# 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=1e-06, jac=None, lband=None, uband=None, vectorized=False, **extraneous)

  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
    fun callable

      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 vectorized argument (see below). The vectorized implementation allows a faster approximation of the Jacobian by finite differences (required for this solver).

    t0 float

      Initial time.

    y0 array_like, shape (n,)

      Initial state.

    t_bound float

      Boundary time - the integration won’t continue beyond it. It also determines the direction of the integration.

    first_step float or None, optional

      Initial step size. Default is `None` which means that the algorithm should choose.

    min_step float, optional

      Minimum allowed step size. Default is 0.0, i.e., the step size is not bounded and determined solely by the solver.

    max_step float, optional

      Maximum allowed step size. Default is np.inf, i.e., the step size is not bounded and determined solely by the solver.

    rtol, atol float 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 1e-3 for rtol and 1e-6 for atol.

    jac None 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, uband int 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.

    vectorized bool, 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. 55-64, 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. 136-148, 1983.

  Attributes
    n int

      Number of equations.

    status string

      Current status of the solver: ‘running’, ‘finished’ or ‘failed’.

    t_bound float

      Boundary time.

    direction float

      Integration direction: +1 or -1.

    t float

      Current time.

    y ndarray

      Current state.

    t_old float

      Previous time. None if no steps were made yet.

    nfev int

      Number of evaluations of the right-hand side.

    njev int

      Number of evaluations of the Jacobian.

  Methods

  `dense_output`() | Compute a local interpolant over the last successful step.
  `step`() | Perform one integration step.

scipy.integrate.BDF.step scipy.integrate.LSODA.dense_output