261 lines
11 KiB
Python
261 lines
11 KiB
Python
# Author: Travis Oliphant
|
|
from __future__ import division, print_function, absolute_import
|
|
|
|
__all__ = ['odeint']
|
|
|
|
import numpy as np
|
|
from . import _odepack
|
|
from copy import copy
|
|
import warnings
|
|
|
|
|
|
class ODEintWarning(Warning):
|
|
pass
|
|
|
|
|
|
_msgs = {2: "Integration successful.",
|
|
1: "Nothing was done; the integration time was 0.",
|
|
-1: "Excess work done on this call (perhaps wrong Dfun type).",
|
|
-2: "Excess accuracy requested (tolerances too small).",
|
|
-3: "Illegal input detected (internal error).",
|
|
-4: "Repeated error test failures (internal error).",
|
|
-5: "Repeated convergence failures (perhaps bad Jacobian or tolerances).",
|
|
-6: "Error weight became zero during problem.",
|
|
-7: "Internal workspace insufficient to finish (internal error).",
|
|
-8: "Run terminated (internal error)."
|
|
}
|
|
|
|
|
|
def odeint(func, y0, t, args=(), Dfun=None, col_deriv=0, full_output=0,
|
|
ml=None, mu=None, rtol=None, atol=None, tcrit=None, h0=0.0,
|
|
hmax=0.0, hmin=0.0, ixpr=0, mxstep=0, mxhnil=0, mxordn=12,
|
|
mxords=5, printmessg=0, tfirst=False):
|
|
"""
|
|
Integrate a system of ordinary differential equations.
|
|
|
|
.. note:: For new code, use `scipy.integrate.solve_ivp` to solve a
|
|
differential equation.
|
|
|
|
Solve a system of ordinary differential equations using lsoda from the
|
|
FORTRAN library odepack.
|
|
|
|
Solves the initial value problem for stiff or non-stiff systems
|
|
of first order ode-s::
|
|
|
|
dy/dt = func(y, t, ...) [or func(t, y, ...)]
|
|
|
|
where y can be a vector.
|
|
|
|
.. note:: By default, the required order of the first two arguments of
|
|
`func` are in the opposite order of the arguments in the system
|
|
definition function used by the `scipy.integrate.ode` class and
|
|
the function `scipy.integrate.solve_ivp`. To use a function with
|
|
the signature ``func(t, y, ...)``, the argument `tfirst` must be
|
|
set to ``True``.
|
|
|
|
Parameters
|
|
----------
|
|
func : callable(y, t, ...) or callable(t, y, ...)
|
|
Computes the derivative of y at t.
|
|
If the signature is ``callable(t, y, ...)``, then the argument
|
|
`tfirst` must be set ``True``.
|
|
y0 : array
|
|
Initial condition on y (can be a vector).
|
|
t : array
|
|
A sequence of time points for which to solve for y. The initial
|
|
value point should be the first element of this sequence.
|
|
This sequence must be monotonically increasing or monotonically
|
|
decreasing; repeated values are allowed.
|
|
args : tuple, optional
|
|
Extra arguments to pass to function.
|
|
Dfun : callable(y, t, ...) or callable(t, y, ...)
|
|
Gradient (Jacobian) of `func`.
|
|
If the signature is ``callable(t, y, ...)``, then the argument
|
|
`tfirst` must be set ``True``.
|
|
col_deriv : bool, optional
|
|
True if `Dfun` defines derivatives down columns (faster),
|
|
otherwise `Dfun` should define derivatives across rows.
|
|
full_output : bool, optional
|
|
True if to return a dictionary of optional outputs as the second output
|
|
printmessg : bool, optional
|
|
Whether to print the convergence message
|
|
tfirst: bool, optional
|
|
If True, the first two arguments of `func` (and `Dfun`, if given)
|
|
must ``t, y`` instead of the default ``y, t``.
|
|
|
|
.. versionadded:: 1.1.0
|
|
|
|
Returns
|
|
-------
|
|
y : array, shape (len(t), len(y0))
|
|
Array containing the value of y for each desired time in t,
|
|
with the initial value `y0` in the first row.
|
|
infodict : dict, only returned if full_output == True
|
|
Dictionary containing additional output information
|
|
|
|
======= ============================================================
|
|
key meaning
|
|
======= ============================================================
|
|
'hu' vector of step sizes successfully used for each time step.
|
|
'tcur' vector with the value of t reached for each time step.
|
|
(will always be at least as large as the input times).
|
|
'tolsf' vector of tolerance scale factors, greater than 1.0,
|
|
computed when a request for too much accuracy was detected.
|
|
'tsw' value of t at the time of the last method switch
|
|
(given for each time step)
|
|
'nst' cumulative number of time steps
|
|
'nfe' cumulative number of function evaluations for each time step
|
|
'nje' cumulative number of jacobian evaluations for each time step
|
|
'nqu' a vector of method orders for each successful step.
|
|
'imxer' index of the component of largest magnitude in the
|
|
weighted local error vector (e / ewt) on an error return, -1
|
|
otherwise.
|
|
'lenrw' the length of the double work array required.
|
|
'leniw' the length of integer work array required.
|
|
'mused' a vector of method indicators for each successful time step:
|
|
1: adams (nonstiff), 2: bdf (stiff)
|
|
======= ============================================================
|
|
|
|
Other Parameters
|
|
----------------
|
|
ml, mu : int, optional
|
|
If either of these are not None or non-negative, then the
|
|
Jacobian is assumed to be banded. These give the number of
|
|
lower and upper non-zero diagonals in this banded matrix.
|
|
For the banded case, `Dfun` should return a matrix whose
|
|
rows contain the non-zero bands (starting with the lowest diagonal).
|
|
Thus, the return matrix `jac` from `Dfun` should have shape
|
|
``(ml + mu + 1, len(y0))`` when ``ml >=0`` or ``mu >=0``.
|
|
The data in `jac` must be stored such that ``jac[i - j + mu, j]``
|
|
holds the derivative of the `i`th equation with respect to the `j`th
|
|
state variable. If `col_deriv` is True, the transpose of this
|
|
`jac` must be returned.
|
|
rtol, atol : float, optional
|
|
The input parameters `rtol` and `atol` determine the error
|
|
control performed by the solver. The solver will control the
|
|
vector, e, of estimated local errors in y, according to an
|
|
inequality of the form ``max-norm of (e / ewt) <= 1``,
|
|
where ewt is a vector of positive error weights computed as
|
|
``ewt = rtol * abs(y) + atol``.
|
|
rtol and atol can be either vectors the same length as y or scalars.
|
|
Defaults to 1.49012e-8.
|
|
tcrit : ndarray, optional
|
|
Vector of critical points (e.g. singularities) where integration
|
|
care should be taken.
|
|
h0 : float, (0: solver-determined), optional
|
|
The step size to be attempted on the first step.
|
|
hmax : float, (0: solver-determined), optional
|
|
The maximum absolute step size allowed.
|
|
hmin : float, (0: solver-determined), optional
|
|
The minimum absolute step size allowed.
|
|
ixpr : bool, optional
|
|
Whether to generate extra printing at method switches.
|
|
mxstep : int, (0: solver-determined), optional
|
|
Maximum number of (internally defined) steps allowed for each
|
|
integration point in t.
|
|
mxhnil : int, (0: solver-determined), optional
|
|
Maximum number of messages printed.
|
|
mxordn : int, (0: solver-determined), optional
|
|
Maximum order to be allowed for the non-stiff (Adams) method.
|
|
mxords : int, (0: solver-determined), optional
|
|
Maximum order to be allowed for the stiff (BDF) method.
|
|
|
|
See Also
|
|
--------
|
|
solve_ivp : Solve an initial value problem for a system of ODEs.
|
|
ode : a more object-oriented integrator based on VODE.
|
|
quad : for finding the area under a curve.
|
|
|
|
Examples
|
|
--------
|
|
The second order differential equation for the angle `theta` of a
|
|
pendulum acted on by gravity with friction can be written::
|
|
|
|
theta''(t) + b*theta'(t) + c*sin(theta(t)) = 0
|
|
|
|
where `b` and `c` are positive constants, and a prime (') denotes a
|
|
derivative. To solve this equation with `odeint`, we must first convert
|
|
it to a system of first order equations. By defining the angular
|
|
velocity ``omega(t) = theta'(t)``, we obtain the system::
|
|
|
|
theta'(t) = omega(t)
|
|
omega'(t) = -b*omega(t) - c*sin(theta(t))
|
|
|
|
Let `y` be the vector [`theta`, `omega`]. We implement this system
|
|
in python as:
|
|
|
|
>>> def pend(y, t, b, c):
|
|
... theta, omega = y
|
|
... dydt = [omega, -b*omega - c*np.sin(theta)]
|
|
... return dydt
|
|
...
|
|
|
|
We assume the constants are `b` = 0.25 and `c` = 5.0:
|
|
|
|
>>> b = 0.25
|
|
>>> c = 5.0
|
|
|
|
For initial conditions, we assume the pendulum is nearly vertical
|
|
with `theta(0)` = `pi` - 0.1, and is initially at rest, so
|
|
`omega(0)` = 0. Then the vector of initial conditions is
|
|
|
|
>>> y0 = [np.pi - 0.1, 0.0]
|
|
|
|
We will generate a solution at 101 evenly spaced samples in the interval
|
|
0 <= `t` <= 10. So our array of times is:
|
|
|
|
>>> t = np.linspace(0, 10, 101)
|
|
|
|
Call `odeint` to generate the solution. To pass the parameters
|
|
`b` and `c` to `pend`, we give them to `odeint` using the `args`
|
|
argument.
|
|
|
|
>>> from scipy.integrate import odeint
|
|
>>> sol = odeint(pend, y0, t, args=(b, c))
|
|
|
|
The solution is an array with shape (101, 2). The first column
|
|
is `theta(t)`, and the second is `omega(t)`. The following code
|
|
plots both components.
|
|
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> plt.plot(t, sol[:, 0], 'b', label='theta(t)')
|
|
>>> plt.plot(t, sol[:, 1], 'g', label='omega(t)')
|
|
>>> plt.legend(loc='best')
|
|
>>> plt.xlabel('t')
|
|
>>> plt.grid()
|
|
>>> plt.show()
|
|
"""
|
|
|
|
if ml is None:
|
|
ml = -1 # changed to zero inside function call
|
|
if mu is None:
|
|
mu = -1 # changed to zero inside function call
|
|
|
|
dt = np.diff(t)
|
|
if not((dt >= 0).all() or (dt <= 0).all()):
|
|
raise ValueError("The values in t must be monotonically increasing "
|
|
"or monotonically decreasing; repeated values are "
|
|
"allowed.")
|
|
|
|
t = copy(t)
|
|
y0 = copy(y0)
|
|
output = _odepack.odeint(func, y0, t, args, Dfun, col_deriv, ml, mu,
|
|
full_output, rtol, atol, tcrit, h0, hmax, hmin,
|
|
ixpr, mxstep, mxhnil, mxordn, mxords,
|
|
int(bool(tfirst)))
|
|
if output[-1] < 0:
|
|
warning_msg = _msgs[output[-1]] + " Run with full_output = 1 to get quantitative information."
|
|
warnings.warn(warning_msg, ODEintWarning)
|
|
elif printmessg:
|
|
warning_msg = _msgs[output[-1]]
|
|
warnings.warn(warning_msg, ODEintWarning)
|
|
|
|
if full_output:
|
|
output[1]['message'] = _msgs[output[-1]]
|
|
|
|
output = output[:-1]
|
|
if len(output) == 1:
|
|
return output[0]
|
|
else:
|
|
return output
|