.. ----------------------------------------------------------------
   Programmer(s): Daniel R. Reynolds @ SMU
                  David J. Gardner @ LLNL
   ----------------------------------------------------------------
   SUNDIALS Copyright Start
   Copyright (c) 2002-2022, Lawrence Livermore National Security
   and Southern Methodist University.
   All rights reserved.

   See the top-level LICENSE and NOTICE files for details.

   SPDX-License-Identifier: BSD-3-Clause
   SUNDIALS Copyright End
   ----------------------------------------------------------------

.. _ARKODE.Usage.UserSupplied:

User-supplied functions
=============================

The user-supplied functions for ARKODE consist of:

* at least one function defining the ODE (required),

* a function that handles error and warning messages (optional),

* a function that provides the error weight vector (optional),

* a function that provides the residual weight vector (optional, ARKStep only),

* a function that handles adaptive time step error control (optional, ARKStep/ERKStep only),

* a function that handles explicit time step stability (optional, ARKStep/ERKStep only),

* a function that updates the implicit stage prediction (optional, ARKStep/MRIStep only),

* a function that defines the root-finding problem(s) to solve (optional),

* one or two functions that provide Jacobian-related information for
  the linear solver, if a component is treated implicitly and a
  Newton-based nonlinear iteration is chosen (optional, ARKStep/MRIStep only),

* one or two functions that define the preconditioner for use in any
  of the Krylov iterative algorithms, if linear systems of equations are to
  be solved using an iterative method (optional, ARKStep/MRIStep only),

* if the problem involves a non-identity mass matrix :math:`M\ne I` with ARKStep:

  * one or two functions that provide mass-matrix-related information
    for the linear and mass matrix solvers (required),

  * one or two functions that define the mass matrix preconditioner
    for use if an iterative mass matrix solver is chosen (optional), and

* a function that handles vector resizing operations, if the
  underlying vector structure supports resizing (as opposed to
  deletion/recreation), and if the user plans to call
  :c:func:`ARKStepResize`, :c:func:`ERKStepResize`, or
  :c:func:`MRIStepResize` (optional).

* MRIStep only: functions to be called before and after each inner integration to
  perform any communication or memory transfers of forcing data supplied
  by the outer integrator to the inner integrator, or state data supplied
  by the inner integrator to the outer integrator.



.. _ARKODE.Usage.ODERHS:

ODE right-hand side
-----------------------------

The user must supply at least one function of type :c:type:`ARKRhsFn` to
specify the explicit and/or implicit portions of the ODE system to ARKStep,
the ODE system function to ERKStep, or the "slow" right-hand side of the
ODE system to MRIStep:


.. c:type:: int (*ARKRhsFn)(realtype t, N_Vector y, N_Vector ydot, void* user_data)

   These functions compute the ODE right-hand side for a given
   value of the independent variable :math:`t` and state vector :math:`y`.

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *y* -- the current value of the dependent variable vector.
      * *ydot* -- the output vector that forms [a portion of] the ODE RHS :math:`f(t,y)`.
      * *user_data* -- the `user_data` pointer that was passed to
        :c:func:`ARKStepSetUserData`, :c:func:`ERKStepSetUserData`, or :c:func:`MRIStepSetUserData`.

   **Return value:**

      An *ARKRhsFn* should return 0 if successful, a positive value if a
      recoverable error occurred (in which case ARKODE will attempt to
      correct), or a negative value if it failed unrecoverably (in which
      case the integration is halted and *ARK_RHSFUNC_FAIL* is returned).

   **Notes:**

      Allocation of memory for `ydot` is handled within ARKODE.

      The vector *ydot* may be uninitialized on input; it is the user's
      responsibility to fill this entire vector with meaningful values.

      A recoverable failure error return from the *ARKRhsFn* is typically
      used to flag a value of the dependent variable :math:`y` that is
      "illegal" in some way (e.g., negative where only a
      non-negative value is physically meaningful).  If such a return is
      made, ARKODE will attempt to recover (possibly repeating the
      nonlinear iteration, or reducing the step size in ARKStep or ERKStep)
      in order to avoid this recoverable error return.  There are some
      situations in which recovery is not possible even if the right-hand
      side function returns a recoverable error flag.  One is when this
      occurs at the very first call to the *ARKRhsFn* (in which case
      ARKODE returns *ARK_FIRST_RHSFUNC_ERR*).  Another is when a
      recoverable error is reported by *ARKRhsFn* after the ARKStep
      integrator completes a successful stage, in which case ARKStep returns
      *ARK_UNREC_RHSFUNC_ERR*).  Similarly, since MRIStep does not currently
      support adaptive time stepping at the slow time scale, it may
      halt on a recoverable error flag that would normally have resulted
      in a stepsize reduction.




.. _ARKODE.Usage.ErrorHandler:

Error message handler function
--------------------------------------

As an alternative to the default behavior of directing error and
warning messages to the file pointed to by `errfp` (see
:c:func:`ARKStepSetErrFile`, :c:func:`ERKStepSetErrFile`, and
:c:func:`MRIStepSetErrFile`), the user may provide a function of type
:c:type:`ARKErrHandlerFn` to process any such messages.



.. c:type:: void (*ARKErrHandlerFn)(int error_code, const char* module, const char* function, char* msg, void* user_data)

   This function processes error and warning messages from
   ARKODE and its sub-modules.

   **Arguments:**
      * *error_code* -- the error code.
      * *module* -- the name of the ARKODE module reporting the error.
      * *function* -- the name of the function in which the error occurred.
      * *msg* -- the error message.
      * *user_data* -- a pointer to user data, the same as the
        *eh_data* parameter that was passed to :c:func:`ARKStepSetErrHandlerFn`,
        :c:func:`ERKStepSetErrHandlerFn`, or :c:func:`MRIStepSetErrHandlerFn`.

   **Return value:**
      An *ARKErrHandlerFn* function has no return value.

   **Notes:**
      *error_code* is negative for errors and positive
      (*ARK_WARNING*) for warnings.  If a function that returns a
      pointer to memory encounters an error, it sets *error_code* to
      0.




.. _ARKODE.Usage.ErrorWeight:

Error weight function
--------------------------------------

As an alternative to providing the relative and absolute tolerances,
the user may provide a function of type :c:type:`ARKEwtFn` to compute a
vector *ewt* containing the weights in the WRMS norm
:math:`\|v\|_{WRMS} = \left(\dfrac{1}{n} \displaystyle \sum_{i=1}^n \left(ewt_i\; v_i\right)^2
\right)^{1/2}`.  These weights will be used in place of those defined
in :numref:`ARKODE.Mathematics.Error.Norm`.



.. c:type:: int (*ARKEwtFn)(N_Vector y, N_Vector ewt, void* user_data)

   This function computes the WRMS error weights for the vector
   :math:`y`.

   **Arguments:**
      * *y* -- the dependent variable vector at which the
        weight vector is to be computed.
      * *ewt* -- the output vector containing the error weights.
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData`,
        :c:func:`ERKStepSetUserData`, or :c:func:`MRIStepSetUserData`.

   **Return value:**
      An *ARKEwtFn* function must return 0 if it
      successfully set the error weights, and -1 otherwise.

   **Notes:**
      Allocation of memory for *ewt* is handled within ARKODE.

      The error weight vector must have all components positive.  It is
      the user's responsibility to perform this test and return -1 if it
      is not satisfied.



.. _ARKODE.Usage.ResidualWeight:

Residual weight function (ARKStep only)
----------------------------------------

As an alternative to providing the scalar or vector absolute residual
tolerances (when the IVP units differ from the solution units), the
user may provide a function of type :c:type:`ARKRwtFn` to compute a
vector *rwt* containing the weights in the WRMS norm
:math:`\|v\|_{WRMS} = \left(\dfrac{1}{n} \displaystyle \sum_{i=1}^n \left(rwt_i\; v_i\right)^2
\right)^{1/2}`.  These weights will be used in place of those defined
in :numref:`ARKODE.Mathematics.Error.Norm`.



.. c:type:: int (*ARKRwtFn)(N_Vector y, N_Vector rwt, void* user_data)

   This function computes the WRMS residual weights for the vector
   :math:`y`.

   **Arguments:**
      * *y* -- the dependent variable vector at which the
        weight vector is to be computed.
      * *rwt* -- the output vector containing the residual weights.
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData()`.

   **Return value:**
      An *ARKRwtFn* function must return 0 if it
      successfully set the residual weights, and -1 otherwise.

   **Notes:**
      Allocation of memory for *rwt* is handled within ARKStep.

      The residual weight vector must have all components positive.  It is
      the user's responsibility to perform this test and return -1 if it
      is not satisfied.



.. _ARKODE.Usage.AdaptivityFn:

Time step adaptivity function (ARKStep and ERKStep only)
--------------------------------------------------------

As an alternative to using one of the built-in time step adaptivity
methods for controlling solution error, the user may provide a
function of type :c:type:`ARKAdaptFn` to compute a target step size
:math:`h` for the next integration step.  These steps should be chosen
such that the error estimate for the next time step remains below 1.



.. c:type:: int (*ARKAdaptFn)(N_Vector y, realtype t, realtype h1, realtype h2, realtype h3, realtype e1, realtype e2, realtype e3, int q, int p, realtype* hnew, void* user_data)

   This function implements a time step adaptivity algorithm
   that chooses :math:`h` to satisfy the error tolerances.

   **Arguments:**
      * *y* -- the current value of the dependent variable vector.
      * *t* -- the current value of the independent variable.
      * *h1* -- the current step size, :math:`t_n - t_{n-1}`.
      * *h2* -- the previous step size, :math:`t_{n-1} - t_{n-2}`.
      * *h3* -- the step size :math:`t_{n-2}-t_{n-3}`.
      * *e1* -- the error estimate from the current step, :math:`n`.
      * *e2* -- the error estimate from the previous step, :math:`n-1`.
      * *e3* -- the error estimate from the step :math:`n-2`.
      * *q* -- the global order of accuracy for the method.
      * *p* -- the global order of accuracy for the embedded method.
      * *hnew* -- the output value of the next step size.
      * *user_data* -- a pointer to user data, the same as the
        *h_data* parameter that was passed to :c:func:`ARKStepSetAdaptivityFn`
        or :c:func:`ERKStepSetAdaptivityFn`.

   **Return value:**
      An *ARKAdaptFn* function should return 0 if it
      successfully set the next step size, and a non-zero value otherwise.




.. _ARKODE.Usage.StabilityFn:

Explicit stability function (ARKStep and ERKStep only)
------------------------------------------------------

A user may supply a function to predict the maximum stable step size
for the explicit portion of the problem, :math:`f^E(t,y)` in ARKStep
or the full :math:`f(t,y)` in ERKStep.  While
the accuracy-based time step adaptivity algorithms may be sufficient
for retaining a stable solution to the ODE system, these may be
inefficient if the explicit right-hand side function contains moderately stiff terms.  In
this scenario, a user may provide a function of type :c:type:`ARKExpStabFn`
to provide this stability information to ARKODE.  This function
must set the scalar step size satisfying the stability restriction for
the upcoming time step.  This value will subsequently be bounded by
the user-supplied values for the minimum and maximum allowed time
step, and the accuracy-based time step.



.. c:type:: int (*ARKExpStabFn)(N_Vector y, realtype t, realtype* hstab, void* user_data)

   This function predicts the maximum stable step size for the
   explicit portion of the ODE system.

   **Arguments:**
      * *y* -- the current value of the dependent variable vector.
      * *t* -- the current value of the independent variable.
      * *hstab* -- the output value with the absolute value of the
        maximum stable step size.
      * *user_data* -- a pointer to user data, the same as the
        *estab_data* parameter that was passed to :c:func:`ARKStepSetStabilityFn`
        or :c:func:`ERKStepSetStabilityFn`.

   **Return value:**
      An *ARKExpStabFn* function should return 0 if it
      successfully set the upcoming stable step size, and a non-zero
      value otherwise.

   **Notes:**
      If this function is not supplied, or if it returns
      *hstab* :math:`\le 0.0`, then ARKODE will assume that there is no explicit
      stability restriction on the time step size.




.. _ARKODE.Usage.StagePredictFn:

Implicit stage prediction function (ARKStep and MRIStep only)
-------------------------------------------------------------

A user may supply a function to update the prediction for each implicit stage solution.
If supplied, this routine will be called *after* any existing ARKStep or MRIStep predictor
algorithm completes, so that the predictor may be modified by the user as desired.
In this scenario, a user may provide a function of type :c:type:`ARKStagePredictFn`
to provide this implicit predictor to ARKODE.  This function takes as input the
already-predicted implicit stage solution and the corresponding "time" for that prediction;
it then updates the prediction vector as desired.  If the user-supplied routine will
construct a full prediction (and thus the ARKODE prediction is irrelevant), it is
recommended that the user *not* call :c:func:`ARKStepSetPredictorMethod` or
:c:func:`MRIStepSetPredictorMethod`, thereby leaving the default trivial predictor in place.



.. c:type:: int (*ARKStagePredictFn)(realtype t, N_Vector zpred, void* user_data)

   This function updates the prediction for the implicit stage solution.

   **Arguments:**
      * *t* -- the current value of the independent variable containing the
        "time" corresponding to the predicted solution.
      * *zpred* -- the ARKStep-predicted stage solution on input, and the
        user-modified predicted stage solution on output.
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData`
        or :c:func:`MRIStepSetUserData`.

   **Return value:**
      An *ARKStagePredictFn* function should return 0 if it
      successfully set the upcoming stable step size, and a non-zero
      value otherwise.

   **Notes:**
      This may be useful if there are bound constraints on the solution,
      and these should be enforced prior to beginning the nonlinear or linear implicit solver
      algorithm.

      This routine is incompatible with the "minimum correction predictor" -- option 5 to the
      routine :c:func:`ARKStepSetPredictorMethod()`.  If both are selected, then ARKStep will
      override its built-in implicit predictor routine to instead use option 0 (trivial predictor).


.. _ARKODE.Usage.RootfindingFn:

Rootfinding function
--------------------------------------

If a rootfinding problem is to be solved during integration of the
ODE system, the user must supply a function of type :c:type:`ARKRootFn`.



.. c:type:: int (*ARKRootFn)(realtype t, N_Vector y, realtype* gout, void* user_data)

   This function implements a vector-valued function
   :math:`g(t,y)` such that roots are sought for the components
   :math:`g_i(t,y)`, :math:`i=0,\ldots,` *nrtfn*-1.

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *y* -- the current value of the dependent variable vector.
      * *gout* -- the output array, of length *nrtfn*, with components :math:`g_i(t,y)`.
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData`,
        :c:func:`ERKStepSetUserData`, or :c:func:`MRIStepSetUserData`.

   **Return value:**
      An *ARKRootFn* function should return 0 if successful
      or a non-zero value if an error occurred (in which case the
      integration is halted and ARKODE returns *ARK_RTFUNC_FAIL*).

   **Notes:**
      Allocation of memory for *gout* is handled within ARKODE.



.. _ARKODE.Usage.JacobianFn:

Jacobian construction (matrix-based linear solvers, ARKStep and MRIStep only)
-----------------------------------------------------------------------------

If a matrix-based linear solver module is used (i.e., a non-NULL ``SUNMatrix``
object was supplied to :c:func:`ARKStepSetLinearSolver` or
:c:func:`MRIStepSetLinearSolver`, the user may provide a function of type
:c:type:`ARKLsJacFn` to provide the Jacobian approximation or
:c:type:`ARKLsLinSysFn` to provide an approximation of the linear system
:math:`\mathcal{A}(t,y) = M(t) - \gamma J(t,y)`.



.. c:type:: int (*ARKLsJacFn)(realtype t, N_Vector y, N_Vector fy, SUNMatrix Jac, void* user_data, N_Vector tmp1, N_Vector tmp2, N_Vector tmp3)

   This function computes the Jacobian matrix :math:`J(t,y) =
   \dfrac{\partial f^I}{\partial y}(t,y)` (or an approximation to it).

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *y* -- the current value of the dependent variable vector, namely
        the predicted value of :math:`y(t)`.
      * *fy* -- the current value of the vector :math:`f^I(t,y)`.
      * *Jac* -- the output Jacobian matrix.
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData`
        or :c:func:`MRIStepSetUserData`.
      * *tmp1*, *tmp2*, *tmp3* -- pointers to memory allocated to
        variables of type ``N_Vector`` which can be used by an
        ARKLsJacFn as temporary storage or work space.

   **Return value:**
      An *ARKLsJacFn* function should return 0 if successful, a positive
      value if a recoverable error occurred (in which case ARKODE will
      attempt to correct, while ARKLS sets *last_flag* to
      *ARKLS_JACFUNC_RECVR*), or a negative value if it failed
      unrecoverably (in which case the integration is halted,
      :c:func:`ARKStepEvolve` or :c:func:`MRIStepEvolve` returns
      *ARK_LSETUP_FAIL* and ARKLS sets *last_flag* to *ARKLS_JACFUNC_UNRECVR*).

   **Notes:**
      Information regarding the specific
      ``SUNMatrix`` structure (e.g.~number of rows, upper/lower
      bandwidth, sparsity type) may be obtained through using the
      implementation-specific ``SUNMatrix`` interface functions
      (see :numref:`SUNMatrix` for details).

      When using a linear solver of type ``SUNLINEARSOLVER_DIRECT``, prior
      to calling the user-supplied Jacobian function, the Jacobian
      matrix :math:`J(t,y)` is zeroed out, so only nonzero elements need
      to be loaded into *Jac*.

      With the default Newton nonlinear solver, each
      call to the user's :c:func:`ARKLsJacFn` function is preceded by a call to the
      implicit :c:func:`ARKRhsFn` user function with the same :math:`(t,y)`
      arguments. Thus, the Jacobian function can use any auxiliary data that is
      computed and saved during the evaluation of :math:`f^I(t,y)`.
      In the case of a user-supplied or external nonlinear solver, this is also
      true if the nonlinear system function is evaluated prior to calling the
      linear solver setup function (see :numref:`SUNNonlinSol.API.SUNSuppliedFn` for more
      information).

      If the user's :c:type:`ARKLsJacFn` function uses difference
      quotient approximations, then it may need to access quantities not
      in the argument list, including the current step size, the
      error weights, etc.  To obtain these, the user will need to add a
      pointer to the ``ark_mem`` structure to their ``user_data``, and
      then use the ``ARKStepGet*`` or ``MRIStepGet*`` functions listed in
      :numref:`ARKODE.Usage.ARKStep.OptionalOutputs` or
      :numref:`ARKODE.Usage.MRIStep.OptionalOutputs`. The unit roundoff can be
      accessed as ``UNIT_ROUNDOFF``, which is defined in the header
      file ``sundials_types.h``.

      **dense** :math:`J(t,y)`:
      A user-supplied dense Jacobian function must load the
      *N* by *N* dense matrix *Jac* with an approximation to the Jacobian
      matrix :math:`J(t,y)` at the point :math:`(t,y)`. Utility routines
      and accessor macros for the SUNMATRIX_DENSE module are documented
      in :numref:`SUNMatrix.Dense`.

      **banded** :math:`J(t,y)`:
      A user-supplied banded Jacobian function must load the band
      matrix *Jac* with the elements of the Jacobian
      :math:`J(t,y)` at the point :math:`(t,y)`. Utility routines
      and accessor macros for the SUNMATRIX_BAND module are
      documented in :numref:`SUNMatrix.Band`.

      **sparse** :math:`J(t,y)`:
      A user-supplied sparse Jacobian function must load the
      compressed-sparse-column (CSC) or compressed-sparse-row (CSR)
      matrix *Jac* with an approximation to the Jacobian matrix
      :math:`J(t,y)` at the point :math:`(t,y)`.  Storage for *Jac*
      already exists on entry to this function, although the user should
      ensure that sufficient space is allocated in *Jac* to hold the
      nonzero values to be set; if the existing space is insufficient the
      user may reallocate the data and index arrays as needed.  Utility
      routines and accessor macros for the SUNMATRIX_SPARSE type are
      documented in :numref:`SUNMatrix.Sparse`.



.. c:type:: int (*ARKLsLinSysFn)(realtype t, N_Vector y, N_Vector fy, SUNMatrix A, SUNMatrix M, booleantype jok, booleantype *jcur, realtype gamma, void *user_data, N_Vector tmp1, N_Vector tmp2, N_Vector tmp3)

   This function computes the linear system matrix :math:`\mathcal{A}(t,y) = M(t) - \gamma J(t,y)` (or
   an approximation to it).

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *y* -- the current value of the dependent variable vector, namely the
        predicted value of :math:`y(t)`.
      * *fy* -- the current value of the vector :math:`f^I(t,y)`.
      * *A* -- the output linear system matrix.
      * *M* -- the current mass matrix (this input is ``NULL`` if :math:`M = I`).
      * *jok* -- is an input flag indicating whether the Jacobian-related data
        needs to be updated. The *jok* argument provides for the reuse of
        Jacobian data. When *jok* = ``SUNFALSE``, the Jacobian-related data
        should be recomputed from scratch. When *jok* = ``SUNTRUE`` the Jacobian
        data, if saved from the previous call to this function, can be reused
        (with the current value of *gamma*). A call with *jok* = ``SUNTRUE`` can
        only occur after a call with *jok* = ``SUNFALSE``.
      * *jcur* -- is a pointer to a flag which should be set to ``SUNTRUE`` if
        Jacobian data was recomputed, or set to ``SUNFALSE`` if Jacobian data
        was not recomputed, but saved data was still reused.
      * *gamma* -- the scalar :math:`\gamma` appearing in the Newton system matrix
        :math:`\mathcal{A}=M(t)-\gamma J(t,y)`.
      * *user_data* -- a pointer to user data, the same as the *user_data*
        parameter that was passed to :c:func:`ARKStepSetUserData` or
        :c:func:`MRIStepSetUserData`.
      * *tmp1*, *tmp2*, *tmp3* -- pointers to memory allocated to variables of
        type ``N_Vector`` which can be used by an ARKLsLinSysFn as temporary
        storage or work space.

   **Return value:**
      An *ARKLsLinSysFn* function should return 0 if successful, a positive value
      if a recoverable error occurred (in which case ARKODE will attempt to
      correct, while ARKLS sets *last_flag* to *ARKLS_JACFUNC_RECVR*), or a
      negative value if it failed unrecoverably (in which case the integration is
      halted, :c:func:`ARKStepEvolve` or :c:func:`MRIStepEvolve` returns
      *ARK_LSETUP_FAIL* and ARKLS sets *last_flag* to *ARKLS_JACFUNC_UNRECVR*).



.. _ARKODE.Usage.JTimesFn:

Jacobian-vector product (matrix-free linear solvers, ARKStep and MRIStep only)
------------------------------------------------------------------------------

When using a matrix-free linear solver module for the implicit
stage solves (i.e., a NULL-valued SUNMATRIX argument was supplied to
:c:func:`ARKStepSetLinearSolver` or :c:func:`MRIStepSetLinearSolver`,
the user may provide a function
of type :c:type:`ARKLsJacTimesVecFn` in the following form, to compute
matrix-vector products :math:`Jv`. If such a function is not supplied,
the default is a difference quotient approximation to these products.


.. c:type:: int (*ARKLsJacTimesVecFn)(N_Vector v, N_Vector Jv, realtype t, N_Vector y, N_Vector fy, void* user_data, N_Vector tmp)

   This function computes the product :math:`Jv` where :math:`J(t,y) \approx
   \dfrac{\partial f^I}{\partial y}(t,y)` (or an approximation to it).

   **Arguments:**
      * *v* -- the vector to multiply.
      * *Jv* -- the output vector computed.
      * *t* -- the current value of the independent variable.
      * *y* -- the current value of the dependent variable vector.
      * *fy* -- the current value of the vector :math:`f^I(t,y)`.
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData`
        or :c:func:`MRIStepSetUserData`.
      * *tmp* -- pointer to memory allocated to a variable of type
        ``N_Vector`` which can be used as temporary storage or work space.

   **Return value:**
      The value to be returned by the Jacobian-vector product
      function should be 0 if successful. Any other return value will
      result in an unrecoverable error of the generic Krylov solver,
      in which case the integration is halted.

   **Notes:**
      If the user's :c:type:`ARKLsJacTimesVecFn` function
      uses difference quotient approximations, it may need to access
      quantities not in the argument list.  These include the current
      step size, the error weights, etc.  To obtain these, the
      user will need to add a pointer to the ``ark_mem`` structure to
      their ``user_data``, and then use the ``ARKStepGet*`` or ``MRIStepGet*``
      functions listed in :numref:`ARKODE.Usage.ARKStep.OptionalOutputs` or
      :numref:`ARKODE.Usage.MRIStep.OptionalOutputs`. The unit roundoff can be
      accessed as ``UNIT_ROUNDOFF``, which is defined in the header
      file ``sundials_types.h``.




.. _ARKODE.Usage.JTSetupFn:

Jacobian-vector product setup (matrix-free linear solvers, ARKStep and MRIStep only)
------------------------------------------------------------------------------------

If the user's Jacobian-times-vector routine requires that any Jacobian-related data
be preprocessed or evaluated, then this needs to be done in a
user-supplied function of type :c:type:`ARKLsJacTimesSetupFn`,
defined as follows:


.. c:type:: int (*ARKLsJacTimesSetupFn)(realtype t, N_Vector y, N_Vector fy, void* user_data)

   This function preprocesses and/or evaluates any Jacobian-related
   data needed by the Jacobian-times-vector routine.

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *y* -- the current value of the dependent variable vector.
      * *fy* -- the current value of the vector :math:`f^I(t,y)`.
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData`
        or :c:func:`MRIStepSetUserData`.

   **Return value:**
      The value to be returned by the Jacobian-vector setup
      function should be 0 if successful, positive for a recoverable
      error (in which case the step will be retried), or negative for an
      unrecoverable error (in which case the integration is halted).

   **Notes:**
      Each call to the Jacobian-vector setup function is
      preceded by a call to the implicit :c:type:`ARKRhsFn` user
      function with the same :math:`(t,y)` arguments.  Thus, the setup
      function can use any auxiliary data that is computed and saved
      during the evaluation of the implicit ODE right-hand side.

      If the user's :c:type:`ARKLsJacTimesSetupFn` function uses
      difference quotient approximations, it may need to access
      quantities not in the argument list.  These include the current
      step size, the error weights, etc.  To obtain these, the
      user will need to add a pointer to the ``ark_mem`` structure to
      their ``user_data``, and then use the ``ARKStepGet*`` or
      ``MRIStepGet*`` functions listed in
      :numref:`ARKODE.Usage.ARKStep.OptionalOutputs` or
      :numref:`ARKODE.Usage.MRIStep.OptionalOutputs`. The unit roundoff can be
      accessed as ``UNIT_ROUNDOFF``, which is defined in the header
      file ``sundials_types.h``.




.. _ARKODE.Usage.PrecSolveFn:

Preconditioner solve (iterative linear solvers, ARKStep and MRIStep only)
-------------------------------------------------------------------------

If a user-supplied preconditioner is to be used with a SUNLinSol
solver module, then the user must provide a function of type
:c:type:`ARKLsPrecSolveFn` to solve the linear system :math:`Pz=r`,
where :math:`P` corresponds to either a left or right
preconditioning matrix.  Here :math:`P` should approximate (at least
crudely) the Newton matrix :math:`\mathcal{A}(t,y)=M(t)-\gamma J(t,y)`,
where :math:`M(t)` is the mass matrix and :math:`J(t,y) = \dfrac{\partial f^I}{\partial
y}(t,y)`  If preconditioning is done on both sides, the product of the two
preconditioner matrices should approximate :math:`\mathcal{A}`.



.. c:type:: int (*ARKLsPrecSolveFn)(realtype t, N_Vector y, N_Vector fy, N_Vector r, N_Vector z, realtype gamma, realtype delta, int lr, void* user_data)

   This function solves the preconditioner system :math:`Pz=r`.

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *y* -- the current value of the dependent variable vector.
      * *fy* -- the current value of the vector :math:`f^I(t,y)`.
      * *r* -- the right-hand side vector of the linear system.
      * *z* -- the computed output solution vector.
      * *gamma* -- the scalar :math:`\gamma` appearing in the Newton
        matrix given by :math:`\mathcal{A}=M(t)-\gamma J(t,y)`.
      * *delta* -- an input tolerance to be used if an iterative method
        is employed in the solution.  In that case, the residual vector
        :math:`Res = r-Pz` of the system should be made to be less than *delta*
        in the weighted :math:`l_2` norm, i.e. :math:`\left(\displaystyle \sum_{i=1}^n
        \left(Res_i * ewt_i\right)^2 \right)^{1/2} < \delta`, where :math:`\delta =`
        `delta`.  To obtain the ``N_Vector`` *ewt*, call
        :c:func:`ARKStepGetErrWeights` or :c:func:`MRIStepGetErrWeights`.
      * *lr* -- an input flag indicating whether the preconditioner
        solve is to use the left preconditioner (*lr* = 1) or the right
        preconditioner (*lr* = 2).
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData`
        or :c:func:`MRIStepSetUserData`.

   **Return value:**
      The value to be returned by the preconditioner solve
      function is a flag indicating whether it was successful. This value
      should be 0 if successful, positive for a recoverable error (in
      which case the step will be retried), or negative for an
      unrecoverable error (in which case the integration is halted).




.. _ARKODE.Usage.PrecSetupFn:

Preconditioner setup (iterative linear solvers, ARKStep and MRIStep only)
-------------------------------------------------------------------------

If the user's preconditioner routine requires that any data be
preprocessed or evaluated, then these actions need to occur within a
user-supplied function of type :c:type:`ARKLsPrecSetupFn`.


.. c:type:: int (*ARKLsPrecSetupFn)(realtype t, N_Vector y, N_Vector fy, booleantype jok, booleantype* jcurPtr, realtype gamma, void* user_data)

   This function preprocesses and/or evaluates Jacobian-related
   data needed by the preconditioner.

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *y* -- the current value of the dependent variable vector.
      * *fy* -- the current value of the vector :math:`f^I(t,y)`.
      * *jok* -- is an input flag indicating whether the Jacobian-related
        data needs to be updated. The *jok* argument provides for the
        reuse of Jacobian data in the preconditioner solve function. When
        *jok* = ``SUNFALSE``, the Jacobian-related data should be recomputed
        from scratch. When *jok* = ``SUNTRUE`` the Jacobian data, if saved from the
        previous call to this function, can be reused (with the current
        value of *gamma*). A call with *jok* = ``SUNTRUE`` can only occur
        after a call with *jok* = ``SUNFALSE``.
      * *jcurPtr* -- is a pointer to a flag which should be set to
        ``SUNTRUE`` if Jacobian data was recomputed, or set to ``SUNFALSE`` if
        Jacobian data was not recomputed, but saved data was still reused.
      * *gamma* -- the scalar :math:`\gamma` appearing in the Newton
        matrix given by :math:`\mathcal{A}=M(t)-\gamma J(t,y)`.
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData`
        or :c:func:`MRIStepSetUserData`.

   **Return value:**
      The value to be returned by the preconditioner setup
      function is a flag indicating whether it was successful. This value
      should be 0 if successful, positive for a recoverable error (in
      which case the step will be retried), or negative for an
      unrecoverable error (in which case the integration is halted).

   **Notes:**
      The operations performed by this function might include
      forming a crude approximate Jacobian, and performing an LU
      factorization of the resulting approximation to :math:`\mathcal{A} = M(t) -
      \gamma J(t,y)`.

      With the default nonlinear solver (the native SUNDIALS Newton method), each
      call to the preconditioner setup function is preceded by a call to the
      implicit :c:type:`ARKRhsFn` user function with the same :math:`(t,y)`
      arguments.  Thus, the preconditioner setup function can use any auxiliary
      data that is computed and saved during the evaluation of the implicit ODE
      right-hand side. In the case of a user-supplied or external nonlinear solver,
      this is also true if the nonlinear system function is evaluated prior to
      calling the linear solver setup function (see
      :numref:`SUNNonlinSol.API.SUNSuppliedFn` for more information).

      This function is not called in advance of every call to the
      preconditioner solve function, but rather is called only as often
      as needed to achieve convergence in the Newton iteration.

      If the user's :c:type:`ARKLsPrecSetupFn` function uses
      difference quotient approximations, it may need to access
      quantities not in the call list. These include the current step
      size, the error weights, etc.  To obtain these, the user will need
      to add a pointer to the ``ark_mem`` structure to their
      ``user_data``, and then use the ``ARKStepGet*`` or ``MRIStepGet*``
      functions listed in :numref:`ARKODE.Usage.ARKStep.OptionalOutputs` or
      :numref:`ARKODE.Usage.MRIStep.OptionalOutputs`. The unit roundoff can be
      accessed as ``UNIT_ROUNDOFF``, which is defined in the header
      file ``sundials_types.h``.



.. _ARKODE.Usage.MassFn:

Mass matrix construction (matrix-based linear solvers, ARKStep only)
--------------------------------------------------------------------

If a matrix-based mass-matrix linear solver is used (i.e., a non-NULL
SUNMATRIX was supplied to :c:func:`ARKStepSetMassLinearSolver`, the
user must provide a function of type :c:type:`ARKLsMassFn` to provide
the mass matrix approximation.



.. c:type:: int (*ARKLsMassFn)(realtype t, SUNMatrix M, void* user_data, N_Vector tmp1, N_Vector tmp2, N_Vector tmp3)

   This function computes the mass matrix :math:`M(t)` (or an approximation to it).

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *M* -- the output mass matrix.
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData()`.
      * *tmp1*, *tmp2*, *tmp3* -- pointers to memory allocated to
        variables of type ``N_Vector`` which can be used by an
        ARKLsMassFn as temporary storage or work space.

   **Return value:**
      An *ARKLsMassFn* function should return 0 if successful, or a
      negative value if it failed unrecoverably (in which case the
      integration is halted, :c:func:`ARKStepEvolve()` returns
      *ARK_MASSSETUP_FAIL* and ARKLS sets *last_flag* to
      *ARKLS_MASSFUNC_UNRECVR*).

   **Notes:**
      Information regarding the structure of the specific
      ``SUNMatrix`` structure (e.g.~number of rows, upper/lower
      bandwidth, sparsity type) may be obtained through using the
      implementation-specific ``SUNMatrix`` interface functions
      (see :numref:`SUNMatrix` for details).

      Prior to calling the user-supplied mass matrix function, the mass
      matrix :math:`M(t)` is zeroed out, so only nonzero elements need to
      be loaded into *M*.

      **dense** :math:`M(t)`:
      A user-supplied dense mass matrix function must load the *N* by *N*
      dense matrix *M* with an approximation to the mass matrix
      :math:`M(t)`. Utility routines and accessor macros for the
      SUNMATRIX_DENSE module are documented in :numref:`SUNMatrix.Dense`.

      **banded** :math:`M(t)`:
      A user-supplied banded mass matrix function must load the band
      matrix *M* with the elements of the mass matrix :math:`M(t)`.
      Utility routines and accessor macros for the SUNMATRIX_BAND module
      are documented in :numref:`SUNMatrix.Band`.

      **sparse** :math:`M(t)`:
      A user-supplied sparse mass matrix function must load the
      compressed-sparse-column (CSR) or compressed-sparse-row (CSR)
      matrix *M* with an approximation to the mass matrix :math:`M(t)`.
      Storage for *M* already exists on entry to this function, although
      the user should ensure that sufficient space is allocated in *M*
      to hold the nonzero values to be set; if the existing space is
      insufficient the user may reallocate the data and row index arrays
      as needed.  Utility routines and accessor macros for the
      SUNMATRIX_SPARSE type are documented in :numref:`SUNMatrix.Sparse`.




.. _ARKODE.Usage.MTimesFn:

Mass matrix-vector product (matrix-free linear solvers, ARKStep only)
---------------------------------------------------------------------

If a matrix-free linear solver is to be used for mass-matrix linear
systems (i.e., a NULL-valued SUNMATRIX argument was supplied to
:c:func:`ARKStepSetMassLinearSolver()` in
:numref:`ARKODE.Usage.ARKStep.Skeleton`), the user *must* provide a
function of type :c:type:`ARKLsMassTimesVecFn` in the following form, to
compute matrix-vector products :math:`M(t)\, v`.



.. c:type:: int (*ARKLsMassTimesVecFn)(N_Vector v, N_Vector Mv, realtype t, void* mtimes_data)

   This function computes the product :math:`M(t)\, v` (or an approximation to it).

   **Arguments:**
      * *v* -- the vector to multiply.
      * *Mv* -- the output vector computed.
      * *t* -- the current value of the independent variable.
      * *mtimes_data* -- a pointer to user data, the same as the
        *mtimes_data* parameter that was passed to :c:func:`ARKStepSetMassTimes()`.

   **Return value:**
      The value to be returned by the mass-matrix-vector product
      function should be 0 if successful. Any other return value will
      result in an unrecoverable error of the generic Krylov solver,
      in which case the integration is halted.



.. _ARKODE.Usage.MTSetupFn:

Mass matrix-vector product setup (matrix-free linear solvers, ARKStep only)
---------------------------------------------------------------------------

If the user's mass-matrix-times-vector routine requires that any mass
matrix-related data be preprocessed or evaluated, then this needs to
be done in a user-supplied function of type
:c:type:`ARKLsMassTimesSetupFn`, defined as follows:



.. c:type:: int (*ARKLsMassTimesSetupFn)(realtype t, void* mtimes_data)

   This function preprocesses and/or evaluates any mass-matrix-related
   data needed by the mass-matrix-times-vector routine.

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *mtimes_data* -- a pointer to user data, the same as the
        *mtimes_data* parameter that was passed to :c:func:`ARKStepSetMassTimes()`.

   **Return value:**
      The value to be returned by the mass-matrix-vector setup
      function should be 0 if successful. Any other return value will
      result in an unrecoverable error of the ARKLS mass matrix solver
      interface, in which case the integration is halted.



.. _ARKODE.Usage.MassPrecSolveFn:

Mass matrix preconditioner solve (iterative linear solvers, ARKStep only)
-------------------------------------------------------------------------

If a user-supplied preconditioner is to be used with a SUNLINEAR
solver module for mass matrix linear systems, then the user must
provide a function of type :c:type:`ARKLsMassPrecSolveFn` to solve the
linear system :math:`Pz=r`, where :math:`P` may be either a left or right
preconditioning matrix.  Here :math:`P` should approximate (at least
crudely) the mass matrix :math:`M(t)`.  If preconditioning is done on
both sides, the product of the two preconditioner matrices should
approximate :math:`M(t)`.


.. c:type:: int (*ARKLsMassPrecSolveFn)(realtype t, N_Vector r, N_Vector z, realtype delta, int lr, void* user_data)

   This function solves the preconditioner system :math:`Pz=r`.

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *r* -- the right-hand side vector of the linear system.
      * *z* -- the computed output solution vector.
      * *delta* -- an input tolerance to be used if an iterative method
        is employed in the solution.  In that case, the residual vector
        :math:`Res = r-Pz` of the system should be made to be less than *delta*
        in the weighted :math:`l_2` norm, i.e. :math:`\left(\displaystyle \sum_{i=1}^n
        \left(Res_i * ewt_i\right)^2 \right)^{1/2} < \delta`, where :math:`\delta =`
        *delta*.  To obtain the ``N_Vector`` *ewt*, call
        :c:func:`ARKStepGetErrWeights()`.
      * *lr* -- an input flag indicating whether the preconditioner
        solve is to use the left preconditioner (*lr* = 1) or the right
        preconditioner (*lr* = 2).
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData()`.

   **Return value:**
      The value to be returned by the preconditioner solve
      function is a flag indicating whether it was successful. This value
      should be 0 if successful, positive for a recoverable error (in
      which case the step will be retried), or negative for an
      unrecoverable error (in which case the integration is halted).




.. _ARKODE.Usage.MassPrecSetupFn:

Mass matrix preconditioner setup (iterative linear solvers, ARKStep only)
-------------------------------------------------------------------------

If the user's mass matrix preconditioner above requires that any
problem data be preprocessed or evaluated, then these actions need to
occur within a user-supplied function of type
:c:type:`ARKLsMassPrecSetupFn`.



.. c:type:: int (*ARKLsMassPrecSetupFn)(realtype t, void* user_data)

   This function preprocesses and/or evaluates mass-matrix-related
   data needed by the preconditioner.

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *user_data* -- a pointer to user data, the same as the
        *user_data* parameter that was passed to :c:func:`ARKStepSetUserData()`.

   **Return value:**
      The value to be returned by the mass matrix preconditioner setup
      function is a flag indicating whether it was successful. This value
      should be 0 if successful, positive for a recoverable error (in
      which case the step will be retried), or negative for an
      unrecoverable error (in which case the integration is halted).

   **Notes:**
      The operations performed by this function might include
      forming a mass matrix and performing an incomplete
      factorization of the result.  Although such operations would
      typically be performed only once at the beginning of a simulation,
      these may be required if the mass matrix can change as a function
      of time.

      If both this function and a :c:type:`ARKLsMassTimesSetupFn` are
      supplied, all calls to this function will be preceded by a call to
      the :c:type:`ARKLsMassTimesSetupFn`, so any setup performed
      there may be reused.


.. _ARKODE.Usage.VecResizeFn:

Vector resize function
--------------------------------------

For simulations involving changes to the number of equations and
unknowns in the ODE system (e.g. when using spatial adaptivity in a
PDE simulation), the ARKODE integrator may be "resized" between
integration steps, through calls to the :c:func:`ARKStepResize`,
:c:func:`ERKStepResize`, or :c:func:`MRIStepResize`
function. Typically, when performing adaptive simulations the solution
is stored in a customized user-supplied data structure, to enable
adaptivity without repeated allocation/deallocation of memory.  In
these scenarios, it is recommended that the user supply a customized
vector kernel to interface between SUNDIALS and their problem-specific
data structure.  If this vector kernel includes a function of type
:c:type:`ARKVecResizeFn` to resize a given vector implementation, then
this function may be supplied to :c:func:`ARKStepResize`,
:c:func:`ERKStepResize`, or :c:func:`MRIStepResize`, so that all
internal ARKODE vectors may be resized, instead of deleting and
re-creating them at each call.  This resize function should have the
following form:


.. c:type:: int (*ARKVecResizeFn)(N_Vector y, N_Vector ytemplate, void* user_data)

   This function resizes the vector *y* to match the dimensions of the
   supplied vector, *ytemplate*.

   **Arguments:**
      * *y* -- the vector to resize.
      * *ytemplate* -- a vector of the desired size.
      * *user_data* -- a pointer to user data, the same as the
        *resize_data* parameter that was passed to :c:func:`ARKStepResize`,
        :c:func:`ERKStepResize`, or :c:func:`MRIStepResize`.

   **Return value:**
      An *ARKVecResizeFn* function should return 0 if it successfully
      resizes the vector *y*, and a non-zero value otherwise.

   **Notes:**
      If this function is not supplied, then ARKODE will
      instead destroy the vector *y* and clone a new vector *y* off of
      *ytemplate*.




.. _ARKODE.Usage.PreInnerFn:

Pre inner integrator communication function (MRIStep only)
----------------------------------------------------------

The user may supply a function of type :c:type:`MRIStepPreInnerFn` that will be
called *before* each inner integration to perform any communication or
memory transfers of forcing data supplied by the outer integrator to the inner
integrator for the inner integration.


.. c:type:: int (*MRIStepPreInnerFn)(realtype t, N_Vector* f, int num_vecs, void* user_data)

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *f* -- an ``N_Vector`` array of outer forcing vectors.
      * *num_vecs* -- the number of vectors in the ``N_Vector`` array.
      * *user_data* -- the `user_data` pointer that was passed to
        :c:func:`MRIStepSetUserData()`.

   **Return value:**
      An *MRIStepPreInnerFn* function should return 0 if successful, a positive value
      if a recoverable error occurred, or a negative value if an unrecoverable
      error occurred. As the MRIStep module only supports fixed step sizes at this
      time any non-zero return value will halt the integration.

   **Notes:**
      In a heterogeneous computing environment if any data copies between the host
      and device vector data are necessary, this is where that should occur.


.. _ARKODE.Usage.PostInnerFn:

Post inner integrator communication function (MRIStep only)
-----------------------------------------------------------

The user may supply a function of type :c:type:`MRIStepPostInnerFn` that will be
called *after* each inner integration to perform any communication or
memory transfers of state data supplied by the inner integrator to the
outer integrator for the outer integration.


.. c:type:: int (*MRIStepPostInnerFn)(realtype t, N_Vector y, void* user_data)

   **Arguments:**
      * *t* -- the current value of the independent variable.
      * *y* -- the current value of the dependent variable vector.
      * *user_data* -- the ``user_data`` pointer that was passed to
        :c:func:`MRIStepSetUserData`.

   **Return value:**
      An :c:func:`MRIStepPostInnerFn` function should return 0 if successful, a
      positive value if a recoverable error occurred, or a negative value if an
      unrecoverable error occurred. As the MRIStep module only supports fixed step
      sizes at this time any non-zero return value will halt the integration.

   **Notes:**
      In a heterogeneous computing environment if any data copies between the host
      and device vector data are necessary, this is where that should occur.
