add double factorized t2 tol and max_vecs test

tests/python/linalg/double_factorized_decomposition_test.py

@@ -396,3 +396,52 @@ def test_double_factorized_t2_alpha_beta_random():
         orbital_rotations[:, 0, 0], orbital_rotations[:, 3, 0].conj(), atol=1e-8
     )
     # TODO add the rest of the relations
+
+
+def test_double_factorized_t2_alpha_beta_tol_max_vecs():
+    """Test double-factorized decomposition alpha-beta error threshold and max vecs."""
+    mol = gto.Mole()
+    mol.build(
+        atom=[["H", (0, 0, 0)], ["O", (0, 0, 1.1)]],
+        basis="6-31g",
+        spin=1,
+        symmetry="Coov",
+    )
+    hartree_fock = scf.ROHF(mol).run()
+
+    ccsd = cc.CCSD(hartree_fock).run()
+    _, t2ab, _ = ccsd.t2
+    nocc_a, nocc_b, nvrt_a, _ = t2ab.shape
+    norb = nocc_a + nvrt_a
+
+    # test max_vecs
+    max_vecs = 25
+    diag_coulomb_mats, orbital_rotations = ffsim.linalg.double_factorized_t2_alpha_beta(
+        t2ab, max_vecs=max_vecs
+    )
+    reconstructed = reconstruct_t2_alpha_beta(
+        diag_coulomb_mats, orbital_rotations, norb=norb, nocc_a=nocc_a, nocc_b=nocc_b
+    )
+    assert len(diag_coulomb_mats) == max_vecs
+    np.testing.assert_allclose(reconstructed, t2ab, atol=1e-4)
+
+    # test error threshold
+    tol = 1e-3
+    diag_coulomb_mats, orbital_rotations = ffsim.linalg.double_factorized_t2_alpha_beta(
+        t2ab, tol=tol
+    )
+    reconstructed = reconstruct_t2_alpha_beta(
+        diag_coulomb_mats, orbital_rotations, norb=norb, nocc_a=nocc_a, nocc_b=nocc_b
+    )
+    assert len(diag_coulomb_mats) <= 23
+    np.testing.assert_allclose(reconstructed, t2ab, atol=tol)
+
+    # test error threshold and max vecs
+    diag_coulomb_mats, orbital_rotations = ffsim.linalg.double_factorized_t2_alpha_beta(
+        t2ab, tol=tol, max_vecs=max_vecs
+    )
+    reconstructed = reconstruct_t2_alpha_beta(
+        diag_coulomb_mats, orbital_rotations, norb=norb, nocc_a=nocc_a, nocc_b=nocc_b
+    )
+    assert len(orbital_rotations) <= 23
+    np.testing.assert_allclose(reconstructed, t2ab, atol=tol)