ó ˽÷Xc @`sÍddlmZmZmZdddddddgZdd lZdd lmZmZmZm Z m Z m Z m Z m Z dd lmZdd lmZd dlmZmZmZd dlmZmZe ddƒZd„Zd„Zd„Zd eeeeeed„Z d eeeeed d ed„ Z!eeedd ded„Z"d eeed„Z#d eeeed d ed„Z$eedd ed„Z%dddgZ&eeed„Z'd S( i(tdivisiontprint_functiontabsolute_importteigteight eig_bandedteigvalsteigvalshteigvals_bandedt hessenbergN(tarraytisfinitetinexacttnonzerot iscomplexobjtcastt flatnonzerotconj(txrange(t_asarray_validatedi(t LinAlgErrort _datacopiedtnorm(tget_lapack_funcst_compute_lworktFyð?cC`sºtj|d|ƒ}|jdk}|dc |jddkO*xrt|ƒD]d}|dd…|df|jdd…|f= %d have converged)gN(sgeevs geev_lwork(RtlenR<R:RR#RNRRR8R1RR;R9RRR(RR"(tatbR?R@RARBROR-R=R>RPRQRRRSR3RRFRGRItwrtwiRMRL((s2/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp.pyRwsf?/ /         c C`st|d| ƒ} t| jƒdksA| jd| jdkrPtdƒ‚n|pbt| |ƒ}t| ƒrzt} nt} |d!k rRt|d| ƒ} |p°t| |ƒ}t| jƒdksâ| jd| jdkrñtdƒ‚n| j| jkr.tdt | jƒt | jƒfƒ‚nt| ƒrCt} qX| pLt} nd!} |rddpgd} |d!k rè|\}}|dks¡|| jdkrÅtd d| jddfƒ‚n|d7}|d7}||f}n|r÷d }nd }| r d }nd }| d!kræt |df| fƒ\}|d!kr‰|| d|d| ddddd| jdd|ƒ\}}}q-|\}}|| d|d| ddd|d|d|ƒ\}}}|d||d!}nG|d!k rzt |df| | fƒ\}|\}}|| | d|d|d|d| d|d|d|ƒ\}}}}|d||d!}n³|rØt |df| | fƒ\}|| | d|d|d| d|d|ƒ\}}}nUt |df| | fƒ\}|| | d|d|d| d|d|ƒ\}}}|dkrP|rC|S||fSnÁ|dkrpt d| ƒ‚n¡|dkr—| d!kr—t dƒ‚nzd|koµ| jdknrö|d!k rãt dt |ƒdƒ‚qt d|ƒ‚nt d || jdƒ‚d!S("sÌ Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix. Find eigenvalues w and optionally eigenvectors v of matrix `a`, where `b` is positive definite:: a v[:,i] = w[i] b v[:,i] v[i,:].conj() a v[:,i] = w[i] v[i,:].conj() b v[:,i] = 1 Parameters ---------- a : (M, M) array_like A complex Hermitian or real symmetric matrix whose eigenvalues and eigenvectors will be computed. b : (M, M) array_like, optional A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed. lower : bool, optional Whether the pertinent array data is taken from the lower or upper triangle of `a`. (Default: lower) eigvals_only : bool, optional Whether to calculate only eigenvalues and no eigenvectors. (Default: both are calculated) turbo : bool, optional Use divide and conquer algorithm (faster but expensive in memory, only for generalized eigenvalue problem and if eigvals=None) eigvals : tuple (lo, hi), optional Indexes of the smallest and largest (in ascending order) eigenvalues and corresponding eigenvectors to be returned: 0 <= lo <= hi <= M-1. If omitted, all eigenvalues and eigenvectors are returned. type : int, optional Specifies the problem type to be solved: type = 1: a v[:,i] = w[i] b v[:,i] type = 2: a b v[:,i] = w[i] v[:,i] type = 3: b a v[:,i] = w[i] v[:,i] 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 Whether to check that the input matrices 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 ------- w : (N,) float ndarray The N (1<=N<=M) selected eigenvalues, in ascending order, each repeated according to its multiplicity. v : (M, N) complex ndarray (if eigvals_only == False) The normalized selected eigenvector corresponding to the eigenvalue w[i] is the column v[:,i]. Normalization: type 1 and 3: v.conj() a v = w type 2: inv(v).conj() a inv(v) = w type = 1 or 2: v.conj() b v = I type = 3: v.conj() inv(b) v = I Raises ------ LinAlgError If eigenvalue computation does not converge, an error occurred, or b matrix is not definite positive. Note that if input matrices are not symmetric or hermitian, no error is reported but results will be wrong. See Also -------- eig : eigenvalues and right eigenvectors for non-symmetric arrays ROiiisexpected square matrixs#wrong b dimensions %s, should be %stNtVsCThe eigenvalue range specified is not valid. Valid range is [%s,%s]tLtUthetsytevrtuplotjobztrangetAtiltiuRAtItgvxtitypeRBtgvdtgvs<illegal value in %i-th argument of internal fortran routine.sunrecoverable internal error.s'the eigenvectors %s failed to converge.sƒinternal fortran routine failed to converge: %i off-diagonal elements of an intermediate tridiagonal form did not converge to zero.s the leading minor of order %i of 'b' is not positive definite. The factorization of 'b' could not be completed and no eigenvalues or eigenvectors were computed.N( RRWR<R:RRtTruetFalseR#tstrRRR (RXRYtlowert eigvals_onlyRARBtturboRttypeROR=tcplxR>t_jobtlothiRctpfxRbRRRItw_totRjtifailRlRm((s2/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp.pyRñsšV/   /%          !"      #  RXcC`sÙ|s |r6t|d|ƒ}|p0t||ƒ}nIt|ƒ}t|jjtƒryt|ƒjƒ ryt dƒ‚nd}t |j ƒdkr£t dƒ‚n|j ƒd$krÄt d ƒ‚n|j ƒd%krL|jj dkrtd&|fƒ\} d} ntd'|fƒ\} d} | |d| d|d|ƒ\} } } n|j ƒd(kr„|j ƒd)krñd}ddt|ƒt|ƒf\}}}}t||ƒdksÑt||ƒ|j dkràt dƒ‚n||d}nLd}t|ƒt|ƒddf\}}}}|dkr=|j d}n|rLd}n|jj dkr‚td*tdddƒfƒ\}n!td+tdddƒfƒ\}d|dƒ}|jj dkrãtd,|fƒ\}d} ntd-|fƒ\}d} |||||d|dd| d|d|d|d|d |ƒ\} } }}} | | } |s„| d!d!…d!|…f} q„n| dkrªt d"| | fƒ‚n| dkrÅtd#ƒ‚n|rÏ| S| | fS(.s Solve real symmetric or complex hermitian band matrix eigenvalue problem. Find eigenvalues w and optionally right eigenvectors v of a:: a v[:,i] = w[i] v[:,i] v.H v = identity The matrix a is stored in a_band either in lower diagonal or upper diagonal ordered form: a_band[u + i - j, j] == a[i,j] (if upper form; i <= j) a_band[ i - j, j] == a[i,j] (if lower form; i >= j) where u is the number of bands above the diagonal. Example of a_band (shape of a is (6,6), u=2):: upper form: * * a02 a13 a24 a35 * a01 a12 a23 a34 a45 a00 a11 a22 a33 a44 a55 lower form: a00 a11 a22 a33 a44 a55 a10 a21 a32 a43 a54 * a20 a31 a42 a53 * * Cells marked with * are not used. Parameters ---------- a_band : (u+1, M) array_like The bands of the M by M matrix a. lower : bool, optional Is the matrix in the lower form. (Default is upper form) eigvals_only : bool, optional Compute only the eigenvalues and no eigenvectors. (Default: calculate also eigenvectors) overwrite_a_band : bool, optional Discard data in a_band (may enhance performance) select : {'a', 'v', 'i'}, optional Which eigenvalues to calculate ====== ======================================== select calculated ====== ======================================== 'a' All eigenvalues 'v' Eigenvalues in the interval (min, max] 'i' Eigenvalues with indices min <= i <= max ====== ======================================== select_range : (min, max), optional Range of selected eigenvalues max_ev : int, optional For select=='v', maximum number of eigenvalues expected. For other values of select, has no meaning. In doubt, leave this parameter untouched. 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 ------- w : (M,) ndarray The eigenvalues, in ascending order, each repeated according to its multiplicity. v : (M, M) float or complex ndarray The normalized eigenvector corresponding to the eigenvalue w[i] is the column v[:,i]. Raises LinAlgError if eigenvalue computation does not converge ROs#array must not contain infs or NaNsiisexpected two-dimensional arrayiRXRR!R(tvaluetindexsinvalid argument for selecttGFDthbevdtsbevdt compute_vRqt overwrite_abgsselect_range out of boundstfFtlamchRRTRVtsthbevxtsbevxtmmaxRetabstolNs.illegal value in %d-th argument of internal %sseig algorithm did not converge( iiiRXRR!sallsvaluesindex(iRXsall(R(R€(iiR!Rsindexsvalue(iR!sindex(slamch(slamch(R†(R‡(RRR t issubclassRRtR R R(R:RWR<RqR;RtmintmaxR(ta_bandRqRrtoverwrite_a_bandtselectt select_rangetmax_evROR=tbevdt internal_nameRRRIRFtvuRgRhR„R‰tbevxR R{((s2/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp.pyRÀstN  ( *1*  $!   "  cC`s.t|d|ddddd|d|d|ƒS(so Compute eigenvalues from an ordinary or generalized eigenvalue problem. Find eigenvalues of a general matrix:: a vr[:,i] = w[i] b vr[:,i] Parameters ---------- a : (M, M) array_like A complex or real matrix whose eigenvalues and eigenvectors will be computed. b : (M, M) array_like, optional Right-hand side matrix in a generalized eigenvalue problem. If omitted, identity matrix is assumed. overwrite_a : bool, optional Whether to overwrite data in a (may improve performance) check_finite : bool, optional Whether to check that the input matrices 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. homogeneous_eigvals : bool, optional If True, return the eigenvalues in homogeneous coordinates. In this case ``w`` is a (2, M) array so that:: w[1,i] a vr[:,i] = w[0,i] b vr[:,i] Default is False. Returns ------- w : (M,) or (2, M) double or complex ndarray The eigenvalues, each repeated according to its multiplicity but not in any specific order. The shape is (M,) unless ``homogeneous_eigvals=True``. Raises ------ LinAlgError If eigenvalue computation does not converge See Also -------- eigvalsh : eigenvalues of symmetric or Hermitian arrays, eig : eigenvalues and right eigenvectors of general arrays. eigh : eigenvalues and eigenvectors of symmetric/Hermitian arrays. RYR?iR@RAROR-(R(RXRYRAROR-((s2/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp.pyR]s3!c C`s@t|d|d|dtd|d|d|d|d|d |ƒ S( s Solve an ordinary or generalized eigenvalue problem for a complex Hermitian or real symmetric matrix. Find eigenvalues w of matrix a, where b is positive definite:: a v[:,i] = w[i] b v[:,i] v[i,:].conj() a v[:,i] = w[i] v[i,:].conj() b v[:,i] = 1 Parameters ---------- a : (M, M) array_like A complex Hermitian or real symmetric matrix whose eigenvalues and eigenvectors will be computed. b : (M, M) array_like, optional A complex Hermitian or real symmetric definite positive matrix in. If omitted, identity matrix is assumed. lower : bool, optional Whether the pertinent array data is taken from the lower or upper triangle of `a`. (Default: lower) turbo : bool, optional Use divide and conquer algorithm (faster but expensive in memory, only for generalized eigenvalue problem and if eigvals=None) eigvals : tuple (lo, hi), optional Indexes of the smallest and largest (in ascending order) eigenvalues and corresponding eigenvectors to be returned: 0 <= lo < hi <= M-1. If omitted, all eigenvalues and eigenvectors are returned. type : int, optional Specifies the problem type to be solved: type = 1: a v[:,i] = w[i] b v[:,i] type = 2: a b v[:,i] = w[i] v[:,i] type = 3: b a v[:,i] = w[i] v[:,i] 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 Whether to check that the input matrices 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 ------- w : (N,) float ndarray The N (1<=N<=M) selected eigenvalues, in ascending order, each repeated according to its multiplicity. Raises ------ LinAlgError If eigenvalue computation does not converge, an error occurred, or b matrix is not definite positive. Note that if input matrices are not symmetric or hermitian, no error is reported but results will be wrong. See Also -------- eigvals : eigenvalues of general arrays eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays eig : eigenvalues and right eigenvectors for non-symmetric arrays RYRqRrRARBRsRRtRO(RRn( RXRYRqRARBRsRRtRO((s2/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp.pyR•sF cC`s.t|d|ddd|d|d|d|ƒS(sI Solve real symmetric or complex hermitian band matrix eigenvalue problem. Find eigenvalues w of a:: a v[:,i] = w[i] v[:,i] v.H v = identity The matrix a is stored in a_band either in lower diagonal or upper diagonal ordered form: a_band[u + i - j, j] == a[i,j] (if upper form; i <= j) a_band[ i - j, j] == a[i,j] (if lower form; i >= j) where u is the number of bands above the diagonal. Example of a_band (shape of a is (6,6), u=2):: upper form: * * a02 a13 a24 a35 * a01 a12 a23 a34 a45 a00 a11 a22 a33 a44 a55 lower form: a00 a11 a22 a33 a44 a55 a10 a21 a32 a43 a54 * a20 a31 a42 a53 * * Cells marked with * are not used. Parameters ---------- a_band : (u+1, M) array_like The bands of the M by M matrix a. lower : bool, optional Is the matrix in the lower form. (Default is upper form) overwrite_a_band : bool, optional Discard data in a_band (may enhance performance) select : {'a', 'v', 'i'}, optional Which eigenvalues to calculate ====== ======================================== select calculated ====== ======================================== 'a' All eigenvalues 'v' Eigenvalues in the interval (min, max] 'i' Eigenvalues with indices min <= i <= max ====== ======================================== select_range : (min, max), optional Range of selected eigenvalues 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 ------- w : (M,) ndarray The eigenvalues, in ascending order, each repeated according to its multiplicity. Raises LinAlgError if eigenvalue computation does not converge See Also -------- eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian band matrices eigvals : eigenvalues of general arrays eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays eig : eigenvalues and right eigenvectors for non-symmetric arrays RqRriRŽRRRO(R(RRqRŽRRRO((s2/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp.pyRásJ R!tlRVcC`s%t|d|ƒ}t|jƒdksA|jd|jdkrPtdƒ‚n|pbt||ƒ}|jddkrœ|r˜|tj|jdƒfS|Std|fƒ\}}}||d dd |ƒ\}} } } } | dkrþtd | ƒ‚nt|ƒ} t||jdd | d | ƒ}||d | d | d|d dƒ\}}} | dkrytd| ƒ‚ntj |dƒ}|s•|Std|fƒ\}}t|| d | d | ƒ}|d|d|d | d | d|d dƒ\}} | dkrtd| ƒ‚n||fS(sú Compute Hessenberg form of a matrix. The Hessenberg decomposition is:: A = Q H Q^H where `Q` is unitary/orthogonal and `H` has only zero elements below the first sub-diagonal. Parameters ---------- a : (M, M) array_like Matrix to bring into Hessenberg form. calc_q : bool, optional Whether to compute the transformation matrix. Default is False. overwrite_a : bool, optional Whether to overwrite `a`; may improve performance. Default is False. 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 ------- H : (M, M) ndarray Hessenberg form of `a`. Q : (M, M) ndarray Unitary/orthogonal similarity transformation matrix ``A = Q H Q^H``. Only returned if ``calc_q=True``. ROiiisexpected square matrixtgehrdtgebalt gehrd_lworktpermuteRAs>illegal value in %d-th argument of internal gebal (hessenberg)RwRxR3s>illegal value in %d-th argument of internal gehrd (hessenberg)iÿÿÿÿtorghrt orghr_lworkRXttaus>illegal value in %d-th argument of internal orghr (hessenberg)(sgehrdsgebals gehrd_lwork(sorghrs orghr_lwork( RRWR<R:RRteyeRRttriu(RXtcalc_qRAROR=R—R˜R™tbaRwRxtpivscaleRItnR3thqRthR›Rœtq((s2/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp.pyR 2s<"/'  "- 3 ((t __future__RRRt__all__RR R R R RRRRtscipy._lib.sixRtscipy._lib._utilRtmiscRRRtlapackRRR9R"R1RNR#RoRnRRRRRRt_double_precisionR (((s2/tmp/pip-build-7oUkmx/scipy/scipy/linalg/decomp.pyts8  :   . y   Í œ  7   JM