class scipy.integrate.LSODA(fun, t0, y0, t_bound, first_step=None, min_step=0.0, max_step=inf, rtol=0.001, atol=1e-06, 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].


Right-hand side of the system: the time derivative of the state y at time t. The calling signature is fun(t, y), where t is a scalar and y is an ndarray with len(y) = len(y0). fun must return an array of the same shape as y. See vectorized for more information.


Initial time.

y0array_like, shape (n,)

Initial state.


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), while atol controls absolute accuracy (number of correct decimal places). To achieve the desired rtol, set atol to be smaller than the smallest value that can be expected from rtol * abs(y) so that 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 rtol such that rtol * abs(y) is always smaller 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 rtol and 1e-6 for atol.

jacNone or callable, optional

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.

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 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.

vectorizedbool, optional

Whether fun may be called in a vectorized fashion. False (default) is recommended for this solver.

If vectorized is False, fun will always be called with y of shape (n,), where n = len(y0).

If vectorized is True, fun may be called with y of shape (n, k), where k is an integer. In this case, fun must behave such that fun(t, y)[:, i] == fun(t, y[:, i]) (i.e. each column of the returned array is the time derivative of the state corresponding with a column of y).

Setting vectorized=True allows for faster finite difference approximation of the Jacobian by methods ‘Radau’ and ‘BDF’, but will result in slower execution for this solver.



A. C. Hindmarsh, “ODEPACK, A Systematized Collection of ODE Solvers,” IMACS Transactions on Scientific Computation, Vol 1., pp. 55-64, 1983.


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. 136-148, 1983.


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.



Compute a local interpolant over the last successful step.


Perform one integration step.