ó î&]\c @`s£ddlmZmZmZdddddddd d d g Zdd lZdd lmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZmZmZmZmZddlmZddlmZddlmZddlmZmZmZddl m!Z!m"Z"e ddƒZ#d„Z$d„Z%d„Z&d e(e)e(e(e)e(d„Z*d e)e(e(e(e)d de)d„ Z+i dd6dd6dd6dd6dd6dd6dd6dd 6dd!6Z,d"„Z-e(e(e(dd de)d#„Z.d e(e)e(d$„Z/d e)e(e(e)d de)d%„Z0e(e(dd e)d&„Z1dd e)d'd(d)„Z2e(dd e)d'd(d*„Z3d+d,„Z4e(e(e)d-„Z5d.„Z6d S(/i(tdivisiontprint_functiontabsolute_importteigteigvalsteighteigvalsht eig_bandedteigvals_bandedteigh_tridiagonalteigvalsh_tridiagonalt hessenbergtcdf2rdfN(tarraytisfinitetinexacttnonzerot iscomplexobjtcastt flatnonzerotconjtasarraytargsorttemptytnewaxistargwheret iscomplexteyetzerosteinsum(txrange(t_asarray_validated(t string_typesi(t LinAlgErrort _datacopiedtnorm(tget_lapack_funcst_compute_lworktFyð?cC`sºtj|d|ƒ}|jdk}|dc |jddkO*xrt|ƒD]d}|dd…|df|jdd…|ft_It _check_infoR5R)R'tcharR/RtshapeR#(ta1tb1tlefttrightt overwrite_at overwrite_bR:R?tcvltcvrtresR@R8R9tvltvrtworktinfoR*talphartalphait only_realttR.((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyt_geneigLsB   59   c C`sát|d|ƒ}t|jƒdksA|jd|jdkrPtdƒ‚n|pbt||ƒ}|dk rt|d|ƒ} |p•t| |ƒ}t| jƒdksÇ| jd| jdkrÖtdƒ‚n| j|jkr÷tdƒ‚nt|| |||||ƒStd|fƒ\} } ||} } t| |jdd | d | ƒ}| j d kr®| |d |d | d | d |ƒ\}}}}t |d|ƒ}nq| |d |d | d | d |ƒ\}}}}}idd6dd6|j j }|t |}t |d|ƒ}t|dddƒtj|jdkƒ}| j d kp\|s¤|j j }|r†t|||ƒ}n|r¤t|||ƒ}q¤n|p­|s´|S|r×|rÍ|||fS||fS||fS(sô Solve an ordinary or generalized eigenvalue problem of a square matrix. Find eigenvalues w and right or left eigenvectors of a general matrix:: a vr[:,i] = w[i] b vr[:,i] a.H vl[:,i] = w[i].conj() b.H vl[:,i] where ``.H`` is the Hermitian conjugation. 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. Default is None, identity matrix is assumed. left : bool, optional Whether to calculate and return left eigenvectors. Default is False. right : bool, optional Whether to calculate and return right eigenvectors. Default is True. overwrite_a : bool, optional Whether to overwrite `a`; may improve performance. Default is False. overwrite_b : bool, optional Whether to overwrite `b`; may improve performance. Default is False. 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. The shape is (M,) unless ``homogeneous_eigvals=True``. vl : (M, M) double or complex ndarray The normalized left eigenvector corresponding to the eigenvalue ``w[i]`` is the column vl[:,i]. Only returned if ``left=True``. vr : (M, M) double or complex ndarray The normalized right eigenvector corresponding to the eigenvalue ``w[i]`` is the column ``vr[:,i]``. Only returned if ``right=True``. Raises ------ LinAlgError If eigenvalue computation does not converge. See Also -------- eigvals : eigenvalues of general arrays eigh : Eigenvalues and right eigenvectors for symmetric/Hermitian arrays. eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian band matrices eigh_tridiagonal : eigenvalues and right eiegenvectors for symmetric/Hermitian tridiagonal matrices Examples -------- >>> from scipy import linalg >>> a = np.array([[0., -1.], [1., 0.]]) >>> linalg.eigvals(a) array([0.+1.j, 0.-1.j]) >>> b = np.array([[0., 1.], [1., 1.]]) >>> linalg.eigvals(a, b) array([ 1.+0.j, -1.+0.j]) >>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]]) >>> linalg.eigvals(a, homogeneous_eigvals=True) array([[3.+0.j, 8.+0.j, 7.+0.j], [1.+0.j, 1.+0.j, 1.+0.j]]) >>> a = np.array([[0., -1.], [1., 0.]]) >>> linalg.eigvals(a) == linalg.eig(a)[0] array([ True, True]) >>> linalg.eig(a, left=True, right=False)[1] # normalized left eigenvector array([[-0.70710678+0.j , -0.70710678-0.j ], [-0. +0.70710678j, -0. -0.70710678j]]) >>> linalg.eig(a, left=False, right=True)[1] # normalized right eigenvector array([[0.70710678+0.j , 0.70710678-0.j ], [0. -0.70710678j, 0. +0.70710678j]]) t check_finiteiiisexpected square matrixs a and b must have the same shapetgeevt geev_lworkt compute_vlt compute_vrRAR@RNR&tftDtdseig algorithm (geev)tpositivesCdid not converge (only eigenvalues with order >= %d have converged)gN(R]R^(RtlenRIt ValueErrorR"R0R[R$R%RER>R'RHRFRGR(R5R)R/(tatbRLRMRNROR\R:RJRKR]R^R_R`R@R*RSRTRVtwrtwiRZRY((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyRvs^`/ /       c C`s t|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!krì|d}t |f| fƒ\}|d!kr|| d|d| ddddd| jdd|ƒ\}}}qE|\}}|| d|d| ddd|d|d|ƒ\}}}|d||d!}nY|d!k r†|d}t |f| | fƒ\}|\}}|| | d|d|d|d| d|d|d|ƒ\}}}}|d||d!}n¿|rê|d}t |f| | fƒ\}|| | d|d|d| d|d|ƒ\}}}n[|d}t |f| | fƒ\}|| | d|d|d| d|d|ƒ\}}}|dkrh|r[|S||fSnt ||dtƒ|dkr¢| d!kr¢t dƒ‚nzd|koÀ| jdknr|d!k rît dt |ƒdƒ‚qt d|ƒ‚nt d || jdƒ‚d!S("sw 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 -------- eigvalsh : eigenvalues of symmetric or Hermitian arrays eig : eigenvalues and right eigenvectors for non-symmetric arrays eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays eigh_tridiagonal : eigenvalues and right eiegenvectors for symmetric/Hermitian tridiagonal matrices Notes ----- This function does not check the input array for being hermitian/symmetric in order to allow for representing arrays with only their upper/lower triangular parts. Examples -------- >>> from scipy.linalg import eigh >>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]]) >>> w, v = eigh(A) >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4))) True R\iiisexpected square matrixs#wrong b dimensions %s, should be %stNtVsCThe eigenvalue range specified is not valid. Valid range is [%s,%s]tLtUthetsytevrtuplotjobztrangetAtiltiuRNtItgvxtitypeROtgvdtgvRdsunrecoverable 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(RReRIRfR"RtTruetFalseR0tstrR$RGR!R(RgRhtlowert eigvals_onlyRNROtturboRttypeR\RJtcplxRKt_jobtlothiRrtpfxtdriverRqR*R,RVtw_totRytifailR{R|((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyRsžh/   /%           !"        #  iR5tvaluetindexRgR,R.c C`s•t|tƒr|jƒ}nyt|}Wntk rKtdƒ‚nXd \}}d}}|dkrt|ƒ}|jdks¬|jdks¬|d|dkr»tdƒ‚n|dkrë|\}}|dkr||}q|q|j j jƒdkr%td |j |j j fƒ‚n|d\}}t ||ƒdks_t ||ƒ|krntd ƒ‚n||d}n||||||fS( s5Check that select is valid, convert to Fortran style.sinvalid argument for selectggð?iiisBselect_range must be a 2-element array-like in nondecreasing orderthilqpsLwhen using select="i", select_range must contain integers, got dtype %s (%s)sselect_range out of bounds(ggð?( t isinstanceR R€t _conv_dicttKeyErrorRfRtndimtsizeR'RHtmintmax( tselectt select_rangetmax_evtmax_lenRStvuRvRwtsr((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyt _check_selectõs0     2    *cC`s|s |r6t|d|ƒ}|p0t||ƒ}nIt|ƒ}t|jjtƒryt|ƒjƒ ryt dƒ‚nd}t |j ƒdkr£t dƒ‚nt ||||j dƒ\}} } } } }~|dkrD|jj dkrûd} nd } t| f|fƒ\}||d | d |d |ƒ\}}}n*|rSd}n|jj d kr‰tdtdddƒfƒ\}n!tdtdddƒfƒ\}d|dƒ}|jj dkrÕd} nd} t| f|fƒ\}||| | | | d | d|d|d |d |d|ƒ\}}}}}|| }|sn|dd…d|…f}nt|| ƒ|r…|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. See Also -------- eigvals_banded : eigenvalues for symmetric/Hermitian band matrices eig : eigenvalues and right eigenvectors of general arrays. eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays eigh_tridiagonal : eigenvalues and right eiegenvectors for symmetric/Hermitian tridiagonal matrices Examples -------- >>> from scipy.linalg import eig_banded >>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]]) >>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]]) >>> w, v = eig_banded(Ab, lower=True) >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4))) True >>> w = eig_banded(Ab, lower=True, eigvals_only=True) >>> w array([-4.26200532, -2.22987175, 3.95222349, 12.53965359]) Request only the eigenvalues between ``[-3, 4]`` >>> w, v = eig_banded(Ab, lower=True, select='v', select_range=[-3, 4]) >>> w array([-2.22987175, 3.95222349]) R\s#array must not contain infs or NaNsiisexpected two-dimensional arrayitGFDthbevdtsbevdt compute_vR€t overwrite_abtfFtlamchR'RaRctsthbevxtsbevxtmmaxRttabstolN(R£(R£(RR"R t issubclassR'RƒRRR5RfReRIRœRHR$RG(ta_bandR€Rtoverwrite_a_bandR–R—R˜R\RJRSRšRvRwt internal_nametbevdR*R,RVR£R¨tbevxR-R‹((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyRsPk  (+   $!   cC`s.t|d|ddddd|d|d|ƒS(s 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 -------- eig : eigenvalues and right eigenvectors of general arrays. eigvalsh : eigenvalues of symmetric or Hermitian arrays eigvals_banded : eigenvalues for symmetric/Hermitian band matrices eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal matrices Examples -------- >>> from scipy import linalg >>> a = np.array([[0., -1.], [1., 0.]]) >>> linalg.eigvals(a) array([0.+1.j, 0.-1.j]) >>> b = np.array([[0., 1.], [1., 1.]]) >>> linalg.eigvals(a, b) array([ 1.+0.j, -1.+0.j]) >>> a = np.array([[3., 0., 0.], [0., 8., 0.], [0., 0., 7.]]) >>> linalg.eigvals(a, homogeneous_eigvals=True) array([[3.+0.j, 8.+0.j, 7.+0.j], [1.+0.j, 1.+0.j, 1.+0.j]]) RhRLiRMRNR\R:(R(RgRhRNR\R:((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyRºsE!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 -------- eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays eigvals : eigenvalues of general arrays eigvals_banded : eigenvalues for symmetric/Hermitian band matrices eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal matrices Notes ----- This function does not check the input array for being hermitian/symmetric in order to allow for representing arrays with only their upper/lower triangular parts. Examples -------- >>> from scipy.linalg import eigvalsh >>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]]) >>> w = eigvalsh(A) >>> w array([-3.74637491, -0.76263923, 6.08502336, 12.42399079]) RhR€RRNROR‚RRƒR\(RR}( RgRhR€RNROR‚RRƒR\((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyRsV cC`s.t|d|ddd|d|d|d|ƒS(s 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 eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal matrices eigvals : eigenvalues of general arrays eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays eig : eigenvalues and right eigenvectors for non-symmetric arrays Examples -------- >>> from scipy.linalg import eigvals_banded >>> A = np.array([[1, 5, 2, 0], [5, 2, 5, 2], [2, 5, 3, 5], [0, 2, 5, 4]]) >>> Ab = np.array([[1, 2, 3, 4], [5, 5, 5, 0], [2, 2, 0, 0]]) >>> w = eigvals_banded(Ab, lower=True) >>> w array([-4.26200532, -2.22987175, 3.95222349, 12.53965359]) R€RiR«R–R—R\(R(RªR€R«R–R—R\((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyR`sW gtautocC`s1t||dtd|d|d|d|d|ƒS(sH Solve eigenvalue problem for a real symmetric tridiagonal matrix. Find eigenvalues `w` of ``a``:: a v[:,i] = w[i] v[:,i] v.H v = identity For a real symmetric matrix ``a`` with diagonal elements `d` and off-diagonal elements `e`. Parameters ---------- d : ndarray, shape (ndim,) The diagonal elements of the array. e : ndarray, shape (ndim-1,) The off-diagonal elements of the array. 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. tol : float The absolute tolerance to which each eigenvalue is required (only used when ``lapack_driver='stebz'``). An eigenvalue (or cluster) is considered to have converged if it lies in an interval of this width. If <= 0. (default), the value ``eps*|a|`` is used where eps is the machine precision, and ``|a|`` is the 1-norm of the matrix ``a``. lapack_driver : str LAPACK function to use, can be 'auto', 'stemr', 'stebz', 'sterf', or 'stev'. When 'auto' (default), it will use 'stemr' if ``select='a'`` and 'stebz' otherwise. 'sterf' and 'stev' can only be used when ``select='a'``. 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 -------- eigh_tridiagonal : eigenvalues and right eiegenvectors for symmetric/Hermitian tridiagonal matrices Examples -------- >>> from scipy.linalg import eigvalsh_tridiagonal, eigvalsh >>> d = 3*np.ones(4) >>> e = -1*np.ones(3) >>> w = eigvalsh_tridiagonal(d, e) >>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1) >>> w2 = eigvalsh(A) # Verify with other eigenvalue routines >>> np.allclose(w - w2, np.zeros(4)) True RR–R—R\ttolt lapack_driver(R R}(RcteR–R—R\R°R±((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyR ¼sKcC`s%t|d|ƒ}t|d|ƒ}xS||fD]E}|jdkrUtdƒ‚n|jjdkr1tdƒ‚q1q1W|j|jdkr¯td|j|jfƒ‚nt||d|jƒ\}} } } } } t|t ƒs÷tdƒ‚nd}||kr"td||fƒ‚n|d krI|dkr@d nd }nt |f||fƒ\}| }|d krË|dkr’tdƒ‚n|s§tdƒ‚n|||ƒ\}}t |ƒ}n†|d kr|dkròtdƒ‚n|||d|ƒ\}}}t |ƒ}n2|d kr t |ƒ}d }t |f||fƒ\}|rddnd}||||| | | | ||ƒ \}}}}}n±t |jd|jƒ}||d*t d ||fƒ\}||||| | | | d|ƒ\}}}t|dƒ||||| | | | d|d|d|ƒ\}}}}t||dƒ|| }|rv|S|d krût d!||fƒ\}||||||ƒ\}}t|dddƒt|ƒ}|||dd…|f}}n|dd…d|…f}||fSdS("sä Solve eigenvalue problem for a real symmetric tridiagonal matrix. Find eigenvalues `w` and optionally right eigenvectors `v` of ``a``:: a v[:,i] = w[i] v[:,i] v.H v = identity For a real symmetric matrix ``a`` with diagonal elements `d` and off-diagonal elements `e`. Parameters ---------- d : ndarray, shape (ndim,) The diagonal elements of the array. e : ndarray, shape (ndim-1,) The off-diagonal elements of the array. 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. tol : float The absolute tolerance to which each eigenvalue is required (only used when 'stebz' is the `lapack_driver`). An eigenvalue (or cluster) is considered to have converged if it lies in an interval of this width. If <= 0. (default), the value ``eps*|a|`` is used where eps is the machine precision, and ``|a|`` is the 1-norm of the matrix ``a``. lapack_driver : str LAPACK function to use, can be 'auto', 'stemr', 'stebz', 'sterf', or 'stev'. When 'auto' (default), it will use 'stemr' if ``select='a'`` and 'stebz' otherwise. When 'stebz' is used to find the eigenvalues and ``eigvals_only=False``, then a second LAPACK call (to ``?STEIN``) is used to find the corresponding eigenvectors. 'sterf' can only be used when ``eigvals_only=True`` and ``select='a'``. 'stev' can only be used when ``select='a'``. Returns ------- w : (M,) ndarray The eigenvalues, in ascending order, each repeated according to its multiplicity. v : (M, M) ndarray The normalized eigenvector corresponding to the eigenvalue ``w[i]`` is the column ``v[:,i]``. Raises ------ LinAlgError If eigenvalue computation does not converge. See Also -------- eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal matrices eig : eigenvalues and right eigenvectors for non-symmetric arrays eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian band matrices Notes ----- This function makes use of LAPACK ``S/DSTEMR`` routines. Examples -------- >>> from scipy.linalg import eigh_tridiagonal >>> d = 3*np.ones(4) >>> e = -1*np.ones(3) >>> w, v = eigh_tridiagonal(d, e) >>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1) >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4))) True R\isexpected one-dimensional arrayRs$Only real arrays currently supporteds-d (%s) must have one more element than e (%s)islapack_driver must be strR¯tstemrtsterftstebztstevs'lapack_driver must be one of %s, got %ss)sterf can only be used when select == "a"s0sterf can only be used when eigvals_only is Trues(stev can only be used when select == "a"R tEtBiÿÿÿÿt stemr_lworkR@tliworks (eigh_tridiagonal)tsteinsstein (eigh_tridiagonal)Rds"%d eigenvectors failed to convergeN(R¯R³R´RµR¶(R¹(R»(RR’RfR'RHt TypeErrorR“RœRR R$RetfloatRRGR(RcR²RR–R—R\R°R±tcheckRSRšRvRwt_tdriverstfuncR R*RVR-R,R¬Rµtordertiblocktisplitte_R¹R@Rº((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyR s|X'          !    $s!did not converge (LAPACK info=%d)cC`sY|dkr&td| |fƒ‚n|dkrU|rUtd|||fƒ‚ndS(sCheck info return value.is+illegal value in argument %d of internal %ss%s N(RfR!(RVR‰Rd((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyRG­s  c C`sþt|d|ƒ}t|jƒdksA|jd|jdkrPtdƒ‚n|pbt||ƒ}|jddkrœ|r˜|tj|jdƒfS|Std|fƒ\}}}||d dd |ƒ\}} } } } t| d d t ƒt|ƒ} t ||jdd | d| ƒ}||d | d| d|d dƒ\}}} t| dd t ƒtj |dƒ}|s{|Std|fƒ\}}t || d | d| ƒ}|d|d|d | d| d|d dƒ\}} t| dd t ƒ||fS(s Compute Hessenberg form of a matrix. The Hessenberg decomposition is:: A = Q H Q^H where `Q` is unitary/orthogonal and `H` has only zero elements below the first sub-diagonal. Parameters ---------- a : (M, M) array_like Matrix to bring into Hessenberg form. calc_q : bool, optional Whether to compute the transformation matrix. Default is False. overwrite_a : bool, optional Whether to overwrite `a`; may improve performance. Default is False. check_finite : bool, optional Whether to check that the input matrix contains only finite numbers. Disabling may give a performance gain, but may result in problems (crashes, non-termination) if the inputs do contain infinities or NaNs. Returns ------- H : (M, M) ndarray Hessenberg form of `a`. Q : (M, M) ndarray Unitary/orthogonal similarity transformation matrix ``A = Q H Q^H``. Only returned if ``calc_q=True``. Examples -------- >>> from scipy.linalg import hessenberg >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]]) >>> H, Q = hessenberg(A, calc_q=True) >>> H array([[ 2. , -11.65843866, 1.42005301, 0.25349066], [ -9.94987437, 14.53535354, -5.31022304, 2.43081618], [ 0. , -1.83299243, 0.38969961, -0.51527034], [ 0. , 0. , -3.83189513, 1.07494686]]) >>> np.allclose(Q @ H @ Q.conj().T - A, np.zeros((4, 4))) True R\iiisexpected square matrixtgehrdtgebalt gehrd_lworktpermuteRNsgebal (hessenberg)RdR†R‡R@sgehrd (hessenberg)iÿÿÿÿtorghrt orghr_lworkRgttausorghr (hessenberg)(RÆRÇRÈ(RÊRË( RReRIRfR"R(RR$RGR~R%ttriu(Rgtcalc_qRNR\RJRÆRÇRÈtbaR†R‡tpivscaleRVtnR@thqRÌthRÊRËtq((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyR ¶s0./' "-3cC`sút|ƒt|ƒ}}|jdkr7tdƒ‚n|jdkrUtdƒ‚n|j|jdkrztdƒ‚n|jd}|jd }|jd|jdkr½tdƒ‚n|jd|krßtd ƒ‚nt|ƒ}|jd dƒ}|dd kjƒs"td ƒ‚nt|ƒ}|d }|d}|d d d…} |dd d…} d} x`|D]X} | d d d…| dd d…kjƒs³tdƒ‚| | d d d…f7} quWt |||fd|j j ƒ} t |ƒ}|j | d||f<|| | fj | | | | f<|| | fj | | | | f>> X = np.array([[1, 2, 3], [0, 4, 5], [0, -5, 4]]) >>> X array([[ 1, 2, 3], [ 0, 4, 5], [ 0, -5, 4]]) >>> from scipy import linalg >>> w, v = linalg.eig(X) >>> w array([ 1.+0.j, 4.+5.j, 4.-5.j]) >>> v array([[ 1.00000+0.j , -0.01906-0.40016j, -0.01906+0.40016j], [ 0.00000+0.j , 0.00000-0.64788j, 0.00000+0.64788j], [ 0.00000+0.j , 0.64788+0.j , 0.64788-0.j ]]) >>> wr, vr = linalg.cdf2rdf(w, v) >>> wr array([[ 1., 0., 0.], [ 0., 4., 5.], [ 0., -5., 4.]]) >>> vr array([[ 1. , 0.40016, -0.01906], [ 0. , 0.64788, 0. ], [ 0. , 0. , 0.64788]]) >>> vr @ wr array([[ 1. , 1.69593, 1.9246 ], [ 0. , 2.59153, 3.23942], [ 0. , -3.23942, 2.59153]]) >>> X @ vr array([[ 1. , 1.69593, 1.9246 ], [ 0. , 2.59153, 3.23942], [ 0. , -3.23942, 2.59153]]) is)expected w to be at least one-dimensionalis)expected v to be at least two-dimensionalsUexpected eigenvectors array to have exactly one dimension more than eigenvalues arrayiÿÿÿÿiþÿÿÿsVexpected v to be a square matrix or stacked square matrices: v.shape[-2] = v.shape[-1]s7expected the same number of eigenvalues as eigenvectorstaxisis/expected complex-conjugate pairs of eigenvaluesNs(Conjugate pair spanned different arrays!R'.gð?yà?gà?yà¿s...ij,...jk->...ik((RR’RfRIRtsumR5RtAssertionErrorRRBR'RtR)R(tcdoubleR(R*R,RÑtMt complex_maskt n_complextidxt idx_stacktidx_elemtjtkt stack_indR.RitdituRT((s2lib/python2.7/site-packages/scipy/linalg/decomp.pyR sNR       8" ""(7t __future__RRRt__all__R(R RRRRRRRRRRRRRRRRtscipy._lib.sixRtscipy._lib._utilRR tmiscR!R"R#tlapackR$R%RFR/R>R[R0R~R}RRRRœRRRRR R RGR R (((s2lib/python2.7/site-packages/scipy/linalg/decomp.pytsN  p   * —   à ¤  I   Z[O   P