# scipy.integrate.Radau

class scipy.integrate.Radau(fun, t0, y0, t_bound, max_step=inf, rtol=0.001, atol=1e-06, jac=None, jac_sparsity=None, vectorized=False, first_step=None, **extraneous)

  Implicit Runge-Kutta method of Radau IIA family of order 5.

  The implementation follows [1]. The error is controlled with a third-order accurate embedded formula. A cubic polynomial which satisfies the collocation conditions is used for the dense output.

  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.

    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, array_like, sparse_matrix, callable}, optional

      Jacobian matrix of the right-hand side of the system with respect to y, required by this method. The Jacobian matrix has shape (n, n) and its element (i, j) is equal to `d f_i / d y_j`. There are three ways to define the Jacobian:

      > - If array_like or sparse_matrix, the Jacobian is assumed to be constant.

      > - If callable, the Jacobian is assumed to depend on both t and y; it will be called as `jac(t, y)` as necessary. For the ‘Radau’ and ‘BDF’ methods, the return value might be a sparse matrix.

      > - 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.

    jac_sparsity{None, array_like, sparse matrix}, optional

      Defines a sparsity structure of the Jacobian matrix for a finite-difference approximation. Its shape must be (n, n). This argument is ignored if jac is not None. If the Jacobian has only few non-zero elements in each row, providing the sparsity structure will greatly speed up the computations [2]. A zero entry means that a corresponding element in the Jacobian is always zero. If None (default), the Jacobian is assumed to be dense.

    vectorized bool, optional

      Whether fun is implemented in a vectorized fashion. Default is False.

  References

  1

    E. Hairer, G. Wanner, “Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems”, Sec. IV.8.

  2

    A. Curtis, M. J. D. Powell, and J. Reid, “On the estimation of sparse Jacobian matrices”, Journal of the Institute of Mathematics and its Applications, 13, pp. 117-120, 1974.

  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.

    step_size float

      Size of the last successful step. 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.

    nlu int

      Number of LU decompositions.

  Methods

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

scipy.integrate.DOP853.step scipy.integrate.Radau.dense_output