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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 time derivative of the state ``y``
at time ``t``. The calling signature is ``fun(t, y)``, where ``t`` is a
scalar and ``y`` is an ndarray with ``len(y) = len(y0)``. Additional
arguments need to be passed if ``args`` is used (see documentation of
``args`` argument). ``fun`` must return an array of the same shape as
``y``. See `vectorized` for more information.
t_span : 2-member sequence
Interval of integration (t0, tf). The solver starts with t=t0 and
integrates until it reaches t=tf. Both t0 and tf must be floats
or values interpretable by the float conversion function.
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)`` where
additional argument have to be passed if ``args`` is used (see
documentation of ``args`` argument). Each function must 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 or int, optional
When boolean, whether to terminate integration if this event occurs.
When integral, termination occurs after the specified the number of
occurrences of this event.
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` can be called in a vectorized fashion. Default is False.
If ``vectorized`` is False, `fun` will always be called with ``y`` of
shape ``(n,)``, where ``n = len(y0)``.
If ``vectorized`` is True, `fun` may be called with ``y`` of shape
``(n, k)``, where ``k`` is an integer. In this case, `fun` must behave
such that ``fun(t, y)[:, i] == fun(t, y[:, i])`` (i.e. each column of
the returned array is the time derivative of the state corresponding
with a column of ``y``).
Setting ``vectorized=True`` allows for faster finite difference
approximation of the Jacobian by methods 'Radau' and 'BDF', but
will result in slower execution for other methods and for 'Radau' and
'BDF' in some circumstances (e.g. small ``len(y0)``).
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), 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 : 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.
Additional arguments have to be passed if ``args`` is
used (see documentation of ``args`` argument).
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.
>>> import numpy as np
>>> 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()
A couple examples of using solve_ivp to solve the differential
equation ``y' = Ay`` with complex matrix ``A``.
>>> A = np.array([[-0.25 + 0.14j, 0, 0.33 + 0.44j],
... [0.25 + 0.58j, -0.2 + 0.14j, 0],
... [0, 0.2 + 0.4j, -0.1 + 0.97j]])
Solving an IVP with ``A`` from above and ``y`` as 3x1 vector:
>>> def deriv_vec(t, y):
... return A @ y
>>> result = solve_ivp(deriv_vec, [0, 25],
... np.array([10 + 0j, 20 + 0j, 30 + 0j]),
... t_eval=np.linspace(0, 25, 101))
>>> print(result.y[:, 0])
[10.+0.j 20.+0.j 30.+0.j]
>>> print(result.y[:, -1])
[18.46291039+45.25653651j 10.01569306+36.23293216j
-4.98662741+80.07360388j]
Solving an IVP with ``A`` from above with ``y`` as 3x3 matrix :
>>> def deriv_mat(t, y):
... return (A @ y.reshape(3, 3)).flatten()
>>> y0 = np.array([[2 + 0j, 3 + 0j, 4 + 0j],
... [5 + 0j, 6 + 0j, 7 + 0j],
... [9 + 0j, 34 + 0j, 78 + 0j]])
>>> result = solve_ivp(deriv_mat, [0, 25], y0.flatten(),
... t_eval=np.linspace(0, 25, 101))
>>> print(result.y[:, 0].reshape(3, 3))
[[ 2.+0.j 3.+0.j 4.+0.j]
[ 5.+0.j 6.+0.j 7.+0.j]
[ 9.+0.j 34.+0.j 78.+0.j]]
>>> print(result.y[:, -1].reshape(3, 3))
[[ 5.67451179 +12.07938445j 17.2888073 +31.03278837j
37.83405768 +63.25138759j]
[ 3.39949503 +11.82123994j 21.32530996 +44.88668871j
53.17531184+103.80400411j]
[ -2.26105874 +22.19277664j -15.1255713 +70.19616341j
-38.34616845+153.29039931j]]
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