egttools.plotting.simplex2d.odeint¶
- 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)[source]¶
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 thescipy.integrate.ode
class and the functionscipy.integrate.solve_ivp
. To use a function with the signaturefunc(t, y, ...)
, the argumenttfirst
must be set toTrue
.- 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 argumenttfirst
must be setTrue
.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 iscallable(t, y, ...)
, then the argumenttfirst
must be setTrue
.col_deriv (bool, optional) – True if
Dfun
defines derivatives down columns (faster), otherwiseDfun
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
(andDfun
, if given) mustt, y
instead of the defaulty, t
.New in version 1.1.0.
ml (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 matrixjac
fromDfun
should have shape(ml + mu + 1, len(y0))
whenml >=0
ormu >=0
. The data injac
must be stored such thatjac[i - j + mu, j]
holds the derivative of thei`th equation with respect to the `j`th state variable. If `col_deriv
is True, the transpose of thisjac
must be returned.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 matrixjac
fromDfun
should have shape(ml + mu + 1, len(y0))
whenml >=0
ormu >=0
. The data injac
must be stored such thatjac[i - j + mu, j]
holds the derivative of thei`th equation with respect to the `j`th state variable. If `col_deriv
is True, the transpose of thisjac
must be returned.rtol (float, optional) – The input parameters
rtol
andatol
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 formmax-norm of (e / ewt) <= 1
, where ewt is a vector of positive error weights computed asewt = rtol * abs(y) + atol
. rtol and atol can be either vectors the same length as y or scalars. Defaults to 1.49012e-8.atol (float, optional) – The input parameters
rtol
andatol
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 formmax-norm of (e / ewt) <= 1
, where ewt is a vector of positive error weights computed asewt = 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.
- 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)
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
andc
are positive constants, and a prime (‘) denotes a derivative. To solve this equation withodeint
, we must first convert it to a system of first order equations. By defining the angular velocityomega(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 andc
= 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, soomega(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 parametersb
andc
topend
, we give them toodeint
using theargs
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 isomega(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()