&j TdZddlZddlZddlmZmZddlmZm Z ddl m Z m Z m Z mZddlmZmZmZddlmZmZdd lmZd Zgd Zd d dddZdddddZidddddddddddddddd d!d"d#d$d%d&d'd(d)d*d+d,d-d.d/d0d1d2d3d4iZeZeZd5ed+<eZ idddd6dddddddddd7dd d!d8d#d$d%d&d'd9d)d*d+d:d-d.d/d0d1d;d3d4iZ!e!Z"idddd<ddddddd!d=d#d$d%d>d'd?d)d@d+d,d-d.d/dAd1dBdCdDdEdFZ#e#Z$dGe$d<dHe$dC<dIe$dE<iddddJddddddKd!d=d#d$d%d>d'dLd)dMd+dNd-d.d/dAd1dBdCdOdEdPZ%e%Z&dQe&dC<dRe&dE<idddddddd7d!d8d#d$d%d>d'dSd)d*d+d:d-d.d/d2dCdTdEdUdVdAdWdXZ'e'Z(dYe(dC<dZe(dW<e!e"d[Z)eeee d\Z*e'e(d[Z+e#e$e%e&d\Z,gd]Z-gd^Z.ddddd_dd`dadbdc Z/dddddZ0Gdedfe1Z2Gdgdhe2Z3diZ4GdjdkZ5Gdldme5Z6Gdndoe5Z7GdpdqeZ8GdrdseZ9dtZ:GdudveZ;GdwdxeZdd|Z? ddZ@ ddZAdS)zn Find a few eigenvectors and eigenvalues of a matrix. Uses ARPACK: https://github.com/opencollab/arpack-ng N)aslinearoperatorLinearOperator)eyeissparse)eigeigh lu_factorlu_solve)convert_pydata_sparse_to_scipyisdenseis_pydata_spmatrix)gmressplu) _arpacklibzrestructuredtext en)eigseigsh ArpackErrorArpackNoConvergencesdcz)frFD z Normal exit.zMaximum number of iterations taken. All possible eigenvalues of OP has been found. IPARAM(5) returns the number of wanted converged Ritz values.zONo longer an informational error. Deprecated starting with release 2 of ARPACK.zNo shifts could be applied during a cycle of the Implicitly restarted Arnoldi iteration. One possibility is to increase the size of NCV relative to NEV. zN must be positive.zNEV must be positive.z)NCV-NEV >= 2 and less than or equal to N.zRThe maximum number of Arnoldi update iterations allowed must be greater than zero.z8 WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'izBMAT must be one of 'I' or 'G'.iz5Length of private work array WORKL is not sufficient.iz0Error return from LAPACK eigenvalue calculation;izStarting vector is zero.izIPARAM(7) must be 1,2,3,4.iz.IPARAM(7) = 1 and BMAT = 'G' are incompatible.iz"IPARAM(1) must be equal to 0 or 1.iz&NEV and WHICH = 'BE' are incompatible.izCould not build an Arnoldi factorization. IPARAM(5) returns the size of the current Arnoldi factorization. The user is advised to check that enough workspace and array storage has been allocated.zIPARAM(7) must be 1,2,3.zRMaximum number of iterations taken. All possible eigenvalues of OP has been found.z9NCV must be greater than NEV and less than or equal to N.z4WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.z`Error return from trid. eigenvalue calculation; Informational error from LAPACK routine dsteqr .zIPARAM(7) must be 1,2,3,4,5.z'NEV and WHICH = 'BE' are incompatible. asThe Schur form computed by LAPACK routine dlahqr could not be reordered by LAPACK routine dtrsen. Re-enter subroutine dneupd with IPARAM(5)NCV and increase the size of the arrays DR and DI to have dimension at least dimension NCV and allocate at least NCV columns for Z. NOTE: Not necessary if Z and V share the same space. Please notify the authors if this erroroccurs.z7WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'z5Length of private work WORKL array is not sufficient.zdError return from calculation of a real Schur form. Informational error from LAPACK routine dlahqr .z^Error return from calculation of eigenvectors. Informational error from LAPACK routine dtrevc.z HOWMNY = 'S' not yet implementedz1HOWMNY must be one of 'A' or 'P' if RVEC = .true.iz= 1 and less than or equal to N.zLError return from LAPACK eigenvalue calculation. This should never happened.z^Error return from calculation of eigenvectors. Informational error from LAPACK routine ztrevc.zIPARAM(7) must be 1,2,3z;ZNAUPD did not find any eigenvalues to sufficient accuracy.zZNEUPD got a different count of the number of converged Ritz values than ZNAUPD got. This indicates the user probably made an error in passing data from ZNAUPD to ZNEUPD or that the data was modified before entering ZNEUPDz;CNAUPD did not find any eigenvalues to sufficient accuracy.zCNEUPD got a different count of the number of converged Ritz values than CNAUPD got. This indicates the user probably made an error in passing data from CNAUPD to CNEUPD or that the data was modified before entering CNEUPDz]Error return from trid. eigenvalue calculation; Information error from LAPACK routine dsteqr.z 669a99:: : ?"fG a<<K'KKLL L V  {2s## N !LMMM >"4r:::DJDDCqcBBII"MMDJD =DJJDJ ;Q--CsA;;1c(B''  + 3+ C+  #+  3 + # + # +  1+  Z&+  A+  D+  A+  s7||+  D+  +  Q+ 48!+ " 1#+ + $ !%+ & Q'+ ( 1)+ * !++ , !-+ . !/+ 0 1+ 2 a3+ 4 !5+ 6 !7+ 8 A9+ : !;+ < !=+ > ?+ @ !A+ B C+ D !E+ + FU+ + + ZrBc Nd}|jd}|jd} |d\}}nP#t$rC}|d|d}tjd}tj|jdf}d}Yd}~nd}~wwxYwt d |d |d |jd ||) Nzz1_SymmetricArpackParams.__init__..qrBIr#matvec must be specified for mode=2%M_matvec must be specified for mode=2(Minv_matvec must be specified for mode=2c,|SrCrMr Minv_matvecmatvecs rArz1_SymmetricArpackParams.__init__.. FF1II 6 6rBGr z'matvec must not be specified for mode=3z(Minv_matvec must be specified for mode=3c|SrCrMrs rArz1_SymmetricArpackParams.__init__..1rBc,|SrCrM)rM_matvecrs rArz1_SymmetricArpackParams.__init__..sKK $<$<rBr/z#matvec must be specified for mode=4z)M_matvec must not be specified for mode=4z(Minv_matvec must be specified for mode=4c@|SrCOPa)rrr<s rArz1_SymmetricArpackParams.__init__..s 3 3rBrz#matvec must be specified for mode=5z(Minv_matvec must be specified for mode=5c8||zzSrCrM)rrrrs rArz1_SymmetricArpackParams.__init__..s"KKq EAI0E$F$FrBc|SrCrMrs rArz1_SymmetricArpackParams.__init__.. rrBcJ||zzSrCrM)rrrrrs rArz1_SymmetricArpackParams.__init__..#s5KKq 27((1++2E1F%G%GrBmode= not implementedzwhich must be one of  zk must be less than ndim(A), k=rrhzncv must be k)rrrrrrrslicerlrkrrrrrrrrrrrrrrr)r<xsliceysliceBxsliceAxs rAiteratez_SymmetricArpackParams.iterateGs; D,dj$&$* J  4 4 4tz!}djmdf&<==tz!}djmdf&<==  E "a ' 'yA~~%)WWTZ-?%@%@ 6"""a%)XXdj.@%A%A 6"%)XXdj.@%A%A 6"""a 1 tz!}tv/EFF]]4:f#566%)XXbDJ48Jw4G5H/I&J&J 6"""  1 tz!}tv/EFF%)XXdj.A%B%B 6"""  e $ ) )!% 6(:!;!;DJv     e $ ) )MNN N  e $ ) ) H,,C376(-<<.rrBrrrrrc,|SrCrMrs rArz3_UnsymmetricArpackParams.__init__..rrBrr r/z*matvec must be specified for mode in (3,4)z/Minv_matvec must be specified for mode in (3,4)DFr zmode=4 invalid for complex Ac>tj|SrC)rprealrrs rArz3_UnsymmetricArpackParams.__init__..Q)@)@rBc>tj|SrC)rpimagrs rArz3_UnsymmetricArpackParams.__init__..rrBc|SrCrMrs rArz3_UnsymmetricArpackParams.__init__..rrBc@|SrCr)rrr<s rArz3_UnsymmetricArpackParams.__init__..sDHHXXa[[$9$9rBrrzParameter which must be one of rz!k must be less than ndim(A)-1, k=rrhzncv must be k+1||j|j|j|j|j|j|jt|jd|jd|j z}t|jd|jd|j z}|jddkr|j dvr*| |j||j|<dSt|jd|jd|j z}| |j||j|<dS|jddkr*||j||j|<dS|jddkrtd|jdd kr|jdvrG|jd d |j g |j|jdd<dS|jd d |j dg t(j|j|jdd<dS|jdd kr*| |j||j|<dSd|_|jddkrdS|jddkr|dSt3|jd|j)Nfdrrrfrrrr rr/rZr[r\rTr=)r=r>)rrrrrrrrr rrlrkrrrrrrrviewrp complex128ravelrrrr)r<rrrs rArz _UnsymmetricArpackParams.iterates 7d??    $*dfdj$*         $*dfdj$* DJ    tz!}djmdf&<==tz!}djmdf&<==  E "a ' 'yF""%)WWTZ-?%@%@ 6""" 1 tz!}tv/EFF%)XXdj.A%B%B 6"""  e $ ) )!% 6(:!;!;DJv     e $ ) )MNN N  e $ ) )w$ $ 0 0T7;vh!1!@!@@Ftw 111 !% 0 0T46ST+ 0!!$r}%%ffTWooeegg 111  e $ ) )!%F);!)-IMyyyyv4B4B4BlNNNNNrBrceZdZdZdZdZdS)SpLuInvzl SpLuInv: helper class to repeatedly solve M*x=b using a sparse LU-decomposition of M ct||_|j|_|j|_t j|jtj |_dSrC)rM_lushaperYrp issubdtypecomplexfloatingisrealr<Ms rAr;zSpLuInv.__init__s@GG W W - B4FGGG rBctj|}|jrtj|jtjr|jtj| |jd|jtj | |jzzS|j| |jS)Nr) rpasarrayr,r*rYr+r(solverrrr<rs rA_matveczSpLuInv._matvecs JqMM ; 92="2DEE 9IOOBGAJJ$5$5dj$A$ABB49??271::+<+ifuncr`)r<r.rFr`rs rAr;zIterInv.__init__ s 1g   'DJJ$$Aa%DJW !88bhtz**..C rBc||j||j\}}|dkr td|jjd|d|S)Nr`rzError in inverting M: function  did not converge (info = ).)rFr.r`rrDr<rr?r=s rAr3zIterInv._matvecsk**TVQDH*554 1995$*2E55,0555 rBN)rDrErFrGrAr;r3rMrBrArCrCsH !, rBrCc:eZdZdZedfdZdZedZdS) IterOpInvzo IterOpInv: helper class to repeatedly solve [A-sigma*M]*x = b using an iterative method rc|_|_|_fd}fd}tjjd}2||j} t|jj|| |_n1||j} t|jj|| |_j|_|dkr&dtj |jjj z}||_ ||_ dS)Nc`||zz SrCr)rr3r.rs rA mult_funcz%IterOpInv.__init__..mult_func2s'88A;;!!44 4rBc:||zz SrCrP)rr3rs rAmult_func_M_Nonez,IterOpInv.__init__..mult_func_M_None5s88A;;* *rBrrrr) r3r.rrprr)rYrrr=r>rFr`) r<r3r.rrFr`rQrSrrYs ``` rAr;zIterOpInv.__init__-s%  5 5 5 5 5 5 5 + + + + + + HQWQZ  9$$Q''-E$TV\%5+0222DGGIaLL&E$TV\%.+0222DGW !88bhtw}--11C rBc||j||j\}}|dkr td|jjd|d|S)NrHrz)Error in inverting [A-sigma*M]: function rIrJ)rFrr`rrDrKs rAr3zIterOpInv._matvecLsk**TWaTX*664 1995DJXrBrMFc|jdkr-|r+tj|jtjs|jSt |r|S|S)z:Convert sparse matrix to CSC (by transposing, if possible)csr)formatrpr*rYr+Tr tocsc)r3 hermitians rA_fast_spmatrix_to_cscr]ZsW EiM!'2+=>> s A  wwyyrBct|rt|jSt|st |r%t ||}t |jSt||jS)Nr\rH)r r6rrr r]r&rC)r.r\r`s rAget_inv_matvecr`fssqzz*Qxx !**1--* !!y 9 9 9qzz  qc"""))rBch|dkrt|||S|'t|rtj|jtjstj|dkrtj|}n|dz}|jdd|j ddzxx|zcc<t|j St|st|rE||t|j dzz }t||}t!|j St#t%||||j St|st|st|r-t|sPt|sAt|s2t#t%|t%|||j St|st|rt|||zz j S|||zz }t||}t!|j S)Nrr\r`yrr_rH)r`r rpr*rYr+rrXflatr)r6rrr rr]r&rMr)r3r.rr\r`rs rAget_OPinv_matvecrdps. zza9#>>>>y 1:: 7 agr'9:: wu~~**GAJJF F##QWQZ!^# $ $ $ - $ $ $88? " a[[ 7.q11 7EC OO++A%a9===A1::$ $-a00300006 7 &HQKK &8J18M8M &QZZ &(0  & 2*k``. Default: ``min(n, max(2*k + 1, 20))`` which : str, ['LM' | 'SM' | 'LR' | 'SR' | 'LI' | 'SI'], optional Which `k` eigenvectors and eigenvalues to find: 'LM' : largest magnitude 'SM' : smallest magnitude 'LR' : largest real part 'SR' : smallest real part 'LI' : largest imaginary part 'SI' : smallest imaginary part When sigma != None, 'which' refers to the shifted eigenvalues w'[i] (see discussion in 'sigma', above). ARPACK is generally better at finding large values than small values. If small eigenvalues are desired, consider using shift-invert mode for better performance. maxiter : int, optional Maximum number of Arnoldi update iterations allowed Default: ``n*10`` tol : float, optional Relative accuracy for eigenvalues (stopping criterion) The default value of 0 implies machine precision. return_eigenvectors : bool, optional Return eigenvectors (True) in addition to eigenvalues Minv : ndarray, sparse matrix or LinearOperator, optional See notes in M, above. OPinv : ndarray, sparse matrix or LinearOperator, optional See notes in sigma, above. OPpart : {'r' or 'i'}, optional See notes in sigma, above rng : `numpy.random.Generator`, optional Pseudorandom number generator state. When `rng` is None, a new `numpy.random.Generator` is created using entropy from the operating system. Types other than `numpy.random.Generator` are passed to `numpy.random.default_rng` to instantiate a ``Generator``. Returns ------- w : ndarray Array of k eigenvalues. v : ndarray An array of `k` eigenvectors. ``v[:, i]`` is the eigenvector corresponding to the eigenvalue w[i]. Raises ------ ArpackNoConvergence When the requested convergence is not obtained. The currently converged eigenvalues and eigenvectors can be found as ``eigenvalues`` and ``eigenvectors`` attributes of the exception object. See Also -------- eigsh : eigenvalues and eigenvectors for symmetric matrix A svds : singular value decomposition for a matrix A Notes ----- This function is a wrapper to the ARPACK [1]_ SNEUPD, DNEUPD, CNEUPD, ZNEUPD, functions which use the Implicitly Restarted Arnoldi Method to find the eigenvalues and eigenvectors [2]_. References ---------- .. [1] ARPACK Software, https://github.com/opencollab/arpack-ng .. [2] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA, 1998. Examples -------- Find 6 eigenvectors of the identity matrix: >>> import numpy as np >>> from scipy.sparse.linalg import eigs >>> id = np.eye(13) >>> vals, vecs = eigs(id, k=6) >>> vals array([ 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j, 1.+0.j]) >>> vecs.shape (13, 6) rrexpected square matrix (shape=)Nwrong M dimensions , should be ZM does not have the same type precision as A. This may adversely affect ARPACK convergencer stacklevelzk=z must be greater than 0.zOk >= N - 1 for N * N square matrix. Attempting to use scipy.linalg.eig instead.zhCannot use scipy.linalg.eig for sparse A with k >= N - 1. Use scipy.linalg.eig(A.toarray()) or reduce k.zACannot use scipy.linalg.eig for LinearOperator A with k >= N - 1.zACannot use scipy.linalg.eig for LinearOperator M with k >= N - 1.)r?right0OPinv should not be specified with sigma = None.z=OPpart should not be specified with sigma = None or complex A+Minv should not be specified with M = None.Trbz;OPpart should not be specified with sigma=None or complex Ar rr"z%OPpart cannot be 'i' if sigma is realr/zOPpart must be one of ('r','i')z4Minv should not be specified when sigma isspecified.F)r r)rrprYcharr warningswarnRuntimeWarningr TypeError isinstancerrrrr`r*r+rrdrrrrrr)r3rQr.rrgrrnrjr`rMinvOPinvOPpartrrlrrkrrparamss rArrsk~ 'q))A&q))AwqzQWQZD!'DDDEEE} 7ag  Q17QQQQRR R 8AG   ! ' ' ) )RXag->->-C-I-I-K-K K K MI%& ( ( ( (  AAvv9a999:::AEzz D$ 4 4 4 4 A;; *)** * a ( ( 2122 2 a ( ( 2122 21!45555 }!!$$+  233 3  9:: : 9DHK "<=== D|,Q$CHHH '--"k '**1HH =""4 5 5 @! "@AAADD ^v||~~44DD \\^^s " "wu~~"" !HIIIDD>?? ?!!$$+  *++ + =*1a5:EEEKK%U++E,K 9HH'**1H )   $ $C %aAGL&$&. U&)2wsCIIF >>- . ..rBnormalctj|jtjrx| dkrt d| d|dkrt d|dkrd}n|dkrd }t |||||||||| | | }| r|d j|d fS|jS|jd |jd krt d |jd||j|jkrt d|jd|jtj|jj tj|jj krtj dd|jd }|d krt d||krtj dtdt|rtdt|t rtdt|t rtdt#||| S|t%|}|j}| t d|d } d}d}| t dned} | t)|d|}nt%| } | j}t%|j}n"| t d| dkrMd } d}| t+|||d|}nt%| } | j}|d}nt%|}|j}n| d!krDd"} | t+|||d|}nt%| j}t%|j}d}nt| d#kr[d$} t%|j}| t+|||d|}nt%| j}|d}n(t%|j}nt d%| d&tj| } t1|||jj|| ||||||||| }|js||j|| S)'a! Find k eigenvalues and eigenvectors of the real symmetric square matrix or complex Hermitian matrix A. Solves ``A @ x[i] = w[i] * x[i]``, the standard eigenvalue problem for w[i] eigenvalues with corresponding eigenvectors x[i]. If M is specified, solves ``A @ x[i] = w[i] * M @ x[i]``, the generalized eigenvalue problem for w[i] eigenvalues with corresponding eigenvectors x[i]. Note that there is no specialized routine for the case when A is a complex Hermitian matrix. In this case, ``eigsh()`` will call ``eigs()`` and return the real parts of the eigenvalues thus obtained. Parameters ---------- A : ndarray, sparse matrix or LinearOperator A square operator representing the operation ``A @ x``, where ``A`` is real symmetric or complex Hermitian. For buckling mode (see below) ``A`` must additionally be positive-definite. k : int, optional The number of eigenvalues and eigenvectors desired. `k` must be smaller than N. It is not possible to compute all eigenvectors of a matrix. Returns ------- w : array Array of k eigenvalues. v : array An array representing the `k` eigenvectors. The column ``v[:, i]`` is the eigenvector corresponding to the eigenvalue ``w[i]``. Other Parameters ---------------- M : An N x N matrix, array, sparse matrix, or linear operator representing the operation ``M @ x`` for the generalized eigenvalue problem A @ x = w * M @ x. M must represent a real symmetric matrix if A is real, and must represent a complex Hermitian matrix if A is complex. For best results, the data type of M should be the same as that of A. Additionally: If sigma is None, M is symmetric positive definite. If sigma is specified, M is symmetric positive semi-definite. In buckling mode, M is symmetric indefinite. If sigma is None, eigsh requires an operator to compute the solution of the linear equation ``M @ x = b``. This is done internally via a (sparse) LU decomposition for an explicit matrix M, or via an iterative solver for a general linear operator. Alternatively, the user can supply the matrix or operator Minv, which gives ``x = Minv @ b = M^-1 @ b``. sigma : real Find eigenvalues near sigma using shift-invert mode. This requires an operator to compute the solution of the linear system ``[A - sigma * M] x = b``, where M is the identity matrix if unspecified. This is computed internally via a (sparse) LU decomposition for explicit matrices A & M, or via an iterative solver if either A or M is a general linear operator. Alternatively, the user can supply the matrix or operator `OPinv`, which gives ``x = OPinv @ b = [A - sigma * M]^-1 @ b``. Regardless of the selected mode (normal, cayley, or buckling), `OPinv` should always be supplied as ``OPinv = [A - sigma * M]^-1``. Note that when sigma is specified, the keyword 'which' refers to the shifted eigenvalues ``w'[i]`` where: if ``mode == 'normal'``: ``w'[i] = 1 / (w[i] - sigma)``. if ``mode == 'cayley'``: ``w'[i] = (w[i] + sigma) / (w[i] - sigma)``. if ``mode == 'buckling'``: ``w'[i] = w[i] / (w[i] - sigma)``. (see further discussion in 'mode' below) v0 : ndarray, optional Starting vector for iteration. Default: random ncv : int, optional The number of Lanczos vectors generated ncv must be greater than k and smaller than n; it is recommended that ``ncv > 2*k``. Default: ``min(n, max(2*k + 1, 20))`` which : str ['LM' | 'SM' | 'LA' | 'SA' | 'BE'] If A is a complex Hermitian matrix, 'BE' is invalid. Which `k` eigenvectors and eigenvalues to find: 'LM' : Largest (in magnitude) eigenvalues. 'SM' : Smallest (in magnitude) eigenvalues. 'LA' : Largest (algebraic) eigenvalues. 'SA' : Smallest (algebraic) eigenvalues. 'BE' : Half (k/2) from each end of the spectrum. When k is odd, return one more (k/2+1) from the high end. When sigma != None, 'which' refers to the shifted eigenvalues ``w'[i]`` (see discussion in 'sigma', above). ARPACK is generally better at finding large values than small values. If small eigenvalues are desired, consider using shift-invert mode for better performance. maxiter : int, optional Maximum number of Arnoldi update iterations allowed. Default: ``n*10`` tol : float Relative accuracy for eigenvalues (stopping criterion). The default value of 0 implies machine precision. Minv : N x N matrix, array, sparse matrix, or LinearOperator See notes in M, above. OPinv : N x N matrix, array, sparse matrix, or LinearOperator See notes in sigma, above. return_eigenvectors : bool Return eigenvectors (True) in addition to eigenvalues. This value determines the order in which eigenvalues are sorted. The sort order is also dependent on the `which` variable. For which = 'LM' or 'SA': If `return_eigenvectors` is True, eigenvalues are sorted by algebraic value. If `return_eigenvectors` is False, eigenvalues are sorted by absolute value. For which = 'BE' or 'LA': eigenvalues are always sorted by algebraic value. For which = 'SM': If `return_eigenvectors` is True, eigenvalues are sorted by algebraic value. If `return_eigenvectors` is False, eigenvalues are sorted by decreasing absolute value. mode : string ['normal' | 'buckling' | 'cayley'] Specify strategy to use for shift-invert mode. This argument applies only for real-valued A and sigma != None. For shift-invert mode, ARPACK internally solves the eigenvalue problem ``OP @ x'[i] = w'[i] * B @ x'[i]`` and transforms the resulting Ritz vectors x'[i] and Ritz values w'[i] into the desired eigenvectors and eigenvalues of the problem ``A @ x[i] = w[i] * M @ x[i]``. The modes are as follows: 'normal' : OP = [A - sigma * M]^-1 @ M, B = M, w'[i] = 1 / (w[i] - sigma) 'buckling' : OP = [A - sigma * M]^-1 @ A, B = A, w'[i] = w[i] / (w[i] - sigma) 'cayley' : OP = [A - sigma * M]^-1 @ [A + sigma * M], B = M, w'[i] = (w[i] + sigma) / (w[i] - sigma) The choice of mode will affect which eigenvalues are selected by the keyword 'which', and can also impact the stability of convergence (see [2] for a discussion). rng : `numpy.random.Generator`, optional Pseudorandom number generator state. When `rng` is None, a new `numpy.random.Generator` is created using entropy from the operating system. Types other than `numpy.random.Generator` are passed to `numpy.random.default_rng` to instantiate a ``Generator``. Raises ------ ArpackNoConvergence When the requested convergence is not obtained. The currently converged eigenvalues and eigenvectors can be found as ``eigenvalues`` and ``eigenvectors`` attributes of the exception object. See Also -------- eigs : eigenvalues and eigenvectors for a general (nonsymmetric) matrix A svds : singular value decomposition for a matrix A Notes ----- This function is a wrapper to the ARPACK [1]_ SSEUPD and DSEUPD functions which use the Implicitly Restarted Lanczos Method to find the eigenvalues and eigenvectors [2]_. References ---------- .. [1] ARPACK Software, https://github.com/opencollab/arpack-ng .. [2] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA, 1998. Examples -------- >>> import numpy as np >>> from scipy.sparse.linalg import eigsh >>> identity = np.eye(13) >>> eigenvalues, eigenvectors = eigsh(identity, k=6) >>> eigenvalues array([1., 1., 1., 1., 1., 1.]) >>> eigenvectors.shape (13, 6) r{rz% cannot be used with complex matrix Ar*z/which='BE' cannot be used with complex matrix Ar(r+r)r,) r.rrgrrnrjr`rrwrxrrrfrgNrhrirjrrkzk must be greater than 0.zLk >= N for N * N square matrix. Attempting to use scipy.linalg.eigh instead.zfCannot use scipy.linalg.eigh for sparse A with k >= N. Use scipy.linalg.eigh(A.toarray()) or reduce k.z>Cannot use scipy.linalg.eigh for LinearOperator A with k >= N.z>Cannot use scipy.linalg.eigh for LinearOperator M with k >= N.)r? eigvals_onlyrnroTrbz5Minv should not be specified when sigma is specified.r bucklingr/cayleyrzunrecognized mode '')rpr*rYr+rrrr)rqr rrrsrtrrurvrrrrr`rdrrrrrr)r3rQr.rrgrrnrjr`rrwrxrkrretrlrrrrzs rArrs@n }QWb011 8  PTPPPQQ Q D==NOO O d]]EE d]]E1a1E2G':     q6;A& &8OwqzQWQZD!'DDDEEE} 7ag  Q17QQQQRR R 8AG   ! ' ' ) )RXag->->-C-I-I-K-K K K MI%& ( ( ( (  AAvv4555Avv E$ 4 4 4 4 A;; *)** * a ( ( .-.. . a ( ( .-.. .A-@)@AAAA } Q    OPP P 9DHK !NOOO D|,Q$CHHH '--"k '**1HH  *++ + 8  DF}.q!U9=3HHH )//#l y$Q''8Z  D}.q!U9=3HHH /u55< %a((/FHHX  D%a((/F}.q!U9=3HHH /u55< y+A..5:4:::;; ; )   $ $C #Aq!',$,k5$'WeS#GGF >>- . ..rB)F)Fr) r0NNr&NNNrTNNNN) r0NNr&NNNrTNNr{N)BrGnumpyrprrscipy.sparse.linalg._interfacerr scipy.sparserr scipy.linalgrrr r scipy.sparse._sputilsr r r scipy.sparse.linalgrrr __docformat____all__r_ndigits DNAUPD_ERRORS SNAUPD_ERRORSrX ZNAUPD_ERRORS CNAUPD_ERRORS DSAUPD_ERRORS SSAUPD_ERRORS DNEUPD_ERRORS SNEUPD_ERRORS ZNEUPD_ERRORS CNEUPD_ERRORS DSEUPD_ERRORS SSEUPD_ERRORSrr8rr rrrr RuntimeErrorrrrRrTrrr&r6rArCrMr]r`rdrrrMrBrArs: JKKKKKKKK&&&&&&&&777777777777,+++++++% A A ASs 5 5 !" - -~= # :3 %B )!"?#$:%&"'( %)* 9+, --. 1/0  D1 < ""$$ / c ~8  + C %>) ?!" ;#&"'( ')* 9+, --. 2/0  D1 < ~3A)?  ;!$ :%( %)* 9+, +-.  ) ? 9" ' 9 1 G <  +!" # 0""$$ 0 c c$#%% ####%% $#%% ####%% .-- 433  1AQaqRS AA&&  8 8 8 8 8, 8 8 8)))))+)))&h+h+h+h+h+h+h+h+Vddddd]dddN@@@@@}@@@D 99999n9990 & & & & &N & & & 777nB11111111h    **** & & & &F59<@15D/D/D/D/N6:=A48[/[/[/[/[/[/rB