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1# -*- coding: utf-8 -*-
2from collections.abc import Iterable
3import numpy as np
5from scipy._lib._util import _asarray_validated
6from scipy.linalg import block_diag, LinAlgError
7from .lapack import _compute_lwork, get_lapack_funcs
9__all__ = ['cossin']
12def cossin(X, p=None, q=None, separate=False,
13 swap_sign=False, compute_u=True, compute_vh=True):
14 """
15 Compute the cosine-sine (CS) decomposition of an orthogonal/unitary matrix.
17 X is an ``(m, m)`` orthogonal/unitary matrix, partitioned as the following
18 where upper left block has the shape of ``(p, q)``::
20 ┌ ┐
21 │ I 0 0 │ 0 0 0 │
22 ┌ ┐ ┌ ┐│ 0 C 0 │ 0 -S 0 │┌ ┐*
23 │ X11 │ X12 │ │ U1 │ ││ 0 0 0 │ 0 0 -I ││ V1 │ │
24 │ ────┼──── │ = │────┼────││─────────┼─────────││────┼────│
25 │ X21 │ X22 │ │ │ U2 ││ 0 0 0 │ I 0 0 ││ │ V2 │
26 └ ┘ └ ┘│ 0 S 0 │ 0 C 0 │└ ┘
27 │ 0 0 I │ 0 0 0 │
28 └ ┘
30 ``U1``, ``U2``, ``V1``, ``V2`` are square orthogonal/unitary matrices of
31 dimensions ``(p,p)``, ``(m-p,m-p)``, ``(q,q)``, and ``(m-q,m-q)``
32 respectively, and ``C`` and ``S`` are ``(r, r)`` nonnegative diagonal
33 matrices satisfying ``C^2 + S^2 = I`` where ``r = min(p, m-p, q, m-q)``.
35 Moreover, the rank of the identity matrices are ``min(p, q) - r``,
36 ``min(p, m - q) - r``, ``min(m - p, q) - r``, and ``min(m - p, m - q) - r``
37 respectively.
39 X can be supplied either by itself and block specifications p, q or its
40 subblocks in an iterable from which the shapes would be derived. See the
41 examples below.
43 Parameters
44 ----------
45 X : array_like, iterable
46 complex unitary or real orthogonal matrix to be decomposed, or iterable
47 of subblocks ``X11``, ``X12``, ``X21``, ``X22``, when ``p``, ``q`` are
48 omitted.
49 p : int, optional
50 Number of rows of the upper left block ``X11``, used only when X is
51 given as an array.
52 q : int, optional
53 Number of columns of the upper left block ``X11``, used only when X is
54 given as an array.
55 separate : bool, optional
56 if ``True``, the low level components are returned instead of the
57 matrix factors, i.e. ``(u1,u2)``, ``theta``, ``(v1h,v2h)`` instead of
58 ``u``, ``cs``, ``vh``.
59 swap_sign : bool, optional
60 if ``True``, the ``-S``, ``-I`` block will be the bottom left,
61 otherwise (by default) they will be in the upper right block.
62 compute_u : bool, optional
63 if ``False``, ``u`` won't be computed and an empty array is returned.
64 compute_vh : bool, optional
65 if ``False``, ``vh`` won't be computed and an empty array is returned.
67 Returns
68 -------
69 u : ndarray
70 When ``compute_u=True``, contains the block diagonal orthogonal/unitary
71 matrix consisting of the blocks ``U1`` (``p`` x ``p``) and ``U2``
72 (``m-p`` x ``m-p``) orthogonal/unitary matrices. If ``separate=True``,
73 this contains the tuple of ``(U1, U2)``.
74 cs : ndarray
75 The cosine-sine factor with the structure described above.
76 If ``separate=True``, this contains the ``theta`` array containing the
77 angles in radians.
78 vh : ndarray
79 When ``compute_vh=True`, contains the block diagonal orthogonal/unitary
80 matrix consisting of the blocks ``V1H`` (``q`` x ``q``) and ``V2H``
81 (``m-q`` x ``m-q``) orthogonal/unitary matrices. If ``separate=True``,
82 this contains the tuple of ``(V1H, V2H)``.
84 Examples
85 --------
86 >>> from scipy.linalg import cossin
87 >>> from scipy.stats import unitary_group
88 >>> x = unitary_group.rvs(4)
89 >>> u, cs, vdh = cossin(x, p=2, q=2)
90 >>> np.allclose(x, u @ cs @ vdh)
91 True
93 Same can be entered via subblocks without the need of ``p`` and ``q``. Also
94 let's skip the computation of ``u``
96 >>> ue, cs, vdh = cossin((x[:2, :2], x[:2, 2:], x[2:, :2], x[2:, 2:]),
97 ... compute_u=False)
98 >>> print(ue)
99 []
100 >>> np.allclose(x, u @ cs @ vdh)
101 True
103 References
104 ----------
105 .. [1] : Brian D. Sutton. Computing the complete CS decomposition. Numer.
106 Algorithms, 50(1):33-65, 2009.
108 """
110 if p or q:
111 p = 1 if p is None else int(p)
112 q = 1 if q is None else int(q)
113 X = _asarray_validated(X, check_finite=True)
114 if not np.equal(*X.shape):
115 raise ValueError("Cosine Sine decomposition only supports square"
116 " matrices, got {}".format(X.shape))
117 m = X.shape[0]
118 if p >= m or p <= 0:
119 raise ValueError("invalid p={}, 0<p<{} must hold"
120 .format(p, X.shape[0]))
121 if q >= m or q <= 0:
122 raise ValueError("invalid q={}, 0<q<{} must hold"
123 .format(q, X.shape[0]))
125 x11, x12, x21, x22 = X[:p, :q], X[:p, q:], X[p:, :q], X[p:, q:]
126 elif not isinstance(X, Iterable):
127 raise ValueError("When p and q are None, X must be an Iterable"
128 " containing the subblocks of X")
129 else:
130 if len(X) != 4:
131 raise ValueError("When p and q are None, exactly four arrays"
132 " should be in X, got {}".format(len(X)))
134 x11, x12, x21, x22 = [np.atleast_2d(x) for x in X]
135 for name, block in zip(["x11", "x12", "x21", "x22"],
136 [x11, x12, x21, x22]):
137 if block.shape[1] == 0:
138 raise ValueError("{} can't be empty".format(name))
139 p, q = x11.shape
140 mmp, mmq = x22.shape
142 if x12.shape != (p, mmq):
143 raise ValueError("Invalid x12 dimensions: desired {}, "
144 "got {}".format((p, mmq), x12.shape))
146 if x21.shape != (mmp, q):
147 raise ValueError("Invalid x21 dimensions: desired {}, "
148 "got {}".format((mmp, q), x21.shape))
150 if p + mmp != q + mmq:
151 raise ValueError("The subblocks have compatible sizes but "
152 "don't form a square array (instead they form a"
153 " {}x{} array). This might be due to missing "
154 "p, q arguments.".format(p + mmp, q + mmq))
156 m = p + mmp
158 cplx = any([np.iscomplexobj(x) for x in [x11, x12, x21, x22]])
159 driver = "uncsd" if cplx else "orcsd"
160 csd, csd_lwork = get_lapack_funcs([driver, driver + "_lwork"],
161 [x11, x12, x21, x22])
162 lwork = _compute_lwork(csd_lwork, m=m, p=p, q=q)
163 lwork_args = ({'lwork': lwork[0], 'lrwork': lwork[1]} if cplx else
164 {'lwork': lwork})
165 *_, theta, u1, u2, v1h, v2h, info = csd(x11=x11, x12=x12, x21=x21, x22=x22,
166 compute_u1=compute_u,
167 compute_u2=compute_u,
168 compute_v1t=compute_vh,
169 compute_v2t=compute_vh,
170 trans=False, signs=swap_sign,
171 **lwork_args)
173 method_name = csd.typecode + driver
174 if info < 0:
175 raise ValueError('illegal value in argument {} of internal {}'
176 .format(-info, method_name))
177 if info > 0:
178 raise LinAlgError("{} did not converge: {}".format(method_name, info))
180 if separate:
181 return (u1, u2), theta, (v1h, v2h)
183 U = block_diag(u1, u2)
184 VDH = block_diag(v1h, v2h)
186 # Construct the middle factor CS
187 c = np.diag(np.cos(theta))
188 s = np.diag(np.sin(theta))
189 r = min(p, q, m - p, m - q)
190 n11 = min(p, q) - r
191 n12 = min(p, m - q) - r
192 n21 = min(m - p, q) - r
193 n22 = min(m - p, m - q) - r
194 Id = np.eye(np.max([n11, n12, n21, n22, r]), dtype=theta.dtype)
195 CS = np.zeros((m, m), dtype=theta.dtype)
197 CS[:n11, :n11] = Id[:n11, :n11]
199 xs = n11 + r
200 xe = n11 + r + n12
201 ys = n11 + n21 + n22 + 2 * r
202 ye = n11 + n21 + n22 + 2 * r + n12
203 CS[xs: xe, ys:ye] = Id[:n12, :n12] if swap_sign else -Id[:n12, :n12]
205 xs = p + n22 + r
206 xe = p + n22 + r + + n21
207 ys = n11 + r
208 ye = n11 + r + n21
209 CS[xs:xe, ys:ye] = -Id[:n21, :n21] if swap_sign else Id[:n21, :n21]
211 CS[p:p + n22, q:q + n22] = Id[:n22, :n22]
212 CS[n11:n11 + r, n11:n11 + r] = c
213 CS[p + n22:p + n22 + r, r + n21 + n22:2 * r + n21 + n22] = c
215 xs = n11
216 xe = n11 + r
217 ys = n11 + n21 + n22 + r
218 ye = n11 + n21 + n22 + 2 * r
219 CS[xs:xe, ys:ye] = s if swap_sign else -s
221 CS[p + n22:p + n22 + r, n11:n11 + r] = -s if swap_sign else s
223 return U, CS, VDH