code: def _LDLdecomposition_sparse(M, hermitian=True):
      """
      Returns the LDL Decomposition (matrices ``L`` and ``D``) of matrix
      ``A``, such that ``L * D * L.T == A``. ``A`` must be a square,
      symmetric, positive-definite and non-singular.
  
      This method eliminates the use of square root and ensures that all
      the diagonal entries of L are 1.
  
      Examples
      ========
  
      >>> from sympy import SparseMatrix
      >>> A = SparseMatrix(((25, 15, -5), (15, 18, 0), (-5, 0, 11)))
      >>> L, D = A.LDLdecomposition()
      >>> L
      Matrix([
      [   1,   0, 0],
      [ 3/5,   1, 0],
      [-1/5, 1/3, 1]])
      >>> D
      Matrix([
      [25, 0, 0],
      [ 0, 9, 0],
      [ 0, 0, 9]])
      >>> L * D * L.T == A
      True
  
      """
  
      from .dense import MutableDenseMatrix
  
      if not M.is_square:
          raise NonSquareMatrixError("Matrix must be square.")
      if hermitian and not M.is_hermitian:
          raise ValueError("Matrix must be Hermitian.")
      if not hermitian and not M.is_symmetric():
          raise ValueError("Matrix must be symmetric.")
  
      dps = _get_intermediate_simp(expand_mul, expand_mul)
      Lrowstruc = M.row_structure_symbolic_cholesky()
      L = MutableDenseMatrix.eye(M.rows)
      D = MutableDenseMatrix.zeros(M.rows, M.cols)
  
      for i in range(len(Lrowstruc)):
          for j in Lrowstruc[i]:
              if i != j:
                  L[i, j] = M[i, j]
                  summ = 0
  
                  for p1 in Lrowstruc[i]:
                      if p1 < j:
                          for p2 in Lrowstruc[j]:
                              if p2 < j:
                                  if p1 == p2:
                                      if hermitian:
                                          summ += (
                                              L[i, p1]
                                              * L[j, p1].conjugate()
                                              * D[p1, p1]
                                          )
                                      else:
                                          summ += (
                                              L[i, p1]
                                              * L[j, p1]
                                              * D[p1, p1]
                                          )
                              else:
                                  break
                      else:
                          break
  
                  L[i, j] = dps((L[i, j] - summ) / D[j, j])
  
              else:  # i == j
                  D[i, i] = M[i, i]
                  summ = 0
  
                  for k in Lrowstruc[i]:
                      if k < i:
                          if hermitian:
                              summ += (
                                  L[i, k]
                                  * L[i, k].conjugate()
                                  * D[k, k]
                              )
                          else:
                              summ += L[i, k] ** 2 * D[k, k]
                      else:
                          break
  
                  D[i, i] = dps(D[i, i] - summ)
  
                  if hermitian and D[i, i].is_positive is False:
                      raise NonPositiveDefiniteMatrixError(
                          "Matrix must be positive-definite"
                      )
  
      return M._new(L), M._new(D)
  
imports: import copy
  from sympy.core import S
  from sympy.core.function import expand_mul
  from sympy.functions.elementary.miscellaneous import Min, sqrt
  from sympy.functions.elementary.complexes import sign
  from .exceptions import NonSquareMatrixError, NonPositiveDefiniteMatrixError
  from .utilities import _get_intermediate_simp, _iszero
  from .determinant import _find_reasonable_pivot_naive
label_desc: Use image node_id 7 for calling a global function with example usage: _LDLdecomposition_sparse(M, hermitian) and returns: M,  M