You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
664 lines
27 KiB
Python
664 lines
27 KiB
Python
import inspect
|
|
import numpy as np
|
|
from .bdf import BDF
|
|
from .radau import Radau
|
|
from .rk import RK23, RK45, DOP853
|
|
from .lsoda import LSODA
|
|
from scipy.optimize import OptimizeResult
|
|
from .common import EPS, OdeSolution
|
|
from .base import OdeSolver
|
|
|
|
|
|
METHODS = {'RK23': RK23,
|
|
'RK45': RK45,
|
|
'DOP853': DOP853,
|
|
'Radau': Radau,
|
|
'BDF': BDF,
|
|
'LSODA': LSODA}
|
|
|
|
|
|
MESSAGES = {0: "The solver successfully reached the end of the integration interval.",
|
|
1: "A termination event occurred."}
|
|
|
|
|
|
class OdeResult(OptimizeResult):
|
|
pass
|
|
|
|
|
|
def prepare_events(events):
|
|
"""Standardize event functions and extract is_terminal and direction."""
|
|
if callable(events):
|
|
events = (events,)
|
|
|
|
if events is not None:
|
|
is_terminal = np.empty(len(events), dtype=bool)
|
|
direction = np.empty(len(events))
|
|
for i, event in enumerate(events):
|
|
try:
|
|
is_terminal[i] = event.terminal
|
|
except AttributeError:
|
|
is_terminal[i] = False
|
|
|
|
try:
|
|
direction[i] = event.direction
|
|
except AttributeError:
|
|
direction[i] = 0
|
|
else:
|
|
is_terminal = None
|
|
direction = None
|
|
|
|
return events, is_terminal, direction
|
|
|
|
|
|
def solve_event_equation(event, sol, t_old, t):
|
|
"""Solve an equation corresponding to an ODE event.
|
|
|
|
The equation is ``event(t, y(t)) = 0``, here ``y(t)`` is known from an
|
|
ODE solver using some sort of interpolation. It is solved by
|
|
`scipy.optimize.brentq` with xtol=atol=4*EPS.
|
|
|
|
Parameters
|
|
----------
|
|
event : callable
|
|
Function ``event(t, y)``.
|
|
sol : callable
|
|
Function ``sol(t)`` which evaluates an ODE solution between `t_old`
|
|
and `t`.
|
|
t_old, t : float
|
|
Previous and new values of time. They will be used as a bracketing
|
|
interval.
|
|
|
|
Returns
|
|
-------
|
|
root : float
|
|
Found solution.
|
|
"""
|
|
from scipy.optimize import brentq
|
|
return brentq(lambda t: event(t, sol(t)), t_old, t,
|
|
xtol=4 * EPS, rtol=4 * EPS)
|
|
|
|
|
|
def handle_events(sol, events, active_events, is_terminal, t_old, t):
|
|
"""Helper function to handle events.
|
|
|
|
Parameters
|
|
----------
|
|
sol : DenseOutput
|
|
Function ``sol(t)`` which evaluates an ODE solution between `t_old`
|
|
and `t`.
|
|
events : list of callables, length n_events
|
|
Event functions with signatures ``event(t, y)``.
|
|
active_events : ndarray
|
|
Indices of events which occurred.
|
|
is_terminal : ndarray, shape (n_events,)
|
|
Which events are terminal.
|
|
t_old, t : float
|
|
Previous and new values of time.
|
|
|
|
Returns
|
|
-------
|
|
root_indices : ndarray
|
|
Indices of events which take zero between `t_old` and `t` and before
|
|
a possible termination.
|
|
roots : ndarray
|
|
Values of t at which events occurred.
|
|
terminate : bool
|
|
Whether a terminal event occurred.
|
|
"""
|
|
roots = [solve_event_equation(events[event_index], sol, t_old, t)
|
|
for event_index in active_events]
|
|
|
|
roots = np.asarray(roots)
|
|
|
|
if np.any(is_terminal[active_events]):
|
|
if t > t_old:
|
|
order = np.argsort(roots)
|
|
else:
|
|
order = np.argsort(-roots)
|
|
active_events = active_events[order]
|
|
roots = roots[order]
|
|
t = np.nonzero(is_terminal[active_events])[0][0]
|
|
active_events = active_events[:t + 1]
|
|
roots = roots[:t + 1]
|
|
terminate = True
|
|
else:
|
|
terminate = False
|
|
|
|
return active_events, roots, terminate
|
|
|
|
|
|
def find_active_events(g, g_new, direction):
|
|
"""Find which event occurred during an integration step.
|
|
|
|
Parameters
|
|
----------
|
|
g, g_new : array_like, shape (n_events,)
|
|
Values of event functions at a current and next points.
|
|
direction : ndarray, shape (n_events,)
|
|
Event "direction" according to the definition in `solve_ivp`.
|
|
|
|
Returns
|
|
-------
|
|
active_events : ndarray
|
|
Indices of events which occurred during the step.
|
|
"""
|
|
g, g_new = np.asarray(g), np.asarray(g_new)
|
|
up = (g <= 0) & (g_new >= 0)
|
|
down = (g >= 0) & (g_new <= 0)
|
|
either = up | down
|
|
mask = (up & (direction > 0) |
|
|
down & (direction < 0) |
|
|
either & (direction == 0))
|
|
|
|
return np.nonzero(mask)[0]
|
|
|
|
|
|
def solve_ivp(fun, t_span, y0, method='RK45', t_eval=None, dense_output=False,
|
|
events=None, vectorized=False, args=None, **options):
|
|
"""Solve an initial value problem for a system of ODEs.
|
|
|
|
This function numerically integrates a system of ordinary differential
|
|
equations given an initial value::
|
|
|
|
dy / dt = f(t, y)
|
|
y(t0) = y0
|
|
|
|
Here t is a 1-D independent variable (time), y(t) is an
|
|
N-D vector-valued function (state), and an N-D
|
|
vector-valued function f(t, y) determines the differential equations.
|
|
The goal is to find y(t) approximately satisfying the differential
|
|
equations, given an initial value y(t0)=y0.
|
|
|
|
Some of the solvers support integration in the complex domain, but note
|
|
that for stiff ODE solvers, the right-hand side must be
|
|
complex-differentiable (satisfy Cauchy-Riemann equations [11]_).
|
|
To solve a problem in the complex domain, pass y0 with a complex data type.
|
|
Another option always available is to rewrite your problem for real and
|
|
imaginary parts separately.
|
|
|
|
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 stiff solvers).
|
|
t_span : 2-tuple of floats
|
|
Interval of integration (t0, tf). The solver starts with t=t0 and
|
|
integrates until it reaches t=tf.
|
|
y0 : array_like, shape (n,)
|
|
Initial state. For problems in the complex domain, pass `y0` with a
|
|
complex data type (even if the initial value is purely real).
|
|
method : string or `OdeSolver`, optional
|
|
Integration method to use:
|
|
|
|
* 'RK45' (default): Explicit Runge-Kutta method of order 5(4) [1]_.
|
|
The error is controlled assuming accuracy of the fourth-order
|
|
method, 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.
|
|
* 'RK23': Explicit Runge-Kutta method of order 3(2) [3]_. 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.
|
|
* 'DOP853': Explicit Runge-Kutta method of order 8 [13]_.
|
|
Python implementation of the "DOP853" algorithm originally
|
|
written in Fortran [14]_. A 7-th order interpolation polynomial
|
|
accurate to 7-th order is used for the dense output.
|
|
Can be applied in the complex domain.
|
|
* 'Radau': Implicit Runge-Kutta method of the Radau IIA family of
|
|
order 5 [4]_. 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.
|
|
* 'BDF': Implicit multi-step variable-order (1 to 5) method based
|
|
on a backward differentiation formula for the derivative
|
|
approximation [5]_. The implementation follows the one described
|
|
in [6]_. A quasi-constant step scheme is used and accuracy is
|
|
enhanced using the NDF modification. Can be applied in the
|
|
complex domain.
|
|
* 'LSODA': Adams/BDF method with automatic stiffness detection and
|
|
switching [7]_, [8]_. This is a wrapper of the Fortran solver
|
|
from ODEPACK.
|
|
|
|
Explicit Runge-Kutta methods ('RK23', 'RK45', 'DOP853') should be used
|
|
for non-stiff problems and implicit methods ('Radau', 'BDF') for
|
|
stiff problems [9]_. Among Runge-Kutta methods, 'DOP853' is recommended
|
|
for solving with high precision (low values of `rtol` and `atol`).
|
|
|
|
If not sure, first try to run 'RK45'. If it makes unusually many
|
|
iterations, diverges, or fails, your problem is likely to be stiff and
|
|
you should use 'Radau' or 'BDF'. 'LSODA' can also be a good universal
|
|
choice, but it might be somewhat less convenient to work with as it
|
|
wraps old Fortran code.
|
|
|
|
You can also pass an arbitrary class derived from `OdeSolver` which
|
|
implements the solver.
|
|
t_eval : array_like or None, optional
|
|
Times at which to store the computed solution, must be sorted and lie
|
|
within `t_span`. If None (default), use points selected by the solver.
|
|
dense_output : bool, optional
|
|
Whether to compute a continuous solution. Default is False.
|
|
events : callable, or list of callables, optional
|
|
Events to track. If None (default), no events will be tracked.
|
|
Each event occurs at the zeros of a continuous function of time and
|
|
state. Each function must have the signature ``event(t, y)`` and return
|
|
a float. The solver will find an accurate value of `t` at which
|
|
``event(t, y(t)) = 0`` using a root-finding algorithm. By default, all
|
|
zeros will be found. The solver looks for a sign change over each step,
|
|
so if multiple zero crossings occur within one step, events may be
|
|
missed. Additionally each `event` function might have the following
|
|
attributes:
|
|
|
|
terminal: bool, optional
|
|
Whether to terminate integration if this event occurs.
|
|
Implicitly False if not assigned.
|
|
direction: float, optional
|
|
Direction of a zero crossing. If `direction` is positive,
|
|
`event` will only trigger when going from negative to positive,
|
|
and vice versa if `direction` is negative. If 0, then either
|
|
direction will trigger event. Implicitly 0 if not assigned.
|
|
|
|
You can assign attributes like ``event.terminal = True`` to any
|
|
function in Python.
|
|
vectorized : bool, optional
|
|
Whether `fun` is implemented in a vectorized fashion. Default is False.
|
|
args : tuple, optional
|
|
Additional arguments to pass to the user-defined functions. If given,
|
|
the additional arguments are passed to all user-defined functions.
|
|
So if, for example, `fun` has the signature ``fun(t, y, a, b, c)``,
|
|
then `jac` (if given) and any event functions must have the same
|
|
signature, and `args` must be a tuple of length 3.
|
|
options
|
|
Options passed to a chosen solver. All options available for already
|
|
implemented solvers are listed below.
|
|
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 or 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`.
|
|
jac : array_like, sparse_matrix, callable or None, optional
|
|
Jacobian matrix of the right-hand side of the system with respect
|
|
to y, required by the 'Radau', 'BDF' and 'LSODA' 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. Not supported by 'LSODA'.
|
|
* If callable, the Jacobian is assumed to depend on both
|
|
t and y; it will be called as ``jac(t, y)``, as necessary.
|
|
For '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 : array_like, sparse matrix or None, 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 [10]_. A zero entry means that
|
|
a corresponding element in the Jacobian is always zero. If None
|
|
(default), the Jacobian is assumed to be dense.
|
|
Not supported by 'LSODA', see `lband` and `uband` instead.
|
|
lband, uband : int or None, optional
|
|
Parameters defining the bandwidth of the Jacobian for the 'LSODA'
|
|
method, i.e., ``jac[i, j] != 0 only for i - lband <= j <= i + uband``.
|
|
Default is None. Setting these requires your jac routine to return the
|
|
Jacobian in the packed format: the returned array must have ``n``
|
|
columns and ``uband + lband + 1`` rows in which Jacobian diagonals are
|
|
written. Specifically ``jac_packed[uband + i - j , j] = jac[i, j]``.
|
|
The same format is used in `scipy.linalg.solve_banded` (check for an
|
|
illustration). These parameters can be also used with ``jac=None`` to
|
|
reduce the number of Jacobian elements estimated by finite differences.
|
|
min_step : float, optional
|
|
The minimum allowed step size for 'LSODA' method.
|
|
By default `min_step` is zero.
|
|
|
|
Returns
|
|
-------
|
|
Bunch object with the following fields defined:
|
|
t : ndarray, shape (n_points,)
|
|
Time points.
|
|
y : ndarray, shape (n, n_points)
|
|
Values of the solution at `t`.
|
|
sol : `OdeSolution` or None
|
|
Found solution as `OdeSolution` instance; None if `dense_output` was
|
|
set to False.
|
|
t_events : list of ndarray or None
|
|
Contains for each event type a list of arrays at which an event of
|
|
that type event was detected. None if `events` was None.
|
|
y_events : list of ndarray or None
|
|
For each value of `t_events`, the corresponding value of the solution.
|
|
None if `events` was None.
|
|
nfev : int
|
|
Number of evaluations of the right-hand side.
|
|
njev : int
|
|
Number of evaluations of the Jacobian.
|
|
nlu : int
|
|
Number of LU decompositions.
|
|
status : int
|
|
Reason for algorithm termination:
|
|
|
|
* -1: Integration step failed.
|
|
* 0: The solver successfully reached the end of `tspan`.
|
|
* 1: A termination event occurred.
|
|
|
|
message : string
|
|
Human-readable description of the termination reason.
|
|
success : bool
|
|
True if the solver reached the interval end or a termination event
|
|
occurred (``status >= 0``).
|
|
|
|
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.
|
|
.. [3] P. Bogacki, L.F. Shampine, "A 3(2) Pair of Runge-Kutta Formulas",
|
|
Appl. Math. Lett. Vol. 2, No. 4. pp. 321-325, 1989.
|
|
.. [4] E. Hairer, G. Wanner, "Solving Ordinary Differential Equations II:
|
|
Stiff and Differential-Algebraic Problems", Sec. IV.8.
|
|
.. [5] `Backward Differentiation Formula
|
|
<https://en.wikipedia.org/wiki/Backward_differentiation_formula>`_
|
|
on Wikipedia.
|
|
.. [6] L. F. Shampine, M. W. Reichelt, "THE MATLAB ODE SUITE", SIAM J. SCI.
|
|
COMPUTE., Vol. 18, No. 1, pp. 1-22, January 1997.
|
|
.. [7] A. C. Hindmarsh, "ODEPACK, A Systematized Collection of ODE
|
|
Solvers," IMACS Transactions on Scientific Computation, Vol 1.,
|
|
pp. 55-64, 1983.
|
|
.. [8] L. Petzold, "Automatic selection of methods for solving stiff and
|
|
nonstiff systems of ordinary differential equations", SIAM Journal
|
|
on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136-148,
|
|
1983.
|
|
.. [9] `Stiff equation <https://en.wikipedia.org/wiki/Stiff_equation>`_ on
|
|
Wikipedia.
|
|
.. [10] 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.
|
|
.. [11] `Cauchy-Riemann equations
|
|
<https://en.wikipedia.org/wiki/Cauchy-Riemann_equations>`_ on
|
|
Wikipedia.
|
|
.. [12] `Lotka-Volterra equations
|
|
<https://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equations>`_
|
|
on Wikipedia.
|
|
.. [13] E. Hairer, S. P. Norsett G. Wanner, "Solving Ordinary Differential
|
|
Equations I: Nonstiff Problems", Sec. II.
|
|
.. [14] `Page with original Fortran code of DOP853
|
|
<http://www.unige.ch/~hairer/software.html>`_.
|
|
|
|
Examples
|
|
--------
|
|
Basic exponential decay showing automatically chosen time points.
|
|
|
|
>>> from scipy.integrate import solve_ivp
|
|
>>> def exponential_decay(t, y): return -0.5 * y
|
|
>>> sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8])
|
|
>>> print(sol.t)
|
|
[ 0. 0.11487653 1.26364188 3.06061781 4.81611105 6.57445806
|
|
8.33328988 10. ]
|
|
>>> print(sol.y)
|
|
[[2. 1.88836035 1.06327177 0.43319312 0.18017253 0.07483045
|
|
0.03107158 0.01350781]
|
|
[4. 3.7767207 2.12654355 0.86638624 0.36034507 0.14966091
|
|
0.06214316 0.02701561]
|
|
[8. 7.5534414 4.25308709 1.73277247 0.72069014 0.29932181
|
|
0.12428631 0.05403123]]
|
|
|
|
Specifying points where the solution is desired.
|
|
|
|
>>> sol = solve_ivp(exponential_decay, [0, 10], [2, 4, 8],
|
|
... t_eval=[0, 1, 2, 4, 10])
|
|
>>> print(sol.t)
|
|
[ 0 1 2 4 10]
|
|
>>> print(sol.y)
|
|
[[2. 1.21305369 0.73534021 0.27066736 0.01350938]
|
|
[4. 2.42610739 1.47068043 0.54133472 0.02701876]
|
|
[8. 4.85221478 2.94136085 1.08266944 0.05403753]]
|
|
|
|
Cannon fired upward with terminal event upon impact. The ``terminal`` and
|
|
``direction`` fields of an event are applied by monkey patching a function.
|
|
Here ``y[0]`` is position and ``y[1]`` is velocity. The projectile starts
|
|
at position 0 with velocity +10. Note that the integration never reaches
|
|
t=100 because the event is terminal.
|
|
|
|
>>> def upward_cannon(t, y): return [y[1], -0.5]
|
|
>>> def hit_ground(t, y): return y[0]
|
|
>>> hit_ground.terminal = True
|
|
>>> hit_ground.direction = -1
|
|
>>> sol = solve_ivp(upward_cannon, [0, 100], [0, 10], events=hit_ground)
|
|
>>> print(sol.t_events)
|
|
[array([40.])]
|
|
>>> print(sol.t)
|
|
[0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02
|
|
1.11088891e-01 1.11098890e+00 1.11099890e+01 4.00000000e+01]
|
|
|
|
Use `dense_output` and `events` to find position, which is 100, at the apex
|
|
of the cannonball's trajectory. Apex is not defined as terminal, so both
|
|
apex and hit_ground are found. There is no information at t=20, so the sol
|
|
attribute is used to evaluate the solution. The sol attribute is returned
|
|
by setting ``dense_output=True``. Alternatively, the `y_events` attribute
|
|
can be used to access the solution at the time of the event.
|
|
|
|
>>> def apex(t, y): return y[1]
|
|
>>> sol = solve_ivp(upward_cannon, [0, 100], [0, 10],
|
|
... events=(hit_ground, apex), dense_output=True)
|
|
>>> print(sol.t_events)
|
|
[array([40.]), array([20.])]
|
|
>>> print(sol.t)
|
|
[0.00000000e+00 9.99900010e-05 1.09989001e-03 1.10988901e-02
|
|
1.11088891e-01 1.11098890e+00 1.11099890e+01 4.00000000e+01]
|
|
>>> print(sol.sol(sol.t_events[1][0]))
|
|
[100. 0.]
|
|
>>> print(sol.y_events)
|
|
[array([[-5.68434189e-14, -1.00000000e+01]]), array([[1.00000000e+02, 1.77635684e-15]])]
|
|
|
|
As an example of a system with additional parameters, we'll implement
|
|
the Lotka-Volterra equations [12]_.
|
|
|
|
>>> def lotkavolterra(t, z, a, b, c, d):
|
|
... x, y = z
|
|
... return [a*x - b*x*y, -c*y + d*x*y]
|
|
...
|
|
|
|
We pass in the parameter values a=1.5, b=1, c=3 and d=1 with the `args`
|
|
argument.
|
|
|
|
>>> sol = solve_ivp(lotkavolterra, [0, 15], [10, 5], args=(1.5, 1, 3, 1),
|
|
... dense_output=True)
|
|
|
|
Compute a dense solution and plot it.
|
|
|
|
>>> t = np.linspace(0, 15, 300)
|
|
>>> z = sol.sol(t)
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> plt.plot(t, z.T)
|
|
>>> plt.xlabel('t')
|
|
>>> plt.legend(['x', 'y'], shadow=True)
|
|
>>> plt.title('Lotka-Volterra System')
|
|
>>> plt.show()
|
|
|
|
"""
|
|
if method not in METHODS and not (
|
|
inspect.isclass(method) and issubclass(method, OdeSolver)):
|
|
raise ValueError("`method` must be one of {} or OdeSolver class."
|
|
.format(METHODS))
|
|
|
|
t0, tf = float(t_span[0]), float(t_span[1])
|
|
|
|
if args is not None:
|
|
# Wrap the user's fun (and jac, if given) in lambdas to hide the
|
|
# additional parameters. Pass in the original fun as a keyword
|
|
# argument to keep it in the scope of the lambda.
|
|
fun = lambda t, x, fun=fun: fun(t, x, *args)
|
|
jac = options.get('jac')
|
|
if callable(jac):
|
|
options['jac'] = lambda t, x: jac(t, x, *args)
|
|
|
|
if t_eval is not None:
|
|
t_eval = np.asarray(t_eval)
|
|
if t_eval.ndim != 1:
|
|
raise ValueError("`t_eval` must be 1-dimensional.")
|
|
|
|
if np.any(t_eval < min(t0, tf)) or np.any(t_eval > max(t0, tf)):
|
|
raise ValueError("Values in `t_eval` are not within `t_span`.")
|
|
|
|
d = np.diff(t_eval)
|
|
if tf > t0 and np.any(d <= 0) or tf < t0 and np.any(d >= 0):
|
|
raise ValueError("Values in `t_eval` are not properly sorted.")
|
|
|
|
if tf > t0:
|
|
t_eval_i = 0
|
|
else:
|
|
# Make order of t_eval decreasing to use np.searchsorted.
|
|
t_eval = t_eval[::-1]
|
|
# This will be an upper bound for slices.
|
|
t_eval_i = t_eval.shape[0]
|
|
|
|
if method in METHODS:
|
|
method = METHODS[method]
|
|
|
|
solver = method(fun, t0, y0, tf, vectorized=vectorized, **options)
|
|
|
|
if t_eval is None:
|
|
ts = [t0]
|
|
ys = [y0]
|
|
elif t_eval is not None and dense_output:
|
|
ts = []
|
|
ti = [t0]
|
|
ys = []
|
|
else:
|
|
ts = []
|
|
ys = []
|
|
|
|
interpolants = []
|
|
|
|
events, is_terminal, event_dir = prepare_events(events)
|
|
|
|
if events is not None:
|
|
if args is not None:
|
|
# Wrap user functions in lambdas to hide the additional parameters.
|
|
# The original event function is passed as a keyword argument to the
|
|
# lambda to keep the original function in scope (i.e., avoid the
|
|
# late binding closure "gotcha").
|
|
events = [lambda t, x, event=event: event(t, x, *args)
|
|
for event in events]
|
|
g = [event(t0, y0) for event in events]
|
|
t_events = [[] for _ in range(len(events))]
|
|
y_events = [[] for _ in range(len(events))]
|
|
else:
|
|
t_events = None
|
|
y_events = None
|
|
|
|
status = None
|
|
while status is None:
|
|
message = solver.step()
|
|
|
|
if solver.status == 'finished':
|
|
status = 0
|
|
elif solver.status == 'failed':
|
|
status = -1
|
|
break
|
|
|
|
t_old = solver.t_old
|
|
t = solver.t
|
|
y = solver.y
|
|
|
|
if dense_output:
|
|
sol = solver.dense_output()
|
|
interpolants.append(sol)
|
|
else:
|
|
sol = None
|
|
|
|
if events is not None:
|
|
g_new = [event(t, y) for event in events]
|
|
active_events = find_active_events(g, g_new, event_dir)
|
|
if active_events.size > 0:
|
|
if sol is None:
|
|
sol = solver.dense_output()
|
|
|
|
root_indices, roots, terminate = handle_events(
|
|
sol, events, active_events, is_terminal, t_old, t)
|
|
|
|
for e, te in zip(root_indices, roots):
|
|
t_events[e].append(te)
|
|
y_events[e].append(sol(te))
|
|
|
|
if terminate:
|
|
status = 1
|
|
t = roots[-1]
|
|
y = sol(t)
|
|
|
|
g = g_new
|
|
|
|
if t_eval is None:
|
|
ts.append(t)
|
|
ys.append(y)
|
|
else:
|
|
# The value in t_eval equal to t will be included.
|
|
if solver.direction > 0:
|
|
t_eval_i_new = np.searchsorted(t_eval, t, side='right')
|
|
t_eval_step = t_eval[t_eval_i:t_eval_i_new]
|
|
else:
|
|
t_eval_i_new = np.searchsorted(t_eval, t, side='left')
|
|
# It has to be done with two slice operations, because
|
|
# you can't slice to 0th element inclusive using backward
|
|
# slicing.
|
|
t_eval_step = t_eval[t_eval_i_new:t_eval_i][::-1]
|
|
|
|
if t_eval_step.size > 0:
|
|
if sol is None:
|
|
sol = solver.dense_output()
|
|
ts.append(t_eval_step)
|
|
ys.append(sol(t_eval_step))
|
|
t_eval_i = t_eval_i_new
|
|
|
|
if t_eval is not None and dense_output:
|
|
ti.append(t)
|
|
|
|
message = MESSAGES.get(status, message)
|
|
|
|
if t_events is not None:
|
|
t_events = [np.asarray(te) for te in t_events]
|
|
y_events = [np.asarray(ye) for ye in y_events]
|
|
|
|
if t_eval is None:
|
|
ts = np.array(ts)
|
|
ys = np.vstack(ys).T
|
|
else:
|
|
ts = np.hstack(ts)
|
|
ys = np.hstack(ys)
|
|
|
|
if dense_output:
|
|
if t_eval is None:
|
|
sol = OdeSolution(ts, interpolants)
|
|
else:
|
|
sol = OdeSolution(ti, interpolants)
|
|
else:
|
|
sol = None
|
|
|
|
return OdeResult(t=ts, y=ys, sol=sol, t_events=t_events, y_events=y_events,
|
|
nfev=solver.nfev, njev=solver.njev, nlu=solver.nlu,
|
|
status=status, message=message, success=status >= 0)
|