+ LV-j]D^RIt^RIHtHt^RIHtHtHt^RI H t ^RI H t ^RI HtHtHtHtHtHtHtHt^RIHtHt^t^tRt^ tR tR tR t!R R ]4t !RR]4t!R#)N) lu_factorlu_solve)issparse csc_matrixeye)splu) group_columns)validate_max_step validate_tolselect_initial_stepnormEPSnum_jacvalidate_first_stepwarn_extraneous) OdeSolver DenseOutputg?cN\P!^V^,4R,p\P!^V^,4p\P!V^,V^,34pV^, W,, V, VR&^V^&\P!V^R7#)z6Compute the matrix for changing the differences array.axisNNNN)NNr)nparangezeroscumprod)orderfactorIJMs&& i/Volumes/fast/ai/experiments/ui-tars-smoke/.venv/lib/python3.14/site-packages/scipy/integrate/_ivp/bdf.py compute_Rr%s} !UQY(A !UQYA %!)UQY'(AQ#q(AfI AaD ::aa  c\W4p\V^4pVPV4p\P!VPVRV^,4VRV^,%R#)z " Mvvbkk!n%%  ba%#+/ *rz"  D[(D  $!)!+,D9GCcI    qL TQX%6%@3%FI  3#6 a!eQ !!r&cnaa]tRt^HtoRt]P RRRRRR3V3RlltRtRt R t R t Vt V;t #) BDFa9Implicit method based on backward-differentiation formulas. This is a variable order method with the order varying automatically from 1 to 5. The general framework of the BDF algorithm is described in [1]_. This class implements a quasi-constant step size as explained in [2]_. The error estimation strategy for the constant-step BDF is derived in [3]_. An accuracy enhancement using modified formulas (NDF) [2]_ is also implemented. Can be applied in the complex domain. Parameters ---------- fun : callable Right-hand side of the system: the time derivative of the state ``y`` at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must return an array of the same shape as ``y``. See `vectorized` for more information. t0 : float Initial time. y0 : array_like, shape (n,) Initial state. t_bound : float Boundary time - the integration won't continue beyond it. It also determines the direction of the integration. first_step : float or None, optional Initial step size. Default is ``None`` which means that the algorithm should choose. max_step : float, optional Maximum allowed step size. Default is np.inf, i.e., the step size is not bounded and determined solely by the solver. rtol, atol : float and array_like, optional Relative and absolute tolerances. The solver keeps the local error estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a relative accuracy (number of correct digits), while `atol` controls absolute accuracy (number of correct decimal places). To achieve the desired `rtol`, set `atol` to be smaller than the smallest value that can be expected from ``rtol * abs(y)`` so that `rtol` dominates the allowable error. If `atol` is larger than ``rtol * abs(y)`` the number of correct digits is not guaranteed. Conversely, to achieve the desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller than `atol`. If components of y have different scales, it might be beneficial to set different `atol` values for different components by passing array_like with shape (n,) for `atol`. Default values are 1e-3 for `rtol` and 1e-6 for `atol`. jac : {None, array_like, sparse_matrix, callable}, optional Jacobian matrix of the right-hand side of the system with respect to y, required by this method. The Jacobian matrix has shape (n, n) and its element (i, j) is equal to ``d f_i / d y_j``. There are three ways to define the Jacobian: * If array_like or sparse_matrix, the Jacobian is assumed to be constant. * If callable, the Jacobian is assumed to depend on both t and y; it will be called as ``jac(t, y)`` as necessary. For the 'Radau' and 'BDF' methods, the return value might be a sparse matrix. * If None (default), the Jacobian will be approximated by finite differences. It is generally recommended to provide the Jacobian rather than relying on a finite-difference approximation. jac_sparsity : {None, array_like, sparse matrix}, optional Defines a sparsity structure of the Jacobian matrix for a finite-difference approximation. Its shape must be (n, n). This argument is ignored if `jac` is not `None`. If the Jacobian has only few non-zero elements in *each* row, providing the sparsity structure will greatly speed up the computations [4]_. A zero entry means that a corresponding element in the Jacobian is always zero. If None (default), the Jacobian is assumed to be dense. vectorized : bool, optional Whether `fun` can be called in a vectorized fashion. Default is False. If ``vectorized`` is False, `fun` will always be called with ``y`` of shape ``(n,)``, where ``n = len(y0)``. If ``vectorized`` is True, `fun` may be called with ``y`` of shape ``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of the returned array is the time derivative of the state corresponding with a column of ``y``). Setting ``vectorized=True`` allows for faster finite difference approximation of the Jacobian by this method, but may result in slower execution overall in some circumstances (e.g. small ``len(y0)``). Attributes ---------- n : int Number of equations. status : string Current status of the solver: 'running', 'finished' or 'failed'. t_bound : float Boundary time. direction : float Integration direction: +1 or -1. t : float Current time. y : ndarray Current state. t_old : float Previous time. None if no steps were made yet. step_size : float Size of the last successful step. None if no steps were made yet. nfev : int Number of evaluations of the right-hand side. njev : int Number of evaluations of the Jacobian. nlu : int Number of LU decompositions. References ---------- .. [1] G. D. Byrne, A. C. Hindmarsh, "A Polyalgorithm for the Numerical Solution of Ordinary Differential Equations", ACM Transactions on Mathematical Software, Vol. 1, No. 1, pp. 71-96, March 1975. .. [2] L. F. Shampine, M. W. Reichelt, "THE MATLAB ODE SUITE", SIAM J. SCI. COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997. .. [3] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations I: Nonstiff Problems", Sec. III.2. .. [4] A. Curtis, M. J. D. Powell, and J. Reid, "On the estimation of sparse Jacobian matrices", Journal of the Institute of Mathematics and its Applications, 13, pp. 117-120, 1974. gMbP?gư>NFc <a\V 4\SS` WW4V RR7\V4Sn\ WgSP 4wSnSnSPSPSP4p V fV\SPSPSPWEV SP^SPSP4 SnM\WV4SnRSnRSn\%^ \&,V, \)RVR,44SnRSnSP/W4wSnSn\5SP24'd7V3RlpRp\7SP RSPP8R7pM?V3R lpR p\:P<!SP SPP8R 7pVSnVSn VSn!\:PD!.R O4p\:PF!^\:PH!^\:PJ!^\L^,4, 434Sn'^V, SPN,Sn(VSPN,^\:PJ!^\L^,4, ,Sn)\:PT!\L^,SP 3SPP8R 7pSPV^&V SP,SP,V^&VSn+^Sn,^Sn-RSn.R#) T)support_complexNgQ??cL<S;P^, un\V4#r)nlurAselfs&r$luBDF.__init__..lusA Awr&c$VPV4#N)solver:bs&&r$r;BDF.__init__..solve_lusxx{"r&csc)formatdtypecP<S;P^, un\VRR7#)rT) overwrite_a)rOrrPs&r$rSrTsA  55r&c\WRR7#)T) overwrite_b)rrXs&&r$r;rZs488r&r])rgGzǿgqqgugsh|?r)/rsuper__init__r max_stepr nrtolatolr5tr?r directionh_absr h_abs_olderror_norm_oldmaxrmin newton_tol jac_factor _validate_jacjacr"rrr]ridentityrSr;r!arrayhstackcumsumr MAX_ORDERgammaalpha error_constemptyr*r n_equal_stepsr:)rRr5t0y0t_boundrergrhrs jac_sparsity vectorized first_step extraneousrCrSr;r!kappar* __class__sf&&&&&&&&&&&, r$rd BDF.__init__s_  # "z)-  /)(3 +D? 49 HHTVVTVV $  ,TXXtvvtvv-4-1^^Q-1YY CDJ -ZWEDJ"b3hos4/EF--c@$& DFF    #DFF5 =A 6 9 DFF$&&,,7A  @AYY299Q1i!m1L-L#MNO %i4::-  4::-BIIaQ4O0OO HHi!mTVV,DFFLL Avv!4::~.! r&caaaaSPpSPoSfCSe,\S4'd \S4o\ S4pSV3oVV3RlpV!VS4pWV3#\ S4'dS!VS4pS;P ^, un\V4'd!\VSPR7pVVV3RlpM*\P!VSPR7pVVV3RlpVPSPSP38wd3\RSPSP3 RVP R24hWV3#\S4'd\SSPR7pM"\P!SSPR7pVPSPSP38wd3\RSPSP3 RVP R24hRpWV3#)Nc <S;P^, unSPW4p\SPWVSPSP S4wpSnV#rN)njev fun_singlerfun_vectorizedrhrq)rir?rCr"rRsparsitys&& r$ jac_wrapped&BDF._validate_jac..jac_wrapped sR Q OOA)%,T-@-@!-1YY-5&7"4?r&rbcp<S;P^, un\S!W4SPR7#rrb)rrr]rir?rsrRrs&&r$rrs%IINI%c!irxx@@r&c<S;P^, un\P!S!W4SPR7#r)rrasarrayr]rs&&r$rr!s)IINI::c!irxx@@r&z `jac` is expected to have shape z, but actually has .) rir?rrr callablerr]rrshaperf ValueError)rRrsrr~groupsrr"rsfff @r$rrBDF._validate_jacs VV VV ;#H%%)(3H&x0$f- B#A>~=c]]B A IINI{{q1AAJJq1Aww466466** #CTVVTVVDTCUV667ggYa"ABB+~}}s"((3JJs"((3ww466466** #CTVVTVVDTCUV667ggYa"ABBK~r&c  VPpVPpVPp^ \P!\P !WP \P,4V, 4,pVPV8d1Tp\W PW0P, 4^Vn MMVPV8d1Tp\W PW@P, 4^Vn M VPpVPpVPpVPpVPp VPp VP p VP"p VP$p VP&RJpRpV'EgWT8dRVP(3#WPP ,pVV,pVP VVP*, ,^8dDVP*p\W(\P!VV, 4V, 4^Vn Rp VV, p\P!V4p\P,!VRV^,^R7pWg\P!V4,,p\P.!V^V^,P0V ^V^,4W,, pRpVW,, pV'gV f*VP3VP4VV ,, 4p \7VP8VVVVWP:VVP<4 wppppV'dKtV'dMVP'VV4p Rp RpKV'g$RpVV,p\W(V4^Vn Rp EKR^\>,^,,^\>,X,, pWg\P!X4,,pW,X,p\AVV, 4pV^8dL\C\DVVR V^,, ,,4pVV,p\W(V4^Vn EKRpEKV;P^, un XVnXVn#WPnWnWnXW(^,,, W(^,&VW(^,&\I\KV^,44F&pVV;;,VV^,,, uu&K( VPV^,8dR #V^8d0W^, ,W(,,p\AVX, 4p M\Pp V\L8d7W^,,W(^,,,p!\AV!X, 4p"M\Pp"\PN!V XV".4p#\PP!RR7;_uu_4V#R \PR!W^,4, ,p$RRR4\PT!X$4^, p%VV%, pWn \W\XX\PB!V$4,4pV;PV,un\W(V4^Vn RVnR # +'giL;i) NFrTrLg?ignore)divide)TN)-rir*rerabs nextafterrjinfrkr.rr}rhrgrzryr{r"r:rsTOO_SMALL_STEPrsumr(r)rSr!rGr5r;rpr2r rn MIN_FACTORr?reversedr1rxruerrstaterargmaxro MAX_FACTOR)&rRrir*remin_steprkrhrgrrzryr{r"r: current_jac step_acceptedhr6r7r<r9rAr8n_itery_newr>r safetyerror error_normierror_m error_m_normerror_p error_p_norm error_normsfactors delta_orders&& r$ _step_implBDF._step_impl5s FF FF==r||A~~/FG!KLL :: E Q Hzz$9 :!"D  ZZ( "E Q Hzz$9 :!"D JJEyyyy   && FF WWhh$&  -d1111&AEE~~!56: 266%!)#4u#<=%&" AFF1IEq%!)}15I"&&"333E&&1eai**E!UQY,?@5<OCIEL A:!a%0B.>HHeY3MM4??/,+ 65!!y" 2AB"&K6*%&"A.23q>7I9?8@AF"&&-//E&*Eeem,JA~Z#jR5195E&FFH6*%&"!%  a  1QY<'!) !) % *+A aDAa!eH D,    )  19!!),qx7G%0L66L 9 !!),q|;G%0L66Lhh j,GH [[ ) )!b299UAI+F&FGG*ii(1,   Z"&&/!9: f 6"* )s 5,Z55 [ c \VPVPVPVP,VP VP RVP ^,P44#rV)BdfDenseOutputt_oldrirkrjrr*r0)rRs&r$_dense_output_implBDF._dense_output_implsQdjj$&&$**t~~2M"jj$&&$**q.*A*F*F*HJ Jr&)r*r!r"r:rzrhr{rmryrkrlrsrqrSrer}rprrgr;rir?)__name__ __module__ __qualname____firstlineno____doc__rrrdrrrr__static_attributes____classdictcell__ __classcell__r __classdict__s@@r$rIrIHsB{z79ff4d!d:x1fM^JJr&rIc8aa]tRtRtoV3RltRtRtVtV;t#)ric<\SV`W4W@nVPV\P !VP4,, VnV^\P !VP4,,VnWPnR#)rN) rcrdrrirrt_shiftdenomr*)rRrrirrr*rs&&&&&&r$rdBdfDenseOutput.__init__sZ " vvBIIdjj$9 99 !bii 334 r&cVP^8Xd;WP, VP, p\P!V4pMIWPR,, VPR,, p\P!V^R7p\P !VP R,PV4pVP^8XdW@P ^,, pV#W@P R,, pV#)rrrr)rrN)ndimrrrrr(r*r))rRrixpr?s&& r$ _call_implBdfDenseOutput._call_impls 66Q;\\!TZZ/A 1 A\\'**djj.AAA 11%A FF466":<< # 66Q; NA  # #Ar&)r*rrr) rrrrrdrrrrrs@@r$rrsr&r)"numpyr scipy.linalgrr scipy.sparserrrscipy.sparse.linalgrscipy.optimize._numdiffr commonr r r r rrrrbaserrrxr2rrr%r.rGrIrr&r$rsn,22$1&&&)     !0!"H~J)~JB [r&