(ph/ :SrSSKrSSKJrJrJrJr SSKJr SSK J r J r SSK J r SSKJr S S /r/S QrSS jr\R&"\5R*r\R&"\5R*rSSSSSSSSSS . rSSSSSSS.rSS/SS//rSrSrSSjrg)zSchur decomposition functions.N)asarray_chkfinitesingleasarrayarray)norm) LinAlgError _datacopied)get_lapack_funcs)eigvalsschurrsf2csf)ildc2US;a [S5eU(a [U5nO [U5n[R"UR [R 5(a[U[R "S5S9n[UR5S:wd URSURS:wa [S5eUR RnUS ;a3US ;a-U[;aURS 5nOURS 5nURS:Xa[[R"SUR S95upUc<[R"XhR S9[R"XiR S94$[R"XhR S9[R"XiR S9S4$U=(d [!X`5n[#S U45un UbUS:Xa8U "SUSS9n U SSR$R[R&5nUcSn S#Sjn OPSn [)U5(aUn O;US:XaS#Sjn O/US:XaS#Sjn O#US:XaS#Sjn OUS:XaS#Sjn O [S5eU "XX#U S9n U SnUS:a[SU*S35eXRSS-:Xa [+S5eXRSS-:Xa [+S 5eUS:a [+S!5eUc U SU S"4$U SU S"U S4$)$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 When the dtype of `a` is real, this specifies whether to compute the real or complex Schur decomposition. When the dtype of `a` is complex, this argument is ignored, and the complex Schur decomposition is computed. 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 an eigenvalue, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). - If ``output='complex'`` OR the dtype of `a` is complex, the callable should have one argument: the eigenvalue expressed as a complex number. - If ``output='real'`` AND the dtype of `a` is real, the callable should have two arguments: the real and imaginary parts of the eigenvalue, respectively. Alternatively, string parameters may be used:: 'lhp' Left-hand plane (real(eigenvalue) < 0.0) 'rhp' Right-hand plane (real(eigenvalue) >= 0.0) 'iuc' Inside the unit circle (abs(eigenvalue) <= 1.0) 'ouc' Outside the unit circle (abs(eigenvalue) > 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. Note that complex conjugate pairs for which the condition is true for either eigenvalue count as 2. 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]) # may vary A custom eigenvalue-sorting condition that sorts by positive imaginary part is satisfied by only one eigenvalue. >>> _, _, sdim = schur(A, output='complex', sort=lambda x: x.imag > 1e-15) >>> sdim 1 When ``output='real'`` and the array `a` is real, the `sort` callable must accept the real and imaginary parts as separate arguments. Note that now the complex eigenvalues ``-0.32948354+0.80225456j`` and ``-0.32948354-0.80225456j`` will be treated as a complex conjugate pair, and according to the `sdim` documentation, complex conjugate pairs for which the condition is True for *either* eigenvalue increase `sdim` by *two*. >>> _, _, sdim = schur(A, output='real', sort=lambda x, y: y > 1e-15) >>> sdim 2 )realcomplexrcz%argument must be 'real', or 'complex'longdtyperrzexpected square matrix)rr)FDrr)geescgN)xs M/var/www/html/venv/lib/python3.13/site-packages/scipy/linalg/_decomp_schur.pyschur..s)lworkcgr r!r"ys r# sfunctionschur..sfunctionsr&lhpc URS:$Nrr*s r#r,r-svv|#r&rhpc URS:$r0r2r*s r#r,r-svv}$r&iucc6UcUOXS--n[U5S:*$Ny?g?absr"r+zs r#r,r-s!AbD1v}$r&oucc6UcUOXS--n[U5S:$r7r8r:s r#r,r-s!AbD1v|#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. Possibly ill-conditioned.r ) ValueErrorrrnp issubdtyperintegerlenshapechar_double_precisionastypesizer eye empty_liker r rint_callabler )aoutputr'r>sort check_finitea1typt0z0rresultr?r,infos r#r r st22@AA q ! QZ }}RXXrzz** Qbhhv. / 288}bhhqkRXXa[8122 ((--C !!c&; # #3B3B ww!|rvvarxx01 <MM"HH5MM"HH57 7MM"HH5MM"HH5q: :5+b"4K Y .ED } nb3r 1 ""))"''2 |  D>>I U] $ U] % U] %U] $MN N)u!F ":D ax,dUG3PQRR !q NOO !q NOO KLL |ay&*$$ay&*fQi//r&) bhBrrfrrr)rrr\rrrr\rrrcSnSnUH?nURRn[U[U5n[U[U5nMA [ UU$)Nr)rrGmax _array_kind_array_precision _array_type)arrayskind precisionrOts r# _commonTyperfsW DI  GGLL4Q( #3A#67  t Y ''r&cSnUHFnURRU:XaX#R54-nM2X#RU54-nMH [ U5S:XaUS$U$)Nr!rr)rrGcopyrIrE)typerb cast_arraysrOs r# _castCopyrkscK  77<<4 % 3K%$(99K   ;11~r&c <U(a[[X45upO[[X45up[X/5HGup4URS:wd"UR SUR S:wdM7[ SSUS35e UR SUR S:wa'SUR SUR 3n[ U5eUR Sn[X[S /S 55n[XqU5up[US- SS 5GHn[XUS- 45[[XS- US- 45[XU45--:Ga7[XS- US-2US- US-245XU4- n [U SXUS- 4/5n U SU - n XUS- 4U - n [U R5U /U *U //US 9n U R!XS- US-2US- S 245XS- US-2US- S 24'US US-2US- US-24R!U R5R"5US US-2US- US-24'US S 2US- US-24R!U R5R"5US S 2US- US-24'SXUS- 4'GM X4$)a Convert real Schur form to complex Schur form. Convert a quasi-diagonal real-valued Schur form to the upper-triangular complex-valued Schur form. Parameters ---------- T : (M, M) array_like Real Schur form of the original array Z : (M, M) array_like Schur transformation matrix check_finite : bool, optional Whether to check that the input arrays contain only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- T : (M, M) ndarray Complex Schur form of the original array Z : (M, M) ndarray Schur transformation matrix corresponding to the complex form See Also -------- schur : Schur decomposition of an array Examples -------- >>> import numpy as np >>> from scipy.linalg import schur, rsf2csf >>> A = np.array([[0, 2, 2], [0, 1, 2], [1, 0, 1]]) >>> T, Z = schur(A) >>> T array([[ 2.65896708, 1.42440458, -1.92933439], [ 0. , -0.32948354, -0.49063704], [ 0. , 1.31178921, -0.32948354]]) >>> Z array([[0.72711591, -0.60156188, 0.33079564], [0.52839428, 0.79801892, 0.28976765], [0.43829436, 0.03590414, -0.89811411]]) >>> T2 , Z2 = rsf2csf(T, Z) >>> T2 array([[2.65896708+0.j, -1.64592781+0.743164187j, -1.21516887+1.00660462j], [0.+0.j , -0.32948354+8.02254558e-01j, -0.82115218-2.77555756e-17j], [0.+0.j , 0.+0.j, -0.32948354-0.802254558j]]) >>> Z2 array([[0.72711591+0.j, 0.28220393-0.31385693j, 0.51319638-0.17258824j], [0.52839428+0.j, 0.24720268+0.41635578j, -0.68079517-0.15118243j], [0.43829436+0.j, -0.76618703+0.01873251j, -0.03063006+0.46857912j]]) rrrzInput 'ZTz' must be square.z"Input array shapes must match: Z: z vs. T: g@rrrNr1)maprr enumeratendimrFrArfrrkranger9epsr rconjdotT)ruZrRindXmessageNremmurrsGs r#rrsl$qf-17QF#QF# 66Q;!''!* 2wtCyk1BCD D$ wwqzQWWQZ6qwwixyQ!!  AA%s+,A Q1 DA 1Q32  qAaCy>CQsAaCx[!1CQ$L!@A AQ3qs7AaC!G+,-Q$7BbeQ!A#vY'(A1 AQqS& A A!}r1g.a8A uuQs1Q3w!}%56Ac!A#gqstm  !A#qs1Q3w/33AFFHJJ?AdqsdAaC!Gm a1QqSjM--affhjj9Aa1QqSjMQqS&  4Kr&)rNFNT)T)__doc__numpyrBrrrr numpy.linalgr_miscr r lapackr _decompr __all__rHr finfofloatrrfepsr_r`rarfrkrr!r&r#rs$;;,$ I #AEG0T hhuo xxAAAAAAA/ CSzC:& ( Sr&