U Hv f°#ć@sTddlmZddlZddlmZddlmZmZddl m Z m Z dgZ d d d„Z dS) é)ŚIterableN)Ś_asarray_validated)Ś block_diagŚ LinAlgErroré)Ś_compute_lworkŚget_lapack_funcsŚcossinFTc- Cs|s |r|dkrdnt|ƒ}|dkr*dnt|ƒ}t|dd}tj|jŽsZtd |j”ƒ‚|jd}||kst|dkrŠtd ||jd”ƒ‚||ksš|dkr°td ||jd”ƒ‚|d|…d|…f|d|…|d…f||d…d|…f||d…|d…ff\}} } } n$t|tƒs td ƒ‚n t |ƒd kr@td  t |ƒ”ƒ‚d d „|Dƒ\}} } } t ddddg|| | | gƒD](\} } | jddkrptd | ”ƒ‚qp|j\}}| j\}}| j||fkrŌtd ||f| j”ƒ‚| j||fkrśtd ||f| j”ƒ‚||||kr$td ||||”ƒ‚||}t dd „|| | | fDƒƒ}|rPdnd}t ||dg|| | | gƒ\}}t ||||d}|rš|d|ddœnd|i}|f|| | | ||||d|dœ |—Ž^}}}}}}}|j|}|dkrütd | |”ƒ‚|dkrtd  ||”ƒ‚|r.||f|||ffSt||ƒ}t||ƒ}t t |””} t t |””}!t||||||ƒ}"t||ƒ|"}#t|||ƒ|"}$t|||ƒ|"}%t||||ƒ|"}&tjt |#|$|%|&|"g”|jd!}'tj||f|jd!}(|'d|#…d|#…f|(d|#…d|#…f<|#|"})|#|"|$}*|#|%|&d"|"}+|#|%|&d"|"|$},|rr|'d|$…d|$…fn|'d|$…d|$…f |(|)|*…|+|,…f<||&|"})||&|"|% }*|#|"}+|#|"|%},|rę|'d|%…d|%…f n|'d|%…d|%…f|(|)|*…|+|,…f<|'d|&…d|&…f|(|||&…|||&…f<| |(|#|#|"…|#|#|"…f<| |(||&||&|"…|"|%|&d"|"|%|&…f<|#})|#|"}*|#|%|&|"}+|#|%|&d"|"},|rĄ|!n|! |(|)|*…|+|,…f<|rā|! n|!|(||&||&|"…|#|#|"…f<||(|fS)#u» 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)``:: ā”Œ ā” ā”‚ I 0 0 ā”‚ 0 0 0 ā”‚ ā”Œ ā” ā”Œ ā”ā”‚ 0 C 0 ā”‚ 0 -S 0 ā”‚ā”Œ ā”* ā”‚ X11 ā”‚ X12 ā”‚ ā”‚ U1 ā”‚ ā”‚ā”‚ 0 0 0 ā”‚ 0 0 -I ā”‚ā”‚ V1 ā”‚ ā”‚ ā”‚ ā”€ā”€ā”€ā”€ā”¼ā”€ā”€ā”€ā”€ ā”‚ = ā”‚ā”€ā”€ā”€ā”€ā”¼ā”€ā”€ā”€ā”€ā”‚ā”‚ā”€ā”€ā”€ā”€ā”€ā”€ā”€ā”€ā”€ā”¼ā”€ā”€ā”€ā”€ā”€ā”€ā”€ā”€ā”€ā”‚ā”‚ā”€ā”€ā”€ā”€ā”¼ā”€ā”€ā”€ā”€ā”‚ ā”‚ X21 ā”‚ X22 ā”‚ ā”‚ ā”‚ U2 ā”‚ā”‚ 0 0 0 ā”‚ I 0 0 ā”‚ā”‚ ā”‚ V2 ā”‚ ā”” ā”˜ ā”” ā”˜ā”‚ 0 S 0 ā”‚ 0 C 0 ā”‚ā”” ā”˜ ā”‚ 0 0 I ā”‚ 0 0 0 ā”‚ ā”” ā”˜ ``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 NrT)Z check_finitez?Cosine Sine decomposition only supports square matrices, got {}rzinvalid p={}, 0‡szcossin..Śx11Śx12Śx21Śx22z{} can't be emptyz*Invalid x12 dimensions: desired {}, got {}z*Invalid x21 dimensions: desired {}, got {}z‘The subblocks have compatible sizes but don't form a square array (instead they form a {}x{} array). 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