Ug) dZddlZddlmZmZmZmZddlmZddl m Z m Z ddl m Z ddlmZd d gZgd Z ddZejejZejejZdddddddddd ZdddddddZddgddggZdZdZddZdS)zSchur decomposition functions.N)asarray_chkfinitesingleasarrayarray)norm) LinAlgError _datacopied)get_lapack_funcs)eigvalsschurrsf2csf)ildrealFTcB|dvrtd|rt|}nt|}tj|jtjr#t|tjd}t|jdks|jd|jdkrtd|jj }|d vr8|d vr4|tvr| d }n| d }|j dkrttjd|j\}} |6tj||jtj|| jfStj||jtj|| jdfS|pt!||}t#d|f\} ||dkr?| d|d} | ddj tj}|d} d} nKd} t)|r|} n7|dkrd} n-|dkrd} n#|dkrd} n|dkrd} ntd| | |||| } | d}|dkrtd| d||jddzkrt+d ||jddzkrt+d!|dkrt+d"|| d| d#fS| d| d#| dfS)$a Compute Schur decomposition of a matrix. The Schur decomposition is:: A = Z T Z^H where Z is unitary and T is either upper-triangular, or for real Schur decomposition (output='real'), quasi-upper triangular. In the quasi-triangular form, 2x2 blocks describing complex-valued eigenvalue pairs may extrude from the diagonal. Parameters ---------- a : (M, M) array_like Matrix to decompose output : {'real', 'complex'}, optional Construct the real or complex Schur decomposition (for real matrices). lwork : int, optional Work array size. If None or -1, it is automatically computed. overwrite_a : bool, optional Whether to overwrite data in a (may improve performance). sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional Specifies whether the upper eigenvalues should be sorted. A callable may be passed that, given a eigenvalue, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). Alternatively, string parameters may be used:: 'lhp' Left-hand plane (x.real < 0.0) 'rhp' Right-hand plane (x.real > 0.0) 'iuc' Inside the unit circle (x*x.conjugate() <= 1.0) 'ouc' Outside the unit circle (x*x.conjugate() > 1.0) Defaults to None (no sorting). check_finite : bool, optional Whether to check that the input matrix contains 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 ------- T : (M, M) ndarray Schur form of A. It is real-valued for the real Schur decomposition. Z : (M, M) ndarray An unitary Schur transformation matrix for A. It is real-valued for the real Schur decomposition. sdim : int If and only if sorting was requested, a third return value will contain the number of eigenvalues satisfying the sort condition. Raises ------ LinAlgError Error raised under three conditions: 1. The algorithm failed due to a failure of the QR algorithm to compute all eigenvalues. 2. If eigenvalue sorting was requested, the eigenvalues could not be reordered due to a failure to separate eigenvalues, usually because of poor conditioning. 3. If eigenvalue sorting was requested, roundoff errors caused the leading eigenvalues to no longer satisfy the sorting condition. See Also -------- rsf2csf : Convert real Schur form to complex Schur form Examples -------- >>> import numpy as np >>> from scipy.linalg import schur, eigvals >>> A = np.array([[0, 2, 2], [0, 1, 2], [1, 0, 1]]) >>> T, Z = schur(A) >>> T array([[ 2.65896708, 1.42440458, -1.92933439], [ 0. , -0.32948354, -0.49063704], [ 0. , 1.31178921, -0.32948354]]) >>> Z array([[0.72711591, -0.60156188, 0.33079564], [0.52839428, 0.79801892, 0.28976765], [0.43829436, 0.03590414, -0.89811411]]) >>> T2, Z2 = schur(A, output='complex') >>> T2 array([[ 2.65896708, -1.22839825+1.32378589j, 0.42590089+1.51937378j], # may vary [ 0. , -0.32948354+0.80225456j, -0.59877807+0.56192146j], [ 0. , 0. , -0.32948354-0.80225456j]]) >>> eigvals(T2) array([2.65896708, -0.32948354+0.80225456j, -0.32948354-0.80225456j]) An arbitrary custom eig-sorting condition, having positive imaginary part, which is satisfied by only one eigenvalue >>> T3, Z3, sdim = schur(A, output='complex', sort=lambda x: x.imag > 0) >>> sdim 1 )rcomplexrcz%argument must be 'real', or 'complex'longdtyperrzexpected square matrix)rr)FDrrN)geescdSNxs Y/var/www/surfInsights/venv3-11/lib/python3.11/site-packages/scipy/linalg/_decomp_schur.pyzschur..s)lworkcdSr r!r"s r$ sfunctionzschur..sfunctions4r&lhpc|jdkSNrr"s r$r*zschur..sfunctionsv|#r&rhpc|jdkSr-r/r"s r$r*zschur..sfunctionsv}$r&iucc(t|dkSNg?absr"s r$r*zschur..sfunctions1vv}$r&oucc(t|dkSr4r5r"s r$r*zschur..sfunctions1vv|#r&zZ'sort' parameter must either be 'None', or a callable, or one of ('lhp','rhp','iuc','ouc'))r' overwrite_asort_tzillegal value in z-th argument of internal geesz2Eigenvalues could not be separated for reordering.z2Leading eigenvalues do not satisfy sort condition.z/Schur form not found. 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Convert a quasi-diagonal real-valued Schur form to the upper-triangular complex-valued Schur form. Parameters ---------- T : (M, M) array_like Real Schur form of the original array Z : (M, M) array_like Schur transformation matrix check_finite : bool, optional Whether to check that the input arrays 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 ------- T : (M, M) ndarray Complex Schur form of the original array Z : (M, M) ndarray Schur transformation matrix corresponding to the complex form See Also -------- schur : Schur decomposition of an array Examples -------- >>> import numpy as np >>> from scipy.linalg import schur, rsf2csf >>> A = np.array([[0, 2, 2], [0, 1, 2], [1, 0, 1]]) >>> T, Z = schur(A) >>> T array([[ 2.65896708, 1.42440458, -1.92933439], [ 0. , -0.32948354, -0.49063704], [ 0. , 1.31178921, -0.32948354]]) >>> Z array([[0.72711591, -0.60156188, 0.33079564], [0.52839428, 0.79801892, 0.28976765], [0.43829436, 0.03590414, -0.89811411]]) >>> T2 , Z2 = rsf2csf(T, Z) >>> T2 array([[2.65896708+0.j, -1.64592781+0.743164187j, -1.21516887+1.00660462j], [0.+0.j , -0.32948354+8.02254558e-01j, -0.82115218-2.77555756e-17j], [0.+0.j , 0.+0.j, -0.32948354-0.802254558j]]) >>> Z2 array([[0.72711591+0.j, 0.28220393-0.31385693j, 0.51319638-0.17258824j], [0.52839428+0.j, 0.24720268+0.41635578j, -0.68079517-0.15118243j], [0.43829436+0.j, -0.76618703+0.01873251j, -0.03063006+0.46857912j]]) rrrzInput '{}' must be square.ZTz"Input array shapes must match: Z: z vs. T: g@rrrNr.)maprr enumeratendimrAr<formatrarrfranger6epsr rconjdotT)rqZrMindXmessageNr`mmurrsGs r$rrsCl$$q!f--117QF##1QF##MMQ 6Q;;!'!* 229@@cKKLL L3 wqzQWQZQqwQQQQ!!!  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