code: def _matrix_inverse_root_newton(
      A,
      root: int,
      epsilon: float = 0.0,
      max_iterations: int = 1000,
      tolerance: float = 1e-6,
  ) -> Tuple[Tensor, Tensor, NewtonConvergenceFlag, int, Tensor]:
      """Compute matrix square root using coupled inverse Newton iteration.
  
          alpha <- -1 / p
          X <- 1/c * I
          M <- 1/c^p * A
          repeat until convergence
              M' <- (1 - alpha) * I + alpha * M
              X <- X * M'
              M <- M'^p * M
  
      where c = (2 |A|_F / (p + 1))^{1/p}. This ensures that |A|_2 <= |A|_F < (p + 1) c^p, which guarantees convergence.
      We will instead use z = (p + 1) / (2 * |A|_F).
  
      Args:
          A (Tensor): Matrix of interest.
          root (int): Root of interest. Any natural number.
          epsilon (float): Adds epsilon * I to matrix before taking matrix root. (Default: 0.0)
          max_iterations (int): Maximum number of iterations. (Default: 1000)
          tolerance (float): Tolerance. (Default: 1e-6)
  
      Returns:
          A_root (Tensor): Inverse square root of matrix.
          M (Tensor): Coupled matrix.
          termination_flag (NewtonConvergenceFlag): Specifies convergence.
          iteration (int): Number of iterations.
          error (Tensor): Final error between M and I.
  
      """
  
      # initialize iteration, dimension, and alpha
      iteration = 0
      dim = A.shape[0]
      alpha = -1 / root
      identity = torch.eye(dim, dtype=A.dtype, device=A.device)
  
      # add regularization
      A.add_(identity, alpha=epsilon)
  
      # initialize matrices
      A_nrm = torch.linalg.norm(A)
      z = (root + 1) / (2 * A_nrm)
      X = z ** (-alpha) * identity
      M = z * A
      error = torch.dist(M, identity, p=torch.inf)
  
      # main for loop
      while error > tolerance and iteration < max_iterations:
          iteration += 1
          M_p = M.mul(alpha).add_(identity, alpha=(1 - alpha))
          X = X @ M_p
          M = torch.linalg.matrix_power(M_p, root) @ M
          error = torch.dist(M, identity, p=torch.inf)
  
      # determine convergence flag
      if error <= tolerance:
          termination_flag = NewtonConvergenceFlag.CONVERGED
      else:
          termination_flag = NewtonConvergenceFlag.REACHED_MAX_ITERS
  
      return X, M, termination_flag, iteration, error
  
file_path: bert-ultrafast/training/cramming/backend/optimizers/shampoo/matrix_functions.py
imports: import enum
  import logging
  from typing import Tuple
  import torch
  from torch import Tensor
label_desc: Use image node_id 3 for calling a global function with example usage: _matrix_inverse_root_newton(A, root, epsilon, max_iterations, tolerance) and returns: X,  M,  termination_flag,  iteration,  error