code: def cossin(
      X,
      p=None,
      q=None,
      separate=False,
      swap_sign=False,
      compute_u=True,
      compute_vh=True,
  ):
      """
      Compute the cosine-sine (CS) decomposition of an orthogonal/unitary matrix.
  
      X is an ``(m, m)`` orthogonal/unitary matrix, partitioned as the following
      where upper left block has the shape of ``(p, q)``::
  
                                     ┌                   ┐
                                     │ I  0  0 │ 0  0  0 │
          ┌           ┐   ┌         ┐│ 0  C  0 │ 0 -S  0 │┌         ┐*
          │ X11 │ X12 │   │ U1 │    ││ 0  0  0 │ 0  0 -I ││ V1 │    │
          │ ────┼──── │ = │────┼────││─────────┼─────────││────┼────│
          │ X21 │ X22 │   │    │ U2 ││ 0  0  0 │ I  0  0 ││    │ V2 │
          └           ┘   └         ┘│ 0  S  0 │ 0  C  0 │└         ┘
                                     │ 0  0  I │ 0  0  0 │
                                     └                   ┘
  
      ``U1``, ``U2``, ``V1``, ``V2`` are square orthogonal/unitary matrices of
      dimensions ``(p,p)``, ``(m-p,m-p)``, ``(q,q)``, and ``(m-q,m-q)``
      respectively, and ``C`` and ``S`` are ``(r, r)`` nonnegative diagonal
      matrices satisfying ``C^2 + S^2 = I`` where ``r = min(p, m-p, q, m-q)``.
  
      Moreover, the rank of the identity matrices are ``min(p, q) - r``,
      ``min(p, m - q) - r``, ``min(m - p, q) - r``, and ``min(m - p, m - q) - r``
      respectively.
  
      X can be supplied either by itself and block specifications p, q or its
      subblocks in an iterable from which the shapes would be derived. See the
      examples below.
  
      Parameters
      ----------
      X : array_like, iterable
          complex unitary or real orthogonal matrix to be decomposed, or iterable
          of subblocks ``X11``, ``X12``, ``X21``, ``X22``, when ``p``, ``q`` are
          omitted.
      p : int, optional
          Number of rows of the upper left block ``X11``, used only when X is
          given as an array.
      q : int, optional
          Number of columns of the upper left block ``X11``, used only when X is
          given as an array.
      separate : bool, optional
          if ``True``, the low level components are returned instead of the
          matrix factors, i.e. ``(u1,u2)``, ``theta``, ``(v1h,v2h)`` instead of
          ``u``, ``cs``, ``vh``.
      swap_sign : bool, optional
          if ``True``, the ``-S``, ``-I`` block will be the bottom left,
          otherwise (by default) they will be in the upper right block.
      compute_u : bool, optional
          if ``False``, ``u`` won't be computed and an empty array is returned.
      compute_vh : bool, optional
          if ``False``, ``vh`` won't be computed and an empty array is returned.
  
      Returns
      -------
      u : ndarray
          When ``compute_u=True``, contains the block diagonal orthogonal/unitary
          matrix consisting of the blocks ``U1`` (``p`` x ``p``) and ``U2``
          (``m-p`` x ``m-p``) orthogonal/unitary matrices. If ``separate=True``,
          this contains the tuple of ``(U1, U2)``.
      cs : ndarray
          The cosine-sine factor with the structure described above.
           If ``separate=True``, this contains the ``theta`` array containing the
           angles in radians.
      vh : ndarray
          When ``compute_vh=True`, contains the block diagonal orthogonal/unitary
          matrix consisting of the blocks ``V1H`` (``q`` x ``q``) and ``V2H``
          (``m-q`` x ``m-q``) orthogonal/unitary matrices. If ``separate=True``,
          this contains the tuple of ``(V1H, V2H)``.
  
      References
      ----------
      .. [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
             Algorithms, 50(1):33-65, 2009.
  
      Examples
      --------
      >>> import numpy as np
      >>> from scipy.linalg import cossin
      >>> from scipy.stats import unitary_group
      >>> x = unitary_group.rvs(4)
      >>> u, cs, vdh = cossin(x, p=2, q=2)
      >>> np.allclose(x, u @ cs @ vdh)
      True
  
      Same can be entered via subblocks without the need of ``p`` and ``q``. Also
      let's skip the computation of ``u``
  
      >>> ue, cs, vdh = cossin((x[:2, :2], x[:2, 2:], x[2:, :2], x[2:, 2:]),
      ...                      compute_u=False)
      >>> print(ue)
      []
      >>> np.allclose(x, u @ cs @ vdh)
      True
  
      """
  
      if p or q:
          p = 1 if p is None else int(p)
          q = 1 if q is None else int(q)
          X = _asarray_validated(X, check_finite=True)
          if not np.equal(*X.shape):
              raise ValueError(
                  "Cosine Sine decomposition only supports square"
                  f" matrices, got {X.shape}"
              )
          m = X.shape[0]
          if p >= m or p <= 0:
              raise ValueError(
                  f"invalid p={p}, 0<p<{X.shape[0]} must hold"
              )
          if q >= m or q <= 0:
              raise ValueError(
                  f"invalid q={q}, 0<q<{X.shape[0]} must hold"
              )
  
          x11, x12, x21, x22 = (
              X[:p, :q],
              X[:p, q:],
              X[p:, :q],
              X[p:, q:],
          )
      elif not isinstance(X, Iterable):
          raise ValueError(
              "When p and q are None, X must be an Iterable"
              " containing the subblocks of X"
          )
      else:
          if len(X) != 4:
              raise ValueError(
                  "When p and q are None, exactly four arrays"
                  f" should be in X, got {len(X)}"
              )
  
          x11, x12, x21, x22 = (np.atleast_2d(x) for x in X)
          for name, block in zip(
              ["x11", "x12", "x21", "x22"], [x11, x12, x21, x22]
          ):
              if block.shape[1] == 0:
                  raise ValueError(f"{name} can't be empty")
          p, q = x11.shape
          mmp, mmq = x22.shape
  
          if x12.shape != (p, mmq):
              raise ValueError(
                  f"Invalid x12 dimensions: desired {(p, mmq)}, "
                  f"got {x12.shape}"
              )
  
          if x21.shape != (mmp, q):
              raise ValueError(
                  f"Invalid x21 dimensions: desired {(mmp, q)}, "
                  f"got {x21.shape}"
              )
  
          if p + mmp != q + mmq:
              raise ValueError(
                  "The subblocks have compatible sizes but "
                  "don't form a square array (instead they form a"
                  " {}x{} array). This might be due to missing "
                  "p, q arguments.".format(p + mmp, q + mmq)
              )
  
          m = p + mmp
  
      cplx = any([np.iscomplexobj(x) for x in [x11, x12, x21, x22]])
      driver = "uncsd" if cplx else "orcsd"
      csd, csd_lwork = get_lapack_funcs(
          [driver, driver + "_lwork"], [x11, x12, x21, x22]
      )
      lwork = _compute_lwork(csd_lwork, m=m, p=p, q=q)
      lwork_args = (
          {"lwork": lwork[0], "lrwork": lwork[1]}
          if cplx
          else {"lwork": lwork}
      )
      *_, theta, u1, u2, v1h, v2h, info = csd(
          x11=x11,
          x12=x12,
          x21=x21,
          x22=x22,
          compute_u1=compute_u,
          compute_u2=compute_u,
          compute_v1t=compute_vh,
          compute_v2t=compute_vh,
          trans=False,
          signs=swap_sign,
          **lwork_args,
      )
  
      method_name = csd.typecode + driver
      if info < 0:
          raise ValueError(
              f"illegal value in argument {-info} of internal {method_name}"
          )
      if info > 0:
          raise LinAlgError(f"{method_name} did not converge: {info}")
  
      if separate:
          return (u1, u2), theta, (v1h, v2h)
  
      U = block_diag(u1, u2)
      VDH = block_diag(v1h, v2h)
  
      # Construct the middle factor CS
      c = np.diag(np.cos(theta))
      s = np.diag(np.sin(theta))
      r = min(p, q, m - p, m - q)
      n11 = min(p, q) - r
      n12 = min(p, m - q) - r
      n21 = min(m - p, q) - r
      n22 = min(m - p, m - q) - r
      Id = np.eye(np.max([n11, n12, n21, n22, r]), dtype=theta.dtype)
      CS = np.zeros((m, m), dtype=theta.dtype)
  
      CS[:n11, :n11] = Id[:n11, :n11]
  
      xs = n11 + r
      xe = n11 + r + n12
      ys = n11 + n21 + n22 + 2 * r
      ye = n11 + n21 + n22 + 2 * r + n12
      CS[xs:xe, ys:ye] = (
          Id[:n12, :n12] if swap_sign else -Id[:n12, :n12]
      )
  
      xs = p + n22 + r
      xe = p + n22 + r + +n21
      ys = n11 + r
      ye = n11 + r + n21
      CS[xs:xe, ys:ye] = (
          -Id[:n21, :n21] if swap_sign else Id[:n21, :n21]
      )
  
      CS[p : p + n22, q : q + n22] = Id[:n22, :n22]
      CS[n11 : n11 + r, n11 : n11 + r] = c
      CS[p + n22 : p + n22 + r, r + n21 + n22 : 2 * r + n21 + n22] = c
  
      xs = n11
      xe = n11 + r
      ys = n11 + n21 + n22 + r
      ye = n11 + n21 + n22 + 2 * r
      CS[xs:xe, ys:ye] = s if swap_sign else -s
  
      CS[p + n22 : p + n22 + r, n11 : n11 + r] = -s if swap_sign else s
  
      return U, CS, VDH
  
imports: from collections.abc import Iterable
  import numpy
  from scipy._lib._util import _asarray_validated
  from scipy.linalg import block_diag, LinAlgError
  from .lapack import _compute_lwork, get_lapack_funcs
label_desc: Use image node_id 1 for calling a global function with example usage: cossin(X, p, q, separate, swap_sign, compute_u, compute_vh) and returns: U,  CS,  VDH, u1,  u2,  theta,  v1h,  v2h