
    """
    The inverse of fftshift.

    Parameters
    ----------
    x : array_like
        Input array.
    axes : int or shape tuple, optional
        Axes over which to calculate.  Defaults to None, which shifts all axes.

    Returns
    -------
    y : ndarray
        The shifted array.

    See Also
    --------
    fftshift : Shift zero-frequency component to the center of the spectrum.

    Examples
    --------
    >>> freqs = np.fft.fftfreq(9, d=1./9).reshape(3, 3)
    >>> freqs
    array([[ 0.,  1.,  2.],
           [ 3.,  4., -4.],
           [-3., -2., -1.]])
    >>> np.fft.ifftshift(np.fft.fftshift(freqs))
    array([[ 0.,  1.,  2.],
           [ 3.,  4., -4.],
           [-3., -2., -1.]])

    """
    tmp = asarray(x)
    ndim = len(tmp.shape)
    if axes is None:
        axes = range(ndim)
    y = tmp
    for k in axes:
        n = tmp.shape[k]
        p2 = n - (n + 1) / 2
        mylist = concatenate((arange(p2, n), arange(p2)))
        y = take(y, mylist, k)
    return y

def fftfreq(n,d=1.0):
    """ fftfreq(n, d=1.0) -> f

    DFT sample frequencies

    The returned float array contains the frequency bins in
    cycles/unit (with zero at the start) given a window length n and a
    sample spacing d:

      f = [0,1,...,n/2-1,-n/2,...,-1]/(d*n)         if n is even
      f = [0,1,...,(n-1)/2,-(n-1)/2,...,-1]/(d*n)   if n is odd
    """
    assert isinstance(n,types.IntType) or isinstance(n, integer)
    val = 1.0/(n*d)
    results = empty(n, int)
    N = (n-1)//2 + 1
    p1 = arange(0,N,dtype=int)
    results[:N] = p1
    p2 = arange(-(n//2),0,dtype=int)
    results[N:] = p2
    return results * val

def fftshift(x,axes=None):
    """ fftshift(x, axes=None) -> y

    Shift zero-frequency component to center of spectrum.

    This function swaps half-spaces for all axes listed (defaults to all).

    Notes:
      If len(x) is even then the Nyquist component is y[0].
    """
    tmp = asarray(x)
    ndim = len(tmp.shape)
    if axes is None:
        axes = range(ndim)
    y = tmp
    for k in axes:
        n = tmp.shape[k]
        p2 = (n+1)/2
        mylist = concatenate((arange(p2,n),arange(p2)))
        y = take(y,mylist,k)
    return y

def test_concatenate_axis_None():
    a = arange(4, dtype=float64).reshape((2,2))
    b = list(range(3))
    c = ['x']
    r = concatenate((a, a), axis=None)
    assert_equal(r.dtype, a.dtype)
    assert_equal(r.ndim, 1)
    r = concatenate((a, b), axis=None)
    assert_equal(r.size, a.size + len(b))
    assert_equal(r.dtype, a.dtype)
    r = concatenate((a, b, c), axis=None)
    d = array(['0', '1', '2', '3', 
               '0', '1', '2', 'x'])
    assert_array_equal(r,d)

def rfftfreq(n, d=1.0):
    """
    Return the Discrete Fourier Transform sample frequencies 
    (for usage with rfft, irfft).

    The returned float array `f` contains the frequency bin centers in cycles 
    per unit of the sample spacing (with zero at the start).  For instance, if 
    the sample spacing is in seconds, then the frequency unit is cycles/second.

    Given a window length `n` and a sample spacing `d`::

      f = [0, 1, ...,     n/2-1,     n/2] / (d*n)   if n is even
      f = [0, 1, ..., (n-1)/2-1, (n-1)/2] / (d*n)   if n is odd

    Unlike `fftfreq` (but like `scipy.fftpack.rfftfreq`)
    the Nyquist frequency component is considered to be positive.

    Parameters
    ----------
    n : int
        Window length.
    d : scalar, optional
        Sample spacing (inverse of the sampling rate). Defaults to 1.

    Returns
    -------
    f : ndarray
        Array of length ``n//2 + 1`` containing the sample frequencies.

    Examples
    --------
    >>> signal = np.array([-2, 8, 6, 4, 1, 0, 3, 5, -3, 4], dtype=float)
    >>> fourier = np.fft.rfft(signal)
    >>> n = signal.size
    >>> sample_rate = 100
    >>> freq = np.fft.fftfreq(n, d=1./sample_rate)
    >>> freq
    array([  0.,  10.,  20.,  30.,  40., -50., -40., -30., -20., -10.])
    >>> freq = np.fft.rfftfreq(n, d=1./sample_rate)
    >>> freq
    array([  0.,  10.,  20.,  30.,  40.,  50.])

    """
    if not (isinstance(n,types.IntType) or isinstance(n, integer)):
        raise ValueError("n should be an integer")
    val = 1.0/(n*d)
    N = n//2 + 1
    results = arange(0, N, dtype=int)
    return results * val

def det(a):
    """
    Compute the determinant of an array.

    Parameters
    ----------
    a : array_like, shape (M, M)
        Input array.

    Returns
    -------
    det : ndarray
        Determinant of `a`.

    Notes
    -----
    The determinant is computed via LU factorization using the LAPACK
    routine z/dgetrf.

    Examples
    --------
    The determinant of a 2-D array [[a, b], [c, d]] is ad - bc:

    >>> a = np.array([[1, 2], [3, 4]])
    >>> np.linalg.det(a)
    -2.0

    """
    a = asarray(a)
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    a = _fastCopyAndTranspose(t, a)
    n = a.shape[0]
    if isComplexType(t):
        lapack_routine = lapack_lite.zgetrf
    else:
        lapack_routine = lapack_lite.dgetrf
    pivots = zeros((n,), fortran_int)
    results = lapack_routine(n, n, a, n, pivots, 0)
    info = results['info']
    if (info < 0):
        raise TypeError, "Illegal input to Fortran routine"
    elif (info > 0):
        return 0.0
    sign = add.reduce(pivots != arange(1, n+1)) % 2
    return (1.-2.*sign)*multiply.reduce(diagonal(a), axis=-1)

def test_concatenate():
    # Test concatenate function
    # No arrays raise ValueError
    assert_raises(ValueError, concatenate, ())
    # Scalars cannot be concatenated
    assert_raises(ValueError, concatenate, (0,))
    assert_raises(ValueError, concatenate, (array(0),))
    # One sequence returns unmodified (but as array)
    r4 = list(range(4))
    assert_array_equal(concatenate((r4,)), r4)
    # Any sequence
    assert_array_equal(concatenate((tuple(r4),)), r4)
    assert_array_equal(concatenate((array(r4),)), r4)
    # 1D default concatenation
    r3 = list(range(3))
    assert_array_equal(concatenate((r4, r3)), r4 + r3)
    # Mixed sequence types
    assert_array_equal(concatenate((tuple(r4), r3)), r4 + r3)
    assert_array_equal(concatenate((array(r4), r3)), r4 + r3)
    # Explicit axis specification
    assert_array_equal(concatenate((r4, r3), 0), r4 + r3)
    # Including negative
    assert_array_equal(concatenate((r4, r3), -1), r4 + r3)
    # 2D
    a23 = array([[10, 11, 12], [13, 14, 15]])
    a13 = array([[0, 1, 2]])
    res = array([[10, 11, 12], [13, 14, 15], [0, 1, 2]])
    assert_array_equal(concatenate((a23, a13)), res)
    assert_array_equal(concatenate((a23, a13), 0), res)
    assert_array_equal(concatenate((a23.T, a13.T), 1), res.T)
    assert_array_equal(concatenate((a23.T, a13.T), -1), res.T)
    # Arrays much match shape
    assert_raises(ValueError, concatenate, (a23.T, a13.T), 0)
    # 3D
    res = arange(2 * 3 * 7).reshape((2, 3, 7))
    a0 = res[..., :4]
    a1 = res[..., 4:6]
    a2 = res[..., 6:]
    assert_array_equal(concatenate((a0, a1, a2), 2), res)
    assert_array_equal(concatenate((a0, a1, a2), -1), res)
    assert_array_equal(concatenate((a0.T, a1.T, a2.T), 0), res.T)

def det(a):
    "The determinant of the 2-d array a"
    a = asarray(a)
    _assertRank2(a)
    _assertSquareness(a)
    t, result_t = _commonType(a)
    a = _fastCopyAndTranspose(t, a)
    n = a.shape[0]
    if isComplexType(t):
        lapack_routine = lapack_lite.zgetrf
    else:
        lapack_routine = lapack_lite.dgetrf
    pivots = zeros((n,), fortran_int)
    results = lapack_routine(n, n, a, n, pivots, 0)
    info = results['info']
    if (info < 0):
        raise TypeError, "Illegal input to Fortran routine"
    elif (info > 0):
        return 0.0
    sign = add.reduce(pivots != arange(1, n+1)) % 2
    return (1.-2.*sign)*multiply.reduce(diagonal(a), axis=-1)

 def test_byte_bounds(self):
     # pointer difference matches size * itemsize
     # due to contiguity
     a = arange(12).reshape(3, 4)
     low, high = utils.byte_bounds(a)
     assert_equal(high - low, a.size * a.itemsize)