Number of equations.
Current status of the solver: 'running', 'finished' or 'failed'.
Boundary time.
Integration direction: +1 or -1.
Current time.
Current state.
Previous time. None if no steps were made yet.
Number of evaluations of the right-hand side.
Number of evaluations of the Jacobian.
This is a wrapper to the Fortran solver from ODEPACK . It switches automatically between the nonstiff Adams method and the stiff BDF method. The method was originally detailed in .
Right-hand side of the system. The calling signature is fun(t, y)
. Here t
is a scalar, and there are two options for the ndarray y
: It can either have shape (n,); then fun
must return array_like with shape (n,). Alternatively it can have shape (n, k); then fun
must return an array_like with shape (n, k), i.e. each column corresponds to a single column in y
. The choice between the two options is determined by :None:None:`vectorized` argument (see below). The vectorized implementation allows a faster approximation of the Jacobian by finite differences (required for this solver).
Initial time.
Initial state.
Boundary time - the integration won't continue beyond it. It also determines the direction of the integration.
Initial step size. Default is None
which means that the algorithm should choose.
Minimum allowed step size. Default is 0.0, i.e., the step size is not bounded and determined solely by the solver.
Maximum allowed step size. Default is np.inf, i.e., the step size is not bounded and determined solely by the solver.
Relative and absolute tolerances. The solver keeps the local error estimates less than atol + rtol * abs(y)
. Here :None:None:`rtol` controls a relative accuracy (number of correct digits), while atol
controls absolute accuracy (number of correct decimal places). To achieve the desired :None:None:`rtol`, set atol
to be lower than the lowest value that can be expected from rtol * abs(y)
so that :None:None:`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 :None:None:`rtol` such that rtol * abs(y)
is always lower 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 :None:None:`rtol` and 1e-6 for atol
.
Jacobian matrix of the right-hand side of the system with respect to y
. The Jacobian matrix has shape (n, n) and its element (i, j) is equal to d f_i / d y_j
. The function will be called as jac(t, y)
. 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.
Parameters defining the bandwidth of the Jacobian, i.e., jac[i, j] != 0 only for i - lband <= j <= i + uband
. 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.
Whether :None:None:`fun` is implemented in a vectorized fashion. A vectorized implementation offers no advantages for this solver. Default is False.
Adams/BDF method with automatic stiffness detection and switching.
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