docstr: Check that spline coefficients satisfy the continuity and boundary
  conditions.
docstr_tok: ["Check","that","spline","coefficients","satisfy","the","continuity","and","boundary","conditions","."]
code: def check_correctness(
      S, bc_start="not-a-knot", bc_end="not-a-knot", tol=1e-14
  ):
      """Check that spline coefficients satisfy the continuity and boundary
      conditions."""
      x = S.x
      c = S.c
      dx = np.diff(x)
      dx = dx.reshape([dx.shape[0]] + [1] * (c.ndim - 2))
      dxi = dx[:-1]
  
      # Check C2 continuity.
      assert_allclose(
          c[3, 1:],
          c[0, :-1] * dxi**3
          + c[1, :-1] * dxi**2
          + c[2, :-1] * dxi
          + c[3, :-1],
          rtol=tol,
          atol=tol,
      )
      assert_allclose(
          c[2, 1:],
          3 * c[0, :-1] * dxi**2 + 2 * c[1, :-1] * dxi + c[2, :-1],
          rtol=tol,
          atol=tol,
      )
      assert_allclose(
          c[1, 1:], 3 * c[0, :-1] * dxi + c[1, :-1], rtol=tol, atol=tol
      )
  
      # Check that we found a parabola, the third derivative is 0.
      if (
          x.size == 3
          and bc_start == "not-a-knot"
          and bc_end == "not-a-knot"
      ):
          assert_allclose(c[0], 0, rtol=tol, atol=tol)
          return
  
      # Check periodic boundary conditions.
      if bc_start == "periodic":
          assert_allclose(S(x[0], 0), S(x[-1], 0), rtol=tol, atol=tol)
          assert_allclose(S(x[0], 1), S(x[-1], 1), rtol=tol, atol=tol)
          assert_allclose(S(x[0], 2), S(x[-1], 2), rtol=tol, atol=tol)
          return
  
      # Check other boundary conditions.
      if bc_start == "not-a-knot":
          if x.size == 2:
              slope = (S(x[1]) - S(x[0])) / dx[0]
              assert_allclose(S(x[0], 1), slope, rtol=tol, atol=tol)
          else:
              assert_allclose(c[0, 0], c[0, 1], rtol=tol, atol=tol)
      elif bc_start == "clamped":
          assert_allclose(S(x[0], 1), 0, rtol=tol, atol=tol)
      elif bc_start == "natural":
          assert_allclose(S(x[0], 2), 0, rtol=tol, atol=tol)
      else:
          order, value = bc_start
          assert_allclose(S(x[0], order), value, rtol=tol, atol=tol)
  
      if bc_end == "not-a-knot":
          if x.size == 2:
              slope = (S(x[1]) - S(x[0])) / dx[0]
              assert_allclose(S(x[1], 1), slope, rtol=tol, atol=tol)
          else:
              assert_allclose(c[0, -1], c[0, -2], rtol=tol, atol=tol)
      elif bc_end == "clamped":
          assert_allclose(S(x[-1], 1), 0, rtol=tol, atol=tol)
      elif bc_end == "natural":
          assert_allclose(S(x[-1], 2), 0, rtol=2 * tol, atol=2 * tol)
      else:
          order, value = bc_end
          assert_allclose(S(x[-1], order), value, rtol=tol, atol=tol)
  
imports: import warnings
  import io
  import numpy
  from numpy.testing import assert_almost_equal, assert_array_equal, assert_array_almost_equal, assert_allclose, assert_equal, assert_
  from pytest import raises
  from scipy.interpolate import KroghInterpolator, krogh_interpolate, BarycentricInterpolator, barycentric_interpolate, approximate_taylor_polynomial, CubicHermiteSpline, pchip, PchipInterpolator, pchip_interpolate, Akima1DInterpolator, CubicSpline, make_interp_spline
label_desc: Use image node_id 1 for calling the TestCubicSpline obj's underlying member method code with example usage: obj.check_correctness(S, bc_start, bc_end, tol) without return types