Ei SrSSKrSSKrSSKJrJr SSKJrJ r SSK J r J r J r Jr SSKJrJrJr SSKJrJr SS KJr S r/S QrS S SSS.rSSSSS.r0SS_SS_SS_SS_SS_SS_SS_SS _S!S"_S#S$_S%S&_S'S(_S)S*_S+S,_S-S._S/S0_S1S2_S3S40Er\r\R=5rS5\S+'\r 0SS_SS6_SS_SS_SS_SS_SS7_SS _S!S8_S#S$_S%S&_S'S9_S)S*_S+S:_S-S._S/S0_S1S;_S3S40Er!\!r"0SS_SS<_SS_SS_SS_S!S=_S#S$_S%S>_S'S?_S)S@_S+S,_S-S._S/SA_S1SB_SCSD_SESF_r#\#R=5r$SG\$S'SH\$SC'SI\$SE'0SS_SSJ_SS_SS_SSK_S!S=_S#S$_S%S>_S'SL_S)SM_S+SN_S-S._S/SA_S1SB_SCSO_SESP_r%\%R=5r&SQ\&SC'SR\&SE'0SS_SS_SS_SS7_S!S8_S#S$_S%S>_S'SS_S)S*_S+S:_S-S._S/S2_SCST_SESU_SVSA_SWSX_r'\'R=5r(SY\(SC'SZ\(SW'\!\"S[.r)\\\\ S\.r*\'\(S[.r+\#\$\%\&S\.r,/S]Qr-/S^Qr.SSSSS_SS`SaSbSc. r/SSSSd.r0"SeSf\15r2"SgSh\25r3Sir4"SjSk5r5"SlSm\55r6"SnSo\55r7"SpSq\5r8"SrSs\5r9Str:"SuSv\5r;"SwSx\5rSS{jr?SS|jr@SS}jrAg)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.zUc[nURUS5n[TU] SUSU35 g)Nz Unknown errorz ARPACK error z: ) _NAUPD_ERRORSgetsuper__init__)selfinfoinfodictmsg __class__s e/home/mande/repo/quber/.venv/lib/python3.13/site-packages/scipy/sparse/linalg/_eigen/arpack/arpack.pyr;ArpackError.__init__s8  $Hll41 =b67N)__name__ __module__ __qualname____firstlineno____doc__r;__static_attributes__ __classcell__)r@s@rArrs88rCrc\rSrSrSrSrSrg)ri%z ARPACK iteration did not converge Attributes ---------- eigenvalues : ndarray Partial result. Converged eigenvalues. eigenvectors : ndarray Partial result. Converged eigenvectors. cN[RUSSU05 X lX0lg)Nr!)rr; eigenvalues eigenvectors)r<r?rOrPs rAr;ArpackNoConvergence.__init__2s$T2Cy1&(rC)rOrPN)rFrGrHrIrJr;rKrDrCrArr%s  )rCrc&[SU-S-S5$)zm Choose number of lanczos vectors based on target number of singular/eigen values and vectors to compute, k. rr)max)ks rA choose_ncvrV8s q1uqy" rCc(\rSrSrSSjrSrSrg) _ArpackParamsi@Nc `US::a[SU35eU cUS-n U S::a[SU 35eUS;a*[R"US5(aSnO [S5eUb[R"USUS 9UlS n O(UR S S U/S 9R U5UlSn UcSUlOX`lUc [U5n[Xq5Ul Xl [R"X4U5Ul XlXPl0SU _SS_SS_SS_SS_SS_SS_S[ U _SS_SU _SS_S[#U 5_SU_SU_SS_SUR_SU_0S S_S!S _S"S_S#S_S$S_S%S_S&S_S'S_S(S_S)S_S*S_S+S_S,S_S-S_S.S_S/S_S0S_ESSSSSSSSS1.EUlX lX0lS2Ulg)3Nrzk must be positive, k= z"maxiter must be positive, maxiter=fdFDrz)matrix type must be 'f', 'd', 'F', or 'D'T)copydtyper?lowhighsizetol getv0_rnorm0g aitr_betaj aitr_rnorm1 aitr_wnorm aup2_rnormidowhichbmatr=itermaxitermodennconvncvnevnpshift getv0_first getv0_iter getv0_itry getv0_orth aitr_iteraitr_j aitr_orth1 aitr_orth2 aitr_restart aitr_step3 aitr_step4 aitr_ierr aup2_initv aup2_iter aup2_getv0) aup2_cnorm aup2_kpluspaup2_nev aup2_nev0aup2_np0 aup2_numcnv aup2_update aup2_ushiftF) ValueErrorrtcan_castarrayresiduniformastypesigmarVminrrrpzerosvrkro WHICH_DICTint arpack_dictrUtp converged) r<rprUrrngrorrrv0rnrkrdr=s rAr;_ArpackParams.__init__As 65aS9: : ?"fG a<A'KL L V {{2s## !LMM >"4r:DJDCqcBII"MDJD =DJJ ;Q-Cs;1(B'  + 3+ C+  #+  3 + # + # +  1+  Z&+  A+  D+  A+  s7|+  D+  +  Q+ 488!+ " 1#+ $ !%+ & Q'+ ( 1)+ * !++ , !-+ . !/+ 0 1+ 2 a3+ 4 !5+ 6 !7+ 8 A9+ : !;+ < !=+ > ?+ @ !A+ B C+ D !E+ FU+ ZrCc \SnURSnURSnURS5upE[ S US US URS 3XE5e![aLnUSUS3n[R"S5n[R"UR S45nSnSnANqSnAff=f) Nz1_SymmetricArpackParams.__init__..qrCIr#matvec must be specified for mode=2%M_matvec must be specified for mode=2(Minv_matvec must be specified for mode=2c >T"T"U55$rErDr Minv_matvecmatvecs rArr F1I 6rCGr z'matvec must not be specified for mode=3z(Minv_matvec must be specified for mode=3cU$rErDrs rArr1rCc >T"T"U55$rErD)rM_matvecrs rArrsK $TRT"U55$rEOPa)rrr<s rArrs 3rCrz#matvec must be specified for mode=5z(Minv_matvec must be specified for mode=5c,>T"T"U5TU--5$rErD)rrrrs rArrsKq EAI0E$FrCcU$rErDrs rArr rrCc8>T"T"U5TT"U5--5$rErD)rrrrrs rArr#s#Kq 27(1+2E1F%GrCmode= not implementedzwhich must be one of  zk must be less than ndim(A), k=rrlzncv must be kqcBC C99((-tQ2txxuc&s 4)- S(81a 88a<488q=9$((DE EXXa!e4773 XXdhh$((Q,7twwG ! j ?@ @(11# 2DE)2233EF -c 2 -c 2XXb1 rCcURURURURURUR UR 5 [URSURSUR-5n[URSURSUR-5nURSS:XGaURS:Xa,URUR U5UR U'gURS:XaWURUR U5UR U'URUR U5UR U'gURS:Xa[URSURSUR-5nURUR U5nURX@RUR U--5UR U'g[URSURSUR-5nURUR U5UR U'gURSS:Xa,URUR U5UR U'gURSS:Xa [!S5eURSS:XaMUR"R%S S UR/S 9R'UR(5URSS&gURSS:Xa,URUR U5UR U'gS UlURS S:XagURS S:XaUR-5 g[/URS UR0S9e)Nrrrjrrr .ARPACK requested user shifts. Assure ISHIFT==0r/r^r_r`Tr=r>)rrrrrrrslicerprorrrrrrrrrrrrrrr)r<xsliceysliceBxsliceAxs rAiterate_SymmetricArpackParams.iterateGs  D,,djj$&&$** JJ  4tzz!}djjmdff&<=tzz!}djjmdff&<=   E "a 'yyA~%)WWTZZ-?%@ 6"a%)XXdjj.@%A 6"%)XXdjj.@%A 6"a 1 tzz!}tvv/EF]]4::f#56%)XXbJJ48JJw4G5H/I&J 6"  1 tzz!}tvv/EF%)XXdjj.A%B 6"   e $ )!% 6(:!;DJJv    e $ )MN N   e $ ) HH,,C3766(-<XXXU X5 TRS:XaSOSTRS'TR U:dTR US-::a[STR 35e[R""SU-TR$S9Tl[R""STR -TR S--TR$S9Tl[*TR$n[,R.US -Tl[,R.US!-Tl[4UTlTl[R""S"[R<S9TlTR$S#;a=[R""TR TR$RA5S9Tl!gSTl!g)$NrrrrcU$rErDrs rAr3_UnsymmetricArpackParams.__init__..rrCrrrrrc >T"T"U55$rErDrs rArrrrCrr 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<>[R"T"U55$rE)rtrealrrs rArrQ)@rCc<>[R"T"U55$rE)rtimagr s rArrr rCcU$rErDrs rArrrrCc2>TRT"U55$rEr)rrr<s rArrsDHHXa[$9rCrrzParameter which must be one of rz!k must be less than ndim(A)-1, k=rrlzncv must be k+1)rrrrrrrrrrrprorrrrrrrviewrt complex128ravelrrrr)r<rrrs rAr _UnsymmetricArpackParams.iterates 77d?     $**dffdjj$**       $**dffdjj$** DJJ  tzz!}djjmdff&<=tzz!}djjmdff&<=   E "a 'yyF"%)WWTZZ-?%@ 6" 1 tzz!}tvv/EF%)XXdjj.A%B 6"   e $ )!% 6(:!;DJJv    e $ )MN N   e $ )ww$ $ 0 0T7;vvh!1!@@Ftww 1 !% 0 0T466ST+ 0!$r}}%ffTWWoeeg 1   e $ )!%F);!5 URSnUS:wa['X@R(S 9eURS nUSUnUSS2SU24nU(aX4$U$)Nrr=r3rr rrrrrrq?rr)r+r,)r-r.)r+r&r-r!)r,r'r.) rUrprrrtrrrrfloatrrr rrrrrrrrrrupperabs conjugatedotrconjrorkargsortr)r<rrUrprrrsigmarsigmaiworkevdrdizr nreturnedrrirdindrs rAr _UnsymmetricArpackParams.extractDsvvtvv1#$ S!((4882884rwwtzz*+rwwtzz*+!dhh,0 77d?1q5'1B1q5'1B1!e*DGG3?B    "5wBFDJJ  DJJ  ##F+Dqy!$1F1FGG((1ITBYA $''--/*A{1f1Q499~*q5&(Ah1a!e8 1D&DAadG*+AqD'*;*;*=AaQhKFA &NIFA1f*1f1Q499~*!vvbAh Bq!tH0EF!q5&(Ah1a!e8 1D&DAadG*+AqD'*;*;*=AaQhK%'VVBq!tH,0KK1a4,A&C')vvbAEl.2kk"QAX,.G(I&I')BFF2ad837;;r!QU(|3L-N.0ffR1q5\59[[AqD5J/L-L'M %MAaD()tyy{Aa!eHFA &NIFA/1f8~jyMa)m$99&BYY&(a**n-B::-**RWW-CZZ</**S\2C**SW-C::!33qbc(4R4.CZZ#55bq'CcFafI!DGG,A!twwc:A    "5w1 FJJ DJJ DJJ ( ##F+Dqy!$1F1FGG##G,D%4A!UdU( A 4KHrC)rrrrrrrlrrrrrrrrrrrrDrCrArrs:>)-IMyv4BlNrCrc$\rSrSrSrSrSrSrg)SpLuInviz\ SpLuInv: helper class to repeatedly solve M*x=b using a sparse LU-decomposition of M c[U5UlURUlURUl[R "UR[R 5(+UlgrE)rM_lushaper]rt issubdtypecomplexfloatingisrealr<Ms rAr;SpLuInv.__init__sBG WW WW -- B4F4FGG rCcH[R"U5nUR(a[R"UR[R 5(aUR R[R"U5RUR55SUR R[R"U5RUR55--$UR RURUR55$)Nr ) rtasarrayr:r8r]r9r6solverrr r<rs rA_matvecSpLuInv._matvecs JJqM ;;2=="2D2DEEIIOOBGGAJ$5$5djj$AB499??2771:+<+TRU5TTRU5-- $rEr)rr3r<rs rA mult_func%IterOpInv.__init__..mult_func2s#88A;!!44 4rCc2>TRU5TU-- $rErd)rr3rs rAmult_func_M_None,IterOpInv.__init__..mult_func_M_None5s88A;* *rCrrrr) r3r<rrtrr7r]rrrOrPrXrd) r<r3r<rrXrdrerhrr]s ``` rAr;IterOpInv.__init__-s  5 + HHQWWQZ  9$Q'--E$TVV\\%5+02DGaL&&E$TVV\\%.+02DGWW !8bhhtww}}-111C rCcURURXRS9up#US:wa&[SURRSUS35eU$)Nr[rz)Error in inverting [A-sigma*M]: function r\r])rXrrdrrFr^s rArBIterOpInv._matvecLs]**TWWaXX*6 19;DJJrCracURS:XaGU(a@[R"UR[R5(d UR $[ U5(aU$UR5$)z:Convert sparse matrix to CSC (by transposing, if possible)csr)formatrtr8r]r9Tr tocsc)r3 hermitians rA_fast_spmatrix_to_cscrvZsP EiMM!''2+=+=>>ss A  wwyrCc[U5(a[U5R$[U5(d[ U5(a[ XS9n[ U5R$[XS9R$)Nrur[)r rFrrr rvr4rU)r<rurds rAget_inv_matvecryfsXqzzQx !*1-- !! 9qz   q")))rCcZUS:Xa [XUS9$UGc2[U5(a[R"UR[R 5(d[R "U5S:Xa[R"U5nOUS-nURSSURSS-2==U-ss'[U5R$[U5(d[U5(a;X[URS5-- n[XS9n[!U5R$[#[%U5XUS9R$[U5(d [U5(d[U5(a0[U5(dG[U5(d7[U5(d'[#[%U5[%U5X$S9R$[U5(d[U5(a[XU-- 5R$XU-- n[XSS9n[!U5R$)Nrrurdyrrxr[)ryr rtr8r]r9r r\flatr7rFrrr rrvr4rar)r3r<rrurdrs rAget_OPinv_matvecr}ps za#>>y 1:: aggr'9'9::wwu~*GGAJF FF#QWWQZ!^# $ - $8?? " a[[.q11C O++A%a=A1:$$ $-a03006 7HQKK8J18M8MQZZ 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.)rQright0OPinv 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.Tr{z;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 r7rrtr]charrwarningswarnRuntimeWarningr TypeError isinstancerrrrryr8r9r r}rrrrrr)r3rUr<rrkrrrrnrdrMinvOPinvOPpartrrprrorrparamss rArrs~ 'q)A&q)AwwqzQWWQZ9!''!DEE} 77agg 2177)<yQR R 88AGG  ! ! ' ' )RXXagg->-C-C-I-I-K K MMI%& (  AAv2aS 89::Ez D$ 4 A;;)* * a ( (12 2 a ( (12 21!455 }!!$++  23 3  9: : 9DHK "<== D|,Q$CH '-"kk '*11H ==""4"4 5 5! "@AAD ^v||~4D \\^s "wwu~" !HIID>? ?!!$++  *+ + =*15:EK%U+E,,K 9H'*11H ))   $C %aAGGLL&$&. U&)wsIF >>- ..rCc[R"UR[R5(amU S:wa[ SU S35eUS:Xa [ S5eUS:XaSnOUS:XaS n[ XX#XEXgUXU S 9 nU (aUS R US 4$UR $URS URS :wa[ S URS35eUbURUR:wa%[ SURSUR35e[R"UR5RR5[R"UR5RR5:wa[R"SSS9 URS nUS ::a [ S5eX:a[R"S[SS9 [U5(a [S5e[U[ 5(a [S5e[U[ 5(a [S5e[#XU (+S9$Uc}[%U5nUR&nU b [ S5eUcS n SnSnU b [ S5eGONSn U c [)USUS9nO[%U 5n U R&n[%U5R&nGOU b [ S5eU S:XaIS n SnU c [+XUSUS9nO[%U 5n U R&nUcSnO[%U5nUR&nOU S!:Xa?S"n U c [+XUSUS9nO[%U 5R&n[%U5R&nSnOmU S#:XaXS$n [%U5R&nU c [+XUSUS9nO[%U 5R&nUcSnO%[%U5R&nO[ S%U S&35e[R,R/U 5n [1XURRUU UUUXeXtX5nUR2(d#UR55 UR2(dM#UR7U 5$)'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) normalrz% cannot be used with complex matrix Ar*z/which='BE' cannot be used with complex matrix Ar(r+r)r,) r<rrkrrrrnrdrrrrrrrNrrrrrzk 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. 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