code: def _slogdet_lu(a: Array) -> tuple[Array, Array]:
      dtype = lax.dtype(a)
      lu, pivot, _ = lax_linalg.lu(a)
      diag = jnp.diagonal(lu, axis1=-2, axis2=-1)
      is_zero = reductions.any(
          diag == jnp.array(0, dtype=dtype), axis=-1
      )
      iota = lax.expand_dims(
          jnp.arange(a.shape[-1], dtype=pivot.dtype),
          range(pivot.ndim - 1),
      )
      parity = reductions.count_nonzero(pivot != iota, axis=-1)
      if jnp.iscomplexobj(a):
          sign = reductions.prod(
              diag / ufuncs.abs(diag).astype(diag.dtype), axis=-1
          )
      else:
          sign = jnp.array(1, dtype=dtype)
          parity = parity + reductions.count_nonzero(diag < 0, axis=-1)
      sign = jnp.where(
          is_zero,
          jnp.array(0, dtype=dtype),
          sign * jnp.array(-2 * (parity % 2) + 1, dtype=dtype),
      )
      logdet = jnp.where(
          is_zero,
          jnp.array(-jnp.inf, dtype=dtype),
          reductions.sum(
              ufuncs.log(ufuncs.abs(diag)).astype(dtype), axis=-1
          ),
      )
      return sign, ufuncs.real(logdet)
  
imports: from __future__ import annotations
  from collections.abc import Sequence
  from functools import partial
  import numpy
  import textwrap
  import operator
  from typing import Literal, cast, overload
  import jax
  from jax import jit, custom_jvp
  from jax import lax
  from jax._src.lax import lax
  from jax._src.lax import linalg
  from jax._src.numpy import lax_numpy
  from jax._src.numpy import reductions, ufuncs
  from jax._src.numpy.util import _wraps, promote_dtypes_inexact, check_arraylike
  from jax._src.util import canonicalize_axis
  from jax._src.typing import ArrayLike, Array
label_desc: Use image node_id 12 for calling a global function with example usage: _slogdet_lu(a) and returns: sign,  ufuncs