code: def solve_banded(
      l_and_u,
      ab,
      b,
      overwrite_ab=False,
      overwrite_b=False,
      check_finite=True,
  ):
      """
      Solve the equation a x = b for x, assuming a is banded matrix.
  
      The matrix a is stored in `ab` using the matrix diagonal ordered form::
  
          ab[u + i - j, j] == a[i,j]
  
      Example of `ab` (shape of a is (6,6), `u` =1, `l` =2)::
  
          *    a01  a12  a23  a34  a45
          a00  a11  a22  a33  a44  a55
          a10  a21  a32  a43  a54   *
          a20  a31  a42  a53   *    *
  
      Parameters
      ----------
      (l, u) : (integer, integer)
          Number of non-zero lower and upper diagonals
      ab : (`l` + `u` + 1, M) array_like
          Banded matrix
      b : (M,) or (M, K) array_like
          Right-hand side
      overwrite_ab : bool, optional
          Discard data in `ab` (may enhance performance)
      overwrite_b : bool, optional
          Discard data in `b` (may enhance performance)
      check_finite : bool, optional
          Whether to check that the input matrices contain only finite numbers.
          Disabling may give a performance gain, but may result in problems
          (crashes, non-termination) if the inputs do contain infinities or NaNs.
  
      Returns
      -------
      x : (M,) or (M, K) ndarray
          The solution to the system a x = b. Returned shape depends on the
          shape of `b`.
  
      Examples
      --------
      Solve the banded system a x = b, where::
  
              [5  2 -1  0  0]       [0]
              [1  4  2 -1  0]       [1]
          a = [0  1  3  2 -1]   b = [2]
              [0  0  1  2  2]       [2]
              [0  0  0  1  1]       [3]
  
      There is one nonzero diagonal below the main diagonal (l = 1), and
      two above (u = 2). The diagonal banded form of the matrix is::
  
               [*  * -1 -1 -1]
          ab = [*  2  2  2  2]
               [5  4  3  2  1]
               [1  1  1  1  *]
  
      >>> import numpy as np
      >>> from scipy.linalg import solve_banded
      >>> ab = np.array([[0,  0, -1, -1, -1],
      ...                [0,  2,  2,  2,  2],
      ...                [5,  4,  3,  2,  1],
      ...                [1,  1,  1,  1,  0]])
      >>> b = np.array([0, 1, 2, 2, 3])
      >>> x = solve_banded((1, 2), ab, b)
      >>> x
      array([-2.37288136,  3.93220339, -4.        ,  4.3559322 , -1.3559322 ])
  
      """
  
      a1 = _asarray_validated(
          ab, check_finite=check_finite, as_inexact=True
      )
      b1 = _asarray_validated(
          b, check_finite=check_finite, as_inexact=True
      )
      # Validate shapes.
      if a1.shape[-1] != b1.shape[0]:
          raise ValueError("shapes of ab and b are not compatible.")
      (nlower, nupper) = l_and_u
      if nlower + nupper + 1 != a1.shape[0]:
          raise ValueError(
              "invalid values for the number of lower and upper "
              "diagonals: l+u+1 (%d) does not equal ab.shape[0] "
              "(%d)" % (nlower + nupper + 1, ab.shape[0])
          )
  
      overwrite_b = overwrite_b or _datacopied(b1, b)
      if a1.shape[-1] == 1:
          b2 = np.array(b1, copy=(not overwrite_b))
          b2 /= a1[1, 0]
          return b2
      if nlower == nupper == 1:
          overwrite_ab = overwrite_ab or _datacopied(a1, ab)
          (gtsv,) = get_lapack_funcs(("gtsv",), (a1, b1))
          du = a1[0, 1:]
          d = a1[1, :]
          dl = a1[2, :-1]
          du2, d, du, x, info = gtsv(
              dl,
              d,
              du,
              b1,
              overwrite_ab,
              overwrite_ab,
              overwrite_ab,
              overwrite_b,
          )
      else:
          (gbsv,) = get_lapack_funcs(("gbsv",), (a1, b1))
          a2 = np.zeros(
              (2 * nlower + nupper + 1, a1.shape[1]), dtype=gbsv.dtype
          )
          a2[nlower:, :] = a1
          lu, piv, x, info = gbsv(
              nlower,
              nupper,
              a2,
              b1,
              overwrite_ab=True,
              overwrite_b=overwrite_b,
          )
      if info == 0:
          return x
      if info > 0:
          raise LinAlgError("singular matrix")
      raise ValueError(
          "illegal value in %d-th argument of internal "
          "gbsv/gtsv" % -info
      )
  
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 4 for calling a global function with example usage: solve_banded(l_and_u, ab, b, overwrite_ab, overwrite_b, check_finite) and returns: b2, x