ó î&]\c@`s¬dZddlmZmZmZddlZddlmZmZmZm Z ddl m Z ddl m Z ddlmZmZdd lmZdd lmZd d gZd ddgZddeded„ZejeƒjZejeƒjZi dd6dd6dd6dd 6dd6dd6dd6dd6dd6Zidd 6dd6dd6dd6dd6dd6Z ddgddggZ!d„Z"d„Z#ed„Z$dS(sSchur decomposition functions.i(tdivisiontprint_functiontabsolute_importN(tasarray_chkfinitetsingletasarraytarray(tnorm(tcallablei(t LinAlgErrort _datacopied(tget_lapack_funcs(teigvalstschurtrsf2csftitltdtrealc C`së|d#krtdƒ‚n|r0t|ƒ}n t|ƒ}t|jƒdksk|jd|jdkrztd ƒ‚n|jj}|d$krÚ|d%krÚ|tkrÂ|jd ƒ}d }qÚ|jd ƒ}d }n|pìt ||ƒ}t d&|fƒ\}|d"ks|d krT|d„|dd ƒ} | ddj jt jƒ}n|d"krrd} d„} n‡d} t|ƒr|} nl|dkr¥d„} nT|dkr½d„} n<|dkrÕd„} n$|dkríd„} n tdƒ‚|| |d|d|d| ƒ} | d } | dkrItdj| ƒƒ‚ng| |jddkrotdƒ‚nA| |jddkr•tdƒ‚n| dkr°td ƒ‚n| dkrÎ| d| d!fS| d| d!| dfSd"S('sÅ 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 -------- >>> 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 Rtcomplextrtcs%argument must be 'real', or 'complex'iiisexpected square matrixtFtDtgeesiÿÿÿÿcS`sdS(N(tNone(tx((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyt‹stlworkiþÿÿÿcS`sdS(N(R(R((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyRstlhpcS`s |jdkS(Ng(R(R((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyR–strhpcS`s |jdkS(Ng(R(R((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyR˜stiuccS`st|ƒdkS(Ngð?(tabs(R((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyRšstouccS`st|ƒdkS(Ngð?(R (R((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyRœssZ'sort' parameter must either be 'None', or a callable, or one of ('lhp','rhp','iuc','ouc')t overwrite_atsort_ts0illegal value in {}-th argument of internal geess2Eigenvalues could not be separated for reordering.s2Leading eigenvalues do not satisfy sort condition.s0Schur form not found. Possibly ill-conditioned.iýÿÿÿN(RRRR(RR(RR(R(t ValueErrorRRtlentshapetdtypetchart_double_precisiontastypeR R RRtnumpytintRtformatR ( tatoutputRR"tsortt check_finiteta1ttypRtresultR#t sfunctiontinfo((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyR sbc  /                        tbthtBtfRRcG`s[d}d}x@|D]8}|jj}t|t|ƒ}t|t|ƒ}qWt||S(Ni(R'R(tmaxt _array_kindt_array_precisiont _array_type(tarraystkindt precisionR.tt((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyt _commonType¾s  cG`swd}xL|D]D}|jj|kr;||jƒf}q ||j|ƒf}q Wt|ƒdkro|dS|SdS(Nii((R'R(tcopyR*R%(ttypeR?t cast_arraysR.((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyt _castCopyÈs c C`so|r$tt||fƒ\}}ntt||fƒ\}}xet||gƒD]Q\}}|jdks‡|jd|jdkrRtdjd|ƒƒ‚qRqRW|jd|jdkrâtdj|j|jƒƒ‚n|jd}t||t dgdƒƒ}t |||ƒ\}}x=t |ddd ƒD]%}t |||dfƒt t ||d|dfƒt |||fƒkrMt||d|d…|d|d…fƒ|||f}t|d|||dfgƒ} |d| } |||df| } t | jƒ| g| | ggd |ƒ} | j||d|d…|dd …fƒ||d|d…|dd …f<|d |d…|d|d…fj| jƒjƒ|d |d…|d|d…f<|d d …|d|d…fj| jƒjƒ|d d …|d|d…f>> 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]]) iiisInput '{}' must be square.tZTs.Input array shapes must match: Z: {} vs. T: {}g@RiÿÿÿÿR'Ng(tmapRRt enumeratetndimR&R$R-RCRRGtrangeR tepsR RtconjtdottT( RPtZR1tindtXtNRBtmtmuRRtstG((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyRÕs05)   P@$+S\W(%t__doc__t __future__RRRR+RRRRt numpy.linalgRtscipy._lib.sixRtmiscR R tlapackR tdecompR t__all__R)RtFalsetTrueR tfinfotfloatRMtfepsR<R=R>RCRGR(((s8lib/python2.7/site-packages/scipy/linalg/decomp_schur.pyts* "    &0