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