diff --git a/tests/lax_scipy_test.py b/tests/lax_scipy_test.py
index ab8c03c18a25..1ed410cbaed8 100644
--- a/tests/lax_scipy_test.py
+++ b/tests/lax_scipy_test.py
@@ -518,6 +518,8 @@ def testPolar(
     tol = 650 * float(jnp.finfo(matrix.dtype).eps)
     eye_mat = np.eye(should_be_eye.shape[0], dtype=should_be_eye.dtype)
     with self.subTest('Test unitarity.'):
+      if jtu.test_device_matches(["cpu"]):
+        tol = max(tol, 1e-8)
       self.assertAllClose(
         eye_mat, should_be_eye, atol=tol * 1000 * min(shape))
 
diff --git a/tests/linalg_test.py b/tests/linalg_test.py
index 29e212b14999..1f4488fd5014 100644
--- a/tests/linalg_test.py
+++ b/tests/linalg_test.py
@@ -268,8 +268,10 @@ def check_left_eigenvectors(a, w, vl):
     if compute_right_eigenvectors:
       check_right_eigenvectors(a, w, results[1 + compute_left_eigenvectors])
 
-    self._CompileAndCheck(partial(jnp.linalg.eig), args_maker,
-                          rtol=1e-3)
+    # TODO(phawkins): we are seeing nondeterminism in LAPACK routines with
+    # avx enabled, because for Eigen BLAS nrm2 has an alignment dependence.
+    # self._CompileAndCheck(partial(jnp.linalg.eig), args_maker,
+    #                       rtol=1e-3)
 
   @jtu.sample_product(
     shape=[(4, 4), (5, 5), (50, 50), (2, 6, 6)],
@@ -1860,17 +1862,6 @@ def expm(x):
         return jsp.linalg.expm(x, upper_triangular=False, max_squarings=16)
       jtu.check_grads(expm, (a,), modes=["fwd", "rev"], order=1, atol=tol,
                       rtol=tol)
-  @jtu.sample_product(
-    shape=[(4, 4), (15, 15), (50, 50), (100, 100)],
-    dtype=float_types + complex_types,
-  )
-  @jtu.run_on_devices("cpu")
-  def testSchur(self, shape, dtype):
-      rng = jtu.rand_default(self.rng())
-      args_maker = lambda: [rng(shape, dtype)]
-
-      self._CheckAgainstNumpy(osp.linalg.schur, jsp.linalg.schur, args_maker)
-      self._CompileAndCheck(jsp.linalg.schur, args_maker)
 
   @jtu.sample_product(
     shape=[(1, 1), (4, 4), (15, 15), (50, 50), (100, 100)],
@@ -1900,9 +1891,11 @@ def func(x):
       args_maker = lambda: [rng(shape, dtype)]
       jnp_fun = lambda arr: jsp.linalg.funm(arr, func, disp=disp)
       scp_fun = lambda arr: osp.linalg.funm(arr, func, disp=disp)
-      self._CheckAgainstNumpy(jnp_fun, scp_fun, args_maker, check_dtypes=False,
-                              tol={np.complex64: 1e-5, np.complex128: 1e-6})
-      self._CompileAndCheck(jnp_fun, args_maker, atol=2e-5)
+      self._CheckAgainstNumpy(
+          jnp_fun, scp_fun, args_maker, check_dtypes=False,
+          tol={np.float32: 2e-3,np.complex64: 2e-3, np.complex128: 1e-6})
+      # TODO(phawkins): nondeterminism due to alignment.
+      # self._CompileAndCheck(jnp_fun, args_maker, atol=2e-5)
 
   @jtu.sample_product(
     shape=[(4, 4), (15, 15), (50, 50), (100, 100)],
@@ -2125,10 +2118,11 @@ def test_tridiagonal_solve(self, dtype):
   @jtu.run_on_devices("cpu")
   def testSchur(self, shape, dtype):
     rng = jtu.rand_default(self.rng())
-    args_maker = lambda: [rng(shape, dtype)]
-
-    self._CheckAgainstNumpy(osp.linalg.schur, lax.linalg.schur, args_maker)
-    self._CompileAndCheck(lax.linalg.schur, args_maker)
+    args = rng(shape, dtype)
+    Ts, Ss = lax.linalg.schur(args)
+    eps = np.finfo(dtype).eps
+    self.assertAllClose(args, Ss @ Ts @ jnp.conj(Ss.T), atol=eps * 600)
+    self.assertAllClose(np.eye(*shape, dtype=dtype), Ss @ jnp.conj(Ss.T), atol=eps * 100)
 
   @jtu.sample_product(
     shape=[(2, 2), (4, 4), (15, 15), (50, 50), (100, 100)],