code: def test_concat(self):
      A = self.piv_chol.expand(2, 2)
      B = self.piv_chol.expand(1, 2)
      C = B.concat(B, dim=0)
      for key in ("tril", "perm", "diag"):
          self.assertTrue(getattr(A, key).equal(getattr(C, key)))
  
      B.step = A.step + 1
      error_msg = (
          "Cannot conncatenate decompositions at different steps."
      )
      with self.assertRaisesRegex(ValueError, error_msg):
          A.concat(B, dim=0)
  
      B.step = A.step
      B.perm = None
      error_msg = "Types of field perm do not match."
      with self.assertRaisesRegex(NotImplementedError, error_msg):
          A.concat(B, dim=0)
  
file_path: botorch/test/utils/probability/test_linalg.py
imports: from __future__ import annotations
  from copy import deepcopy
  import torch
  from botorch.utils.probability.linalg import augment_cholesky, PivotedCholesky
  from botorch.utils.testing import BotorchTestCase
label_desc: Use image node_id 4 for calling the TestPivotedCholesky obj's underlying member method code with example usage: obj.test_concat() without return types