scipy.integrate.LSODA¶

class
scipy.integrate.
LSODA
(fun, t0, y0, t_bound, first_step=None, min_step=0.0, max_step=inf, rtol=0.001, atol=1e06, jac=None, lband=None, uband=None, vectorized=False, **extraneous)[source]¶ Adams/BDF method with automatic stiffness detection and switching.
This is a wrapper to the Fortran solver from ODEPACK [1]. It switches automatically between the nonstiff Adams method and the stiff BDF method. The method was originally detailed in [2].
 Parameters
 funcallable
Righthand side of the system. The calling signature is
fun(t, y)
. Heret
is a scalar, and there are two options for the ndarrayy
: It can either have shape (n,); thenfun
must return array_like with shape (n,). Alternatively it can have shape (n, k); thenfun
must return an array_like with shape (n, k), i.e. each column corresponds to a single column iny
. The choice between the two options is determined by vectorized argument (see below). The vectorized implementation allows a faster approximation of the Jacobian by finite differences (required for this solver). t0float
Initial time.
 y0array_like, shape (n,)
Initial state.
 t_boundfloat
Boundary time  the integration won’t continue beyond it. It also determines the direction of the integration.
 first_stepfloat or None, optional
Initial step size. Default is
None
which means that the algorithm should choose. min_stepfloat, optional
Minimum allowed step size. Default is 0.0, i.e., the step size is not bounded and determined solely by the solver.
 max_stepfloat, optional
Maximum allowed step size. Default is np.inf, i.e., the step size is not bounded and determined solely by the solver.
 rtol, atolfloat and array_like, optional
Relative and absolute tolerances. The solver keeps the local error estimates less than
atol + rtol * abs(y)
. Here rtol controls a relative accuracy (number of correct digits). But if a component of y is approximately below atol, the error only needs to fall within the same atol threshold, and the number of correct digits is not guaranteed. 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 1e3 for rtol and 1e6 for atol. jacNone or callable, optional
Jacobian matrix of the righthand side of the system with respect to
y
. The Jacobian matrix has shape (n, n) and its element (i, j) is equal tod f_i / d y_j
. The function will be called asjac(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 finitedifference approximation. lband, ubandint or None
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 haven
columns anduband + lband + 1
rows in which Jacobian diagonals are written. Specificallyjac_packed[uband + i  j , j] = jac[i, j]
. The same format is used inscipy.linalg.solve_banded
(check for an illustration). These parameters can be also used withjac=None
to reduce the number of Jacobian elements estimated by finite differences. vectorizedbool, optional
Whether fun is implemented in a vectorized fashion. A vectorized implementation offers no advantages for this solver. Default is False.
References
 1
A. C. Hindmarsh, “ODEPACK, A Systematized Collection of ODE Solvers,” IMACS Transactions on Scientific Computation, Vol 1., pp. 5564, 1983.
 2
L. Petzold, “Automatic selection of methods for solving stiff and nonstiff systems of ordinary differential equations”, SIAM Journal on Scientific and Statistical Computing, Vol. 4, No. 1, pp. 136148, 1983.
 Attributes
 nint
Number of equations.
 statusstring
Current status of the solver: ‘running’, ‘finished’ or ‘failed’.
 t_boundfloat
Boundary time.
 directionfloat
Integration direction: +1 or 1.
 tfloat
Current time.
 yndarray
Current state.
 t_oldfloat
Previous time. None if no steps were made yet.
 nfevint
Number of evaluations of the righthand side.
 njevint
Number of evaluations of the Jacobian.
Methods
dense_output
(self)Compute a local interpolant over the last successful step.
step
(self)Perform one integration step.