code: def orthogonality(A, g):
      """Measure orthogonality between a vector and the null space of a matrix.
  
      Compute a measure of orthogonality between the null space
      of the (possibly sparse) matrix ``A`` and a given vector ``g``.
  
      The formula is a simplified (and cheaper) version of formula (3.13)
      from [1]_.
      ``orth =  norm(A g, ord=2)/(norm(A, ord='fro')*norm(g, ord=2))``.
  
      References
      ----------
      .. [1] Gould, Nicholas IM, Mary E. Hribar, and Jorge Nocedal.
             "On the solution of equality constrained quadratic
              programming problems arising in optimization."
              SIAM Journal on Scientific Computing 23.4 (2001): 1376-1395.
      """
      # Compute vector norms
      norm_g = np.linalg.norm(g)
      # Compute Froebnius norm of the matrix A
      if issparse(A):
          norm_A = scipy.sparse.linalg.norm(A, ord="fro")
      else:
          norm_A = np.linalg.norm(A, ord="fro")
  
      # Check if norms are zero
      if norm_g == 0 or norm_A == 0:
          return 0
  
      norm_A_g = np.linalg.norm(A.dot(g))
      # Orthogonality measure
      orth = norm_A_g / (norm_A * norm_g)
      return orth
  
file_path: catboost/contrib/python/scipy/py3/scipy/optimize/_trustregion_constr/projections.py
imports: from scipy.sparse import bmat, csc_matrix, eye, issparse
  from scipy.sparse.linalg import LinearOperator
  import scipy.linalg
  import scipy.sparse.linalg
  import numpy
  from warnings import warn
label_desc: Use image node_id 1 for calling a global function with example usage: orthogonality(A, g) and returns: orth, int