xfs( 6dZddlZddlmZmZmZmZddlmZddlm Z m Z ddl m Z ddl mZd d gZgd Z dd Zej$ej(Zej$ej(Zdddddddddd ZdddddddZddgddggZdZdZddZy)zSchur decomposition functions.N)asarray_chkfinitesingleasarrayarray)norm) LinAlgError _datacopied)get_lapack_funcs)eigvalsschurrsf2csf)ildc|dvr td|r t|}n t|}t|jdk7s|jd|jdk7r td|j j }|dvr3|dvr/|tvr|jd }d }n|jd }d }|xs t||}td |f\}||d k(r;|d |d } | ddjjtj}|d} d} n?d} t|r|} n/|dk(rd} n&|dk(rd} n|dk(rd} n|dk(rd} n td|| |||| } | d } | dkrtdj| | |jddzk(r t!d| |jddzk(r t!d| dkDr t!d| dk(r | d| dfS| 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], [ 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 )realcomplexrcz%argument must be 'real', or 'complex'rrzexpected square matrix)rr)FDrr)geescyNxs zschur..s)lworkcyrrrs r! sfunctionzschur..sfunctionsr#lhpc |jdkSNrrs r!r'zschur..sfunctionsvv|#r#rhpc |jdk\Sr*r,rs r!r'zschur..sfunctionsvv}$r#iucct|dkSNg?absrs r!r'zschur..sfunctions1v}$r#oucct|dkDSr1r2rs r!r'zschur..sfunctions1v|#r#zZ'sort' parameter must either be 'None', or a callable, or one of ('lhp','rhp','iuc','ouc'))r$ overwrite_asort_tz0illegal value in {}-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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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: {} vs. T: {}g@rr)r<Nr+)maprr enumeratendimr;r9rCrYrr^ranger3epsr rconjdotT) rhZrGindXNrXmmurrsGs r!rrsl$q!f-17QF#1QF#MQ 66Q;!''!* 29@@cKL LM wwqzQWWQZ"F177AGG46 6  AAq%s+,A Q1 DAq 1Q32   qAaCy>CQqsAaCx[!1C!Q$L!@A A1Q3qs7AaC!G+,-!Q$7BbeQq!A#vY'(A1 A!QqS& A A!}r1g.a8A uuQqs1Q3w!}%56Aac!A#gqstm  !A#qs1Q3w/33AFFHJJ?AdqsdAaC!Gm a1QqSjM--affhjj9Aa1QqSjM!QqS&   a4Kr#)rNFNT)T)__doc__r@rrrr numpy.linalgr_miscr r lapackr _decompr __all__r>r finfofloatrefepsrRrSrTrYr^rrr#r!rzs$ ;;,$ I #AEd0Nekk%u{{6AAAAAAA/ CSzC:& ( Sr#