225 lines
9.7 KiB
Python
225 lines
9.7 KiB
Python
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import numpy as np
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from scipy.integrate import ode
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from .common import validate_tol, validate_first_step, warn_extraneous
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from .base import OdeSolver, DenseOutput
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class LSODA(OdeSolver):
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"""Adams/BDF method with automatic stiffness detection and switching.
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This is a wrapper to the Fortran solver from ODEPACK [1]_. It switches
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automatically between the nonstiff Adams method and the stiff BDF method.
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The method was originally detailed in [2]_.
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Parameters
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----------
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fun : callable
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Right-hand side of the system: the time derivative of the state ``y``
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at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a
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scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. ``fun`` must
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return an array of the same shape as ``y``. See `vectorized` for more
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information.
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t0 : float
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Initial time.
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y0 : array_like, shape (n,)
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Initial state.
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t_bound : float
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Boundary time - the integration won't continue beyond it. It also
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determines the direction of the integration.
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first_step : float or None, optional
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Initial step size. Default is ``None`` which means that the algorithm
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should choose.
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min_step : float, optional
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Minimum allowed step size. Default is 0.0, i.e., the step size is not
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bounded and determined solely by the solver.
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max_step : float, optional
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Maximum allowed step size. Default is np.inf, i.e., the step size is not
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bounded and determined solely by the solver.
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rtol, atol : float and array_like, optional
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Relative and absolute tolerances. The solver keeps the local error
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estimates less than ``atol + rtol * abs(y)``. Here `rtol` controls a
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relative accuracy (number of correct digits), while `atol` controls
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absolute accuracy (number of correct decimal places). To achieve the
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desired `rtol`, set `atol` to be smaller than the smallest value that
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can be expected from ``rtol * abs(y)`` so that `rtol` dominates the
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allowable error. If `atol` is larger than ``rtol * abs(y)`` the
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number of correct digits is not guaranteed. Conversely, to achieve the
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desired `atol` set `rtol` such that ``rtol * abs(y)`` is always smaller
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than `atol`. If components of y have different scales, it might be
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beneficial to set different `atol` values for different components by
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passing array_like with shape (n,) for `atol`. Default values are
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1e-3 for `rtol` and 1e-6 for `atol`.
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jac : None or callable, optional
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Jacobian matrix of the right-hand side of the system with respect to
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``y``. The Jacobian matrix has shape (n, n) and its element (i, j) is
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equal to ``d f_i / d y_j``. The function will be called as
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``jac(t, y)``. If None (default), the Jacobian will be
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approximated by finite differences. It is generally recommended to
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provide the Jacobian rather than relying on a finite-difference
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approximation.
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lband, uband : int or None
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Parameters defining the bandwidth of the Jacobian,
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i.e., ``jac[i, j] != 0 only for i - lband <= j <= i + uband``. Setting
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these requires your jac routine to return the Jacobian in the packed format:
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the returned array must have ``n`` columns and ``uband + lband + 1``
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rows in which Jacobian diagonals are written. Specifically
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``jac_packed[uband + i - j , j] = jac[i, j]``. The same format is used
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in `scipy.linalg.solve_banded` (check for an illustration).
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These parameters can be also used with ``jac=None`` to reduce the
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number of Jacobian elements estimated by finite differences.
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vectorized : bool, optional
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Whether `fun` may be called in a vectorized fashion. False (default)
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is recommended for this solver.
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If ``vectorized`` is False, `fun` will always be called with ``y`` of
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shape ``(n,)``, where ``n = len(y0)``.
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If ``vectorized`` is True, `fun` may be called with ``y`` of shape
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``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
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such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
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the returned array is the time derivative of the state corresponding
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with a column of ``y``).
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Setting ``vectorized=True`` allows for faster finite difference
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approximation of the Jacobian by methods 'Radau' and 'BDF', but
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will result in slower execution for this solver.
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Attributes
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----------
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n : int
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Number of equations.
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status : string
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Current status of the solver: 'running', 'finished' or 'failed'.
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t_bound : float
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Boundary time.
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direction : float
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Integration direction: +1 or -1.
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t : float
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Current time.
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y : ndarray
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Current state.
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t_old : float
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Previous time. None if no steps were made yet.
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nfev : int
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Number of evaluations of the right-hand side.
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njev : int
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Number of evaluations of the Jacobian.
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References
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----------
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.. [1] A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE
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Solvers," IMACS Transactions on Scientific Computation, Vol 1.,
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pp. 55-64, 1983.
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.. [2] L. Petzold, "Automatic selection of methods for solving stiff and
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nonstiff systems of ordinary differential equations", SIAM Journal
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on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136-148,
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1983.
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"""
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def __init__(self, fun, t0, y0, t_bound, first_step=None, min_step=0.0,
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max_step=np.inf, rtol=1e-3, atol=1e-6, jac=None, lband=None,
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uband=None, vectorized=False, **extraneous):
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warn_extraneous(extraneous)
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super().__init__(fun, t0, y0, t_bound, vectorized)
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if first_step is None:
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first_step = 0 # LSODA value for automatic selection.
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else:
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first_step = validate_first_step(first_step, t0, t_bound)
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first_step *= self.direction
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if max_step == np.inf:
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max_step = 0 # LSODA value for infinity.
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elif max_step <= 0:
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raise ValueError("`max_step` must be positive.")
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if min_step < 0:
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raise ValueError("`min_step` must be nonnegative.")
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rtol, atol = validate_tol(rtol, atol, self.n)
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solver = ode(self.fun, jac)
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solver.set_integrator('lsoda', rtol=rtol, atol=atol, max_step=max_step,
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min_step=min_step, first_step=first_step,
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lband=lband, uband=uband)
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solver.set_initial_value(y0, t0)
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# Inject t_bound into rwork array as needed for itask=5.
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solver._integrator.rwork[0] = self.t_bound
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solver._integrator.call_args[4] = solver._integrator.rwork
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self._lsoda_solver = solver
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def _step_impl(self):
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solver = self._lsoda_solver
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integrator = solver._integrator
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# From lsoda.step and lsoda.integrate itask=5 means take a single
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# step and do not go past t_bound.
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itask = integrator.call_args[2]
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integrator.call_args[2] = 5
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solver._y, solver.t = integrator.run(
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solver.f, solver.jac or (lambda: None), solver._y, solver.t,
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self.t_bound, solver.f_params, solver.jac_params)
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integrator.call_args[2] = itask
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if solver.successful():
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self.t = solver.t
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self.y = solver._y
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# From LSODA Fortran source njev is equal to nlu.
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self.njev = integrator.iwork[12]
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self.nlu = integrator.iwork[12]
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return True, None
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else:
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return False, 'Unexpected istate in LSODA.'
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def _dense_output_impl(self):
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iwork = self._lsoda_solver._integrator.iwork
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rwork = self._lsoda_solver._integrator.rwork
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# We want to produce the Nordsieck history array, yh, up to the order
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# used in the last successful iteration. The step size is unimportant
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# because it will be scaled out in LsodaDenseOutput. Some additional
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# work may be required because ODEPACK's LSODA implementation produces
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# the Nordsieck history in the state needed for the next iteration.
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# iwork[13] contains order from last successful iteration, while
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# iwork[14] contains order to be attempted next.
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order = iwork[13]
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# rwork[11] contains the step size to be attempted next, while
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# rwork[10] contains step size from last successful iteration.
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h = rwork[11]
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# rwork[20:20 + (iwork[14] + 1) * self.n] contains entries of the
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# Nordsieck array in state needed for next iteration. We want
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# the entries up to order for the last successful step so use the
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# following.
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yh = np.reshape(rwork[20:20 + (order + 1) * self.n],
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(self.n, order + 1), order='F').copy()
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if iwork[14] < order:
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# If the order is set to decrease then the final column of yh
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# has not been updated within ODEPACK's LSODA
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# implementation because this column will not be used in the
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# next iteration. We must rescale this column to make the
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# associated step size consistent with the other columns.
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yh[:, -1] *= (h / rwork[10]) ** order
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return LsodaDenseOutput(self.t_old, self.t, h, order, yh)
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class LsodaDenseOutput(DenseOutput):
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def __init__(self, t_old, t, h, order, yh):
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super().__init__(t_old, t)
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self.h = h
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self.yh = yh
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self.p = np.arange(order + 1)
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def _call_impl(self, t):
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if t.ndim == 0:
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x = ((t - self.t) / self.h) ** self.p
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else:
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x = ((t - self.t) / self.h) ** self.p[:, None]
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return np.dot(self.yh, x)
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