ó î&]\c @`s€dZddlmZmZmZdZdddddgZd d lmZdd l Z dd l Z dd l m Z mZdd lmZmZmZmZddlmZmZmZmZddlmZddlmZmZddlmZddl m!Z!idd6dd6dd6dd6Z"idd6dd6dd6dd6Z#idd6dd 6dd6d d!6d"d#6d$d%6d&d'6d(d)6d*d+6d,d-6d.d/6d0d16d2d36d4d56d6d76d8d96d:d;6d<d=6Z$e$Z%e$j&ƒZ'd>e'd5dv„Z?dwefdx„ƒYZ@dyefdz„ƒYZAd{„ZBd|efd}„ƒYZCd~efd„ƒYZDeEdd€„ZFeEdd„ZGe!d‚ƒZHdƒeIeIddeIeIeIdeJeIeIeId„„ ZKdƒeIeIddeIeIeIdeJeIeId…d†„ ZLd‡„ZMdˆ„ZNd‰„ZOdƒeIdddeIeIeJdŠ„ZPd S(‹sp Find a few eigenvectors and eigenvalues of a matrix. Uses ARPACK: http://www.caam.rice.edu/software/ARPACK/ i(tdivisiontprint_functiontabsolute_importsrestructuredtext enteigsteigshtsvdst ArpackErrortArpackNoConvergencei(t_arpackN(taslinearoperatortLinearOperator(teyetissparset isspmatrixtisspmatrix_csr(teigteight lu_factortlu_solve(tisdense(tgmrestsplu(t_aligned_zeros(tReentrancyLocktstftdtctFtztDii s Normal exit.sMaximum number of iterations taken. All possible eigenvalues of OP has been found. IPARAM(5) returns the number of wanted converged Ritz values.sONo longer an informational error. Deprecated starting with release 2 of ARPACK.is™No 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. isN must be positive.iÿÿÿÿsNEV must be positive.iþÿÿÿs)NCV-NEV >= 2 and less than or equal to N.iýÿÿÿsRThe maximum number of Arnoldi update iterations allowed must be greater than zero.iüÿÿÿs8 WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI'iûÿÿÿsBMAT must be one of 'I' or 'G'.iúÿÿÿs5Length of private work array WORKL is not sufficient.iùÿÿÿs0Error return from LAPACK eigenvalue calculation;iøÿÿÿsStarting vector is zero.i÷ÿÿÿsIPARAM(7) must be 1,2,3,4.iöÿÿÿs.IPARAM(7) = 1 and BMAT = 'G' are incompatible.iõÿÿÿs"IPARAM(1) must be equal to 0 or 1.iôÿÿÿs&NEV and WHICH = 'BE' are incompatible.ióÿÿÿsÃCould 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.iñØÿÿsIPARAM(7) must be 1,2,3.sRMaximum number of iterations taken. All possible eigenvalues of OP has been found.s9NCV must be greater than NEV and less than or equal to N.s4WHICH must be one of 'LM', 'SM', 'LA', 'SA' or 'BE'.s`Error return from trid. eigenvalue calculation; Informational error from LAPACK routine dsteqr .sIPARAM(7) must be 1,2,3,4,5.s'NEV and WHICH = 'BE' are incompatible. ssThe 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.s7WHICH must be one of 'LM', 'SM', 'LR', 'SR', 'LI', 'SI's5Length of private work WORKL array is not sufficient.sdError return from calculation of a real Schur form. Informational error from LAPACK routine dlahqr .s^Error return from calculation of eigenvectors. Informational error from LAPACK routine dtrevc.s HOWMNY = 'S' not yet implementeds1HOWMNY must be one of 'A' or 'P' if RVEC = .true.s<DNAUPD did not find any eigenvalues to sufficient accuracy.iòÿÿÿsßDNEUPD got a different count of the number of converged Ritz values than DNAUPD got. This indicates the user probably made an error in passing data from DNAUPD to DNEUPD or that the data was modified before entering DNEUPDiñÿÿÿsvThe Schur form computed by LAPACK routine slahqr could not be reordered by LAPACK routine strsen . 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 error occurs.s;SNAUPD did not find any eigenvalues to sufficient accuracy.sßSNEUPD got a different count of the number of converged Ritz values than SNAUPD got. This indicates the user probably made an error in passing data from SNAUPD to SNEUPD or that the data was modified before entering SNEUPDskThe Schur form computed by LAPACK routine csheqr could not be reordered by LAPACK routine ztrsen. Re-enter subroutine zneupd with IPARAM(5)=NCV and increase the size of the array D 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 error occurs.s)NCV-NEV >= 1 and less than or equal to N.sLError return from LAPACK eigenvalue calculation. This should never happened.s^Error return from calculation of eigenvectors. Informational error from LAPACK routine ztrevc.sIPARAM(7) must be 1,2,3s;ZNAUPD did not find any eigenvalues to sufficient accuracy.sßZNEUPD 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 ZNEUPDs;CNAUPD did not find any eigenvalues to sufficient accuracy.sßCNEUPD 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 CNEUPDs]Error return from trid. eigenvalue calculation; Information error from LAPACK routine dsteqr.s<DSAUPD did not find any eigenvalues to sufficient accuracy.s1HOWMNY must be one of 'A' or 'S' if RVEC = .true.iðÿÿÿsåDSEUPD got a different count of the number of converged Ritz values than DSAUPD got. This indicates the user probably made an error in passing data from DSAUPD to DSEUPD or that the data was modified before entering DSEUPD.iïÿÿÿs<SSAUPD did not find any eigenvalues to sufficient accuracy.såSSEUPD got a different count of the number of converged Ritz values than SSAUPD got. This indicates the user probably made an error in passing data from SSAUPD to SSEUPD or that the data was modified before entering SSEUPD.tLMtSMtLAtSAtBEtLRtSRtLItSIcB`seZdZed„ZRS(s ARPACK error cC`s0|j|dƒ}tj|d||fƒdS(Ns Unknown errorsARPACK error %d: %s(tgett RuntimeErrort__init__(tselftinfotinfodicttmsg((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR*s(t__name__t __module__t__doc__t _NAUPD_ERRORSR*(((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyRscB`seZdZd„ZRS(sß ARPACK iteration did not converge Attributes ---------- eigenvalues : ndarray Partial result. Converged eigenvalues. eigenvectors : ndarray Partial result. Converged eigenvectors. cC`s0tj|di|d6ƒ||_||_dS(Niÿÿÿÿ(RR*t eigenvaluest eigenvectors(R+R.R3R4((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR*'s (R/R0R1R*(((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyRs cC`std|ddƒS(sy Choose number of lanczos vectors based on target number of singular/eigen values and vectors to compute, k. iii(tmax(tk((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt choose_ncv-st _ArpackParamscB`s/eZdddddddd„Zd„ZRS(iRic C`sÖ|dkrtd|ƒ‚n|dkr8|d}n|dkrWtd|ƒ‚n|dkrrtdƒ‚n|dk rŸtj|dtƒ|_d} ntj||ƒ|_d} |dkrÒd|_n ||_|dkröt|ƒ}nt ||ƒ}tj||f|ƒ|_ tjd d ƒ|_ d} ||_ | |j d<||j d RBRHRR6(R+R.tk_oktnum_itertevtvecterr((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt_raise_no_convergencems   N(R/R0R=R*RY(((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR85s6t_SymmetricArpackParamsc B`s>eZdddddddddd„ Zd„Zd„ZRS(iRic `s‡|dkr~ˆdkr'tdƒ‚nˆdk rBtdƒ‚nˆdk r]tdƒ‚nˆˆ_d„ˆ_dˆ_n¤|dkrˆdkr¥tdƒ‚nˆdkrÀtd ƒ‚nˆdkrÛtd ƒ‚n‡‡fd †ˆ_ˆˆ_ˆˆ_ˆˆ_d ˆ_n |d krˆdk r>tdƒ‚nˆdkrYtdƒ‚nˆdkrˆˆ_ˆˆ_d„ˆ_dˆ_q"‡‡fd†ˆ_ˆˆ_ˆˆ_d ˆ_n`|dkrRˆdkrétdƒ‚nˆdk rtdƒ‚nˆdkrtdƒ‚nˆˆ_‡‡fd†ˆ_ˆˆ_d ˆ_nÐ|dkrˆdkrytdƒ‚nˆdkr”tdƒ‚nˆˆ_ˆˆ_ˆdkr⇇‡fd†ˆ_d„ˆ_dˆ_q"‡‡‡‡fd†ˆ_ˆˆ_d ˆ_ntd|ƒ‚| tkrJtddj tƒƒ‚n||kritd |ƒ‚nt j ˆ||||ˆ| | | | | ƒ ˆj |ks²ˆj |krÈtd!ˆj ƒ‚nt d |ˆjƒˆ_t ˆj ˆj d"ˆjƒˆ_tˆj}|d*kr,td%ƒ‚ntj|d&ˆ_tj|d'ˆ_t|ˆ_t|ˆ_tjd(d)ƒˆ_dS(+Nis#matvec must be specified for mode=1s'M_matvec cannot be specified for mode=1s*Minv_matvec cannot be specified for mode=1cS`s|S(N((tx((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt»stIis#matvec must be specified for mode=2s%M_matvec must be specified for mode=2s(Minv_matvec must be specified for mode=2c`sˆˆ|ƒƒS(N((R[(t Minv_matvectmatvec(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\ÅstGis'matvec must not be specified for mode=3s(Minv_matvec must be specified for mode=3cS`s|S(N((R[((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\Ósc`sˆˆ|ƒƒS(N((R[(tM_matvecR^(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\Ösis#matvec must be specified for mode=4s)M_matvec must not be specified for mode=4s(Minv_matvec must be specified for mode=4c`sˆjˆ|ƒƒS(N(tOPa(R[(R_R+(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\âsis#matvec must be specified for mode=5s(Minv_matvec must be specified for mode=5c`sˆˆ|ƒˆ|ƒS(N((R[(R^R_RC(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\ïscS`s|S(N((R[((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\ðsc`sˆˆ|ƒˆˆ|ƒƒS(N((R[(RaR^R_RC(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\ós smode=%i not implementedswhich must be one of %st s!k must be less than ndim(A), k=%dsncv must be kRBtipntr(R+RHR6RMR_RGRaR^RCRKRQRJRLRItltr((RaR^R_R+RCsFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR*|s¢6                                             #    c C`s÷|j|j|j|j|j|j|j|j|j|j |j |j |j ƒ \|_|_|_|_|_|_ |_ t |j dd|j dd|jƒ}t |j dd|j dd|jƒ}|jdkr|j|j |ƒ|j |RBRKRrRCRhRLR6RIRARERFRwRmRHRnRRv( R+treturn_eigenvectorstrvectierrthowmnytsselectRRRT((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyRS>s$    N(R/R0R=R*R~RS(((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyRZ{s ˜ (t_UnsymmetricArpackParamsc B`s>eZdddddddddd„ Zd„Zd„ZRS(iRic `sÁ|dkr~ˆdkr'tdƒ‚nˆdk rBtdƒ‚nˆdk r]tdƒ‚nˆˆ_d„ˆ_dˆ_n´|dkrˆdkr¥tdƒ‚nˆdkrÀtd ƒ‚nˆdkrÛtd ƒ‚n‡‡fd †ˆ_ˆˆ_ˆˆ_ˆˆ_d ˆ_n|d!kr"ˆdkr>tdƒ‚nˆdkrYtdƒ‚nˆˆ_|dkr•|d kr†ˆˆ_qÈtdƒ‚n3|d kr¶‡fd†ˆ_n‡fd†ˆ_ˆdkrød„ˆ_dˆ_ˆjˆ_q2ˆˆ_d ˆ_‡‡fd†ˆ_ntd|ƒ‚| tkrZtddj tƒƒ‚n||dkr}td|ƒ‚nt j ˆ|||||| | | | | ƒ ˆj |ksʈj |dkràtdˆj ƒ‚nt d |ˆjƒˆ_t d ˆj ˆj dˆjƒˆ_tˆj}tj|dˆ_tj|dˆ_t|ˆ_t|ˆ_tjddƒˆ_ˆjd kr´t ˆj ˆjjƒƒˆ_n dˆ_dS("Nis#matvec must be specified for mode=1s'M_matvec cannot be specified for mode=1s*Minv_matvec cannot be specified for mode=1cS`s|S(N((R[((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\„sR]is#matvec must be specified for mode=2s%M_matvec must be specified for mode=2s(Minv_matvec must be specified for mode=2c`sˆˆ|ƒƒS(N((R[(R^R_(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\ŽsR`iis*matvec must be specified for mode in (3,4)s/Minv_matvec must be specified for mode in (3,4)tDFsmode=4 invalid for complex Ac`stjˆ|ƒƒS(N(R>treal(R[(R^(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\£sc`stjˆ|ƒƒS(N(R>timag(R[(R^(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\¥scS`s|S(N((R[((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\§sc`sˆjˆ|ƒƒS(N(Rb(R[(RaR+(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR\­ssmode=%i not implementeds!Parameter which must be one of %sRcs#k must be less than ndim(A)-1, k=%dsncv must be k+1RBRwtlowertrwork(R+RHR6RMR_RGRaR^RCRKRQRJRLRIRx((RaR^R_R+sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR*Vs~%                             "'   !cC`sÅ|jdkr|j|j|j|j|j|j|j|j|j |j |j |j |j ƒ \|_|_|_|_|_ |_ |_ n„|j|j|j|j|j|j|j|j|j |j |j |j |j|j ƒ \|_|_|_|_|_ |_ |_ t|j dd|j dd|jƒ}t|j dd|j dd|jƒ}|jdkr›|j|j |ƒ|j |RBRKR‡RCRˆRMRrRhRLRIRARERFRwRmRnR,RRvtastypetuppertabst conjugatetdotR_tconjtroundt_ndigitstargsortR(R+R€R6RHR‚RƒR„tsigmartsigmaitworkevtdrtditzrt nreturnedRRtitrdtindRT((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyRSøsœ $    B0  B B0$#)      $    N(R/R0R=R*R~RS(((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR…Us x (cC`sHt|ƒ}t|dƒsDtj|jdƒ}||j|_n|S(Ntdtypei(R thasattrR>RBtshapeR¥(tmR[((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt_aslinearoperator_with_dtype}s  tSpLuInvcB`s eZdZd„Zd„ZRS(sl SpLuInv: helper class to repeatedly solve M*x=b using a sparse LU-decopposition of M cC`sGt|ƒ|_|j|_|j|_tj|jtjƒ |_dS(N(RtM_luR§R¥R>t issubdtypetcomplexfloatingtisreal(R+tM((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR*‹s  cC`sžtj|ƒ}|jr~tj|jtjƒr~|jjtj|ƒj |jƒƒd|jjtj |ƒj |jƒƒS|jj|j |jƒƒSdS(Nyð?( R>tasarrayR®R¬R¥R­R«tsolveR‡R’Rˆ(R+R[((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt_matvec‘s !$*(R/R0R1R*R²(((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyRª…s tLuInvcB`s eZdZd„Zd„ZRS(sd LuInv: helper class to repeatedly solve M*x=b using an LU-decomposition of M cC`s+t|ƒ|_|j|_|j|_dS(N(RR«R§R¥(R+R¯((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR*¢s cC`st|j|ƒS(N(RR«(R+R[((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR²§s(R/R0R1R*R²(((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR³œs cC`sZtj|ƒ}dtj|jƒtj|jƒj}t||dt||ƒddƒS(s2 gmres with looser termination condition. ièRItatoli( R>R°tsqrttsizetfinfoR¥tepsRR5(RtbRItmin_tol((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt gmres_loose«s)tIterInvcB`s&eZdZedd„Zd„ZRS(sb IterInv: helper class to repeatedly solve M*x=b using an iterative method. icC`s—||_t|dƒr'|j|_n&tj|jdƒ}||j|_|j|_|dkrdtj|jƒj}n||_||_ dS(NR¥iii( R¯R¦R¥R>RBR§R·R¸tifuncRI(R+R¯R½RIR[((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR*ºs    cC`sS|j|j|d|jƒ\}}|dkrOtd|jj|fƒ‚n|S(NRIis?Error in inverting M: function %s did not converge (info = %i).(R½R¯RIR<R/(R+R[R¹R,((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR²Ês $ (R/R0R1R»R*R²(((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR¼´st IterOpInvcB`s5eZdZedd„Zd„Zed„ƒZRS(so IterOpInv: helper class to repeatedly solve [A-sigma*M]*x = b using an iterative method ic `sˆ|_ˆ|_ˆ|_‡‡‡fd†}‡‡fd†}tjˆjdƒ}ˆdkr”||ƒj} t|jj|d| ƒ|_ n-||ƒj} t|jj|d| ƒ|_ ˆj|_|dkrødtj |j jƒj }n||_ ||_ dS(Nc`sˆj|ƒˆˆj|ƒS(N(R_(R[(RR¯RC(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt mult_funcÞsc`sˆj|ƒˆ|S(N(R_(R[(RRC(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pytmult_func_M_NoneásiR¥ii(RR¯RCR>RBR§R=R¥R RfR·R¸R½RI( R+RR¯RCR½RIR¿RÀR[R¥((RR¯RCsFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR*Ùs(          cC`sS|j|j|d|jƒ\}}|dkrOtd|jj|fƒ‚n|S(NRIisIError in inverting [A-sigma*M]: function %s did not converge (info = %i).(R½RfRIR<R/(R+R[R¹R,((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR²øs $ cC`s |jjS(N(RfR¥(R+((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR¥s(R/R0R1R»R*R²tpropertyR¥(((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR¾Ós cC`sgt|ƒrt|ƒjSt|ƒrPt|ƒrC|rC|j}nt|ƒjSt|d|ƒjSdS(NRI(RR³R_R RtTRªR¼(R¯t symmetricRI((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pytget_inv_matvecs     cC`sú|dkr"t|d|d|ƒS|dkr1t|ƒr·tj|jtjƒsgtj|ƒdkrytj|ƒ}n |d}|j dd|j dd…c|8R¬R¥R­RˆR:tflatR§R³R_R R RRÂRªttocscR¾R©(RR¯RCRÃRIRf((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pytget_OPinv_matvecs8    '       s@Nested calls to eigs/eighs not allowed: ARPACK is not re-entrantic C`sÿ|jd|jdkr3td|jfƒ‚n|dk r¹|j|jkrptd|j|jfƒ‚ntj|jƒjjƒtj|jƒjjƒkr¹tjdƒq¹n|jd} |dkråtd|ƒ‚n|| dkrrtjdt ƒt |ƒr t dƒ‚nt |t ƒr>t d ƒ‚nt |t ƒr\t d ƒ‚nt|d |d | ƒS|dkrSt|ƒj}| dk r¨td ƒ‚n| dk rÃtdƒ‚n|dkrÿd}d}d}| dk rPtdƒ‚qPq‘d}| dkr,t|dtd|ƒ}nt| ƒ} | j}t|ƒj}n>tj|jtjƒr| dk r†tdƒ‚nd}nr| dks­| jƒdkr¶d}nK| jƒdkrõtj|ƒdkrìtdƒ‚nd}n tdƒ‚t|ƒj}| dk r+tdƒ‚n| dkrXt|||dtd|ƒ}nt| ƒ} | j}|dkr‚d}nt|ƒj}t| ||jj||||||||||ƒ }t,x|jsç|jƒqÑW|j| ƒSWdQXdS(s3 Find k eigenvalues and eigenvectors of the square 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] Parameters ---------- A : ndarray, sparse matrix or LinearOperator An array, sparse matrix, or LinearOperator representing the operation ``A * x``, where A is a real or complex square matrix. k : int, optional The number of eigenvalues and eigenvectors desired. `k` must be smaller than N-1. It is not possible to compute all eigenvectors of a matrix. M : ndarray, sparse matrix or LinearOperator, optional An array, sparse matrix, or LinearOperator 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 positive definite If sigma is specified, M is positive semi-definite If sigma is None, eigs 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 or complex, optional 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``. For a real matrix A, shift-invert can either be done in imaginary mode or real mode, specified by the parameter OPpart ('r' or 'i'). Note that when sigma is specified, the keyword 'which' (below) refers to the shifted eigenvalues ``w'[i]`` where: If A is real and OPpart == 'r' (default), ``w'[i] = 1/2 * [1/(w[i]-sigma) + 1/(w[i]-conj(sigma))]``. If A is real and OPpart == 'i', ``w'[i] = 1/2i * [1/(w[i]-sigma) - 1/(w[i]-conj(sigma))]``. If A is complex, ``w'[i] = 1/(w[i]-sigma)``. v0 : ndarray, optional Starting vector for iteration. Default: random ncv : int, optional The number of Lanczos vectors generated `ncv` must be greater than `k`; it is recommended that ``ncv > 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 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, http://www.caam.rice.edu/software/ARPACK/ .. [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: >>> 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) iis!expected square matrix (shape=%s)s#wrong M dimensions %s, should be %ssZM does not have the same type precision as A. This may adversely affect ARPACK convergencesk=%d must be greater than 0.sOk >= N - 1 for N * N square matrix. Attempting to use scipy.linalg.eig instead.shCannot use scipy.linalg.eig for sparse A with k >= N - 1. Use scipy.linalg.eig(A.toarray()) or reduce k.sACannot use scipy.linalg.eig for LinearOperator A with k >= N - 1.sACannot use scipy.linalg.eig for LinearOperator M with k >= N - 1.R¹trights0OPinv should not be specified with sigma = None.s=OPpart should not be specified with sigma = None or complex As+Minv should not be specified with M = None.iRÃRIs;OPpart should not be specified with sigma=None or complex AitrR¢s%OPpart cannot be 'i' if sigma is realisOPpart must be one of ('r','i')s*Minv should not be specified when sigma isN(R§R<R=R>R¥tcharRŽtwarningstwarntRuntimeWarningR t TypeErrort isinstanceR RR©R_RÄR@R¬R­RˆRÇRNR…t _ARPACK_LOCKROR~RS(RR6R¯RCRLRQRKRJRIR€tMinvtOPinvtOPpartRHR_RGRaR^tparams((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR;sˆ™ 6                         tnormalc C`s-tj|jtjƒrì| dkr7td| ƒ‚n|dkrRtdƒ‚n*|dkrgd}n|dkr|d}nt||d |d |d |d |d |d|d|d| d| d| ƒ } | râ| dj| dfS| jSn|jd|jdkrtd|jfƒ‚n|d*k r¥|j|jkr\td|j|jfƒ‚ntj|jƒj j ƒtj|jƒj j ƒkr¥t j dƒq¥n|jd}|dkrÍtdƒ‚n||krWt j dt ƒt|ƒrtdƒ‚nt|tƒr"tdƒ‚nt|tƒr@tdƒ‚nt|d|d| ƒS|d*kr#t|ƒ}|j}| d*k r“tdƒ‚n|d*krÏd} d*}d*}| d*k r td ƒ‚q q¿d!} | d*krüt|d"td|ƒ}nt| ƒ} | j}t|ƒj}nœ| d*k r>td#ƒ‚n| dkrÅd$} d*}| d*krƒt|||d"td|ƒ}nt| ƒ} | j}|d*kr­d*}q¿t|ƒ}|j}nú| d%kr+d&} | d*krt|||d"td|ƒ}nt| ƒj}t|ƒj}d*}n”| d'kr¯d(} t|ƒj}| d*kryt|||d"td|ƒ}nt| ƒj}|d*krd*}q¿t|ƒj}ntd)| ƒ‚t|||jj || ||||||||ƒ }t,x|js|jƒqÿW|j| ƒSWd*QXd*S(+sž 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]. 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``. 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). 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, http://www.caam.rice.edu/software/ARPACK/ .. [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 -------- >>> 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) RÕs,mode=%s cannot be used with complex matrix AR#s/which='BE' cannot be used with complex matrix AR!R$R"R%R¯RCRLRQRKRJRIR€RÑRÒiis!expected square matrix (shape=%s)s#wrong M dimensions %s, should be %ssZM does not have the same type precision as A. This may adversely affect ARPACK convergencesk must be greater than 0.sLk >= N for N * N square matrix. Attempting to use scipy.linalg.eigh instead.sfCannot use scipy.linalg.eigh for sparse A with k >= N. Use scipy.linalg.eigh(A.toarray()) or reduce k.s>Cannot use scipy.linalg.eigh for LinearOperator A with k >= N.s>Cannot use scipy.linalg.eigh for LinearOperator M with k >= N.R¹t eigvals_onlys0OPinv should not be specified with sigma = None.s+Minv should not be specified with M = None.iRÃs*Minv should not be specified when sigma isitbucklingitcayleyisunrecognized mode '%s'N(R>R¬R¥R­R<RR‡R§R=RÊRŽRËRÌRÍR RÎRÏR RR©R_RÄR@RÇRZRÐROR~RS(RR6R¯RCRLRQRKRJRIR€RÑRÒRGtretRHR_RaR^RÔ((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR9s¾Å       $    6                                c C`sL|j\}}tj|||fd|jƒ}||dd…d|…ftemptyR¥trangetrandomtrandnt iscomplexobjR–R—Rµ( R[R6RHR¨tyR¢REtjtu((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt_augmented_orthonormal_colsˆs":%cC`st|j|ƒjS(N(RâRÂ(R[R6((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt_augmented_orthonormal_rows scC`s |jjƒS(N(RÂR—(R[((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt_herm¤sc`s2tˆtƒptˆƒs-tjˆƒ‰nˆj\}‰|dks]|t|ˆƒkrptd|ƒ‚ntˆtƒr|ˆkr©ˆj‰ˆj } ˆj ‰q\ˆj ‰ˆj‰t ˆddƒ‰ˆdkrýˆj tjˆdgƒƒj‰n‡‡‡fd†} nG|ˆkr@ˆj ‰} tˆƒj ‰nˆj ‰tˆƒj ‰} ‡‡fd†} td| dˆjdtˆjƒtˆjƒfƒ} t| d |d |d d |d |d|d|ƒ\} } |dkr€tj| jdƒ} | jjjƒ}idd6dd6}||tj|ƒj}|tj| ƒ}| |k}|jƒ}||}tj| |ƒ}tj| ƒ}|||*|sª|S|ˆkrý| dd…|f}|dkrè| |ƒ|nd}t|ƒ}n>| dd…|f}|dkr5t| |ƒ|ƒnd}|dk rVt||ƒnd}|dk rwt||ƒnd}n¥|dkrtj| ƒ}|s¥|S|ˆkrè| }|dkrÓ| |ƒ|nd}t|ƒ}q%| }|dkrt| |ƒ|ƒnd}n tdƒ‚|||fS(s Compute the largest k singular values/vectors for a sparse matrix. Parameters ---------- A : {sparse matrix, LinearOperator} Array to compute the SVD on, of shape (M, N) k : int, optional Number of singular values and vectors to compute. Must be 1 <= k < min(A.shape). ncv : int, optional The number of Lanczos vectors generated ncv must be greater than k+1 and smaller than n; it is recommended that ncv > 2*k Default: ``min(n, max(2*k + 1, 20))`` tol : float, optional Tolerance for singular values. Zero (default) means machine precision. which : str, ['LM' | 'SM'], optional Which `k` singular values to find: - 'LM' : largest singular values - 'SM' : smallest singular values .. versionadded:: 0.12.0 v0 : ndarray, optional Starting vector for iteration, of length min(A.shape). Should be an (approximate) left singular vector if N > M and a right singular vector otherwise. Default: random .. versionadded:: 0.12.0 maxiter : int, optional Maximum number of iterations. .. versionadded:: 0.12.0 return_singular_vectors : bool or str, optional - True: return singular vectors (True) in addition to singular values. .. versionadded:: 0.12.0 - "u": only return the u matrix, without computing vh (if N > M). - "vh": only return the vh matrix, without computing u (if N <= M). .. versionadded:: 0.16.0 Returns ------- u : ndarray, shape=(M, k) Unitary matrix having left singular vectors as columns. If `return_singular_vectors` is "vh", this variable is not computed, and None is returned instead. s : ndarray, shape=(k,) The singular values. vt : ndarray, shape=(k, N) Unitary matrix having right singular vectors as rows. If `return_singular_vectors` is "u", this variable is not computed, and None is returned instead. Notes ----- This is a naive implementation using ARPACK as an eigensolver on A.H * A or A * A.H, depending on which one is more efficient. Examples -------- >>> from scipy.sparse import csc_matrix >>> from scipy.sparse.linalg import svds, eigs >>> A = csc_matrix([[1, 0, 0], [5, 0, 2], [0, -1, 0], [0, 0, 3]], dtype=float) >>> u, s, vt = svds(A, k=2) >>> s array([ 2.75193379, 5.6059665 ]) >>> np.sqrt(eigs(A.dot(A.T), k=2)[0]).real array([ 5.6059665 , 2.75193379]) is*k must be between 1 and min(A.shape), k=%dR¥ic`swtj|jdˆfdˆƒ}xKt|jƒD]:\}}ˆj|jddƒƒj||dd…fRÚR§t enumerateRÂtrmatvectreshape(tVtoutR¢tcol(RR¥R¨(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pytX_matmat s"2c`sˆˆ|ƒƒS(N((R[(tXH_dottX_dot(sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyt matvec_XH_XsR_R§R6RIiRJRKRLRQRg@@Rg€„.ARNtvhRáR s"which must be either 'LM' or 'SM'.(RÏR R R>R°R§RDR<R_tmatmatRætgetattrR=R–RBR¥RäRtmaximumR‡RÊRŽR·R¸R5tsumRµt zeros_likeRâRã(RR6RKRIRLRQRJtreturn_singular_vectorsRHRëRîtXH_Xteigvalsteigvectttfactortcondtcutofft above_cutofftnlargetnsmalltslargeRtvlargetulargetvhlargeRáRïRE((RRìRíR¥R¨sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyR¨sxL!       $   !      "(!$  "+ (QR1t __future__RRRt __docformat__t__all__tRtnumpyR>RËtscipy.sparse.linalg.interfaceR R t scipy.sparseR R R Rt scipy.linalgRRRRtscipy.sparse.sputilsRtscipy.sparse.linalgRRtscipy._lib._utilRtscipy._lib._threadsafetyRRoR™t DNAUPD_ERRORSt SNAUPD_ERRORSR:t ZNAUPD_ERRORSt CNAUPD_ERRORSt DSAUPD_ERRORSt SSAUPD_ERRORSt DNEUPD_ERRORSt SNEUPD_ERRORSt ZNEUPD_ERRORSt CNEUPD_ERRORSt DSEUPD_ERRORSt SSEUPD_ERRORSRsR2RuRRkRŒR)RRR7tobjectR8RZR…R©RªR³R»R¼R¾RNRÄRÇRÐR=R@RRRâRãRäR(((sFlib/python2.7/site-packages/scipy/sparse/linalg/eigen/arpack/arpack.pyts@   """"                            FÚÿ)  2 '  ü ÿN