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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:
    funcallable

      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.

    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), 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.

  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 right-hand side.

    njevint

      Number of evaluations of the Jacobian.

  Methods

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

  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.