o EbV@sddlZddlmZmZddlmZmZmZm Z m Z m Z ddl m Z dZdZdZd d ZGd d d eZGd ddeZGdddeZGdddeZGdddeZGdddeZdS)N) OdeSolver DenseOutput)validate_max_step validate_tolselect_initial_stepnormwarn_extraneousvalidate_first_step)dop853_coefficientsg?皙? c Cs||d<tt|dd|ddddD]$\} \} } t|d| j| d| |} ||| ||| || <q||t|ddj|} |||| }||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_newr%9/usr/lib/python3/dist-packages/scipy/integrate/_ivp/rk.pyrk_steps/."r'cseZdZUdZeZejed<eZ ejed<eZ ejed<eZ ejed<eZ ejed<eZ eed<eZeed<eZeed <ejd d d d ffdd ZddZddZddZddZZS) RungeKuttaz,Base class for explicit Runge-Kutta methods.rrrEPordererror_estimator_ordern_stagesMbP?ư>FNc st| tj|||||ddd|_t||_t|||j\|_|_ | |j |j |_ | durEt|j |j |j |j |j|j|j|j |_nt| |||_tj|jd|jf|j jd|_d|jd|_d|_dS)NT)Zsupport_complexrdtyper)r super__init__y_oldrmax_steprnrtolatolrrrrr directionr,h_absr remptyr-r1rerror_exponent h_previous selfrZt0Zy0t_boundr5r7r8Z vectorizedZ first_stepZ extraneous __class__r%r&r3Us"    zRungeKutta.__init__cCst|j|j|SN)rrrr))r?rrr%r%r&_estimate_errorizRungeKutta._estimate_errorcCst||||SrC)rrD)r?rrscaler%r%r&_estimate_error_normlrEzRungeKutta._estimate_error_normc Cs|j}|j}|j}|j}|j}dtt||jtj |}|j |kr(|}n |j |kr0|}n|j }d}d} |s||krBd|j fS||j} || } |j| |j dkrX|j } | |} t| }t |j|||j| |j|j|j|j \} } |tt|t| |}||j| |}|dkr|dkrt}n ttt||j}| rtd|}||9}d}n|ttt||j9}d} |r9| |_||_| |_| |_||_ | |_dS)Nr FrrT)TN)rrr5r7r8rabsZ nextafterr9infr:ZTOO_SMALL_STEPr@r'rrrrrrZmaximumrG MAX_FACTORminSAFETYr<max MIN_FACTORr=r4)r?rrr5r7r8Zmin_stepr:Z step_acceptedZ step_rejectedrZt_newr#r$rFZ error_normZfactorr%r%r& _step_implosb"          $zRungeKutta._step_implcCs$|jj|j}t|j|j|j|SrC)rrrr* RkDenseOutputt_oldrr4)r?Qr%r%r&_dense_output_implszRungeKutta._dense_output_impl)__name__ __module__ __qualname____doc__NotImplementedrrZndarray__annotations__rrr)r*r+intr,r-rIr3rDrGrOrS __classcell__r%r%rAr&r(Js$    Cr(c@seZdZdZdZdZdZegdZ egdgdgdgZ egdZ egd Z egd gd gd gd gZ dS)RK23a Explicit 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 calling signature is ``fun(t, y)``. Here ``t`` is a scalar and there are two options for ndarray ``y``. It can either have shape (n,), then ``fun`` must return array_like with shape (n,). Or alternatively it can have shape (n, k), then ``fun`` must return 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 lower than the lowest 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 lower 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] 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)r_rr)rr`r)gqq?gUUUUUU?gqq?)grqDZ?gUUUUUUgqqg?)rgUUUUUUgrq?)rrgUUUUUU)rgUUUUUU?gqq)rrrNrTrUrVrWr+r,r-rZarrayrrrr)r*r%r%r%r&r\s$O  r\c @seZdZdZdZdZdZegdZ egdgdgdgd gd gd gZ egd Z egd Z egdgdgdgdgdgdgdgZ 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 lower than the lowest 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 lower 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 `_. 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