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norm, warn_extraneous)
# Multiply steps computed from asymptotic behaviour of errors by this.
"""Perform 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 : list of ndarray, length n_stages - 1 Coefficients for combining previous RK stages to compute the next stage. For explicit methods the coefficients above the main diagonal are zeros, so `A` is stored as a list of arrays of increasing lengths. The first stage is always just `f`, thus no coefficients for it are required. B : ndarray, shape (n_stages,) Coefficients for combining RK stages for computing the final prediction. C : ndarray, shape (n_stages - 1,) Coefficients for incrementing time for consecutive RK stages. The value for the first stage is always zero, thus it is not stored. E : ndarray, shape (n_stages + 1,) Coefficients for estimating the error of a less accurate method. They are computed as the difference between b's in an extended tableau. K : ndarray, shape (n_stages + 1, n) Storage array for putting RK stages here. Stages are stored in rows.
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)``. error : ndarray, shape (n,) Error estimate of a less accurate method.
References ---------- .. [1] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential Equations I: Nonstiff Problems", Sec. II.4. """ K[0] = f for s, (a, c) in enumerate(zip(A, C)): dy = np.dot(K[:s + 1].T, a) * h K[s + 1] = fun(t + c * h, y + dy)
y_new = y + h * np.dot(K[:-1].T, B) f_new = fun(t + h, y_new)
K[-1] = f_new error = np.dot(K.T, E) * h
return y_new, f_new, error
"""Base class for explicit Runge-Kutta methods."""
rtol=1e-3, atol=1e-6, vectorized=False, **extraneous): warn_extraneous(extraneous) super(RungeKutta, self).__init__(fun, t0, y0, t_bound, vectorized, support_complex=True) self.y_old = None self.max_step = validate_max_step(max_step) self.rtol, self.atol = validate_tol(rtol, atol, self.n) self.f = self.fun(self.t, self.y) self.h_abs = select_initial_step( self.fun, self.t, self.y, self.f, self.direction, self.order, self.rtol, self.atol) self.K = np.empty((self.n_stages + 1, self.n), dtype=self.y.dtype)
t = self.t y = self.y
max_step = self.max_step rtol = self.rtol atol = self.atol
min_step = 10 * np.abs(np.nextafter(t, self.direction * np.inf) - t)
if self.h_abs > max_step: h_abs = max_step elif self.h_abs < min_step: h_abs = min_step else: h_abs = self.h_abs
order = self.order step_accepted = False
while not step_accepted: if h_abs < min_step: return False, self.TOO_SMALL_STEP
h = h_abs * self.direction t_new = t + h
if self.direction * (t_new - self.t_bound) > 0: t_new = self.t_bound
h = t_new - t h_abs = np.abs(h)
y_new, f_new, error = rk_step(self.fun, t, y, self.f, h, self.A, self.B, self.C, self.E, self.K) scale = atol + np.maximum(np.abs(y), np.abs(y_new)) * rtol error_norm = norm(error / scale)
if error_norm == 0.0: h_abs *= MAX_FACTOR step_accepted = True elif error_norm < 1: h_abs *= min(MAX_FACTOR, max(1, SAFETY * error_norm ** (-1 / (order + 1)))) step_accepted = True else: h_abs *= max(MIN_FACTOR, SAFETY * error_norm ** (-1 / (order + 1)))
self.y_old = y
self.t = t_new self.y = y_new
self.h_abs = h_abs self.f = f_new
return True, None
Q = self.K.T.dot(self.P) return RkDenseOutput(self.t_old, self.t, self.y_old, Q)
"""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. 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. """ np.array([0, 3/4])] [0, 1, -2/3], [0, 4/3, -8/9], [0, -1, 1]])
"""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. 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. """ np.array([3/40, 9/40]), np.array([44/45, -56/15, 32/9]), np.array([19372/6561, -25360/2187, 64448/6561, -212/729]), np.array([9017/3168, -355/33, 46732/5247, 49/176, -5103/18656])] 1/40]) # Corresponds to the optimum value of c_6 from [2]_. [1, -8048581381/2820520608, 8663915743/2820520608, -12715105075/11282082432], [0, 0, 0, 0], [0, 131558114200/32700410799, -68118460800/10900136933, 87487479700/32700410799], [0, -1754552775/470086768, 14199869525/1410260304, -10690763975/1880347072], [0, 127303824393/49829197408, -318862633887/49829197408, 701980252875 / 199316789632], [0, -282668133/205662961, 2019193451/616988883, -1453857185/822651844], [0, 40617522/29380423, -110615467/29380423, 69997945/29380423]])
super(RkDenseOutput, self).__init__(t_old, t) self.h = t - t_old self.Q = Q self.order = Q.shape[1] - 1 self.y_old = y_old
x = (t - self.t_old) / self.h if t.ndim == 0: p = np.tile(x, self.order + 1) p = np.cumprod(p) else: p = np.tile(x, (self.order + 1, 1)) p = np.cumprod(p, axis=0) y = self.h * np.dot(self.Q, p) if y.ndim == 2: y += self.y_old[:, None] else: y += self.y_old
return y |