^Mh#XddlmZddlZddlmZddlmZmZddl m Z m Z dgZ d d Z dS) )IterableN)_asarray_validated) block_diag LinAlgError)_compute_lworkget_lapack_funcscossinFTc4 |s|r|dnt|}|dnt|}t|d}tj|jst d|j|jd}||ks|dkr!t d|d|jdd ||ks|dkr!t d |d |jdd |d|d|f|d||df||dd|f||d|dff\}} } } n9t |tst d t|d krt dt|d|D\}} } } tgd|| | | gD](\} } | jddkrt | d)|j\}}| j\}}| j||fkrt d||fd| j| j||fkrt d||fd| j||z||zkrt d||zd||zd||z}td|| | | fD}|rdnd}t||dzg|| | | g\}}t||||}|r|d|ddnd|i}|d&|| | | ||||d|d |^}}}}}}}|j |z}|dkrt d!| d"||dkrt|d#||r ||f|||ffSt||}t||}tjtj|} tjtj|}!t'||||z ||z }"t'|||"z }#t'|||z |"z }$t'||z ||"z }%t'||z ||z |"z }&tjtj|#|$|%|&|"g|j$}'tj||f|j$}(|'d|#d|#f|(d|#d|#f<|#|"z})|#|"z|$z}*|#|%z|&zd%|"zz}+|#|%z|&zd%|"zz|$z},|r|'d|$d|$fn|'d|$d|$f |(|)|*|+|,f<||&z|"z})||&z|"z|% z}*|#|"z}+|#|"z|%z},|r|'d|%d|%f n |'d|%d|%f|(|)|*|+|,f<|'d|&d|&f|(|||&z|||&zf<| |(|#|#|"z|#|#|"zf<| |(||&z||&z|"z|#|"z|%z|&zd%|"z|#z|%z|&zf<|#})|#|"z}*|#|%z|&z|"z}+|#|%z|&zd%|"zz},|r|!n|! |(|)|*|+|,f<|r|! n|!|(||&z||&z|"z|#|#|"zf<||(|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)``:: ┌ ┐ │ 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 NrT) check_finitez=Cosine Sine decomposition only supports square matrices, got rz invalid p=z, 0K|]}tj|VdS)N)np atleast_2d.0xs [/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/scipy/linalg/_decomp_cossin.py zcossin..s,::1bmA..::::::)x11x12x21x22z can't be emptyz Invalid x12 dimensions: desired z, got z Invalid x21 dimensions: desired zWThe subblocks have compatible sizes but don't form a square array (instead they form a rz5 array). 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