^Mh ,dZddlZddlmZdgZddZdS) z+ Solve the orthogonal Procrustes problem. N)svdorthogonal_procrustesTc|r)tj|}tj|}n(tj|}tj|}|jdkrt d|j|j|jkr t d|jd|jdt |jtj|zj\}}}||z}| }||fS)au Compute the matrix solution of the orthogonal (or unitary) Procrustes problem. Given matrices `A` and `B` of the same shape, find an orthogonal (or unitary in the case of complex input) matrix `R` that most closely maps `A` to `B` using the algorithm given in [1]_. Parameters ---------- A : (M, N) array_like Matrix to be mapped. B : (M, N) array_like Target matrix. check_finite : bool, optional Whether to check that the input matrices contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- R : (N, N) ndarray The matrix solution of the orthogonal Procrustes problem. Minimizes the Frobenius norm of ``(A @ R) - B``, subject to ``R.conj().T @ R = I``. scale : float Sum of the singular values of ``A.conj().T @ B``. Raises ------ ValueError If the input array shapes don't match or if check_finite is True and the arrays contain Inf or NaN. Notes ----- Note that unlike higher level Procrustes analyses of spatial data, this function only uses orthogonal transformations like rotations and reflections, and it does not use scaling or translation. .. versionadded:: 0.15.0 References ---------- .. [1] Peter H. Schonemann, "A generalized solution of the orthogonal Procrustes problem", Psychometrica -- Vol. 31, No. 1, March, 1966. :doi:`10.1007/BF02289451` Examples -------- >>> import numpy as np >>> from scipy.linalg import orthogonal_procrustes >>> A = np.array([[ 2, 0, 1], [-2, 0, 0]]) Flip the order of columns and check for the anti-diagonal mapping >>> R, sca = orthogonal_procrustes(A, np.fliplr(A)) >>> R array([[-5.34384992e-17, 0.00000000e+00, 1.00000000e+00], [ 0.00000000e+00, 1.00000000e+00, 0.00000000e+00], [ 1.00000000e+00, 0.00000000e+00, -7.85941422e-17]]) >>> sca 9.0 As an example of the unitary Procrustes problem, generate a random complex matrix ``A``, a random unitary matrix ``Q``, and their product ``B``. >>> shape = (4, 4) >>> rng = np.random.default_rng(589234981235) >>> A = rng.random(shape) + rng.random(shape)*1j >>> Q = rng.random(shape) + rng.random(shape)*1j >>> Q, _ = np.linalg.qr(Q) >>> B = A @ Q `orthogonal_procrustes` recovers the unitary matrix ``Q`` from ``A`` and ``B``. >>> R, _ = orthogonal_procrustes(A, B) >>> np.allclose(R, Q) True z$expected ndim to be 2, but observed zthe shapes of A and B differ (z vs )) npasarray_chkfinite asanyarrayndim ValueErrorshaperT conjugatesum)AB check_finiteuwvtRscales X/var/www/html/test/jupyter/venv/lib/python3.11/site-packages/scipy/linalg/_procrustes.pyrr sf   # #   # # M!   M!  v{{HHHIIIw!'Q!'QQqwQQQRRRAC",q//),--HAq" BA EEGGE e8O)T)__doc__numpyr _decomp_svdr__all__rrrr!s] # #ccccccr