code: def test_cache_root_decomposition(self):
      tkwargs = {"device": self.device}
      constructors = [
          DeprecatedCachedCholeskyAcqf,
          DummyCachedCholeskyAcqf,
      ]
      for constructor in constructors:
          for dtype in (torch.float, torch.double):
              tkwargs["dtype"] = dtype
              # test mt-mvn
              train_x = torch.rand(2, 1, **tkwargs)
              train_y = torch.rand(2, 2, **tkwargs)
              test_x = torch.rand(2, 1, **tkwargs)
              model = SingleTaskGP(train_x, train_y)
              sampler = IIDNormalSampler(sample_shape=torch.Size([1]))
              with torch.no_grad():
                  posterior = model.posterior(test_x)
              acqf = constructor(
                  model=model,
                  sampler=sampler,
                  objective=GenericMCObjective(lambda Y, _: Y[..., 0]),
              )
              baseline_L = torch.eye(2, **tkwargs)
              with mock.patch(
                  EXTRACT_BATCH_COVAR_PATH, wraps=extract_batch_covar
              ) as mock_extract_batch_covar:
                  with mock.patch(
                      CHOLESKY_PATH, return_value=baseline_L
                  ) as mock_cholesky:
                      baseline_L_acqf = (
                          acqf._compute_root_decomposition(
                              posterior=posterior
                          )
                      )
                      mock_extract_batch_covar.assert_called_once_with(
                          posterior.distribution
                      )
                      mock_cholesky.assert_called_once()
              # test mvn
              model = SingleTaskGP(train_x, train_y[:, :1])
              with torch.no_grad():
                  posterior = model.posterior(test_x)
              with mock.patch(
                  EXTRACT_BATCH_COVAR_PATH
              ) as mock_extract_batch_covar:
                  with mock.patch(
                      CHOLESKY_PATH, return_value=baseline_L
                  ) as mock_cholesky:
                      baseline_L_acqf = (
                          acqf._compute_root_decomposition(
                              posterior=posterior
                          )
                      )
                      mock_extract_batch_covar.assert_not_called()
                      mock_cholesky.assert_called_once()
              self.assertTrue(torch.equal(baseline_L_acqf, baseline_L))
  
imports: import warnings
  from typing import Optional
  from unittest import mock
  import torch
  from botorch import settings
  from botorch.acquisition.cached_cholesky import CachedCholeskyMCAcquisitionFunction, CachedCholeskyMCSamplerMixin
  from botorch.acquisition.monte_carlo import MCAcquisitionFunction
  from botorch.acquisition.objective import GenericMCObjective, MCAcquisitionObjective
  from botorch.exceptions.warnings import BotorchWarning
  from botorch.models import SingleTaskGP
  from botorch.models.deterministic import GenericDeterministicModel
  from botorch.models.higher_order_gp import HigherOrderGP
  from botorch.models.model import Model, ModelList
  from botorch.models.transforms.outcome import Log
  from botorch.sampling.normal import IIDNormalSampler, MCSampler
  from botorch.utils.low_rank import extract_batch_covar
  from botorch.utils.testing import BotorchTestCase, MockModel, MockPosterior
  from linear_operator.utils.errors import NanError, NotPSDError
label_desc: Use image node_id 2 for calling the TestCachedCholeskyMCSamplerMixin obj's underlying member method code with example usage: obj.test_cache_root_decomposition() without return types