code: def slogdet(a):
      """Returns sign and logarithm of the determinant of an array.
  
      It calculates the natural logarithm of the determinant of a given value.
  
      Args:
          a (cupy.ndarray): The input matrix with dimension ``(..., N, N)``.
  
      Returns:
          tuple of :class:`~cupy.ndarray`:
              It returns a tuple ``(sign, logdet)``. ``sign`` represents each
              sign of the determinant as a real number ``0``, ``1`` or ``-1``.
              'logdet' represents the natural logarithm of the absolute of the
              determinant.
              If the determinant is zero, ``sign`` will be ``0`` and ``logdet``
              will be ``-inf``.
              The shapes of both ``sign`` and ``logdet`` are equal to
              ``a.shape[:-2]``.
  
      .. warning::
          This function calls one or more cuSOLVER routine(s) which may yield
          invalid results if input conditions are not met.
          To detect these invalid results, you can set the `linalg`
          configuration to a value that is not `ignore` in
          :func:`cupyx.errstate` or :func:`cupyx.seterr`.
  
      .. warning::
          To produce the same results as :func:`numpy.linalg.slogdet` for
          singular inputs, set the `linalg` configuration to `raise`.
  
      .. seealso:: :func:`numpy.linalg.slogdet`
      """
      _util._assert_stacked_2d(a)
      _util._assert_stacked_square(a)
  
      dtype, sign_dtype = _util.linalg_common_type(a)
      logdet_dtype = numpy.dtype(sign_dtype.char.lower())
  
      a_shape = a.shape
      shape = a_shape[:-2]
      n = a_shape[-2]
  
      if a.size == 0:
          # empty batch (result is empty, too) or empty matrices det([[]]) == 1
          sign = cupy.ones(shape, sign_dtype)
          logdet = cupy.zeros(shape, logdet_dtype)
          return sign, logdet
  
      lu, ipiv, dev_info = _decomposition._lu_factor(a, dtype)
  
      # dev_info < 0 means illegal value (in dimensions, strides, and etc.) that
      # should never happen even if the matrix contains nan or inf.
      # TODO(kataoka): assert dev_info >= 0 if synchronization is allowed for
      # debugging purposes.
  
      diag = cupy.diagonal(lu, axis1=-2, axis2=-1)
  
      logdet = cupy.log(cupy.abs(diag)).sum(axis=-1)
  
      # ipiv is 1-origin
      non_zero = cupy.count_nonzero(
          ipiv != cupy.arange(1, n + 1), axis=-1
      )
      if dtype.kind == "f":
          non_zero += cupy.count_nonzero(diag < 0, axis=-1)
  
      # Note: sign == -1 ** (non_zero % 2)
      sign = (non_zero % 2) * -2 + 1
      if dtype.kind == "c":
          sign = sign * cupy.prod(diag / cupy.abs(diag), axis=-1)
  
      sign = sign.astype(dtype)
      logdet = logdet.astype(logdet_dtype, copy=False)
      singular = dev_info > 0
      return (
          cupy.where(singular, sign_dtype.type(0), sign).reshape(shape),
          cupy.where(
              singular, logdet_dtype.type("-inf"), logdet
          ).reshape(shape),
      )
  
file_path: cupy/cupy/linalg/_norms.py
imports: import numpy
  import cupy
  from cupy import _core
  from cupy.linalg import _decomposition
  from cupy.linalg import _util
  import functools
label_desc: Use image node_id 5 for calling a global function with example usage: slogdet(a) and returns: cupy,  cupy, sign,  logdet