original_docstring: 
      Compute the N-dimensional discrete Fourier Transform for real input.
  
      This function computes the N-dimensional discrete Fourier Transform over
      any number of axes in an M-dimensional real array by means of the Fast
      Fourier Transform (FFT).  By default, all axes are transformed, with the
      real transform performed over the last axis, while the remaining
      transforms are complex.
  
      Parameters
      ----------
      a : array_like
          Input array, taken to be real.
      s : sequence of ints, optional
          Shape (length along each transformed axis) to use from the input.
          (``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.).
          The final element of `s` corresponds to `n` for ``rfft(x, n)``, while
          for the remaining axes, it corresponds to `n` for ``fft(x, n)``.
          Along any axis, if the given shape is smaller than that of the input,
          the input is cropped.  If it is larger, the input is padded with zeros.
          if `s` is not given, the shape of the input along the axes specified
          by `axes` is used.
      axes : sequence of ints, optional
          Axes over which to compute the FFT.  If not given, the last ``len(s)``
          axes are used, or all axes if `s` is also not specified.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      Returns
      -------
      out : complex ndarray
          The truncated or zero-padded input, transformed along the axes
          indicated by `axes`, or by a combination of `s` and `a`,
          as explained in the parameters section above.
          The length of the last axis transformed will be ``s[-1]//2+1``,
          while the remaining transformed axes will have lengths according to
          `s`, or unchanged from the input.
  
      Raises
      ------
      ValueError
          If `s` and `axes` have different length.
      IndexError
          If an element of `axes` is larger than than the number of axes of `a`.
  
      See Also
      --------
      irfftn : The inverse of `rfftn`, i.e. the inverse of the n-dimensional FFT
           of real input.
      fft : The one-dimensional FFT, with definitions and conventions used.
      rfft : The one-dimensional FFT of real input.
      fftn : The n-dimensional FFT.
      rfft2 : The two-dimensional FFT of real input.
  
      Notes
      -----
      The transform for real input is performed over the last transformation
      axis, as by `rfft`, then the transform over the remaining axes is
      performed as by `fftn`.  The order of the output is as for `rfft` for the
      final transformation axis, and as for `fftn` for the remaining
      transformation axes.
  
      See `fft` for details, definitions and conventions used.
  
      Examples
      --------
      >>> a = np.ones((2, 2, 2))
      >>> np.fft.rfftn(a)
      array([[[ 8.+0.j,  0.+0.j],
              [ 0.+0.j,  0.+0.j]],
             [[ 0.+0.j,  0.+0.j],
              [ 0.+0.j,  0.+0.j]]])
  
      >>> np.fft.rfftn(a, axes=(2, 0))
      array([[[ 4.+0.j,  0.+0.j],
              [ 4.+0.j,  0.+0.j]],
             [[ 0.+0.j,  0.+0.j],
              [ 0.+0.j,  0.+0.j]]])
  
      
docstring: Compute the N-dimensional discrete Fourier Transform for real input.
  This function computes the N-dimensional discrete Fourier Transform over
  any number of axes in an M-dimensional real array by means of the Fast
  Fourier Transform (FFT).  By default, all axes are transformed, with the
  real transform performed over the last axis, while the remaining
  transforms are complex.
  
  Parameters
  
  a : array_like
  Input array, taken to be real.
  s : sequence of ints, optional
  Shape (length along each transformed axis) to use from the input.
  
  Returns
  
  
  
  Raises
  
  ValueError
  If `s` and `axes` have different length.
  IndexError
  If an element of `axes` is larger than than the number of axes of `a`.
  
  See Also
  
  irfftn : The inverse of `rfftn`, i.e. the inverse of the n-dimensional FFT
  of real input.
  fft : The one-dimensional FFT, with definitions and conventions used.
  rfft : The one-dimensional FFT of real input.
  fftn : The n-dimensional FFT.
  rfft2 : The two-dimensional FFT of real input.
  
  Notes
  
  The transform for real input is performed over the last transformation
  axis, as by `rfft`, then the transform over the remaining axes is
  performed as by `fftn`.  The order of the output is as for `rfft` for the
  final transformation axis, and as for `fftn` for the remaining
  transformation axes.
  
  See `fft` for details, definitions and conventions used.
  
  Examples
  
  
  
  
docstring_tokens: Compute the N - dimensional discrete Fourier Transform for real input . This function computes the N - dimensional discrete Fourier Transform over any number of axes in an M - dimensional real array by means of the Fast Fourier Transform ( FFT ) . By default all axes are transformed with the real transform performed over the last axis while the remaining transforms are complex . Parameters a : array_like Input array taken to be real . s : sequence of ints optional Shape ( length along each transformed axis ) to use from the input . Returns Raises ValueError If ` s ` and ` axes ` have different length . IndexError If an element of ` axes ` is larger than than the number of axes of ` a ` . See Also irfftn : The inverse of ` rfftn ` i . e . the inverse of the n - dimensional FFT of real input . fft : The one - dimensional FFT with definitions and conventions used . rfft : The one - dimensional FFT of real input . fftn : The n - dimensional FFT . rfft2 : The two - dimensional FFT of real input . Notes The transform for real input is performed over the last transformation axis as by ` rfft ` then the transform over the remaining axes is performed as by ` fftn ` . The order of the output is as for ` rfft ` for the final transformation axis and as for ` fftn ` for the remaining transformation axes . See ` fft ` for details definitions and conventions used . Examples
code: def rfftn(a, s=None, axes=None, norm=None):
      """
      Compute the N-dimensional discrete Fourier Transform for real input.
  
      This function computes the N-dimensional discrete Fourier Transform over
      any number of axes in an M-dimensional real array by means of the Fast
      Fourier Transform (FFT).  By default, all axes are transformed, with the
      real transform performed over the last axis, while the remaining
      transforms are complex.
  
      Parameters
      ----------
      a : array_like
          Input array, taken to be real.
      s : sequence of ints, optional
          Shape (length along each transformed axis) to use from the input.
          (``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.).
          The final element of `s` corresponds to `n` for ``rfft(x, n)``, while
          for the remaining axes, it corresponds to `n` for ``fft(x, n)``.
          Along any axis, if the given shape is smaller than that of the input,
          the input is cropped.  If it is larger, the input is padded with zeros.
          if `s` is not given, the shape of the input along the axes specified
          by `axes` is used.
      axes : sequence of ints, optional
          Axes over which to compute the FFT.  If not given, the last ``len(s)``
          axes are used, or all axes if `s` is also not specified.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      Returns
      -------
      out : complex ndarray
          The truncated or zero-padded input, transformed along the axes
          indicated by `axes`, or by a combination of `s` and `a`,
          as explained in the parameters section above.
          The length of the last axis transformed will be ``s[-1]//2+1``,
          while the remaining transformed axes will have lengths according to
          `s`, or unchanged from the input.
  
      Raises
      ------
      ValueError
          If `s` and `axes` have different length.
      IndexError
          If an element of `axes` is larger than than the number of axes of `a`.
  
      See Also
      --------
      irfftn : The inverse of `rfftn`, i.e. the inverse of the n-dimensional FFT
           of real input.
      fft : The one-dimensional FFT, with definitions and conventions used.
      rfft : The one-dimensional FFT of real input.
      fftn : The n-dimensional FFT.
      rfft2 : The two-dimensional FFT of real input.
  
      Notes
      -----
      The transform for real input is performed over the last transformation
      axis, as by `rfft`, then the transform over the remaining axes is
      performed as by `fftn`.  The order of the output is as for `rfft` for the
      final transformation axis, and as for `fftn` for the remaining
      transformation axes.
  
      See `fft` for details, definitions and conventions used.
  
      Examples
      --------
      >>> a = np.ones((2, 2, 2))
      >>> np.fft.rfftn(a)
      array([[[ 8.+0.j,  0.+0.j],
              [ 0.+0.j,  0.+0.j]],
             [[ 0.+0.j,  0.+0.j],
              [ 0.+0.j,  0.+0.j]]])
  
      >>> np.fft.rfftn(a, axes=(2, 0))
      array([[[ 4.+0.j,  0.+0.j],
              [ 4.+0.j,  0.+0.j]],
             [[ 0.+0.j,  0.+0.j],
              [ 0.+0.j,  0.+0.j]]])
  
      """
      # The copy may be required for multithreading.
      a = array(a, copy=True, dtype=float)
      s, axes = _cook_nd_args(a, s, axes)
      a = rfft(a, s[-1], axes[-1], norm)
      for ii in range(len(axes)-1):
          a = fft(a, s[ii], axes[ii], norm)
      return a
code_tokens: def rfftn ( a , s = None , axes = None , norm = None ) : a = array ( a , copy = True , dtype = float ) s , axes = _cook_nd_args ( a , s , axes ) a = rfft ( a , s [ - 1 ] , axes [ - 1 ] , norm ) for ii in range ( len ( axes ) - 1 ) : a = fft ( a , s [ ii ] , axes [ ii ] , norm ) return a
short_docstring: Compute the N-dimensional discrete Fourier Transform for real input.
short_docstring_tokens: Compute the N - dimensional discrete Fourier Transform for real input .
comment: """
      Compute the N-dimensional discrete Fourier Transform for real input.
  
      This function computes the N-dimensional discrete Fourier Transform over
      any number of axes in an M-dimensional real array by means of the Fast
      Fourier Transform (FFT).  By default, all axes are transformed, with the
      real transform performed over the last axis, while the remaining
      transforms are complex.
  
      Parameters
      ----------
      a : array_like
          Input array, taken to be real.
      s : sequence of ints, optional
          Shape (length along each transformed axis) to use from the input.
          (``s[0]`` refers to axis 0, ``s[1]`` to axis 1, etc.).
          The final element of `s` corresponds to `n` for ``rfft(x, n)``, while
          for the remaining axes, it corresponds to `n` for ``fft(x, n)``.
          Along any axis, if the given shape is smaller than that of the input,
          the input is cropped.  If it is larger, the input is padded with zeros.
          if `s` is not given, the shape of the input along the axes specified
          by `axes` is used.
      axes : sequence of ints, optional
          Axes over which to compute the FFT.  If not given, the last ``len(s)``
          axes are used, or all axes if `s` is also not specified.
      norm : {None, "ortho"}, optional
          .. versionadded:: 1.10.0
          Normalization mode (see `numpy.fft`). Default is None.
  
      Returns
      -------
      out : complex ndarray
          The truncated or zero-padded input, transformed along the axes
          indicated by `axes`, or by a combination of `s` and `a`,
          as explained in the parameters section above.
          The length of the last axis transformed will be ``s[-1]//2+1``,
          while the remaining transformed axes will have lengths according to
          `s`, or unchanged from the input.
  
      Raises
      ------
      ValueError
          If `s` and `axes` have different length.
      IndexError
          If an element of `axes` is larger than than the number of axes of `a`.
  
      See Also
      --------
      irfftn : The inverse of `rfftn`, i.e. the inverse of the n-dimensional FFT
           of real input.
      fft : The one-dimensional FFT, with definitions and conventions used.
      rfft : The one-dimensional FFT of real input.
      fftn : The n-dimensional FFT.
      rfft2 : The two-dimensional FFT of real input.
  
      Notes
      -----
      The transform for real input is performed over the last transformation
      axis, as by `rfft`, then the transform over the remaining axes is
      performed as by `fftn`.  The order of the output is as for `rfft` for the
      final transformation axis, and as for `fftn` for the remaining
      transformation axes.
  
      See `fft` for details, definitions and conventions used.
  
      Examples
      --------
      >>> a = np.ones((2, 2, 2))
      >>> np.fft.rfftn(a)
      array([[[ 8.+0.j,  0.+0.j],
              [ 0.+0.j,  0.+0.j]],
             [[ 0.+0.j,  0.+0.j],
              [ 0.+0.j,  0.+0.j]]])
  
      >>> np.fft.rfftn(a, axes=(2, 0))
      array([[[ 4.+0.j,  0.+0.j],
              [ 4.+0.j,  0.+0.j]],
             [[ 0.+0.j,  0.+0.j],
              [ 0.+0.j,  0.+0.j]]])
  
      """$SEP$# The copy may be required for multithreading.