code: def power(L, A, v=None):
      """Compute A^L and the scan sum_i A^i v_i.
  
      A: (..., N, N)
      v: (..., N, L)
      """
      E = torch.eye(A.shape[-1]).to(
          A
      )  # , dtype=A.dtype, device=A.device)
  
      powers = [A]
      length = 1
      while True:
          if L % 2 == 1:
              E = powers[-1] @ E
          L //= 2
          if L == 0:
              break
          length *= 2
          powers.append(powers[-1] @ powers[-1])
  
      if v is None:
          return E
  
      # Invariants:
      # powers[-1] := A^length
      # length := largest po2 at most L
  
      # Note that an alternative divide and conquer to compute the reduction is possible
      # and can be embedded into the above loop without caching intermediate powers of A
      # We do this reverse divide-and-conquer for efficiency reasons:
      # 1) it involves fewer padding steps for non-po2 L
      # 2) it involves more contiguous arrays
  
      # Take care of edge case for non-po2 arrays
      # Note that this initial step is a no-op for the case of power of 2 (length == L)
      k = v.size(-1) - length
      v_ = powers.pop() @ v[..., length:]
      v = v[..., :length]
      v[..., :k] = v[..., :k] + v_
  
      # Handle reduction for power of 2
      while v.size(-1) > 1:
          v = rearrange(v, "... (z l) -> ... z l", z=2)
          v = v[..., 0, :] + powers.pop() @ v[..., 1, :]
      return E, v.squeeze(-1)
  
imports: import logging
  import math
  import os
  from functools import wraps
  from typing import Any, Callable, Optional
  import numpy
  import opt_einsum
  import torch
  import torch.nn
  import torch.nn.functional
  from einops import rearrange, repeat
  from espnet2.asr.state_spaces.components import Activation, DropoutNd, LinearActivation
label_desc: Use image node_id 5 for calling a global function with example usage: power(L, A, v) and returns: E,  v, E