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