| odeint(func, y0, t[, args, Dfun, col_deriv, ...]) | Integrate a system of ordinary differential equations. |
Integrate a system of ordinary differential equations.
Description:
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,t0,...) where y can be a vector.
Inputs:
func – func(y,t0,...) computes the derivative of y at t0. y0 – initial condition on y (can be a vector). t – a sequence of time points for which to solve for y. The intial
value point should be the first element of this sequence.args – extra arguments to pass to function. Dfun – the gradient (Jacobian) of func (same input signature as func). col_deriv – non-zero implies that Dfun defines derivatives down
columns (faster), otherwise Dfun should define derivatives across rows.
- full_output – non-zero to return a dictionary of optional outputs as
- the second output.
printmessg – print the convergence message.
Outputs: (y, {infodict,})
- y – a rank-2 array containing the value of y in each row for each
- desired time in t (with the initial value y0 in the first row).
- infodict – a dictionary of optional outputs:
‘hu’ : a vector of step sizes successfully used for each time step. ‘tcur’ : a vector with the value of t reached for each time step.
(will always be at least as large as the input times).
- ‘tolsf’ : a vector of tolerance scale factors, greater than 1.0,
- computed when a request for too much accuracy was detected.
- ‘tsw’ : the value of t at the time of the last method switch
- (given for each time step).
‘nst’ : the cumulative number of time steps. ‘nfe’ : the cumulative number of function evaluations for eadh
time step.
- ‘nje’ : the 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.‘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)
- ‘istate’: Exit status:
- 1 – nothing was done, as tout was equal to t with
- istate = 1 on input. (however, an internal counter was set to detect and prevent repeated calls of this type.)
2 – the integration was performed successfully.
- -1 – an excessive amount of work (more than mxstep
- steps) was done on this call, before completing the requested task, but the integration was otherwise successful as far as t. (mxstep is an optional input and is normally 500.) to continue, the user may simply reset istate to a value .gt. 1 and call again (the excess work step counter will be reset to 0). in addition, the user may increase mxstep to avoid this error return (see below on optional inputs).
- -2 – too much accuracy was requested for the precision
- of the machine being used. this was detected before completing the requested task, but the integration was successful as far as t. to continue, the tolerance parameters must be reset, and istate must be set to 3. the optional output tolsf may be used for this purpose. (note.. if this condition is detected before taking any steps, then an illegal input return (istate = -3) occurs instead.)
- -3 – illegal input was detected, before taking any
- integration steps. see written message for details. note.. if the solver detects an infinite loop of calls to the solver with illegal input, it will cause the run to stop.
- -4 – there were repeated error test failures on
- one attempted step, before completing the requested task, but the integration was successful as far as t. the problem may have a singularity, or the input may be inappropriate.
- -5 – there were repeated convergence test failures on
- one attempted step, before completing the requested task, but the integration was successful as far as t. this may be caused by an inaccurate jacobian matrix, if one is being used.
- -6 – ewt(i) became zero for some i during the
- integration. pure relative error control (atol(i)=0.0) was requested on a variable which has now vanished. the integration was successful as far as t.
- -7 – the length of rwork and/or iwork was too small to
- proceed, but the integration was successful as far as t. this happens when lsoda chooses to switch methods but lrw and/or liw is too small for the new method.
Additional Inputs:
- ml, mu – 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 columns contain the non-zero bands (starting with the lowest diagonal). Thus, the return matrix from Dfun should have shape len(y0) x (ml + mu + 1) when ml >=0 or mu >=0
rtol – The input parameters rtol and atol determine the error atol 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.
- tcrit – a vector of critical points (e.g. singularities) where
(For the next inputs a zero default means the solver determines it).
h0 – the step size to be attempted on the first step. hmax – the maximum absolute step size allowed. hmin – the minimum absolute step size allowed. ixpr – non-zero to generate extra printing at method switches. mxstep – maximum number of (internally defined) steps allowed
for each integration point in t.mxhnil – maximum number of messages printed. mxordn – maximum order to be allowed for the nonstiff (Adams) method. mxords – maximum order to be allowed for the stiff (BDF) method.