code: def multistep_uni_pc_bh_update(
      self,
      x,
      model_prev_list,
      t_prev_list,
      t,
      order,
      x_t=None,
      use_corrector=True,
  ):
      # print(f'using unified predictor-corrector with order {order} (solver type: B(h))')
      ns = self.noise_schedule
      assert order <= len(model_prev_list)
      dims = x.dim()
  
      # first compute rks
      t_prev_0 = t_prev_list[-1]
      lambda_prev_0 = ns.marginal_lambda(t_prev_0)
      lambda_t = ns.marginal_lambda(t)
      model_prev_0 = model_prev_list[-1]
      sigma_prev_0, sigma_t = ns.marginal_std(
          t_prev_0
      ), ns.marginal_std(t)
      log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(
          t_prev_0
      ), ns.marginal_log_mean_coeff(t)
      alpha_t = torch.exp(log_alpha_t)
  
      h = lambda_t - lambda_prev_0
  
      rks = []
      D1s = []
      for i in range(1, order):
          t_prev_i = t_prev_list[-(i + 1)]
          model_prev_i = model_prev_list[-(i + 1)]
          lambda_prev_i = ns.marginal_lambda(t_prev_i)
          rk = ((lambda_prev_i - lambda_prev_0) / h)[0]
          rks.append(rk)
          D1s.append((model_prev_i - model_prev_0) / rk)
  
      rks.append(1.0)
      rks = torch.tensor(rks, device=x.device)
  
      R = []
      b = []
  
      hh = -h[0] if self.predict_x0 else h[0]
      h_phi_1 = torch.expm1(hh)  # h\phi_1(h) = e^h - 1
      h_phi_k = h_phi_1 / hh - 1
  
      factorial_i = 1
  
      if self.variant == "bh1":
          B_h = hh
      elif self.variant == "bh2":
          B_h = torch.expm1(hh)
      else:
          raise NotImplementedError()
  
      for i in range(1, order + 1):
          R.append(torch.pow(rks, i - 1))
          b.append(h_phi_k * factorial_i / B_h)
          factorial_i *= i + 1
          h_phi_k = h_phi_k / hh - 1 / factorial_i
  
      R = torch.stack(R)
      b = torch.tensor(b, device=x.device)
  
      # now predictor
      use_predictor = len(D1s) > 0 and x_t is None
      if len(D1s) > 0:
          D1s = torch.stack(D1s, dim=1)  # (B, K)
          if x_t is None:
              # for order 2, we use a simplified version
              if order == 2:
                  rhos_p = torch.tensor([0.5], device=b.device)
              else:
                  rhos_p = torch.linalg.solve(R[:-1, :-1], b[:-1])
      else:
          D1s = None
  
      if use_corrector:
          # print('using corrector')
          # for order 1, we use a simplified version
          if order == 1:
              rhos_c = torch.tensor([0.5], device=b.device)
          else:
              rhos_c = torch.linalg.solve(R, b)
  
      model_t = None
      if self.predict_x0:
          x_t_ = (
              expand_dims(sigma_t / sigma_prev_0, dims) * x
              - expand_dims(alpha_t * h_phi_1, dims) * model_prev_0
          )
  
          if x_t is None:
              if use_predictor:
                  pred_res = torch.einsum("k,bkchw->bchw", rhos_p, D1s)
              else:
                  pred_res = 0
              x_t = x_t_ - expand_dims(alpha_t * B_h, dims) * pred_res
  
          if use_corrector:
              model_t = self.model_fn(x_t, t)
              if D1s is not None:
                  corr_res = torch.einsum(
                      "k,bkchw->bchw", rhos_c[:-1], D1s
                  )
              else:
                  corr_res = 0
              D1_t = model_t - model_prev_0
              x_t = x_t_ - expand_dims(alpha_t * B_h, dims) * (
                  corr_res + rhos_c[-1] * D1_t
              )
      else:
          x_t_ = (
              expand_dims(
                  torch.exp(log_alpha_t - log_alpha_prev_0), dims
              )
              * x
              - expand_dims(sigma_t * h_phi_1, dims) * model_prev_0
          )
          if x_t is None:
              if use_predictor:
                  pred_res = torch.einsum("k,bkchw->bchw", rhos_p, D1s)
              else:
                  pred_res = 0
              x_t = x_t_ - expand_dims(sigma_t * B_h, dims) * pred_res
  
          if use_corrector:
              model_t = self.model_fn(x_t, t)
              if D1s is not None:
                  corr_res = torch.einsum(
                      "k,bkchw->bchw", rhos_c[:-1], D1s
                  )
              else:
                  corr_res = 0
              D1_t = model_t - model_prev_0
              x_t = x_t_ - expand_dims(sigma_t * B_h, dims) * (
                  corr_res + rhos_c[-1] * D1_t
              )
      return x_t, model_t
  
imports: import torch
  import math
  import tqdm
label_desc: Use image node_id 12 for calling the UniPC obj's underlying member method code with example usage: obj.multistep_uni_pc_bh_update(x, model_prev_list, t_prev_list, t, order, x_t, use_corrector) and returns: x_t,  model_t