B 0dFS@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?g? c Cs||d<xntt|dd|ddddD]H\} \} } t|d| j| d| |} ||| ||| || <q,W||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. rrN)start) enumeratezipnpdotT)funtyfhABCKsacdyy_newf_newr#I/var/www/html/venv/lib/python3.7/site-packages/scipy/integrate/_ivp/rk.pyrk_steps/0"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_stagesgMbP?gư>FNc st| tj|||||ddd|_t||_t|||j\|_|_ | |j |j |_ | dkrt|j |j |j |j |j|j|j|j |_nt| |||_tj|jd|jf|j jd|_d|jd|_d|_dS)NT)Zsupport_complexr)dtyper)r super__init__y_oldrmax_steprnrtolatolrrrrr directionr*h_absr remptyr+r,rerror_exponent h_previous) selfrt0y0t_boundr0r2r3 vectorized first_step extraneous) __class__r#r$r.Us  zRungeKutta.__init__cCst|j|j|S)N)rrrr')r9rrr#r#r$_estimate_erroriszRungeKutta._estimate_errorcCst||||S)N)rrA)r9rrscaler#r#r$_estimate_error_normlszRungeKutta._estimate_error_normc Cs|j}|j}|j}|j}|j}dtt||jtj |}|j |krP|}n|j |kr`|}n|j }d}d} x|s||krd|j fS||j} || } |j| |j dkr|j } | |} t| }t |j|||j| |j|j|j|j \} } |tt|t| |}||j| |}|dkrl|dkr:t}nttt||j}| r^td|}||9}d}qr|ttt||j9}d} qrW| |_||_| |_| |_||_ | |_dS)Nr FrrT)TN)rrr0r2r3rabsZ nextafterr4infr5ZTOO_SMALL_STEPr<r%rrrrrrmaximumrC MAX_FACTORminSAFETYr7max MIN_FACTORr8r/)r9rrr0r2r3Zmin_stepr5Z step_acceptedZ step_rejectedrZt_newr!r"rBZ error_normfactorr#r#r$ _step_implosZ"          zRungeKutta._step_implcCs$|jj|j}t|j|j|j|S)N)rrrr( RkDenseOutputt_oldrr/)r9Qr#r#r$_dense_output_implszRungeKutta._dense_output_impl)__name__ __module__ __qualname____doc__NotImplementedrrZndarray__annotations__rrr'r(r)intr*r+rEr.rArCrMrQ __classcell__r#r#)r@r$r&Js    Cr&c@seZdZdZdZdZdZedddgZ edddgdddgdddggZ eddd gZ ed d d d gZ edddgdddgdddgdddggZ 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). But if a component of `y` is approximately below `atol`, the error only needs to fall within the same `atol` threshold, and the number of correct digits is not guaranteed. 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. rg?g?gqq?gUUUUUU?gqq?grqDZ?gUUUUUUgqqg?rgUUUUUUgrq?gUUUUUUgUUUUUU?gqqrN)rRrSrTrUr)r*r+rarrayrrrr'r(r#r#r#r$rZsK rZc @seZdZdZdZdZdZeddddd d gZ edddddgdddddgd d dddgd ddddgdddddgdddddggZ eddddddgZ edddd d!d"d#gZ ed d$d%d&gddddgdd'd(d)gdd*d+d,gdd-d.d/gdd0d1d2gdd3d4d5ggZ d6S)7RK45a Explicit 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). But if a component of `y` is approximately below `atol`, the error only needs to fall within the same `atol` threshold, and the number of correct digits is not guaranteed. 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. rg?g333333?g?gqq?rg333333?g?gII?g gqq @gq@g 1'gR<6R#@gE3ҿg+@g>%gr!@gE]t?g/pѿgUUUUUU?gVI?gUUUUU?gϡԿg1 0 ?g2TgĿ UZkq?ggX ?g{tg?g# !gJ<@gFCgF@gFj'NgDg@gdD gaP#$@g 2g `_. 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