U Gv fK@sddlZddlmZmZddlmZmZmZddl m Z ddl m Z ddl mZmZmZmZmZmZmZmZddlmZmZd Zed ed d ed dgZed d ed d edgdZdZdZedddgdddgdddggZedddgdddgddd ggZ e dZ!e dd!e d"Z"ed#d edd$d%edd&d'egd#d edd$d%edd&d'egd(d)d&ggZ#d*Z$d+Z%d Z&d,d-Z'd.d/Z(Gd0d1d1eZ)Gd2d3d3eZ*dS)4N) lu_factorlu_solve) csc_matrixissparseeye)splu) group_columns)validate_max_step validate_tolselect_initial_stepnormnum_jacEPSwarn_extraneousvalidate_first_step) OdeSolver DenseOutputg.! @ igs>H @yrr@Ggg{g]#-?g;@L¿ghm?g }?gQ  ?gmؿgF@gN]?gV?gFgN]Կg!R?g$Z?goNg{??gUUUUUU@gg @gUUUUUU?gUUUUUUg?c Cs|jd} t|} t|} t|} |}td| f}|t}d}t| }d}d}t t D]R}t dD]"}|||||||||<qjt t |sq|j t| | d}|j t| | dd| d}| ||}| ||}||d<|j|d<|j|d<t||}|dk r.||}|dk rf|dks`|t |d|||krfq| |7} t | }|dks|dk r|d|||krd}q|}q\||d||fS) a^Solve the collocation system. Parameters ---------- fun : callable Right-hand side of the system. t : float Current time. y : ndarray, shape (n,) Current state. h : float Step to try. Z0 : ndarray, shape (3, n) Initial guess for the solution. It determines new values of `y` at ``t + h * C`` as ``y + Z0``, where ``C`` is the Radau method constants. scale : ndarray, shape (n) Problem tolerance scale, i.e. ``rtol * abs(y) + atol``. tol : float Tolerance to which solve the system. This value is compared with the normalized by `scale` error. LU_real, LU_complex LU decompositions of the system Jacobians. solve_lu : callable Callable which solves a linear system given a LU decomposition. The signature is ``solve_lu(LU, b)``. Returns ------- converged : bool Whether iterations converged. n_iter : int Number of completed iterations. Z : ndarray, shape (3, n) Found solution. rate : float The rate of convergence. rrNFr rrT)shapeMU_REAL MU_COMPLEXTIdotnpemptyCZ empty_likerangeNEWTON_MAXITERallisfiniteTTI_REAL TI_COMPLEXrealimagr )funtyhZ0scaleZtolLU_real LU_complexsolve_lunZM_realZ M_complexWZFchZ dW_norm_oldZdW convergedratekiZf_realZ f_complexZdW_realZ dW_complexZdW_normrA>/tmp/pip-unpacked-wheel-96ln3f52/scipy/integrate/_ivp/radau.pysolve_collocation_system0sR'     $        rCc Cs`|dks|dks|dkrd}n||||d}tjddtd||d}W5QRX|S)a9Predict by which factor to increase/decrease the step size. The algorithm is described in [1]_. Parameters ---------- h_abs, h_abs_old : float Current and previous values of the step size, `h_abs_old` can be None (see Notes). error_norm, error_norm_old : float Current and previous values of the error norm, `error_norm_old` can be None (see Notes). Returns ------- factor : float Predicted factor. Notes ----- If `h_abs_old` and `error_norm_old` are both not None then a two-step algorithm is used, otherwise a one-step algorithm is used. References ---------- .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems", Sec. IV.8. Nrr g?ignore)dividegп)r#Zerrstatemin)h_abs h_abs_old error_normerror_norm_oldZ multiplierfactorrArArBpredict_factors rLcsReZdZdZejddddddffdd Zdd Zd d Zd d Z ddZ Z S)RadauaZImplicit Runge-Kutta method of Radau IIA family of order 5. The implementation follows [1]_. The error is controlled with a third-order accurate embedded formula. A cubic polynomial which satisfies the collocation conditions is used for the dense output. 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). The vectorized implementation allows a faster approximation of the Jacobian by finite differences (required for this solver). 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)``. HHere `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 [2]_. 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` 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 of evaluations of the right-hand side. njev : int Number of evaluations of the Jacobian. nlu : int Number of LU decompositions. References ---------- .. [1] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems", Sec. IV.8. .. [2] 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. 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