Module Cvode

module Cvode: sig .. end

Variable-step solution of ODE initial value problems with zero-crossing detection.

This module solves numerically problems of the form $\dot{y} = f(t, y)$, $y(t_0) = y_0$.

This documented interface is structured as follows.

  1. Linear solvers
  2. Tolerances
  3. Solver initialization and use
  4. Modifying the solver
  5. Querying the solver
  6. Additional root finding functions
  7. Exceptions

type ('d, 'k) session = ('d, 'k) session 

A session with the CVODE solver.

An example session with Cvode (cvode_skel.ml):

open Sundials

(* 1. Define a right-hand-side function. *)
let f t y yd = yd.{0} <- y.{1}; yd.{1} <- -9.81

(* 2. Optionally define a root function. *)
let g t y gout = gout.{0} <- 1.0 -. y.{0}

(* 3. Set vector of initial values.
      The length of this vector determines the problem size. *)
let yd = RealArray.of_list [ 10.0; 0.0 ]
let y = Nvector_serial.wrap yd

(* 4. Create and initialize a solver session.
      This will initialize a specific linear solver and the root-finding
      mechanism, if necessary. *)
let s = Cvode.(init Adams Functional
                    (SStolerances (1e-4, 1e-8))
                    f ~roots:(1, g) 0.0 y);;

(* 5. Set optional inputs, e.g.,
      call [set_*] functions to change solver parameters. *)
Cvode.set_stop_time s 10.0;;
Cvode.set_all_root_directions s RootDirs.Increasing;;

(* 6. Advance the solution in time,
      by repeatedly calling [solve_normal] or [solve_one_step]. *)
let rec go (t, r) =
  Printf.printf "% .10e\t% .10e\t% .10e\n" t yd.{0} yd.{1};
  match r with
  | Cvode.Success -> go (Cvode.solve_normal s (t +. 0.5) y)
  | Cvode.RootsFound -> begin
        yd.{1} <- -0.8 *. yd.{1};
        Cvode.reinit s t y;
        go (t, Cvode.Success)
      end
  | Cvode.StopTimeReached -> ();;

Printf.printf "time\ty\ty'\n";;
go (0.0, Cvode.Success);;

(* 7. Get optional outputs,
      call the [get_*] functions to examine solver statistics. *)
let ns = Cvode.get_num_steps s
    
type [> Nvector_serial.kind ] serial_session = (Nvector_serial.data, [> Nvector_serial.kind ] as 'a) session 

Alias for sessions based on serial nvectors.

Linear solvers

type ('data, 'kind) session_linear_solver = ('data, 'kind) session_linear_solver 

Linear solvers used by Cvode.

type [> Nvector_serial.kind ] serial_session_linear_solver = (Nvector_serial.data, [> Nvector_serial.kind ] as 'a)
session_linear_solver

Alias for linear solvers that are restricted to serial nvectors.

type 'd double = 'd * 'd 

Workspaces with two temporary vectors.

type 'd triple = 'd * 'd * 'd 

Workspaces with three temporary vectors.

type ('t, 'd) jacobian_arg = ('t, 'd) jacobian_arg = {
   jac_t : float; (*

The independent variable.

*)
   jac_y : 'd; (*

The dependent variable vector.

*)
   jac_fy : 'd; (*

The derivative vector (i.e., $\frac{\mathrm{d}y}{\mathrm{d}t}$).

*)
   jac_tmp : 't; (*

Workspace data.

*)
}

Arguments common to Jacobian callback functions.

module Diag: sig .. end

Diagonal approximation of Jacobians by difference quotients.

module Dls: sig .. end

Direct Linear Solvers operating on dense, banded, and sparse matrices.

module Spils: sig .. end

Scaled Preconditioned Iterative Linear Solvers.

module Alternate: sig .. end

Alternate Linear Solvers.

Tolerances

type 'data error_weight_fun = 'data -> 'data -> unit 

Functions that set the multiplicative error weights for use in the weighted RMS norm. The call efun y ewt takes the dependent variable vector y and fills the error-weight vector ewt with positive values or raises Sundials.NonPositiveEwt. Other exceptions are eventually propagated, but should be avoided (efun is not allowed to abort the solver).

type ('data, 'kind) tolerance = 
| SStolerances of float * float (*

(rel, abs) : scalar relative and absolute tolerances.

*)
| SVtolerances of float * ('data, 'kind) Nvector.t (*

(rel, abs) : scalar relative and vector absolute tolerances.

*)
| WFtolerances of 'data error_weight_fun (*

Set the multiplicative error weights for the weighted RMS norm.

*)

Tolerance specifications.

val default_tolerances : ('data, 'kind) tolerance

A default relative tolerance of 1.0e-4 and absolute tolerance of 1.0e-8.

Solver initialization and use

type ('d, 'kind) iter = 
| Newton of ('d, 'kind) session_linear_solver (*

Newton iteration with a given linear solver.

*)
| Functional (*

Functional iteration (non-stiff systems only).

*)

Choice of method for solving non-linear systems that arise in solver formulas.

type lmm = 
| Adams (*

Adams-Moulton formulas (non-stiff systems).

*)
| BDF (*

Backward Differentiation Formulas (stiff systems).

*)

Choice of linear multistep method.

type 'd rhsfn = float -> 'd -> 'd -> unit 

Right-hand side functions for calculating ODE derivatives. They are passed three arguments:

Within the function, raising a Sundials.RecoverableFailure exception indicates a recoverable error. Any other exception is treated as an unrecoverable error.

y and y' should not be accessed after the function returns.

type 'd rootsfn = float -> 'd -> Sundials.RealArray.t -> unit 

Called by the solver to calculate the values of root functions. These ‘zero-crossings’ are used to detect significant events. The function is passed three arguments:

y and gout should not be accessed after the function has returned.

val init : lmm ->
('data, 'kind) iter ->
('data, 'kind) tolerance ->
'data rhsfn ->
?roots:int * 'data rootsfn ->
float -> ('data, 'kind) Nvector.t -> ('data, 'kind) session

Creates and initializes a session with the solver. The call

init lmm iter tol f ~roots:(nroots, g) t0 y0

has as arguments:

This function does everything necessary to initialize a session, i.e., it makes the calls referenced below. The Cvode.solve_normal and Cvode.solve_one_step functions may be called directly.

val no_roots : int * 'd rootsfn

A convenience value for signalling that there are no roots to monitor.

type solver_result = 
| Success (*

The solution was advanced. (CV_SUCCESS)

*)
| RootsFound (*

A root was found. See Cvode.get_root_info. (CV_ROOT_RETURN)

*)
| StopTimeReached (*

The stop time was reached. See Cvode.set_stop_time. (CV_TSTOP_RETURN)

*)

Values returned by the step functions. Failures are indicated by exceptions.

val solve_normal : ('d, 'k) session ->
float -> ('d, 'k) Nvector.t -> float * solver_result

Integrates an ODE system over an interval. The call tret, r = solve_normal s tout yout has as arguments

It returns tret, the time reached by the solver, which will be equal to tout if no errors occur, and, r, a Cvode.solver_result.

val solve_one_step : ('d, 'k) session ->
float -> ('d, 'k) Nvector.t -> float * solver_result

Like Cvode.solve_normal but returns after one internal solver step.

val get_dky : ('d, 'k) session -> ('d, 'k) Nvector.t -> float -> int -> unit

Returns the interpolated solution or derivatives. get_dky s dky t k computes the kth derivative of the function at time t, i.e., $\frac{d^\mathtt{k}y(\mathtt{t})}{\mathit{dt}^\mathtt{k}}$, and stores it in dky. The arguments must satisfy $t_n - h_u \leq \mathtt{t} \leq t_n$—where $t_n$ denotes Cvode.get_current_time and $h_u$ denotes Cvode.get_last_step,— and $0 \leq \mathtt{k} \leq q_u$—where $q_u$ denotes Cvode.get_last_order.

This function may only be called after a successful return from either Cvode.solve_normal or Cvode.solve_one_step.

val reinit : ('d, 'kind) session ->
?iter:('d, 'kind) iter ->
?roots:int * 'd rootsfn -> float -> ('d, 'kind) Nvector.t -> unit

Reinitializes the solver with new parameters and state values. The values of the independent variable, i.e., the simulation time, and the state variables must be given. If given, iter specifies a new iteration method, and roots specifies a new root finding function; both default to unchanged.

Modifying the solver (optional input functions)

val set_tolerances : ('d, 'k) session -> ('d, 'k) tolerance -> unit

Sets the integration tolerances.

val set_error_file : ('d, 'k) session -> Sundials.Logfile.t -> unit

Configure the default error handler to write messages to a file. By default it writes to Logfile.stderr.

val set_err_handler_fn : ('d, 'k) session -> (Sundials.Util.error_details -> unit) -> unit

Specifies a custom function for handling error messages. The handler must not fail: any exceptions are trapped and discarded.

val clear_err_handler_fn : ('d, 'k) session -> unit

Restores the default error handling function.

val set_max_ord : ('d, 'k) session -> int -> unit

Specifies the maximum order of the linear multistep method.

val set_max_num_steps : ('d, 'k) session -> int -> unit

Specifies the maximum number of steps taken in attempting to reach a given output time.

val set_max_hnil_warns : ('d, 'k) session -> int -> unit

Specifies the maximum number of messages warning that t + h = t on the next internal step.

val set_stab_lim_det : ('d, 'k) session -> bool -> unit

Indicates whether the BDF stability limit detection algorithm should be used.

val set_init_step : ('d, 'k) session -> float -> unit

Specifies the initial step size.

val set_min_step : ('d, 'k) session -> float -> unit

Specifies a lower bound on the magnitude of the step size.

val set_max_step : ('d, 'k) session -> float -> unit

Specifies an upper bound on the magnitude of the step size.

val set_stop_time : ('d, 'k) session -> float -> unit

Limits the value of the independent variable t when solving. By default no stop time is imposed.

val set_max_err_test_fails : ('d, 'k) session -> int -> unit

Specifies the maximum number of error test failures permitted in attempting one step.

val set_max_nonlin_iters : ('d, 'k) session -> int -> unit

Specifies the maximum number of nonlinear solver iterations permitted per step.

val set_max_conv_fails : ('d, 'k) session -> int -> unit

Specifies the maximum number of nonlinear solver convergence failures permitted during one step.

val set_nonlin_conv_coef : ('d, 'k) session -> float -> unit

Specifies the safety factor used in the nonlinear convergence test.

Querying the solver (optional output functions)

val get_work_space : ('d, 'k) session -> int * int

Returns the real and integer workspace sizes.

val get_num_steps : ('d, 'k) session -> int

Returns the cumulative number of internal steps taken by the solver.

val get_num_rhs_evals : ('d, 'k) session -> int

Returns the number of calls to the right-hand side function.

val get_num_lin_solv_setups : ('d, 'k) session -> int

Returns the number of calls made to the linear solver's setup function.

val get_num_err_test_fails : ('d, 'k) session -> int

Returns the number of local error test failures that have occurred.

val get_last_order : ('d, 'k) session -> int

Returns the integration method order used during the last internal step.

val get_current_order : ('d, 'k) session -> int

Returns the integration method order to be used on the next internal step.

val get_last_step : ('d, 'k) session -> float

Returns the integration step size taken on the last internal step.

val get_current_step : ('d, 'k) session -> float

Returns the integration step size to be attempted on the next internal step.

val get_actual_init_step : ('d, 'k) session -> float

Returns the the value of the integration step size used on the first step.

val get_current_time : ('d, 'k) session -> float

Returns the the current internal time reached by the solver.

val get_num_stab_lim_order_reds : ('d, 'k) session -> int

Returns the number of order reductions dictated by the BDF stability limit detection algorithm.

val get_tol_scale_factor : ('d, 'k) session -> float

Returns a suggested factor by which the user's tolerances should be scaled when too much accuracy has been requested for some internal step.

val get_err_weights : ('d, 'k) session -> ('d, 'k) Nvector.t -> unit

Returns the solution error weights at the current time.

val get_est_local_errors : ('d, 'k) session -> ('d, 'k) Nvector.t -> unit

Returns the vector of estimated local errors.

type integrator_stats = {
   num_steps : int; (*

Cumulative number of internal solver steps.

*)
   num_rhs_evals : int; (*

Number of calls to the right-hand side function.

*)
   num_lin_solv_setups : int; (*

Number of setups calls to the linear solver.

*)
   num_err_test_fails : int; (*

Number of local error test failures.

*)
   last_order : int; (*

Integration method order used in the last internal step.

*)
   current_order : int; (*

Integration method order to be used in the next internal step.

*)
   actual_init_step : float; (*

Integration step sized used on the first step.

*)
   last_step : float; (*

Integration step size of the last internal step.

*)
   current_step : float; (*

Integration step size to attempt on the next internal step.

*)
   current_time : float; (*

Current internal time reached by the solver.

*)
}

Summaries of integrator statistics.

val get_integrator_stats : ('d, 'k) session -> integrator_stats

Returns the integrator statistics as a group.

val print_integrator_stats : ('d, 'k) session -> Stdlib.out_channel -> unit

Prints the integrator statistics on the given channel.

val get_num_nonlin_solv_iters : ('d, 'k) session -> int

Returns the number of nonlinear (functional or Newton) iterations performed.

val get_num_nonlin_solv_conv_fails : ('d, 'k) session -> int

Returns the number of nonlinear convergence failures that have occurred.

val get_nonlin_solv_stats : ('d, 'k) session -> int * int

Returns both the numbers of nonlinear iterations performed nniters and nonlinear convergence failures nncfails.

Additional root-finding functions

val set_root_direction : ('d, 'k) session -> Sundials.RootDirs.d array -> unit

set_root_direction s dir specifies the direction of zero-crossings to be located and returned. dir may contain one entry for each root function.

val set_all_root_directions : ('d, 'k) session -> Sundials.RootDirs.d -> unit

Like Cvode.set_root_direction but specifies a single direction for all root functions.

val set_no_inactive_root_warn : ('d, 'k) session -> unit

Disables issuing a warning if some root function appears to be identically zero at the beginning of the integration.

val get_num_roots : ('d, 'k) session -> int

Returns the number of root functions.

val get_root_info : ('d, 'k) session -> Sundials.Roots.t -> unit

Fills an array showing which functions were found to have a root.

val get_num_g_evals : ('d, 'k) session -> int

Returns the cumulative number of calls made to the user-supplied root function g.

Exceptions

exception IllInput

Raised on missing or illegal solver inputs. Also raised if an element of the error weight vector becomes zero during time stepping, or the linear solver initialization function failed, or a root was found both at t and very near t.

exception TooClose

The initial and final times are too close to each other and an initial step size was not specified.

exception TooMuchWork

The requested time could not be reached in mxstep internal steps. See Cvode.set_max_num_steps

exception TooMuchAccuracy

The requested accuracy could not be satisfied.

exception ErrFailure

Too many error test failures within a step or at the minimum step size. See Cvode.set_max_err_test_fails and Cvode.set_min_step.

exception ConvergenceFailure

Too many convergence test failures within a step or at the minimum step size. See Cvode.set_max_conv_fails and Cvode.set_min_step.

exception LinearInitFailure

Linear solver initialization failed.

exception LinearSetupFailure

Linear solver setup failed in an unrecoverable manner.

exception LinearSolveFailure

Linear solver solution failed in an unrecoverable manner.

exception RhsFuncFailure

The right-hand side function failed in an unrecoverable manner.

exception FirstRhsFuncFailure

The right-hand side function had a recoverable error when first called.

exception RepeatedRhsFuncFailure

Too many convergence test failures, or unable to estimate the initial step size, due to repeated recoverable errors in the right-hand side function.

exception UnrecoverableRhsFuncFailure

The right-hand side function had a recoverable error, but no recovery was possible. This error can only occur after an error test failure at order one.

exception RootFuncFailure

The rootfinding function failed.

exception BadK

Raised by Cvode.get_dky for invalid order values.

exception BadT

Raised by Cvode.get_dky for invalid time values.