code: def matrix_balance(
      A, permute=True, scale=True, separate=False, overwrite_a=False
  ):
      """
      Compute a diagonal similarity transformation for row/column balancing.
  
      The balancing tries to equalize the row and column 1-norms by applying
      a similarity transformation such that the magnitude variation of the
      matrix entries is reflected to the scaling matrices.
  
      Moreover, if enabled, the matrix is first permuted to isolate the upper
      triangular parts of the matrix and, again if scaling is also enabled,
      only the remaining subblocks are subjected to scaling.
  
      The balanced matrix satisfies the following equality
  
      .. math::
  
                          B = T^{-1} A T
  
      The scaling coefficients are approximated to the nearest power of 2
      to avoid round-off errors.
  
      Parameters
      ----------
      A : (n, n) array_like
          Square data matrix for the balancing.
      permute : bool, optional
          The selector to define whether permutation of A is also performed
          prior to scaling.
      scale : bool, optional
          The selector to turn on and off the scaling. If False, the matrix
          will not be scaled.
      separate : bool, optional
          This switches from returning a full matrix of the transformation
          to a tuple of two separate 1-D permutation and scaling arrays.
      overwrite_a : bool, optional
          This is passed to xGEBAL directly. Essentially, overwrites the result
          to the data. It might increase the space efficiency. See LAPACK manual
          for details. This is False by default.
  
      Returns
      -------
      B : (n, n) ndarray
          Balanced matrix
      T : (n, n) ndarray
          A possibly permuted diagonal matrix whose nonzero entries are
          integer powers of 2 to avoid numerical truncation errors.
      scale, perm : (n,) ndarray
          If ``separate`` keyword is set to True then instead of the array
          ``T`` above, the scaling and the permutation vectors are given
          separately as a tuple without allocating the full array ``T``.
  
      Notes
      -----
      This algorithm is particularly useful for eigenvalue and matrix
      decompositions and in many cases it is already called by various
      LAPACK routines.
  
      The algorithm is based on the well-known technique of [1]_ and has
      been modified to account for special cases. See [2]_ for details
      which have been implemented since LAPACK v3.5.0. Before this version
      there are corner cases where balancing can actually worsen the
      conditioning. See [3]_ for such examples.
  
      The code is a wrapper around LAPACK's xGEBAL routine family for matrix
      balancing.
  
      .. versionadded:: 0.19.0
  
      References
      ----------
      .. [1] B.N. Parlett and C. Reinsch, "Balancing a Matrix for
         Calculation of Eigenvalues and Eigenvectors", Numerische Mathematik,
         Vol.13(4), 1969, :doi:`10.1007/BF02165404`
      .. [2] R. James, J. Langou, B.R. Lowery, "On matrix balancing and
         eigenvector computation", 2014, :arxiv:`1401.5766`
      .. [3] D.S. Watkins. A case where balancing is harmful.
         Electron. Trans. Numer. Anal, Vol.23, 2006.
  
      Examples
      --------
      >>> import numpy as np
      >>> from scipy import linalg
      >>> x = np.array([[1,2,0], [9,1,0.01], [1,2,10*np.pi]])
  
      >>> y, permscale = linalg.matrix_balance(x)
      >>> np.abs(x).sum(axis=0) / np.abs(x).sum(axis=1)
      array([ 3.66666667,  0.4995005 ,  0.91312162])
  
      >>> np.abs(y).sum(axis=0) / np.abs(y).sum(axis=1)
      array([ 1.2       ,  1.27041742,  0.92658316])  # may vary
  
      >>> permscale  # only powers of 2 (0.5 == 2^(-1))
      array([[  0.5,   0. ,  0. ],  # may vary
             [  0. ,   1. ,  0. ],
             [  0. ,   0. ,  1. ]])
  
      """
  
      A = np.atleast_2d(_asarray_validated(A, check_finite=True))
  
      if not np.equal(*A.shape):
          raise ValueError(
              "The data matrix for balancing should be square."
          )
  
      gebal = get_lapack_funcs(("gebal"), (A,))
      B, lo, hi, ps, info = gebal(
          A, scale=scale, permute=permute, overwrite_a=overwrite_a
      )
  
      if info < 0:
          raise ValueError(
              "xGEBAL exited with the internal error "
              f'"illegal value in argument number {-info}.". See '
              "LAPACK documentation for the xGEBAL error codes."
          )
  
      # Separate the permutations from the scalings and then convert to int
      scaling = np.ones_like(ps, dtype=float)
      scaling[lo : hi + 1] = ps[lo : hi + 1]
  
      # gebal uses 1-indexing
      ps = ps.astype(int, copy=False) - 1
      n = A.shape[0]
      perm = np.arange(n)
  
      # LAPACK permutes with the ordering n --> hi, then 0--> lo
      if hi < n:
          for ind, x in enumerate(ps[hi + 1 :][::-1], 1):
              if n - ind == x:
                  continue
              perm[[x, n - ind]] = perm[[n - ind, x]]
  
      if lo > 0:
          for ind, x in enumerate(ps[:lo]):
              if ind == x:
                  continue
              perm[[x, ind]] = perm[[ind, x]]
  
      if separate:
          return B, (scaling, perm)
  
      # get the inverse permutation
      iperm = np.empty_like(perm)
      iperm[perm] = np.arange(n)
  
      return B, np.diag(scaling)[iperm, :]
  
imports: from warnings import warn
  from itertools import product
  import numpy
  from numpy import atleast_1d, atleast_2d
  from .lapack import get_lapack_funcs, _compute_lwork
  from ._misc import LinAlgError, _datacopied, LinAlgWarning
  from ._decomp import _asarray_validated
  from .None import _decomp, _decomp_svd
  from ._solve_toeplitz import levinson
  from ._cythonized_array_utils import find_det_from_lu
  from scipy._lib.deprecation import _NoValue, _deprecate_positional_args
label_desc: Use image node_id 14 for calling a global function with example usage: matrix_balance(A, permute, scale, separate, overwrite_a) and returns: B,  np, B,  scaling,  perm