ftjYddlZddlmZmZddlmZmZmZm Z m Z m Z ddl m Z dZdZdZd ZGd d eZGd d eZGddeZGddeZGddeZGddeZdS)N) OdeSolver DenseOutput)validate_max_step validate_tolselect_initial_stepnormwarn_extraneousvalidate_first_step)dop853_coefficientsg?皙? c ||d<tt|dd|dddD]M\} \} } tj|d| j| d| |z} ||| |zz|| z|| <N||tj|ddj|zz} |||z| }||d<| |fS)a8Perform a single Runge-Kutta step. This function computes a prediction of an explicit Runge-Kutta method and also estimates the error of a less accurate method. Notation for Butcher tableau is as in [1]_. Parameters ---------- fun : callable Right-hand side of the system. t : float Current time. y : ndarray, shape (n,) Current state. f : ndarray, shape (n,) Current value of the derivative, i.e., ``fun(x, y)``. h : float Step to use. A : ndarray, shape (n_stages, n_stages) Coefficients for combining previous RK stages to compute the next stage. For explicit methods the coefficients at and above the main diagonal are zeros. B : ndarray, shape (n_stages,) Coefficients for combining RK stages for computing the final prediction. C : ndarray, shape (n_stages,) Coefficients for incrementing time for consecutive RK stages. The value for the first stage is always zero. K : ndarray, shape (n_stages + 1, n) Storage array for putting RK stages here. Stages are stored in rows. The last row is a linear combination of the previous rows with coefficients Returns ------- y_new : ndarray, shape (n,) Solution at t + h computed with a higher accuracy. f_new : ndarray, shape (n,) Derivative ``fun(t + h, y_new)``. References ---------- .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential Equations I: Nonstiff Problems", Sec. II.4. rrNstart) enumeratezipnpdotT)funtyfhABCKsacdyy_newf_news U/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/scipy/integrate/_ivp/rk.pyrk_stepr(s^ AaDs1QRR5!ABB%00:::&& 6Aq VAbqbEGQrrU # #a 's1q1u9a"f%%! BF1SbS68Q''' 'E CAu  E AbE %<ceZdZUdZeZejed<eZ ejed<eZ ejed<eZ ejed<eZ ejed<eZ eed<eZeed<eZeed <ejd d d d ffd ZdZdZdZdZxZS) RungeKuttaz,Base class for explicit Runge-Kutta methods.rrrEPordererror_estimator_ordern_stagesMbP?ư>FNc t| t|||||dd|_t ||_t |||j\|_|_ | |j |j |_ | Ft|j |j |j |||j |j|j|j|j |_nt%| |||_t'j|jdz|jf|j j|_d|jdzz |_d|_dS)NT)support_complexrdtyper)r super__init__y_oldrmax_steprnrtolatolrrrrr directionr/h_absr remptyr0r6r error_exponent h_previous selfrt0y0t_boundr:r<r= vectorized first_step extraneous __class__s r'r8zRungeKutta.__init__Us+  ### b"gz)-  / / / )(33 +D$?? 49$&$&))  ,$&$&'8TVT^*DItyBBDJJ-ZWEEDJ4=1,df5TV\JJJ D$>$BCr)cFtj|j|j|zSN)rrrr,)rDr rs r'_estimate_errorzRungeKutta._estimate_errorisvac46""Q&&r)cNt||||z SrM)r rN)rDr rscales r'_estimate_error_normzRungeKutta._estimate_error_normls%D((A..6777r)c @|j}|j}|j}|j}|j}dt jt j||jt j z|z z}|j |kr|}n|j |kr|}n|j }d}d} |sg||kr d|j fS||jz} || z} |j| |j z zdkr|j } | |z } t j| }t|j|||j| |j|j|j|j \} } |t jt j|t j| |zz}||j| |}|dkrM|dkrt,}n%t/t,t0||jzz}| rt/d|}||z}d}n*|t5t6t0||jzzz}d} |g| |_||_| |_| |_||_ | |_dS)NrFrrT)TN)rrr:r<r=rabs nextafterr>infr?TOO_SMALL_STEPrGr(rrrrrr maximumrQ MAX_FACTORminSAFETYrAmax MIN_FACTORrBr9)rDrrr:r<r=min_stepr? step_accepted step_rejectedrt_newr%r&rP error_normfactors r' _step_implzRungeKutta._step_implos> F F=yyr|At~/FGG!KLLL : EE Z( " "EEJE  " %xd111&AEE~!56::  AF1IIE"48Q461df#'64646;;LE52:bfQii??$FFE22461eDDJA~~??'FF !'*8K*K!KMMF!, F^^F $ Z#jD4G&GGIII $ E " %H  zr)c|jj|j}t |j|j|j|SrM)r rrr- RkDenseOutputt_oldrr9)rDQs r'_dense_output_implzRungeKutta._dense_output_impls3 FHLL TZQ???r))__name__ __module__ __qualname____doc__NotImplementedrrndarray__annotations__rrr,r-r.intr/r0rUr8rNrQrcrh __classcell__rKs@r'r+r+Js&66"Arz""""Arz""""Arz""""Arz""""Arz"""E3!/3///"Hc"""68f% ('''888AAAF@@@@@@@r)r+ceZdZdZdZdZdZejgdZ ejgdgdgdgZ ejgdZ ejgd Z ejgd gd gd gd gZ dS)RK23aExplicit Runge-Kutta method of order 3(2). This uses the Bogacki-Shampine pair of formulas [1]_. The error is controlled assuming accuracy of the second-order method, but steps are taken using the third-order accurate formula (local extrapolation is done). A cubic Hermite polynomial is used for the dense output. 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`. vectorized : bool, optional Whether `fun` may be called in a vectorized fashion. False (default) is recommended for this solver. 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 methods 'Radau' and 'BDF', but will result in slower execution for this solver. 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 evaluations of the system's right-hand side. njev : int Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian. nlu : int Number of LU decompositions. Is always 0 for this solver. References ---------- .. [1] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas", Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989. )r??)rrr)rwrr)rrxr)gqq?gUUUUUU?gqq?)grqDZ?gUUUUUUgqqg?)rgUUUUUUgrq?)rrgUUUUUU)rgUUUUUU?gqq)rrrNrirjrkrlr.r/r0rarrayrrrr,r-r)r'rtrts[[x EHA     A !!A)))**A$$$,, ..**  AAAr)rtc eZdZdZdZdZdZejgdZ ejgdgdgdgd gd gd gZ ejgd Z ejgd Z ejgdgdgdgdgdgdgdgZ dS)RK45aExplicit Runge-Kutta method of order 5(4). This uses the Dormand-Prince pair of formulas [1]_. The error is controlled assuming accuracy of the fourth-order method accuracy, but steps are taken using the fifth-order accurate formula (local extrapolation is done). A quartic interpolation polynomial is used for the dense output [2]_. Can be applied in the complex domain. Parameters ---------- fun : callable Right-hand side of the system. The calling signature is ``fun(t, y)``. Here ``t`` is a scalar, and there are two options for the ndarray ``y``: It can either have shape (n,); then ``fun`` must return array_like with shape (n,). Alternatively it can have shape (n, k); then ``fun`` must return an array_like with shape (n, k), i.e., each column corresponds to a single column in ``y``. The choice between the two options is determined by `vectorized` argument (see below). 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`. vectorized : bool, optional Whether `fun` is implemented in a vectorized fashion. Default is False. 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 evaluations of the system's right-hand side. njev : int Number of evaluations of the Jacobian. Is always 0 for this solver as it does not use the Jacobian. nlu : int Number of LU decompositions. Is always 0 for this solver. References ---------- .. [1] J. R. Dormand, P. J. Prince, "A family of embedded Runge-Kutta formulae", Journal of Computational and Applied Mathematics, Vol. 6, No. 1, pp. 19-26, 1980. .. [2] L. W. Shampine, "Some Practical Runge-Kutta Formulas", Mathematics of Computation,, Vol. 46, No. 173, pp. 135-150, 1986. )rr g333333?g?gqq?r)rrrrr)r rrrr)g333333?g?rrr)gII?g gqq @rr)gq@g 1'gR<6R#@gE3ҿr)g+@g>%gr!@gE]t?g/pѿ)gUUUUUU?rgVI?gUUUUU?gϡԿg1 0 ?)g2TrgĿ UZkq?ggX ?g{tg?)rg# !gJ<@gFC)rrrr)rgF@gFj'NgDg@)rgdD gaP#$@g 2)rg `_. Nrr1r2Fc tj||||||||| f i| tjtj|jf|jj|_ |j d|j dz|_ dS)Nr5r) r7r8rr@r N_STAGES_EXTENDEDr;rr6 K_extendedr0r rCs r'r8zDOP853.__init__s b"gxt#Z ? ?3= ? ? ?($7$I$(F$,376<AAA!3$-!"3!34r)ctj|j|j}tj|j|j}tjtj|dtj|z}tj|}|dk}tj||||z ||<||z|zS)Ng?r)rrrE5E3hypotrS ones_like)rDr rerr5err3denomcorrection_factormasks r'rNzDOP853._estimate_errorsvac47##vac47##sRVD\\'9::L..qy"$&d"4"4uT{"B$4x+++r)ctj|j|j|z }tj|j|j|z }tj|dz}tj|dz}|dkr|dkrdS|d|zz}tj||ztj|t|zz S)Nrvrgg{Gz?) rrrrrlinalgr rSsqrtlen) rDr rrPrr err5_norm_2 err3_norm_2rs r'rQzDOP853._estimate_error_normsvac47##e+vac47##e+innT**A- innT**A- !   q 0 03d[00vayy;&U1C)D)DDDr)c|j}|j}tt|j|j|jdzD]a\}\}}tj|d|j |d||z}| |j ||zz|j |z||<btj tj|jf|j j}|d}|j|j z } | |d<||z| z |d<d| z||j|zzz |d<|tj|j|z|dd<t+|j |j|j |S)Nrrr5rrvru)rrBrrA_EXTRAC_EXTRAr0rrrrrfr9r@r INTERPOLATOR_POWERr;r6rrDDop853DenseOutputr) rDr rr!r"r#r$Ff_olddelta_ys r'rhzDOP853._dense_output_implsZ O O"3t|T\#B#B)-):<<< A AIAv1"1"2A2''!+B88DJQ. R@@AaDD H)**A 1 AADJNA.//A 11%%%A FRVDFA&& & 6Q;; AAAtG$ $AA OAr)rirjrkr8rrqrrs@r'rere(sGr)rec$eZdZfdZdZxZS)rc|t||||z |_||_||_dSrM)r7r8rrr9)rDrfrr9rrKs r'r8zDop853DenseOutput.__init__Bs: """U r)c||jz |jz }|jdkrtj|j}nM|dddf}tjt|t|jf|jj}tt|j D]!\}}||z }|dzdkr||z}|d|z z}"||jz }|j S)Nrr5rvr) rfrrr zeros_liker9zerosrr6rreversedrr)rDrrrirs r'rzDop853DenseOutput._call_implHs ^tv % 6Q;; dj))AA!!!T' A#a&&#dj//2$*:JKKKAhtv..//  DAq FA1uzzQQU  TZs r)rrrs@r'rrAsG r)r)numpyrbaserrcommonrrrr r r r rZr\rXr(r+rtr}rrerr{r)r'rs((((((((AAAAAAAAAAAAAAAA!!!!!!    999xj@j@j@j@j@j@j@j@Zkkkkk:kkk\oIoIoIoIoI:oIoIoIdNDNDNDNDNDZNDNDNDbK2 r)