code: def multistep_uni_pc_vary_update(
      self,
      x,
      model_prev_list,
      t_prev_list,
      t,
      order,
      use_corrector=True,
  ):
      # print(f'using unified predictor-corrector with order {order} (solver type: vary coeff)')
      ns = self.noise_schedule
      assert order <= len(model_prev_list)
  
      # 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_t = 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
          rks.append(rk)
          D1s.append((model_prev_i - model_prev_0) / rk)
  
      rks.append(1.0)
      rks = torch.tensor(rks, device=x.device)
  
      K = len(rks)
      # build C matrix
      C = []
  
      col = torch.ones_like(rks)
      for k in range(1, K + 1):
          C.append(col)
          col = col * rks / (k + 1)
      C = torch.stack(C, dim=1)
  
      if len(D1s) > 0:
          D1s = torch.stack(D1s, dim=1)  # (B, K)
          C_inv_p = torch.linalg.inv(C[:-1, :-1])
          A_p = C_inv_p
  
      if use_corrector:
          # print('using corrector')
          C_inv = torch.linalg.inv(C)
          A_c = C_inv
  
      hh = -h if self.predict_x0 else h
      h_phi_1 = torch.expm1(hh)
      h_phi_ks = []
      factorial_k = 1
      h_phi_k = h_phi_1
      for k in range(1, K + 2):
          h_phi_ks.append(h_phi_k)
          h_phi_k = h_phi_k / hh - 1 / factorial_k
          factorial_k *= k + 1
  
      model_t = None
      if self.predict_x0:
          x_t_ = (
              sigma_t / sigma_prev_0 * x
              - alpha_t * h_phi_1 * model_prev_0
          )
          # now predictor
          x_t = x_t_
          if len(D1s) > 0:
              # compute the residuals for predictor
              for k in range(K - 1):
                  x_t = x_t - alpha_t * h_phi_ks[k + 1] * torch.einsum(
                      "bkchw,k->bchw", D1s, A_p[k]
                  )
          # now corrector
          if use_corrector:
              model_t = self.model_fn(x_t, t)
              D1_t = model_t - model_prev_0
              x_t = x_t_
              k = 0
              for k in range(K - 1):
                  x_t = x_t - alpha_t * h_phi_ks[k + 1] * torch.einsum(
                      "bkchw,k->bchw", D1s, A_c[k][:-1]
                  )
              x_t = x_t - alpha_t * h_phi_ks[K] * (D1_t * A_c[k][-1])
      else:
          log_alpha_prev_0, log_alpha_t = ns.marginal_log_mean_coeff(
              t_prev_0
          ), ns.marginal_log_mean_coeff(t)
          x_t_ = (torch.exp(log_alpha_t - log_alpha_prev_0)) * x - (
              sigma_t * h_phi_1
          ) * model_prev_0
          # now predictor
          x_t = x_t_
          if len(D1s) > 0:
              # compute the residuals for predictor
              for k in range(K - 1):
                  x_t = x_t - sigma_t * h_phi_ks[k + 1] * torch.einsum(
                      "bkchw,k->bchw", D1s, A_p[k]
                  )
          # now corrector
          if use_corrector:
              model_t = self.model_fn(x_t, t)
              D1_t = model_t - model_prev_0
              x_t = x_t_
              k = 0
              for k in range(K - 1):
                  x_t = x_t - sigma_t * h_phi_ks[k + 1] * torch.einsum(
                      "bkchw,k->bchw", D1s, A_c[k][:-1]
                  )
              x_t = x_t - sigma_t * h_phi_ks[K] * (D1_t * A_c[k][-1])
      return x_t, model_t
  
imports: import torch
  import math
  import tqdm
label_desc: Use image node_id 11 for calling the UniPC obj's underlying member method code with example usage: obj.multistep_uni_pc_vary_update(x, model_prev_list, t_prev_list, t, order, use_corrector) and returns: x_t,  model_t