code: def krylov(l, a, b, c=None, return_power=False):
      """
      Compute the Krylov matrix (b, ab, a^2b, ...) using the squaring trick.
      If return_power=True, return A^{L-1} as well
      Args:
          a (torch.tensor): torch.tensor
          b (torch.tensor): torch.tensor
          c (torch.tensor): torch.tensor
          return_power (bool): return_power
      """
      # TODO There is an edge case if l=1 where output doesn't get broadcasted, which might be an issue
      #  if caller is expecting broadcasting semantics... can deal with it if it arises
  
      x = b.unsqueeze(-1)  # (..., N, 1)
      a_ = a
  
      al = None
      if return_power:
          al = torch.eye(a.shape[-1], dtype=a.dtype, device=a.device)
          l_ = l - 1
  
      done = l == 1
      # loop invariant: l_ represents how many indices left to compute
      while not done:
          if return_power:
              if l_ % 2 == 1:
                  al = a_ @ al
              l_ //= 2
  
          # Save memory on last iteration
          ll = x.shape[-1]
          if l - ll <= ll:
              done = True
              x_ = x[..., : l - ll]
          else:
              x_ = x
  
          x_ = a_ @ x_
          x = torch.cat(
              [x, x_], dim=-1
          )  # there might be a more efficient way of ordering axes
          if not done:
              a_ = a_ @ a_
  
      assert x.shape[-1] == l
  
      if c is not None:
          x = torch.einsum("...nl, ...n -> ...l", x, c)
      x = x.contiguous()  # WOW!!
      if return_power:
          return x, al
      else:
          return x
  
file_path: towhee/towhee/models/vis4mer/utils.py
imports: import torch
  from einops import rearrange
  from scipy import special
  import numpy
  from opt_einsum import contract
  from pykeops.torch import Genred
label_desc: Use image node_id 4 for calling a global function with example usage: krylov(l, a, b, c, return_power) and returns: x,  al, x