-i= SSKrSSKJr SSKJrJrJrJrJ r J r SSK J r SSK Jr S/rSS jrSS S S SSS SSSSS. Sjjrg)N) LinAlgError)get_blas_funcsqrsolvesvd qr_insertlstsq)_get_atol_rtol) make_systemgcrotmkFc UcSnUcSn[/SQU45uppU/n /nSn[RnU[U5-n[R"[U5U4UR S9n[R "SUR S9n[R"SUR S9n[R"UR 5RnSn[U5GHxnU(aU[U5:a UUunnO_U(aU[U5:Xa U"U5nSnO>U(d*UU[U5- :aUUU[U5- - unnO U"U S 5nSnUcU"U"U55nOUR5nU "U5n[U5H/unnU "UU5nUUUU4'U "UUURS U*5nM1 [R"US -UR S9n[U 5H-unnU "UU5nUUU'U "UUURS U*5nM/ U "U5UWS -'[R"S S S9 S US - nSSS5 [R"W5(a U "UU5nUS UU-:dSnU RU5 URU5 [R"US -US -4UR SS9nUUSUS -2SUS -24'S UUS -US -4'[R"US -U4UR SS9n UU SUS -2SS24'[!UU UUSSSS9unn[#US5nUU:d U(dGMy O [R"UWU45(d [%5e['USUS -2SUS -24US SUS -24R)55un n!USS2SUS -24nUUUXUU4$!,(df  GN=f)a FGMRES Arnoldi process, with optional projection or augmentation Parameters ---------- matvec : callable Operation A*x v0 : ndarray Initial vector, normalized to nrm2(v0) == 1 m : int Number of GMRES rounds atol : float Absolute tolerance for early exit lpsolve : callable Left preconditioner L rpsolve : callable Right preconditioner R cs : list of (ndarray, ndarray) Columns of matrices C and U in GCROT outer_v : list of ndarrays Augmentation vectors in LGMRES prepend_outer_v : bool, optional Whether augmentation vectors come before or after Krylov iterates Raises ------ LinAlgError If nans encountered Returns ------- Q, R : ndarray QR decomposition of the upper Hessenberg H=QR B : ndarray Projections corresponding to matrix C vs : list of ndarray Columns of matrix V zs : list of ndarray Columns of matrix Z y : ndarray Solution to ||H y - e_1||_2 = min! res : float The final (preconditioned) residual norm NcU$Nxs W/var/www/html/venv/lib/python3.13/site-packages/scipy/sparse/linalg/_isolve/_gcrotmk.pylpsolve_fgmres..lpsolve@HcU$rrrs rrpsolve_fgmres..rpsolveCrraxpydotscalnrm2)dtype)r r )r rFrr ignore)overdivideTFr!ordercol)which overwrite_qru check_finite)rr")rnpnanlenzerosr!onesfinfoepsrangecopy enumerateshapeerrstateisfiniteappendrabsrr conj)"matvecv0matolrrcsouter_vprepend_outer_vrrrr vszsyresBQRr4 breakdownjzww_normicalphahcurvQ2R2_s" r_fgmresrYsb  ++JRERDt B B A &&C CLA #b'1RXX.A bhh'A rxx(A ((288  CI1X q3w</1:DAq c'l!2 AA Q!c'l*:%:1CL 012DAq2AA 9q "AAabMDAq1IEAacFQ1771:v.A" xx!177+bMDAq1IEDGQ1771:v.A"GQqS [[hx 8d2hJE9 ;;u  UAAR3<'I !  ! XXqsAaCjs ;4AaC419 1Q3qs7 XXqsAhaggS 94AaC46 Rq'+%A1!D'l : WZ ;;q1v  m$1Q3$t!t) a$1Q3$inn&67KAq!Q !DQqSD& A aBAs ""k9 8s  O O' gh㈵>gioldest) rtolrAmaxiterMcallbackr@kCU discard_Ctruncatec [XX!5upp[R"U5R5(d [ S5eU S;a[ SU <35eUR nUR nU c/n U cUn SunnnUcUR 5nO X"U 5- n[/SQU U45unnnnU"U5n[SUXC5upCUS:XaUn U S4$U (aU VVs/sH unnSU4PM snnU SS&U (GaU RS S 9 [R"URS[U 54URS S 9n/nSnU (aGU RS5unnUcU"U5nUUSS2U4'US - nURU5 U (aMG[!USSSS9unnnA[#UR$5n/n['[U55HnUUUn['U5H'n U"UUU UURSUU U4*5nM) [)UUU45S[)US5-:a O&U"SUUU4- U5nURU5 M [#[+UU55SSS2U SS&U (aW[SS/U45unnU H?unnU"UU5n!U"UXRSU!5n U"UUURSU!*5nMA ['U5GHn"UbU"U 5 U"U5n#[-XCU-5n$U#U$::aU"S:dU (aX"U 5- nU"U5n#U#U$::aSn" GOU[-U [U 5- S5-n%U VVs/sHunnUPM nnn[/UUU#- U%U[-XCU-5U#- US9unnn&n'n(n)n*U)U#-n)U(SU)S-n+[+U(S SU)S S5Hun,n!U"U,U+U+RSU!5n+M U&R3U)5n-[+U U-5H$un.n/U.unnU"UU+U+RSU/*5n+M& [R4"SS9 UR3UR3U)55n0SSS5 U'SW0S-n1[+U'S SU0S S5Hun2n3U"U2U1U1RSU35n1M S U"U15- n4[R"U45(d [75eU"U4U15n1U"U4U+5n+U"U1U5n5U"U1UURSU5*5nU"U+XRSU55n U S:Xa3[U 5U :a"U (aU S [U 5U :a U (aMGOvU S:XGao[U 5U :Ga_U (GaW[;USS2SS24R$U&R$5R$n6[=U65un7n8n9/n:[?U7SS2SU S - 24R$5Hunn;U SunnUU;S-nUU;S-n[+U S SU;S S5H;unn?U"U>UURSU=5nU"U?UURSU=5nM= U:HAun>n?U"U>U5n4U"U>UURSU4*5nU"U?UURSU4*5nMC U"U5n4U"SU4- U5nU"SU4- U5nU:RUU45 M U:U SS&U RU1U+45 GM U RSU R 545 U (aU V@VAs/sH un@nASUA4PM snAn@U SS&U W"S -4$s snnfs snnf![0a  Mff=f!,(df  GN=f![6[84a GMf=fs snAn@f)a Solve ``Ax = b`` with the flexible GCROT(m,k) algorithm. Parameters ---------- A : {sparse array, ndarray, LinearOperator} The real or complex N-by-N matrix of the linear system. Alternatively, `A` can be a linear operator which can produce ``Ax`` using, e.g., `LinearOperator`. b : ndarray Right hand side of the linear system. Has shape (N,) or (N,1). x0 : ndarray Starting guess for the solution. rtol, atol : float, optional Parameters for the convergence test. For convergence, ``norm(b - A @ x) <= max(rtol*norm(b), atol)`` should be satisfied. The default is ``rtol=1e-5`` and ``atol=0.0``. maxiter : int, optional Maximum number of iterations. Iteration will stop after maxiter steps even if the specified tolerance has not been achieved. The default is ``1000``. M : {sparse array, ndarray, LinearOperator}, optional Preconditioner for `A`. The preconditioner should approximate the inverse of `A`. gcrotmk is a 'flexible' algorithm and the preconditioner can vary from iteration to iteration. Effective preconditioning dramatically improves the rate of convergence, which implies that fewer iterations are needed to reach a given error tolerance. callback : function, optional User-supplied function to call after each iteration. It is called as ``callback(xk)``, where ``xk`` is the current solution vector. m : int, optional Number of inner FGMRES iterations per each outer iteration. Default: 20 k : int, optional Number of vectors to carry between inner FGMRES iterations. According to [2]_, good values are around `m`. Default: `m` CU : list of tuples, optional List of tuples ``(c, u)`` which contain the columns of the matrices C and U in the GCROT(m,k) algorithm. For details, see [2]_. The list given and vectors contained in it are modified in-place. If not given, start from empty matrices. The ``c`` elements in the tuples can be ``None``, in which case the vectors are recomputed via ``c = A u`` on start and orthogonalized as described in [3]_. discard_C : bool, optional Discard the C-vectors at the end. Useful if recycling Krylov subspaces for different linear systems. truncate : {'oldest', 'smallest'}, optional Truncation scheme to use. Drop: oldest vectors, or vectors with smallest singular values using the scheme discussed in [1,2]. See [2]_ for detailed comparison. Default: 'oldest' Returns ------- x : ndarray The solution found. info : int Provides convergence information: * 0 : successful exit * >0 : convergence to tolerance not achieved, number of iterations References ---------- .. [1] E. de Sturler, ''Truncation strategies for optimal Krylov subspace methods'', SIAM J. Numer. Anal. 36, 864 (1999). .. [2] J.E. Hicken and D.W. Zingg, ''A simplified and flexible variant of GCROT for solving nonsymmetric linear systems'', SIAM J. Sci. Comput. 32, 172 (2010). .. [3] M.L. Parks, E. de Sturler, G. Mackey, D.D. Johnson, S. Maiti, ''Recycling Krylov subspaces for sequences of linear systems'', SIAM J. Sci. Comput. 28, 1651 (2006). Examples -------- >>> import numpy as np >>> from scipy.sparse import csc_array >>> from scipy.sparse.linalg import gcrotmk >>> R = np.random.randn(5, 5) >>> A = csc_array(R) >>> b = np.random.randn(5) >>> x, exit_code = gcrotmk(A, b, atol=1e-5) >>> print(exit_code) 0 >>> np.allclose(A.dot(x), b) True z$RHS must contain only finite numbers)r[smallestzInvalid value for 'truncate': N)NNNrr rcUSSL$)Nrr)cus rgcrotmk..=s r!uD0r)keyr'r(r Teconomic) overwrite_amodepivotingg-q=)rrg?r"rr)rrArBr$)invalidr[re) r r.r:all ValueErrorr>r6rr sortemptyr8r0r!popr;rlistTr5r<zipmaxrYrrr9FloatingPointErrorZeroDivisionErrorrrr7)BAbx0r\rAr]r^r_r@r`rarbrcrr>psolverrrrr b_normrRuCusrMrJrKPrBnew_usrQycj_outerbetabeta_tolmlrIrErFrGpresuxrNbyrgbychycxrUhycrSgammaDWsigmaVnew_CUrOcupwpcpupczuzsB rr r sx!b#GA ;;q>    ?@@--9(FGG XXF XXF z y &OD#t z FFH q M*+JQPQFSD#tT !WF 64>JD { 1v ')*rtq!$r*1  01 HHaggaj#b'*!'' E  66!9DAqy1IAacF FA IIaLbQDzDI1a !##Ys2wA1Q4A1XAaD1aggaj1QqS6':1QqS6{US3[00S1Q3Z#A MM!  SV_%dd+1 "FE?QD9 cDAqQBQ771:r*AQ1771:s+A >   QKAwtF]+ 8 1F1I A7D 8 G  QR[!$ $ BDAqaB  '.v/0v/17=476k4J44O24 (6 $Aq!RQ IA4U1Q4ZAB12'EAraRXXa["-B( UU1X2r{GBDAqaRXXa[3$/B# [[ *quuQxB+ URU]"QR&"QR&)FAsaRXXa[#.B*  d2hJE;;u%%(**& %_ %_B  Q UF + Q E * x b'Q,2qEb'Q,22  #2w!|!CRCE(**acc*,,!!f 5!%a$1Q3$ikk2DAqa5DAqAaDAAaDA#&r!"vqu#5R!$B Q B7 Q B7$6#)B #B  Q UF; Q UF;#)!GESY*ASY*AMM1a&))3*1 2r({"@IItQVVX*,-"B$"-1 gk>+`    D+ *#$56   j.sB#\1!\74-\=9!]"0]!]9= ]  ]  ] !]65]6)NNrrFr)numpyr. numpy.linalgr scipy.linalgrrrrrr iterativer !scipy.sparse.linalg._isolve.utilsr __all__rYr rrrrsU$KK%9 +MO!g#T4b$$QUDTUXr