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« prev ^ index » next coverage.py v7.3.2, created at 2023-12-12 06:31 +0000
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 References
85 ----------
86 .. [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
87 Algorithms, 50(1):33-65, 2009.
89 Examples
90 --------
91 >>> import numpy as np
92 >>> from scipy.linalg import cossin
93 >>> from scipy.stats import unitary_group
94 >>> x = unitary_group.rvs(4)
95 >>> u, cs, vdh = cossin(x, p=2, q=2)
96 >>> np.allclose(x, u @ cs @ vdh)
97 True
99 Same can be entered via subblocks without the need of ``p`` and ``q``. Also
100 let's skip the computation of ``u``
102 >>> ue, cs, vdh = cossin((x[:2, :2], x[:2, 2:], x[2:, :2], x[2:, 2:]),
103 ... compute_u=False)
104 >>> print(ue)
105 []
106 >>> np.allclose(x, u @ cs @ vdh)
107 True
109 """
111 if p or q:
112 p = 1 if p is None else int(p)
113 q = 1 if q is None else int(q)
114 X = _asarray_validated(X, check_finite=True)
115 if not np.equal(*X.shape):
116 raise ValueError("Cosine Sine decomposition only supports square"
117 " matrices, got {}".format(X.shape))
118 m = X.shape[0]
119 if p >= m or p <= 0:
120 raise ValueError("invalid p={}, 0<p<{} must hold"
121 .format(p, X.shape[0]))
122 if q >= m or q <= 0:
123 raise ValueError("invalid q={}, 0<q<{} must hold"
124 .format(q, X.shape[0]))
126 x11, x12, x21, x22 = X[:p, :q], X[:p, q:], X[p:, :q], X[p:, q:]
127 elif not isinstance(X, Iterable):
128 raise ValueError("When p and q are None, X must be an Iterable"
129 " containing the subblocks of X")
130 else:
131 if len(X) != 4:
132 raise ValueError("When p and q are None, exactly four arrays"
133 " should be in X, got {}".format(len(X)))
135 x11, x12, x21, x22 = [np.atleast_2d(x) for x in X]
136 for name, block in zip(["x11", "x12", "x21", "x22"],
137 [x11, x12, x21, x22]):
138 if block.shape[1] == 0:
139 raise ValueError("{} can't be empty".format(name))
140 p, q = x11.shape
141 mmp, mmq = x22.shape
143 if x12.shape != (p, mmq):
144 raise ValueError("Invalid x12 dimensions: desired {}, "
145 "got {}".format((p, mmq), x12.shape))
147 if x21.shape != (mmp, q):
148 raise ValueError("Invalid x21 dimensions: desired {}, "
149 "got {}".format((mmp, q), x21.shape))
151 if p + mmp != q + mmq:
152 raise ValueError("The subblocks have compatible sizes but "
153 "don't form a square array (instead they form a"
154 " {}x{} array). This might be due to missing "
155 "p, q arguments.".format(p + mmp, q + mmq))
157 m = p + mmp
159 cplx = any([np.iscomplexobj(x) for x in [x11, x12, x21, x22]])
160 driver = "uncsd" if cplx else "orcsd"
161 csd, csd_lwork = get_lapack_funcs([driver, driver + "_lwork"],
162 [x11, x12, x21, x22])
163 lwork = _compute_lwork(csd_lwork, m=m, p=p, q=q)
164 lwork_args = ({'lwork': lwork[0], 'lrwork': lwork[1]} if cplx else
165 {'lwork': lwork})
166 *_, theta, u1, u2, v1h, v2h, info = csd(x11=x11, x12=x12, x21=x21, x22=x22,
167 compute_u1=compute_u,
168 compute_u2=compute_u,
169 compute_v1t=compute_vh,
170 compute_v2t=compute_vh,
171 trans=False, signs=swap_sign,
172 **lwork_args)
174 method_name = csd.typecode + driver
175 if info < 0:
176 raise ValueError('illegal value in argument {} of internal {}'
177 .format(-info, method_name))
178 if info > 0:
179 raise LinAlgError("{} did not converge: {}".format(method_name, info))
181 if separate:
182 return (u1, u2), theta, (v1h, v2h)
184 U = block_diag(u1, u2)
185 VDH = block_diag(v1h, v2h)
187 # Construct the middle factor CS
188 c = np.diag(np.cos(theta))
189 s = np.diag(np.sin(theta))
190 r = min(p, q, m - p, m - q)
191 n11 = min(p, q) - r
192 n12 = min(p, m - q) - r
193 n21 = min(m - p, q) - r
194 n22 = min(m - p, m - q) - r
195 Id = np.eye(np.max([n11, n12, n21, n22, r]), dtype=theta.dtype)
196 CS = np.zeros((m, m), dtype=theta.dtype)
198 CS[:n11, :n11] = Id[:n11, :n11]
200 xs = n11 + r
201 xe = n11 + r + n12
202 ys = n11 + n21 + n22 + 2 * r
203 ye = n11 + n21 + n22 + 2 * r + n12
204 CS[xs: xe, ys:ye] = Id[:n12, :n12] if swap_sign else -Id[:n12, :n12]
206 xs = p + n22 + r
207 xe = p + n22 + r + + n21
208 ys = n11 + r
209 ye = n11 + r + n21
210 CS[xs:xe, ys:ye] = -Id[:n21, :n21] if swap_sign else Id[:n21, :n21]
212 CS[p:p + n22, q:q + n22] = Id[:n22, :n22]
213 CS[n11:n11 + r, n11:n11 + r] = c
214 CS[p + n22:p + n22 + r, r + n21 + n22:2 * r + n21 + n22] = c
216 xs = n11
217 xe = n11 + r
218 ys = n11 + n21 + n22 + r
219 ye = n11 + n21 + n22 + 2 * r
220 CS[xs:xe, ys:ye] = s if swap_sign else -s
222 CS[p + n22:p + n22 + r, n11:n11 + r] = -s if swap_sign else s
224 return U, CS, VDH