(ph?SSKrSSKrSSKJr SSKJrJrJr SSKJ r SS/r /SQr S r S r S rS rS rSSjrSSjrSSjrg)N)asarray_chkfinite) LinAlgError _datacopied LinAlgWarning)get_lapack_funcsqzordqz)ildc[U5(aUnU$US:Xa[nU$US:Xa[nU$US:Xa[nU$US:Xa[nU$[ S5e)NlhprhpiucouczLsort parameter must be None, a callable, or one of ('lhp','rhp','iuc','ouc'))callable_lhp_rhp_iuc_ouc ValueError)sort sfunctions J/var/www/html/venv/lib/python3.13/site-packages/scipy/linalg/_decomp_qz.py_select_functionrs~~             <= =c[R"U[S9nUS:gnSX#)'[R"XX- 5S:X#'U$NdtyperFgnp empty_likeboolrealxyoutnonzeros rrr!G -- &CAvGCMGGAJqz12S8CL Jrc[R"U[S9nUS:gnSX#)'[R"XX- 5S:X#'U$rr"r's rrr*r,rcz[R"U[S9nUS:gnSX#)'[XX- 5S:X#'U$)Nr rF?r#r$r%absr's rrr3sC -- &CAvGCM 1:-.4CL Jrc[R"U[S9nUS:HnUS:HnSX#U-'SX#)U-'[X)X)- 5S:X$)'U$)Nr rFTr/r0)r(r)r*xzeroyzeros rrr<sc -- &C !VE !VEC Cqy6*+c1CK Jrc Ub [S5eUS;a [S5eU(a[U5n[U5n O,[R"U5n[R"U5n URupU RupXs=:Xa U s=:XaU :Xd O [S5eUR R nUS;a7US;a1U[;aURS5nSnOURS5nSnU R R nUS;a7US;a1U[;aU RS5n SnOU RS5n SnU=(d [X5nU=(d [X5n[S X45unUbUS :Xa.U"S XS S 9nUS SRR[5nSnU"UXX5USS9nUS nUS:a[SU*S35eUS:a(UU ::a"[R"SUS- S3[SS9 O_qz..tsr)lworkrcgr>r?r@s rr_qz..sfunctionwsr)rC overwrite_a overwrite_bsort_tIllegal value in argument z of ggesztThe QZ iteration failed. (a,b) are not in Schur form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be correct for J=rz,...,N) stacklevelz(Something other than QZ iteration failedzAfter reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the Generalized Schur form no longer satisfy sort=True. This could also be due to scaling.z#Reordering failed in tgsen)rrr#asarrayshaper!char_double_precisionastyperrr&intwarningswarnrrtypecode)ABoutputrCrrGrH check_finitea1b1a_ma_nb_mb_ntypatypbr;resultrinfos r_qzreFsh KL L22@AA q ! q ! ZZ] ZZ]xxHCxxHC  $# $ $DEE 88==D !!d*&< $ $3BD3BD 88==D !!d*&< $ $3BD3BD5+b"4K5+b"4K Y 1ED } nbB7r 1 ""))#. )R5)!5F ":D ax5teWHEFF dck ''+Avhf67D!" $ QDEE QJK K Q?@@ 4==  rc D[XX#UXVUS9upUSUSUSUS4$)a0 QZ decomposition for generalized eigenvalues of a pair of matrices. The QZ, or generalized Schur, decomposition for a pair of n-by-n matrices (A,B) is:: (A,B) = (Q @ AA @ Z*, Q @ BB @ Z*) where AA, BB is in generalized Schur form if BB is upper-triangular with non-negative diagonal and AA is upper-triangular, or for real QZ decomposition (``output='real'``) block upper triangular with 1x1 and 2x2 blocks. In this case, the 1x1 blocks correspond to real generalized eigenvalues and 2x2 blocks are 'standardized' by making the corresponding elements of BB have the form:: [ a 0 ] [ 0 b ] and the pair of corresponding 2x2 blocks in AA and BB will have a complex conjugate pair of generalized eigenvalues. If (``output='complex'``) or A and B are complex matrices, Z' denotes the conjugate-transpose of Z. Q and Z are unitary matrices. Parameters ---------- A : (N, N) array_like 2-D array to decompose B : (N, N) array_like 2-D array to decompose output : {'real', 'complex'}, optional Construct the real or complex QZ decomposition for real matrices. Default is 'real'. lwork : int, optional Work array size. If None or -1, it is automatically computed. sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional NOTE: THIS INPUT IS DISABLED FOR NOW. Use ordqz instead. 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). For real matrix pairs, the sort function takes three real arguments (alphar, alphai, beta). The eigenvalue ``x = (alphar + alphai*1j)/beta``. For complex matrix pairs or output='complex', the sort function takes two complex arguments (alpha, beta). The eigenvalue ``x = (alpha/beta)``. 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). overwrite_a : bool, optional Whether to overwrite data in a (may improve performance) overwrite_b : bool, optional Whether to overwrite data in b (may improve performance) check_finite : bool, optional If true checks the elements of `A` and `B` are finite numbers. If false does no checking and passes matrix through to underlying algorithm. Returns ------- AA : (N, N) ndarray Generalized Schur form of A. BB : (N, N) ndarray Generalized Schur form of B. Q : (N, N) ndarray The left Schur vectors. Z : (N, N) ndarray The right Schur vectors. See Also -------- ordqz Notes ----- Q is transposed versus the equivalent function in Matlab. .. versionadded:: 0.11.0 Examples -------- >>> import numpy as np >>> from scipy.linalg import qz >>> A = np.array([[1, 2, -1], [5, 5, 5], [2, 4, -8]]) >>> B = np.array([[1, 1, -3], [3, 1, -1], [5, 6, -2]]) Compute the decomposition. The QZ decomposition is not unique, so depending on the underlying library that is used, there may be differences in the signs of coefficients in the following output. >>> AA, BB, Q, Z = qz(A, B) >>> AA array([[-1.36949157, -4.05459025, 7.44389431], [ 0. , 7.65653432, 5.13476017], [ 0. , -0.65978437, 2.4186015 ]]) # may vary >>> BB array([[ 1.71890633, -1.64723705, -0.72696385], [ 0. , 8.6965692 , -0. ], [ 0. , 0. , 2.27446233]]) # may vary >>> Q array([[-0.37048362, 0.1903278 , 0.90912992], [-0.90073232, 0.16534124, -0.40167593], [ 0.22676676, 0.96769706, -0.11017818]]) # may vary >>> Z array([[-0.67660785, 0.63528924, -0.37230283], [ 0.70243299, 0.70853819, -0.06753907], [ 0.22088393, -0.30721526, -0.92565062]]) # may vary Verify the QZ decomposition. With real output, we only need the transpose of ``Z`` in the following expressions. >>> Q @ AA @ Z.T # Should be A array([[ 1., 2., -1.], [ 5., 5., 5.], [ 2., 4., -8.]]) >>> Q @ BB @ Z.T # Should be B array([[ 1., 1., -3.], [ 3., 1., -1.], [ 5., 6., -2.]]) Repeat the decomposition, but with ``output='complex'``. >>> AA, BB, Q, Z = qz(A, B, output='complex') For conciseness in the output, we use ``np.set_printoptions()`` to set the output precision of NumPy arrays to 3 and display tiny values as 0. >>> np.set_printoptions(precision=3, suppress=True) >>> AA array([[-1.369+0.j , 2.248+4.237j, 4.861-5.022j], [ 0. +0.j , 7.037+2.922j, 0.794+4.932j], [ 0. +0.j , 0. +0.j , 2.655-1.103j]]) # may vary >>> BB array([[ 1.719+0.j , -1.115+1.j , -0.763-0.646j], [ 0. +0.j , 7.24 +0.j , -3.144+3.322j], [ 0. +0.j , 0. +0.j , 2.732+0.j ]]) # may vary >>> Q array([[ 0.326+0.175j, -0.273-0.029j, -0.886-0.052j], [ 0.794+0.426j, -0.093+0.134j, 0.402-0.02j ], [-0.2 -0.107j, -0.816+0.482j, 0.151-0.167j]]) # may vary >>> Z array([[ 0.596+0.32j , -0.31 +0.414j, 0.393-0.347j], [-0.619-0.332j, -0.479+0.314j, 0.154-0.393j], [-0.195-0.104j, 0.576+0.27j , 0.715+0.187j]]) # may vary With complex arrays, we must use ``Z.conj().T`` in the following expressions to verify the decomposition. >>> Q @ AA @ Z.conj().T # Should be A array([[ 1.-0.j, 2.-0.j, -1.-0.j], [ 5.+0.j, 5.+0.j, 5.-0.j], [ 2.+0.j, 4.+0.j, -8.+0.j]]) >>> Q @ BB @ Z.conj().T # Should be B array([[ 1.+0.j, 1.+0.j, -3.+0.j], [ 3.-0.j, 1.-0.j, -1.+0.j], [ 5.+0.j, 6.+0.j, -2.+0.j]]) )rYrCrrGrHrZrr)re) rWrXrYrCrrGrHrZrc_s rr r s>RA4 +!-/IF !9fQiVBZ 77rc X[XUSUUUS9uGtpxpp pU S:Xa'U SU S[R"S5--U SpOU S:XaU SU SS--U SpOU up[U5nU"X5n[ S Xx45nU S ;aS UR S-S -OSnU"UXxXSUSS 9Gtnnn nn n nU S:Xa'U SU S[R"S5--U SpOU S:XaU SU SS--U SpOU upUS:a[ SU*S35eUS:Xa [ S5eUUXUU4$)a QZ decomposition for a pair of matrices with reordering. Parameters ---------- A : (N, N) array_like 2-D array to decompose B : (N, N) array_like 2-D array to decompose sort : {callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional Specifies whether the upper eigenvalues should be sorted. A callable may be passed that, given an ordered pair ``(alpha, beta)`` representing the eigenvalue ``x = (alpha/beta)``, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). For the real matrix pairs ``beta`` is real while ``alpha`` can be complex, and for complex matrix pairs both ``alpha`` and ``beta`` can be complex. The callable must be able to accept a NumPy array. 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) With the predefined sorting functions, an infinite eigenvalue (i.e., ``alpha != 0`` and ``beta = 0``) is considered to lie in neither the left-hand nor the right-hand plane, but it is considered to lie outside the unit circle. For the eigenvalue ``(alpha, beta) = (0, 0)``, the predefined sorting functions all return `False`. output : str {'real','complex'}, optional Construct the real or complex QZ decomposition for real matrices. Default is 'real'. overwrite_a : bool, optional If True, the contents of A are overwritten. overwrite_b : bool, optional If True, the contents of B are overwritten. check_finite : bool, optional If true checks the elements of `A` and `B` are finite numbers. If false does no checking and passes matrix through to underlying algorithm. Returns ------- AA : (N, N) ndarray Generalized Schur form of A. BB : (N, N) ndarray Generalized Schur form of B. alpha : (N,) ndarray alpha = alphar + alphai * 1j. See notes. beta : (N,) ndarray See notes. Q : (N, N) ndarray The left Schur vectors. Z : (N, N) ndarray The right Schur vectors. See Also -------- qz Notes ----- On exit, ``(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N``, will be the generalized eigenvalues. ``ALPHAR(j) + ALPHAI(j)*i`` and ``BETA(j),j=1,...,N`` are the diagonals of the complex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of the real generalized Schur form of (A,B) were further reduced to triangular form using complex unitary transformations. If ALPHAI(j) is zero, then the jth eigenvalue is real; if positive, then the ``j``\ th and ``(j+1)``\ st eigenvalues are a complex conjugate pair, with ``ALPHAI(j+1)`` negative. .. versionadded:: 0.17.0 Examples -------- >>> import numpy as np >>> from scipy.linalg import ordqz >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]]) >>> B = np.array([[0, 6, 0, 0], [5, 0, 2, 1], [5, 2, 6, 6], [4, 7, 7, 7]]) >>> AA, BB, alpha, beta, Q, Z = ordqz(A, B, sort='lhp') Since we have sorted for left half plane eigenvalues, negatives come first >>> (alpha/beta).real < 0 array([ True, True, False, False], dtype=bool) N)rYrrGrHrZsrry?rMr tgsensd)ijobrCliworkrJz of tgsenzReordering of (A, B) failed because the transformed matrix pair (A, B) would be too far from generalized Schur form; the problem is very ill-conditioned. 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