Module Ida

module Ida: sig .. end

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

This module solves numerically problems of the form $F(t, y, \dot{y})=0$, $y(t_0) = y_0$, $\dot{y}(t_0)=\dot{y}_0$.

This documented interface is structured as follows.

  1. Linear solvers
  2. Tolerances
  3. Initialization
  4. Solution
  5. Modifying the solver
  6. Querying the solver
  7. Additional root finding functions
  8. Exceptions

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

A session with the IDA solver.

An example session with Ida (ida_skel.ml):

open Sundials

(* 1. Define a residual function. *)
let resf t v v' r =
  r.{0} <- v.{2}  -. v'.{0};
  r.{1} <- v.{3}  -. v'.{1};
  r.{2} <- v'.{2} -. v.{4} *. v.{0};
  r.{3} <- v'.{3} -. v.{4} *. v.{1} +. 9.81;
  r.{4} <- v.{2}*.v.{2} +. v'.{2}*.v.{0} +. v.{3}*.v'.{1} +. v'.{3}*.v.{1}

(* 2. Optionally define a root function. *)
let g t v v' gout = gout.{0} <- v.{0} -. v.{1} *. 0.5774

(* 3. Set vector of initial values.
      The length of this vector determines the problem size. *)
let vd = RealArray.of_list [ 0.9848; 0.1736; 0.0; 0.0; 0.0 ]
let v  = Nvector_serial.wrap vd
let v' = Nvector_serial.make 5 0.0

(* 4. Create and initialize a solver session.
      This will initialize a specific linear solver and the root-finding
      mechanism, if necessary. *)
let m = Matrix.dense 5
let s = Ida.(init Dls.(solver (dense v m))
                  (SStolerances (1e-9, 1e-9))
                  resf ~roots:(1, g) 0.0 v v');;

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

(* 6. Correct initial values *)
let vids = Nvector_serial.make 5 Ida.VarId.differential;;
(Nvector_serial.unwrap vids).{4} <- Ida.VarId.algebraic;

Ida.set_suppress_alg s ~varid:vids true;
Ida.calc_ic_ya_yd' s ~y:v ~y':v' 0.1;;

(* 7. 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\t% .10e\t% .10e\t% .10e\n"
                t vd.{0} vd.{1} vd.{2} vd.{3} vd.{4};
  match r with
  | Ida.Success -> go (Ida.solve_normal s (t +. 0.05) v v')
  | Ida.RootsFound -> begin
         vd.{2} <- -. 0.5 *. vd.{2};
         vd.{3} <- -. 0.5 *. vd.{3};
         Ida.reinit s t v v';
         Ida.calc_ic_ya_yd' ~y:v ~y':v' s ~varid:vids (t +. 0.05);
          go (t, Ida.Success)
      end
  | Ida.StopTimeReached -> ();;

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

(* 8. Get optional outputs,
      call the [get_*] functions to examine solver statistics. *)
let ns = Ida.get_num_steps s
    
type [> Nvector_serial.kind ] serial_session = (Sundials.RealArray.t, [> 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 Ida.

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_y' : 'd; (*

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

*)
   jac_res : 'd; (*

The current value of the residual vector.

*)
   jac_coef : float; (*

The coefficient $c_j$ in $J = \frac{\partial F}{\partial y} + c_j \frac{\partial F}{\partial\dot{y}}$.

*)
   jac_tmp : 't; (*

Workspace data.

*)
}

Arguments common to Jacobian callback functions.

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.

Initialization

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

Residual functions that define a DAE problem. They are passed four * arguments:

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

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

type 'd rootsfn = float -> 'd -> '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 four arguments:

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

val init : ('d, 'kind) session_linear_solver ->
('d, 'kind) tolerance ->
'd resfn ->
?varid:('d, 'kind) Nvector.t ->
?roots:int * 'd rootsfn ->
float ->
('d, 'kind) Nvector.t -> ('d, 'kind) Nvector.t -> ('d, 'kind) session

Creates and initializes a session with the solver. The call

init linsolv tol f ~varid:varid ~roots:(nroots, g) t0 y0 y'0

has as arguments:

This function does everything necessary to initialize a session, i.e., it makes the calls referenced below. The Ida.solve_normal and Ida.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.

Initial Condition Calculation

module VarId: sig .. end

Symbolic names for constants used when calculating initial values or supressing local error tests.

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

Class components of the state vector as either algebraic or differential. These classifications are required by Ida.calc_ic_ya_yd' and Ida.set_suppress_alg. See also Ida.VarId.

val set_suppress_alg : ('d, 'k) session -> ?varid:('d, 'k) Nvector.t -> bool -> unit

Indicates whether or not to ignore algebraic variables in the local error test. When ignoring algebraic variables (true), a varid vector must be specified either in the call or by a prior call to Ida.init or Ida.set_id. Suppressing local error tests for algebraic variables is discouraged for DAE systems of index 1 and encouraged for systems of index 2 or more.

val calc_ic_y : ('d, 'k) session -> ?y:('d, 'k) Nvector.t -> float -> unit

Computes the initial state vector for certain index-one problems. All components of $y$ are computed, using $\dot{y}$, to satisfy the constraint $F(t_0, y_0, \dot{y}_0) = 0$. If given, the ~y vector is filled with the corrected values. The last argument is the first vale of $t$ at which a solution will be requested. A Ida.reinit is required before calling this function after Ida.solve_normal or Ida.solve_one_step.

val calc_ic_ya_yd' : ('d, 'k) session ->
?y:('d, 'k) Nvector.t ->
?y':('d, 'k) Nvector.t -> ?varid:('d, 'k) Nvector.t -> float -> unit

Computes the algebraic components of the initial state and derivative vectors for certain index-one problems. The elements of $y$ and $\dot{y}$ marked as algebraic are computed, using $\dot{y}$, to satisfy the constraint $F(t_0, y_0, \dot{y}_0) = 0$. The variable ids must be given in ~varid or by a prior call to Ida.init or Ida.set_id. If given, the ~y and ~y' vectors are filled with the corrected values. The last argument is the first vale of $t$ at which a solution will be requested. A Ida.reinit is required before calling this function after Ida.solve_normal or Ida.solve_one_step.

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

Specifies the positive constant in the nonlinear convergence test of the initial condition calculation.

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

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

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

Specifies the maximum number of approximate Jacobian or preconditioner evaluations allowed when the Newton iteration appears to be slowly converging.

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

Specifies the maximum number of Newton iterations allowed in any one attempt to calculate initial conditions.

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

Specifies the maximum number of linesearch backtracks allowed in any Newton iteration, when solving the initial conditions calculation problem.

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

Enables (true) or disables (false) the linesearch algorithm in the initial condition calculation.

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

Specifies a positive lower bound on the Newton step in the initial condition calculation.

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

Returns the number of backtrack operations during Ida.calc_ic_ya_yd' or Ida.calc_ic_y.

Solution

type solver_result = 
| Success (*

The solution was advanced. (IDA_SUCCESS)

*)
| RootsFound (*

A root was found. See Ida.get_root_info. (IDA_ROOT_RETURN)

*)
| StopTimeReached (*

The stop time was reached. See Ida.set_stop_time. (IDA_TSTOP_RETURN)

*)

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

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

Integrates a DAE system over an interval. The call tret, r = solve_normal s tout yout y'out 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 Ida.solver_result.

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

Like Ida.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 Ida.get_current_time and $h_u$ denotes Ida.get_last_step,— and $0 \leq \mathtt{k} \leq q_u$—where $q_u$ denotes Ida.get_last_order.

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

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

Reinitializes the solver with new parameters and state values. The values of the independent variable, i.e., the simulation time, the state variables, and the derivatives must be given. If the argument ~linsolv is not given, the current linear solver remains unchanged. The argument ~roots works similarly; pass Ida.no_roots to disable root finding.

Modifying the solver (optional input functions)

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

Set 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_init_step : ('d, 'k) session -> float -> unit

Specifies the initial 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.

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

Specifies a vector defining inequality constraints for each component of the solution vector u. See Sundials.Constraint.

Querying the solver (optional output functions)

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

Returns the sizes of the real and integer workspaces.

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

Returns the cumulative number of internal solver steps.

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

Returns the number of calls to the residual 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_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_res_evals : int; (*

Number of calls to the residual 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 Ida.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 TooMuchWork

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

exception TooMuchAccuracy

The requested accuracy could not be satisfied.

exception ErrFailure

Too many error test failures within a step. See Ida.set_max_err_test_fails.

exception ConvergenceFailure

Too many convergence test failures within a step, or Newton convergence failed. See Ida.set_max_conv_fails.

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 ResFuncFailure

The residual function failed in an unrecoverable manner.

exception FirstResFuncFailure

The residual function had a recoverable error when first called.

exception RepeatedResFuncFailure

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

exception RootFuncFailure

The rootfinding function failed.

exception ConstraintFailure

No solution satisfying the inequality constraints could be found.

exception LinesearchFailure

Linesearch could not find a solution with a step larger than steptol in weighted RMS norm.

exception NoRecovery

A recoverable error occurred in a callback but no recovery was possible.

exception BadEwt

A component of the error weight vector, either for the input value or a corrected value, is zero.

exception BadK

Raised by Ida.get_dky for invalid order values.

exception BadT

Raised by Ida.get_dky for invalid time values.

exception IdNotSet

Variable ids are required but not set. See Ida.set_id.