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Class BDF

Backward differentiation formula (BDF) solver for stiff ODEs.

Inherits From: Solver

Defined in python/math/ode/

Implements the solver described in [Shampine and Reichelt (1997)][1], a variable step size, variable order (VSVO) BDF integrator with order varying between 1 and 5.

Algorithm details

Each step involves solving the following nonlinear equation by Newton's method:

0 = 1/1 * BDF(1, y)[n+1] + ... + 1/k * BDF(k, y)[n+1]
  - h ode_fn(t[n+1], y[n+1])
  - bdf_coefficients[k-1] * (1/1 + ... + 1/k) * (y[n+1] - y[n] - BDF(1, y)[n]
                                                        -  ... - BDF(k, y)[n])

where k >= 1 is the current order of the integrator, h is the current step size, bdf_coefficients is a list of numbers that parameterizes the method, and BDF(m, y) is the m-th order backward difference of the vector y. In particular, BDF(0, y)[n] = y[n] and BDF(m + 1, y)[n] = BDF(m, y)[n] - BDF(m, y)[n - 1] for m >= 0.

Newton's method can fail because * the method has exceeded the maximum number of iterations, * the method is converging too slowly, or * the method is not expected to converge (the last two conditions are determined by approximating the Lipschitz constant associated with the iteration).

When evaluate_jacobian_lazily is True, the solver avoids evaluating the Jacobian of the dynamics function as much as possible. In particular, Newton's method will try to use the Jacobian from a previous integration step. If Newton's method fails with an out-of-date Jacobian, the Jacobian is re-evaluated and Newton's method is restarted. If Newton's method fails and the Jacobian is already up-to-date, then the step size is decreased and Newton's method is restarted.

Even if Newton's method converges, the solution it generates can still be rejected if it exceeds the specified error tolerance due to truncation error. In this case, the step size is decreased and Newton's method is restarted.


[1]: Lawrence F. Shampine and Mark W. Reichelt. The MATLAB ODE Suite. SIAM Journal on Scientific Computing 18(1):1-22, 1997.


    bdf_coefficients=(-0.185, -1.0 / 9.0, -0.0823, -0.0415, 0.0),

Initializes the solver.


  • rtol: Optional float Tensor specifying an upper bound on relative error, per element of the dependent variable. The error tolerance for the next step is tol = atol + rtol * abs(state) where state is the computed state at the current step (see also atol). The next step is rejected if it incurs a local truncation error larger than tol. Default value: 1e-3.
  • atol: Optional float Tensor specifying an upper bound on absolute error, per element of the dependent variable (see also rtol). Default value: 1e-6.
  • first_step_size: Optional scalar float Tensor specifying the size of the first step. If unspecified, the size is chosen automatically. Default value: None.
  • safety_factor: Scalar positive float Tensor. When Newton's method converges, the solver may choose to update the step size by applying a multiplicative factor to the current step size. This factor is factor = clamp(factor_unclamped, min_step_size_factor, max_step_size_factor) where factor_unclamped = error_ratio**(-1. / (order + 1)) * safety_factor (see also min_step_size_factor and max_step_size_factor). A small (respectively, large) value for the safety factor causes the solver to take smaller (respectively, larger) step sizes. A value larger than one, though not explicitly prohibited, is discouraged. Default value: 0.9.
  • min_step_size_factor: Scalar float Tensor (see safety_factor). Default value: 0.1.
  • max_step_size_factor: Scalar float Tensor (see safety_factor). Default value: 10..
  • max_num_steps: Optional scalar integer Tensor specifying the maximum number of steps allowed (including rejected steps). If unspecified, there is no upper bound on the number of steps. Default value: None.
  • max_order: Scalar integer Tensor taking values between 1 and 5 (inclusive) specifying the maximum BDF order. Default value: 5.
  • max_num_newton_iters: Optional scalar integer Tensor specifying the maximum number of iterations per invocation of Newton's method. If unspecified, there is no upper bound on the number iterations. Default value: 4.
  • newton_tol_factor: Scalar float Tensor used to determine the stopping condition for Newton's method. In particular, Newton's method terminates when the distance to the root is estimated to be less than newton_tol_factor * norm(atol + rtol * abs(state)) where state is the computed state at the current step. Default value: 0.1.
  • newton_step_size_factor: Scalar float Tensor specifying a multiplicative factor applied to the size of the integration step when Newton's method fails. Default value: 0.5.
  • bdf_coefficients: 1-D float Tensor with 5 entries that parameterize the solver. The default values are those proposed in [1]. Default value: (-0.1850, -1. / 9., -0.0823, -0.0415, 0.).
  • evaluate_jacobian_lazily: Optional boolean specifying whether the Jacobian should be evaluated at each point in time or as needed (i.e., lazily). Default value: True.
  • use_pfor_to_compute_jacobian: Boolean specifying whether or not to use parallel for in computing the Jacobian when jacobian_fn is not specified. Default value: True.
  • validate_args: Whether to validate input with asserts. If validate_args is False and the inputs are invalid, correct behavior is not guaranteed. Default value: False.
  • name: Python str name prefixed to Ops created by this function. Default value: None (i.e., 'bdf').




See tfp.math.ode.Solver.solve.