code: def _slogdet_qr(a: Array) -> tuple[Array, Array]:
      # Implementation of slogdet using QR decomposition. One reason we might prefer
      # QR decomposition is that it is more amenable to a fast batched
      # implementation on TPU because of the lack of row pivoting.
      if jnp.issubdtype(lax.dtype(a), jnp.complexfloating):
          raise NotImplementedError(
              "slogdet method='qr' not implemented for complex "
              "inputs"
          )
      n = a.shape[-1]
      a, taus = lax_linalg.geqrf(a)
      # The determinant of a triangular matrix is the product of its diagonal
      # elements. We are working in log space, so we compute the magnitude as the
      # the trace of the log-absolute values, and we compute the sign separately.
      a_diag = jnp.diagonal(a, axis1=-2, axis2=-1)
      log_abs_det = reductions.sum(
          ufuncs.log(ufuncs.abs(a_diag)), axis=-1
      )
      sign_diag = reductions.prod(ufuncs.sign(a_diag), axis=-1)
      # The determinant of a Householder reflector is -1. So whenever we actually
      # made a reflection (tau != 0), multiply the result by -1.
      sign_taus = reductions.prod(
          jnp.where(taus[..., : (n - 1)] != 0, -1, 1), axis=-1
      ).astype(sign_diag.dtype)
      return sign_diag * sign_taus, log_abs_det
  
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 13 for calling a global function with example usage: _slogdet_qr(a) and returns: unknown,  log_abs_det