# 8.4. Using KINSOL for the Solution of Nonlinear Systems

This section is concerned with the use of KINSOL for the solution of nonlinear systems.

The following sections treat the header files and the layout of the user’s main program, and provide descriptions of the KINSOL user-callable functions and user-supplied functions. The sample programs described in the companion document [38] may also be helpful. Those codes may be used as templates (with the removal of some lines used in testing) and are included in the KINSOL package.

KINSOL uses various constants for both input and output. These are defined as needed in this chapter, but for convenience are also listed separately in §8.5.

The user should be aware that not all SUNLinearSolver and SUNMatrix objects are compatible with all N_Vector implementations. Details on compatibility are given in the documentation for each SUNMatrix (Chapter §10) and SUNLinearSolver (Chapter §11) implementation. For example, NVECTOR_PARALLEL is not compatible with the dense, banded, or sparse SUNMatrix types, or with the corresponding dense, banded, or sparse SUNLinearSolver objects. Please check Chapters §10 and §11 to verify compatibility between these objects. In addition to that documentation, we note that the KINBBDPRE preconditioner can only be used with NVECTOR_PARALLEL. It is not recommended to use a threaded vector object with SuperLU_MT unless it is the NVECTOR_OPENMP module, and SuperLU_MT is also compiled with OpenMP.

At this point, it is assumed that the installation of KINSOL, following the procedure described in §14, has been completed successfully.

Regardless of where the user’s application program resides, its associated compilation and load commands must make reference to the appropriate locations for the library and header files required by KINSOL. The relevant library files are

<libdir>/libsundials_kinsol.<so|a>
<libdir>/libsundials_nvec*.<so|a>
<libdir>/libsundials_sunmat*.<so|a>
<libdir>/libsundials_sunlinsol*.<so|a>
<libdir>/libsundials_sunnonlinsol*.<so|a>


where the file extension .so is typically for shared libraries and .a for static libraries. The relevant header files are located in the subdirectories

<incdir>/kinsol
<incdir>/sundials
<incdir>/nvector
<incdir>/sunmatrix
<incdir>/sunlinsol
<incdir>/sunnonlinsol


The directories libdir and incdir are the install library and include directories, respectively. For a default installation, these are <instdir>/lib or <instdir>/lib64 and <instdir>/include, respectively, where instdir is the directory where SUNDIALS was installed (see §14).

## 8.4.2. Data Types

The header file sundials_types.h contains the definition of the types:

### 8.4.2.1. Floating point types

type realtype

The type realtype can be float, double, or long double, with the default being double. The user can change the precision of the arithmetic used in the SUNDIALS solvers at the configuration stage (see SUNDIALS_PRECISION).

Additionally, based on the current precision, sundials_types.h defines BIG_REAL to be the largest value representable as a realtype, SMALL_REAL to be the smallest value representable as a realtype, and UNIT_ROUNDOFF to be the difference between $$1.0$$ and the minimum realtype greater than $$1.0$$.

Within SUNDIALS, real constants are set by way of a macro called RCONST. It is this macro that needs the ability to branch on the definition of realtype. In ANSI C, a floating-point constant with no suffix is stored as a double. Placing the suffix “F” at the end of a floating point constant makes it a float, whereas using the suffix “L” makes it a long double. For example,

#define A 1.0
#define B 1.0F
#define C 1.0L


defines A to be a double constant equal to $$1.0$$, B to be a float constant equal to $$1.0$$, and C to be a long double constant equal to $$1.0$$. The macro call RCONST(1.0) automatically expands to 1.0 if realtype is double, to 1.0F if realtype is float, or to 1.0L if realtype is long double. SUNDIALS uses the RCONST macro internally to declare all of its floating-point constants.

Additionally, SUNDIALS defines several macros for common mathematical functions e.g., fabs, sqrt, exp, etc. in sundials_math.h. The macros are prefixed with SUNR and expand to the appropriate C function based on the realtype. For example, the macro SUNRabs expands to the C function fabs when realtype is double, fabsf when realtype is float, and fabsl when realtype is long double.

A user program which uses the type realtype, the RCONST macro, and the SUNR mathematical function macros is precision-independent except for any calls to precision-specific library functions. Our example programs use realtype, RCONST, and the SUNR macros. Users can, however, use the type double, float, or long double in their code (assuming that this usage is consistent with the typedef for realtype) and call the appropriate math library functions directly. Thus, a previously existing piece of C or C++ code can use SUNDIALS without modifying the code to use realtype, RCONST, or the SUNR macros so long as the SUNDIALS libraries are built to use the corresponding precision (see §14.1.2).

### 8.4.2.2. Integer types used for indexing

type sunindextype

The type sunindextype is used for indexing array entries in SUNDIALS modules as well as for storing the total problem size (e.g., vector lengths and matrix sizes). During configuration sunindextype may be selected to be either a 32- or 64-bit signed integer with the default being 64-bit (see SUNDIALS_INDEX_SIZE).

When using a 32-bit integer the total problem size is limited to $$2^{31}-1$$ and with 64-bit integers the limit is $$2^{63}-1$$. For users with problem sizes that exceed the 64-bit limit an advanced configuration option is available to specify the type used for sunindextype (see SUNDIALS_INDEX_TYPE).

A user program which uses sunindextype to handle indices will work with both index storage types except for any calls to index storage-specific external libraries. Our C and C++ example programs use sunindextype. Users can, however, use any compatible type (e.g., int, long int, int32_t, int64_t, or long long int) in their code, assuming that this usage is consistent with the typedef for sunindextype on their architecture. Thus, a previously existing piece of C or C++ code can use SUNDIALS without modifying the code to use sunindextype, so long as the SUNDIALS libraries use the appropriate index storage type (for details see §14.1.2).

### 8.4.2.3. Boolean type

type booleantype

As ANSI C89 (ISO C90) does not have a built-in boolean data type, SUNDIALS defines the type booleantype as an int.

The advantage of using the name booleantype (instead of int) is an increase in code readability. It also allows the programmer to make a distinction between int and boolean data. Variables of type booleantype are intended to have only the two values SUNFALSE (0) and SUNTRUE (1).

### 8.4.2.4. Output formatting type

enum SUNOutputFormat

The enumerated type SUNOutputFormat defines the enumeration constants for SUNDIALS output formats

enumerator SUN_OUTPUTFORMAT_TABLE

The output will be a table of values

enumerator SUN_OUTPUTFORMAT_CSV

The output will be a comma-separated list of key and value pairs e.g., key1,value1,key2,value2,...

Note

The file scripts/sundials_csv.py provides python utility functions to read and output the data from a SUNDIALS CSV output file using the key and value pair format.

The calling program must include several header files so that various macros and data types can be used. The header file that is always required is:

• kinsol/kinsol.h the main header file for kinsol, which defines the types and various constants, and includes function prototypes. This includes the header file for KINLS, kinsol/kinsol_ls.h.

Note that kinsol.h includes sundials_types.h, which defines the types, realtype, sunindextype, and booleantype and the constants SUNFALSE and SUNTRUE.

The calling program must also include an N_Vector implementation header file, of the form nvector/nvector_*.h (see §9 for more information). This file in turn includes the header file sundials_nvector.h which defines the abstract vector data type.

If using a Newton or Picard nonlinear solver that requires the solution of a linear system, then a linear solver module header file will be required. If the linear solver is matrix-based, the linear solver header will also include a header file of the from sunmatrix/sunmatrix_*.h where * is the name of the matrix implementation compatible with the linear solver. The matrix header file provides access to the relevant matrix functions/macros and in turn includes the header file sundials_matrix.h which defines the abstract matrix data type.

Other headers may be needed, according to the choice of preconditioner, etc. For example, in the example kinFoodWeb_kry_p (see [38]), preconditioning is done with a block-diagonal matrix. For this, even though the SUNLINSOL_SPGMR linear solver is used, the header sundials/sundials_dense.h is included for access to the underlying generic dense matrix arithmetic routines.

## 8.4.4. A skeleton of the user’s main program

The following is a skeleton of the user’s main program (or calling program) for the solution of a nonlinear system problem.. Most of the steps are independent of the N_Vector, SUNMatrix, and SUNLinearSolver implementations used. For the steps that are not, refer to §9, §10, and §11 for the specific name of the function to be called or macro to be referenced.

1. Initialize parallel or multi-threaded environment (if appropriate)

For example, call MPI_Init to initialize MPI if used.

2. Create the SUNDIALS context object

Call SUNContext_Create() to allocate the SUNContext object.

3. Set the problem dimensions etc.

This generally includes the problem size N, and may include the local vector length Nlocal.

4. Create the vector with the initial guess

Construct an N_Vector of initial guess values using the appropriate functions defined by the particular N_Vector implementation (see §9 for details).

For native SUNDIALS vector implementations, use a call of the form y0 = N_VMake_***(..., ydata) if the array containing the initial values of $$y$$ already exists. Otherwise, create a new vector by making a call of the form N_VNew_***(...), and then set its elements by accessing the underlying data with a call of the form ydata = N_VGetArrayPointer(y0). Here, *** is the name of the vector implementation.

For hypre, PETSc, and Trilinos vector wrappers, first create and initialize the underlying vector, and then create an N_Vector wrapper with a call of the form y0 = N_VMake_***(yvec), where yvec is a hypre, PETSc, or Trilinos vector. Note that calls like N_VNew_***(...) and N_VGetArrayPointer(...) are not available for these vector wrappers.

5. Create matrix object (if appropriate)

If a linear solver is required (e.g., when using the default Newton solver) and the linear solver will be a matrix-based linear solver, then a template Jacobian matrix must be created by calling the appropriate constructor defined by the particular SUNMatrix implementation.

For the native SUNDIALS SUNMatrix implementations, the matrix object may be created using a call of the form SUN***Matrix(...) where *** is the name of the matrix (see §10 for details).

6. Create linear solver object (if appropriate)

If a linear solver is required (e.g., when using the default Newton solver), then the desired linear solver object must be created by calling the appropriate constructor defined by the particular SUNLinearSolver implementation.

For any of the native SUNDIALS SUNLinearSolver implementations, the linear solver object may be created using a call of the form SUNLinearSolver LS = SUNLinSol_***(...); where *** is the name of the linear solver (see §11 for details).

7. Create KINSOL object

Call KINCreate() to create the KINSOL solver object.

8. Initialize KINSOL solver

Call KINInit() to allocate internal memory.

9. Attach the linear solver (if appropriate)

If a linear solver was created above, initialize the KINLS linear solver interface by attaching the linear solver object (and matrix object, if applicable) with KINSetLinearSolver().

10. Set linear solver optional inputs (if appropriate)

See Table 8.1 for KINLS optional inputs and Chapter §11 for linear solver specific optional inputs.

11. Set optional inputs

Call KINSet*** functions to change any optional inputs that control the behavior of KINSOL from their default values. See §8.4.5.4 for details.

12. Solve problem

Call ier = KINSol(...) to solve the nonlinear problem for a given initial guess.

See KINSol() for details.

13. Get optional outputs

Call KINGet*** functions to obtain optional output. See §8.4.5.5 for details.

14. Deallocate memory

Upon completion of the integration call the following, as necessary, to free any objects or memory allocated above:

15. Finalize MPI, if used

Call MPI_Finalize to terminate MPI.

## 8.4.5. User-callable functions

This section describes the KINSOL functions that are called by the user to setup and then solve an IVP. Some of these are required. However, starting with §8.4.5.4, the functions listed involve optional inputs/outputs or restarting, and those paragraphs may be skipped for a casual use of KINSOL. In any case, refer to §8.4.4 for the correct order of these calls.

On an error, each user-callable function returns a negative value and sends an error message to the error handler routine, which prints the message on stderr by default. However, the user can set a file as error output or can provide his own error handler function (see §8.4.5.4).

### 8.4.5.1. KINSOL initialization and deallocation functions

void KINCreate(SUNContext sunctx)

The function KINCreate() instantiates a KINSOL solver object.

Arguments:
Return value:
• void

int KINInit(void *kin_mem, KINSysFn func, N_Vector tmpl)

The function KINInit() specifies the problem-defining function, allocates internal memory, and initializes KINSOL.

Arguments:
• kin_mem – pointer to the KINSOL memory block returned by KINCreate().

• func – is the CC function which computes the system function $$F(u)$$ (or $$G(u)$$ for fixed-point iteration) in the nonlinear problem. This function has the form func(u, fval, user_data). (For full details see §8.4.6.1).

• tmpl – is any N_Vector (e.g. the initial guess vector u) which is used as a template to create (by cloning) necessary vectors in kin_mem.

Return value:
• KIN_SUCCESS – The call to KINInit() was successful.

• KIN_MEM_NULL – The KINSOL memory block was not initialized through a previous call to KINCreate().

• KIN_MEM_FAIL – A memory allocation request has failed.

• KIN_ILL_INPUT – An input argument to KINInit() has an illegal value.

Notes:

If an error occurred, KINInit() sends an error message to the error handler function.

void KINFree(void **kin_mem)

The function KINFree() frees the pointer allocated by a previous call to KINCreate().

Arguments:
• kin_mem – pointer to the KINSOL solver object.

Return value:
• void

### 8.4.5.2. Linear solver specification functions

As previously explained, Newton and Picard iterations require the solution of linear systems of the form $$J\delta = -F$$. Solution of these linear systems is handled using the KINLS linear solver interface. This interface supports all valid SUNLinearSolver modules. Here, matrix-based SUNLinearSolver modules utilize SUNMatrix objects to store the Jacobian matrix $$J = F'(u)$$ and factorizations used throughout the solution process. Conversely, matrix-free SUNLinearSolver modules instead use iterative methods to solve the linear systems of equations, and only require the action of the Jacobian on a vector, $$Jv$$.

With most iterative linear solvers, preconditioning can be done on the left only, on the right only, on both the left and the right, or not at all. However, only right preconditioning is supported within KINLS. If preconditioning is done, user-supplied functions define the linear operator corresponding to a right preconditioner matrix $$P$$, which should approximate the system Jacobian matrix $$J$$. For the specification of a preconditioner, see the iterative linear solver sections in §8.4.5.4 and §8.4.6. A preconditioner matrix $$P$$ must approximate the Jacobian $$J$$, at least crudely.

To specify a generic linear solver to KINSOL, after the call to KINCreate() but before any calls to KINSol(), the user’s program must create the appropriate SUNLinearSolver object and call the function KINSetLinearSolver(), as documented below. To create the SUNLinearSolver object, the user may call one of the SUNDIALS-packaged SUNLinearSolver module constructor routines via a call of the form

SUNLinearSolver LS = SUNLinSol_*(...);


For a current list of such constructor routines see §11.

Alternately, a user-supplied SUNLinearSolver module may be created and used instead. The use of each of the generic linear solvers involves certain constants, functions and possibly some macros, that are likely to be needed in the user code. These are available in the corresponding header file associated with the specific SUNMatrix or SUNLinearSolver module in question, as described in Chapters §10 and §11.

Once this solver object has been constructed, the user should attach it to KINSOL via a call to KINSetLinearSolver(). The first argument passed to this function is the KINSOL memory pointer returned by KINCreate(); the second argument is the desired SUNLinearSolver object to use for solving Newton or Picard systems. The third argument is an optional SUNMatrix object to accompany matrix-based SUNLinearSolver inputs (for matrix-free linear solvers, the third argument should be NULL). A call to this function initializes the KINLS linear solver interface, linking it to the main KINSOL solver, and allows the user to specify additional parameters and routines pertinent to their choice of linear solver.

int KINSetLinearSolver(void *kin_mem, SUNLinearSolver LS, SUNMatrix J)

The function KINSetLinearSolver() attaches a generic SUNLinSol object LS and corresponding template Jacobian SUNMatrix object J (if applicable) to KINSOL, initializing the KINLS linear solver interface.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• LS – SUNLINSOL object to use for solving Newton linear systems.

• J – SUNMATRIX object for used as a template for the Jacobian (or NULL if not applicable).

Return value:
• KINLS_SUCCESS – The KINLS initialization was successful.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_ILL_INPUT – The KINLS interface is not compatible with the LS or J input objects or is incompatible with the current NVECTOR module.

• KINLS_SUNLS_FAIL – A call to the LS object failed.

• KINLS_MEM_FAIL – A memory allocation request failed.

Notes:

If LS is a matrix-based linear solver, then the template Jacobian matrix J will be used in the solve process, so if additional storage is required within the SUNMatrix object (e.g. for factorization of a banded matrix), ensure that the input object is allocated with sufficient size (see the documentation of the particular SUNMatrix type in Chapter §10 for further information).

The previous routines KINDlsSetLinearSolver() and KINSpilsSetLinearSolver() are now wrappers for this routine, and may still be used for backward-compatibility. However, these will be deprecated in future releases, so we recommend that users transition to the new routine name soon.

### 8.4.5.3. KINSOL solver function

This is the central step in the solution process, the call to solve the nonlinear algebraic system.

int KINSol(void *kin_mem, N_Vector u, int strategy, N_Vector u_scale, N_Vector f_scale)

The function KINSol() computes an approximate solution to the nonlinear system.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• u – vector set to initial guess by user before calling KINSol() , but which upon return contains an approximate solution of the nonlinear system $$F(u) = 0$$.

• strategy – strategy used to solve the nonlinear system. It must be of the following:

• KIN_NONE basic Newton iteration

• KIN_LINESEARCH Newton with globalization

• KIN_FP fixed-point iteration with Anderson Acceleration (no linear solver needed)

• KIN_PICARD Picard iteration with Anderson Acceleration (uses a linear solver)

• u_scale – vector containing diagonal elements of scaling matrix $$D_u$$ for vector u chosen so that the components of $$D_u\ u$$ (as a matrix multiplication) all have roughly the same magnitude when u is close to a root of $$F(u)$$.

• f_scale – vector containing diagonal elements of scaling matrix $$D_F$$ for $$F(u)$$ chosen so that the components of $$D_F\ F(u)$$ (as a matrix multiplication) all have roughly the same magnitude when u is not too near a root of $$F(u)$$. In the case of a fixed-point iteration, consider $$F(u) = G(u) - u$$.

Return value:
• KIN_SUCCESSKINSol() succeeded; the scaled norm of $$F(u)$$ is less than fnormtol.

• KIN_INITIAL_GUESS_OK – The guess u $$=u_0$$ satisfied the system $$F(u)=0$$ within the tolerances specified (the scaled norm of $$F(u_0)$$ is less than 0.01*fnormtol).

• KIN_STEP_LT_STPTOL – KINSOL stopped based on scaled step length. This means that the current iterate may be an approximate solution of the given nonlinear system, but it is also quite possible that the algorithm is “stalled” (making insufficient progress) near an invalid solution, or that the scalar scsteptol is too large (see KINSetScaledStepTol() in §8.4.5.4 to change scsteptol from its default value).

• KIN_MEM_NULL – The KINSOL memory block pointer was NULL.

• KIN_ILL_INPUT – An input parameter was invalid.

• KIN_NO_MALLOC – The KINSOL memory was not allocated by a call to KINCreate().

• KIN_MEM_FAIL – A memory allocation failed.

• KIN_LINESEARCH_NONCONV – The line search algorithm was unable to find an iterate sufficiently distinct from the current iterate, or could not find an iterate satisfying the sufficient decrease condition. Failure to satisfy the sufficient decrease condition could mean the current iterate is “close” to an approximate solution of the given nonlinear system, the difference approximation of the matrix-vector product $$J(u)\ v$$ is inaccurate, or the real scalar scsteptol is too large.

• KIN_MAXITER_REACHED – The maximum number of nonlinear iterations has been reached.

• KIN_MXNEWT_5X_EXCEEDED – Five consecutive steps have been taken that satisfy the inequality $$\|D_u p\|_{L2} > 0.99\ \texttt{mxnewtstep}$$ , where $$p$$ denotes the current step and mxnewtstep is a scalar upper bound on the scaled step length. Such a failure may mean that $$\|D_F F(u)\|_{L2}$$ asymptotes from above to a positive value, or the real scalar mxnewtstep is too small.

• KIN_LINESEARCH_BCFAIL – The line search algorithm was unable to satisfy the “beta-condition” for MXNBCF+1 nonlinear iterations (not necessarily consecutive), which may indicate the algorithm is making poor progress.

• KIN_LINSOLV_NO_RECOVERY – The user-supplied routine psolve encountered a recoverable error, but the preconditioner is already current.

• KIN_LINIT_FAIL – The KINLS initialization routine (linit) encountered an error.

• KIN_LSETUP_FAIL – The KINLS setup routine (lsetup) encountered an error; e.g., the user-supplied routine pset (used to set up the preconditioner data) encountered an unrecoverable error.

• KIN_LSOLVE_FAIL – The KINLS solve routine (lsolve) encountered an error; e.g., the user-supplied routine psolve (used to to solve the preconditioned linear system) encountered an unrecoverable error.

• KIN_SYSFUNC_FAIL – The system function failed in an unrecoverable manner.

• KIN_FIRST_SYSFUNC_ERR – The system function failed recoverably at the first call.

• KIN_REPTD_SYSFUNC_ERR – The system function had repeated recoverable errors. No recovery is possible.

Notes:

The components of vectors u_scale and f_scale should be strictly positive. KIN_SUCCESS=0, KIN_INITIAL_GUESS_OK=1, and KIN_STEP_LT_STPTOL=2. All remaining return values are negative and therefore a test flag $$< 0$$ will trap all KINSol() failures.

### 8.4.5.4. Optional input functions

There are numerous optional input parameters that control the behavior of the KINSOL solver. KINSOL provides functions that can be used to change these from their default values. Table 8.1 lists all optional input functions in KINSOL which are then described in detail in the remainder of this section, beginning with those for the main KINSOL solver and continuing with those for the KINLS linear solver interface.

We note that, on error return, all of these functions also send an error message to the error handler function. We also note that all error return values are negative, so a test retval $$<0$$ will catch any error.

Table 8.1 Optional inputs for KINSOL and KINLS

Optional input

Function name

Default

KINSOL main solver

Error handler function

KINSetErrHandlerFn()

internal fn.

Pointer to an error file

KINSetErrFile()

stderr

Info handler function

KINSetInfoHandlerFn()

internal fn.

Pointer to an info file

KINSetInfoFile()

stdout

Data for problem-defining function

KINSetUserData()

NULL

Verbosity level of output

KINSetPrintLevel()

0

Max. number of nonlinear iterations

KINSetNumMaxIters()

200

No initial matrix setup

KINSetNoInitSetup()

SUNFALSE

No residual monitoring

KINSetNoResMon()

SUNFALSE

Max. iterations without matrix setup

KINSetMaxSetupCalls()

10

Max. iterations without residual check

KINSetMaxSubSetupCalls()

5

Form of $$\eta$$ coefficient

KINSetEtaForm()

KIN_ETACHOICE1

Constant value of $$\eta$$

KINSetEtaConstValue()

0.1

Values of $$\gamma$$ and $$\alpha$$

KINSetEtaParams()

0.9 and 2.0

Values of $$\omega_{min}$$ and $$\omega_{max}$$

KINSetResMonParams()

0.00001 and 0.9

Constant value of $$\omega$$

KINSetResMonConstValue()

0.9

Lower bound on $$\epsilon$$

KINSetNoMinEps()

SUNFALSE

Max. scaled length of Newton step

KINSetMaxNewtonStep()

$$1000|D_u u_0|_2$$

Max. number of $$\beta$$-condition failures

KINSetMaxBetaFails()

10

Rel. error for D.Q. $$Jv$$

KINSetRelErrFunc()

$$\sqrt{\text{uround}}$$

Function-norm stopping tolerance

KINSetFuncNormTol()

uround$$^{1/3}$$

Scaled-step stopping tolerance

KINSetScaledStepTol()

$$\text{uround}^{2/3}$$

Inequality constraints on solution

KINSetConstraints()

NULL

Nonlinear system function

KINSetSysFunc()

none

Return the newest fixed point iteration

KINSetReturnNewest()

SUNFALSE

Fixed point/Picard damping parameter

KINSetDamping()

1.0

Anderson Acceleration subspace size

KINSetMAA()

0

Anderson Acceleration damping parameter

KINSetDampingAA()

1.0

Anderson Acceleration delay

KINSetDelayAA()

0

Anderson Acceleration orthogonalization routine

KINSetOrthAA()

KIN_ORTH_MGS

KINLS linear solver interface

Jacobian function

KINSetJacFn()

DQ

Preconditioner functions and data

KINSetPreconditioner()

NULL, NULL, NULL

Jacobian-times-vector function and data

KINSetJacTimesVecFn()

internal DQ, NULL

Jacobian-times-vector system function

KINSetJacTimesVecSysFn()

NULL

int KINSetErrFile(void *kin_mem, FILE *errfp)

The function KINSetErrFile() specifies the pointer to the file where all KINSOL messages should be directed when the default KINSOL error handler function is used.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• errfp – pointer to output file.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

Notes:

The default value for errfp is stderr.

Passing a value of NULL disables all future error message output (except for the case in which the KINSOL memory pointer is NULL). This use of KINSetErrFile() is strongly discouraged.

Warning

If KINSetErrFile() is to be called, it should be called before any other optional input functions, in order to take effect for any later error message.

int KINSetErrHandlerFn(void *kin_mem, KINErrHandlerFn ehfun, void *eh_data)

The function KINSetErrHandlerFn() specifies the optional user-defined function to be used in handling error messages.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• ehfun – is the user’s CC error handler function (see §8.4.6.2).

• eh_data – pointer to user data passed to ehfun every time it is called.

Return value:
• KIN_SUCCESS – The function ehfun and data pointer eh_data have been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

Notes:

The default internal error handler function directs error messages to the file specified by the file pointer errfp (see KINSetErrFile() above).

Error messages indicating that the KINSOL solver memory is NULL will always be directed to stderr.

int KINSetInfoFile(void *kin_mem, FILE *infofp)

The function KINSetInfoFile() specifies the pointer to the file where all informative (non-error) messages should be directed.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• infofp – pointer to output file.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

Notes:

The default value for infofp is stdout.

Deprecated since version 6.2.0: Use SUNLogger_SetInfoFilename() instead.

int KINSetInfoHandlerFn(void *kin_mem, KINInfoHandlerFn ihfun, void *ih_data)

The function KINSetInfoHandlerFn() specifies the optional user-defined function to be used in handling informative (non-error) messages.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• ihfun – is the user’s CC information handler function (see §8.4.6.3).

• ih_data – pointer to user data passed to ihfun every time it is called.

Return value:
• KIN_SUCCESS – The function ihfun and data pointer ih_data have been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

Notes:

The default internal information handler function directs informative (non-error) messages to the file specified by the file pointer infofp (see KINSetInfoFile() above).

int KINSetPrintLevel(void *kin_mem, int printfl)

The function KINSetPrintLevel() specifies the level of verbosity of the output.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• printfl – flag indicating the level of verbosity. Must be one of:

0 – no information is displayed.

1 – for each nonlinear iteration display the following information:

• the scaled Euclidean $$\ell_2$$ norm of the system function evaluated at the current iterate,

• the scaled norm of the Newton step (only if using KIN_NONE), and

• the number of function evaluations performed so far.

2 – display level 1 output and the following values for each iteration:

• $$\|F(u)\|_{D_F}$$ (only for KIN_NONE).

• $$\|F(u)\|_{D_F,\infty}$$ (for KIN_NONE and KIN_LINESEARCH).

3 – display level 2 output plus

• additional values used by the global strategy (only if using KIN_LINESEARCH), and

• statistical information for iterative linear solver modules.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument printfl had an illegal value.

Notes:

The default value for printfl is $$0$$.

int KINSetUserData(void *kin_mem, void *user_data)

The function KINSetUserData() specifies the pointer to user-defined memory that is to be passed to all user-supplied functions.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• user_data – pointer to the user-defined memory.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

Notes:

If specified, the pointer to user_data is passed to all user-supplied functions that have it as an argument. Otherwise, a NULL pointer is passed.

Warning

If user_data is needed in user linear solver or preconditioner functions, the call to KINSetUserData() must be made before the call to specify the linear solver module.

int KINSetNumMaxIters(void *kin_mem, long int mxiter)

The function KINSetNumMaxIters() specifies the maximum number of nonlinear iterations allowed.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• mxiter – maximum number of nonlinear iterations.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The maximum number of iterations was non-positive.

Notes:

The default value for mxiter is MXITER_DEFAULT $$=200$$.

int KINSetNoInitSetup(void *kin_mem, booleantype noInitSetup)

The function KINSetNoInitSetup() specifies whether an initial call to the preconditioner or Jacobian setup function should be made or not.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• noInitSetup – flag controlling whether an initial call to the preconditioner or Jacobian setup function is made (pass SUNFALSE) or not made (pass SUNTRUE).

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

Notes:

The default value for noInitSetup is SUNFALSE, meaning that an initial call to the preconditioner or Jacobian setup function will be made. A call to this function is useful when solving a sequence of problems, in which the final preconditioner or Jacobian value from one problem is to be used initially for the next problem.

int KINSetNoResMon(void *kin_mem, booleantype noNNIResMon)

The function KINSetNoResMon() specifies whether or not the nonlinear residual monitoring scheme is used to control Jacobian updating

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• noNNIResMon – flag controlling whether residual monitoring is used (pass SUNFALSE) or not used (pass SUNTRUE).

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

Notes:

When using a direct solver, the default value for noNNIResMon is SUNFALSE, meaning that the nonlinear residual will be monitored.

Warning

Residual monitoring is only available for use with matrix-based linear solver modules.

int KINSetMaxSetupCalls(void *kin_mem, long int msbset)

The function KINSetMaxSetupCalls() specifies the maximum number of nonlinear iterations that can be performed between calls to the preconditioner or Jacobian setup function.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• msbset – maximum number of nonlinear iterations without a call to the preconditioner or Jacobian setup function. Pass 0 to indicate the default.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument msbset was negative.

Notes:

The default value for msbset is MSBSET_DEFAULT=10. The value of msbset should be a multiple of msbsetsub (see KINSetMaxSubSetupCalls()).

int KINSetMaxSubSetupCalls(void *kin_mem, long int msbsetsub)

The function KINSetMaxSubSetupCalls() specifies the maximum number of nonlinear iterations between checks by the residual monitoring algorithm.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• msbsetsub – maximum number of nonlinear iterations without checking the nonlinear residual. Pass 0 to indicate the default.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument msbsetsub was negative.

Notes:

The default value for msbsetsub is MSBSET_SUB_DEFAULT $$=5$$. The value of msbset (see KINSetMaxSetupCalls()) should be a multiple of msbsetsub.

Warning

Residual monitoring is only available for use with matrix-based linear solver modules.

int KINSetEtaForm(void *kin_mem, int etachoice)

The function KINSetEtaForm() specifies the method for computing the value of the $$\eta$$ coefficient used in the calculation of the linear solver convergence tolerance.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• etachoice – flag indicating the method for computing $$\eta$$. The value must be one of KIN_ETACHOICE1 , KIN_ETACHOICE2 , or KIN_ETACONSTANT (see Chapter §8.2 for details).

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument etachoice had an illegal value.

Notes:

The default value for etachoice is KIN_ETACHOICE1. When using either KIN_ETACHOICE1 or KIN_ETACHOICE2 the safeguard

$\eta_n = \max(\eta_n, \eta_{\text{safe}})$

is applied when $$\eta_{\text{safe}} > 0.1$$. For KIN_ETACHOICE1

$\eta_{\text{safe}} = \eta_{n-1}^{\frac{1+\sqrt{5}}{2}}$

and for KIN_ETACHOICE2

$\eta_{\text{safe}} = \gamma \eta_{n-1}^\alpha$

where $$\gamma$$ and $$\alpha$$ can be set with KINSetEtaParams().

The following safeguards are always applied when using either KIN_ETACHOICE1 or KIN_ETACHOICE2 so that $$\eta_{\text{min}} \leq \eta_n \leq\eta_{\text{max}}$$:

\begin{split}\begin{aligned} \eta_n &= \max(\eta_n, \eta_{\text{min}}) \\ \eta_n &= \min(\eta_n, \eta_{\text{max}}) \end{aligned}\end{split}

where $$\eta_{\text{min}} = 10^{-4}$$ and $$\eta_{\text{max}} = 0.9$$.

int KINSetEtaConstValue(void *kin_mem, realtype eta)

The function KINSetEtaConstValue() specifies the constant value for $$\eta$$ in the case etachoice = KIN_ETACONSTANT.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• eta – constant value for $$\eta$$. Pass $$0.0$$ to indicate the default.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument eta had an illegal value

Notes:

The default value for eta is $$0.1$$. The legal values are $$0.0 <$$ eta $$\le 1.0$$.

int KINSetEtaParams(void *kin_mem, realtype egamma, realtype ealpha)

The function KINSetEtaParams() specifies the parameters $$\gamma$$ and $$\alpha$$ in the formula for $$\eta$$, in the case etachoice = KIN_ETACHOICE2.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• egamma – value of the $$\gamma$$ parameter. Pass $$0.0$$ to indicate the default.

• ealpha – value of the $$\alpha$$ parameter. Pass $$0.0$$ to indicate the default.

Return value:
• KIN_SUCCESS – The optional values have been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – One of the arguments egamma or ealpha had an illegal value.

Notes:

The default values for egamma and ealpha are $$0.9$$ and $$2.0$$, respectively. The legal values are $$0.0 <$$ egamma $$\le 1.0$$ and $$1.0<$$ ealpha $$\le 2.0$$.

int KINSetResMonConstValue(void *kin_mem, realtype omegaconst)

The function KINSetResMonConstValue() specifies the constant value for $$\omega$$ when using residual monitoring.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• omegaconst – constant value for $$\omega$$. Passing $$0.0$$ results in using Eqn. (8.4).

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument omegaconst had an illegal value

Notes:

The default value for omegaconst is $$0.9$$. The legal values are $$0.0 <$$ omegaconst $$< 1.0$$.

int KINSetResMonParams(void *kin_mem, realtype omegamin, realtype omegamax)

The function KINSetResMonParams() specifies the parameters $$\omega_{min}$$ and $$\omega_{max}$$ in the formula (8.4) for $$\omega$$.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• omegamin – value of the $$\omega_{min}$$ parameter. Pass $$0.0$$ to indicate the default.

• omegamax – value of the $$\omega_{max}$$ parameter. Pass $$0.0$$ to indicate the default.

Return value:
• KIN_SUCCESS – The optional values have been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – One of the arguments omegamin or omegamax had an illegal value.

Notes:

The default values for omegamin and omegamax are $$0.00001$$ and $$0.9$$, respectively. The legal values are $$0.0 <$$ omegamin $$<$$ omegamax $$< 1.0$$.

Warning

Residual monitoring is only available for use with matrix-based linear solver modules.

int KINSetNoMinEps(void *kin_mem, booleantype noMinEps)

The function KINSetNoMinEps() specifies a flag that controls whether or not the value of $$\epsilon$$, the scaled linear residual tolerance, is bounded from below.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• noMinEps – flag controlling the bound on $$\epsilon$$. If SUNFALSE is passed the value of $$\epsilon$$ is constrained and if SUNTRUE is passed then $$\epsilon$$ is not constrained.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

Notes:

The default value for noMinEps is SUNFALSE, meaning that a positive minimum value, equal to $$0.01$$, is applied to $$\epsilon$$ (see KINSetFuncNormTol() below).

int KINSetMaxNewtonStep(void *kin_mem, realtype mxnewtstep)

The function KINSetMaxNewtonStep() specifies the maximum allowable scaled length of the Newton step.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• mxnewtstep – maximum scaled step length $$(\geq 0.0)$$. Pass $$0.0$$ to indicate the default.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The input value was negative.

Notes:

The default value of mxnewtstep is $$1000\, \| u_0 \|_{D_u}$$, where $$u_0$$ is the initial guess.

int KINSetMaxBetaFails(void *kin_mem, realtype mxnbcf)

The function KINSetMaxBetaFails() specifies the maximum number of $$\beta$$-condition failures in the linesearch algorithm.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• mxnbcf – maximum number of $$\beta$$ -condition failures. Pass $$0.0$$ to indicate the default.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUTmxnbcf was negative.

Notes:

The default value of mxnbcf is MXNBCF_DEFAULT $$=10$$.

int KINSetRelErrFunc(void *kin_mem, realtype relfunc)

The function KINSetRelErrFunc() specifies the relative error in computing $$F(u)$$, which is used in the difference quotient approximation to the Jacobian matrix [see Eq. (8.6) ] or the Jacobian-vector product [see Eq. (8.8) ]. The value stored is $$\sqrt{\texttt{relfunc}}$$.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• relfunc – relative error in $$F(u)$$ ($$\texttt{relfunc} \geq 0.0$$). Pass $$0.0$$ to indicate the default.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The relative error was negative.

Notes:

The default value for relfunc is $$U$$ = unit roundoff.

int KINSetFuncNormTol(void *kin_mem, realtype fnormtol)

The function KINSetFuncNormTol() specifies the scalar used as a stopping tolerance on the scaled maximum norm of the system function $$F(u)$$.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• fnormtol – tolerance for stopping based on scaled function norm $$(\geq 0.0)$$. Pass $$0.0$$ to indicate the default.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The tolerance was negative.

Notes:

The default value for fnormtol is (unit roundoff) $$^{1/3}$$.

int KINSetScaledStepTol(void *kin_mem, realtype scsteptol)

The function KINSetScaledStepTol() specifies the scalar used as a stopping tolerance on the minimum scaled step length.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• scsteptol – tolerance for stopping based on scaled step length $$(\geq 0.0)$$. Pass $$0.0$$ to indicate the default.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The tolerance was non-positive.

Notes:

The default value for scsteptol is (unit roundoff) $$^{2/3}$$.

int KINSetConstraints(void *kin_mem, N_Vector constraints)

The function KINSetConstraints() specifies a vector that defines inequality constraints for each component of the solution vector $$u$$.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• constraints – vector of constraint flags. If constraints[i] is

• $$0.0$$ then no constraint is imposed on $$u_i$$.

• $$1.0$$ then $$u_i$$ will be constrained to be $$u_i \ge 0.0$$.

• $$-1.0$$ then $$u_i$$ will be constrained to be $$u_i \le 0.0$$.

• $$2.0$$ then $$u_i$$ will be constrained to be $$u_i > 0.0$$.

• $$-2.0$$ then $$u_i$$ will be constrained to be $$u_i < 0.0$$.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The constraint vector contains illegal values.

Notes:

The presence of a non-NULL constraints vector that is not $$0.0$$ in all components will cause constraint checking to be performed. If a NULL vector is supplied, constraint checking will be disabled. The function creates a private copy of the constraints vector. Consequently, the user-supplied vector can be freed after the function call, and the constraints can only be changed by calling this function.

int KINSetSysFunc(void *kin_mem, KINSysFn func)

The function KINSetSysFunc() specifies the user-provided function that evaluates the nonlinear system function $$F(u)$$ or $$G(u)$$.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• func – user-supplied function that evaluates $$F(u)$$ (or $$G(u)$$ for fixed-point iteration).

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument func was NULL.

Notes:

The nonlinear system function is initially specified through KINInit(). The option of changing the system function is provided for a user who wishes to solve several problems of the same size but with different functions.

The function KINSetReturnNewest() specifies if the fixed point iteration should return the newest iteration or the iteration consistent with the last function evaluation.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• ret_newestSUNTRUE – return the newest iteration. SUNFALSE – return the iteration consistent with the last function evaluation.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

Notes:

The default value of ret_newest is SUNFALSE.

int KINSetDamping(void *kin_mem, realtype beta)

The function KINSetDamping() specifies the value of the damping parameter in the fixed point or Picard iteration.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• beta – the damping parameter value $$0 < beta \leq 1.0$$.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument beta was zero or negative.

Notes:

This function sets the damping parameter value, which needs to be greater than zero and less than one if damping is to be used. A value $$\geq 1$$ disables damping. The default value of beta is 1.0, indicating no damping. To set the damping parameter used in Anderson acceleration see KINSetDampingAA(). With the fixed point iteration the difference between successive iterations is used to determine convergence. As such, when damping is enabled, the tolerance used to stop the fixed point iteration is scaled by beta to account for the effects of damping. If beta is extremely small (close to zero), this can lead to an excessively tight tolerance.

int KINSetMAA(void *kin_mem, long int maa)

The function KINSetMAA() specifies the size of the subspace used with Anderson acceleration in conjunction with Picard or fixed-point iteration.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• maa – subspace size for various methods. A value of 0 means no acceleration, while a positive value means acceleration will be done.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument maa was negative.

Notes:

This function sets the subspace size, which needs to be $$> 0$$ if Anderson Acceleration is to be used. It also allocates additional memory necessary for Anderson Acceleration. The default value of maa is 0, indicating no acceleration. The value of maa should always be less than mxiter. This function MUST be called before calling KINInit(). If the user calls the function KINSetNumMaxIters, that call should be made before the call to KINSetMAA, as the latter uses the value of mxiter.

int KINSetDampingAA(void *kin_mem, realtype beta)

The function KINSetDampingAA() specifies the value of the Anderson acceleration damping paramter.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• beta – the damping parameter value $$0 < beta \leq 1.0$$.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument beta was zero or negative.

Notes:

This function sets the damping parameter value, which needs to be greater than zero and less than one if damping is to be used. A value $$\geq 1$$ disables damping. The default value of beta is 1.0, indicating no damping. When delaying the start of Anderson acceleration with KINSetDelayAA(), use KINSetDamping() to set the damping parameter in the fixed point or Picard iterations before Anderson acceleration begins. When using Anderson acceleration without delay, the value provided to KINSetDampingAA() is applied to all iterations and any value provided to KINSetDamping() is ignored.

int KINSetDelayAA(void *kin_mem, long int delay)

The function KINSetDelayAA() specifies the number of iterations to delay the start of Anderson acceleration.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• delay – the number of iterations to delay Anderson acceleration.

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument delay was less than zero.

Notes:

The default value of delay is 0, indicating no delay.

int KINSetOrthAA(void *kin_mem, int orthaa)

The function KINSetOrthAA() specifies the orthogonalization routine to be used in the QR factorization portion of Anderson acceleration.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• orthaa – the orthogonalization routine parameter. Can be set to any of

the following

• KIN_ORTH_MGS – Modified Gram Schmidt (default)

• KIN_ORTH_ICWY – Inverse Compact WY Modified Gram Schmidt

• KIN_ORTH_CGS2 – Classical Gram Schmidt with Reorthogonalization (CGS2)

• KIN_ORTH_DCGS2 – Classical Gram Schmidt with Delayed Reorthogonlization

Return value:
• KIN_SUCCESS – The optional value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – The argument orthaa was not one of the predefined orthogonalization routines defined in KINSOL.

Note

This function must be called before calling KINInit().

An example of how to use this function can be found in examples/kinsol/serial/kinAnalytic_fp.c

#### 8.4.5.4.1. Linear solver interface optional input functions

For matrix-based linear solver modules, the KINLS solver interface needs a function to compute an approximation to the Jacobian matrix $$J(u)$$. This function must be of type KINLsJacFn. The user can supply a Jacobian function, or if using the SUNMATRIX_DENSE or SUNMATRIX_BAND modules for $$J$$ can use the default internal difference quotient approximation that comes with the KINLS solver. To specify a user-supplied Jacobian function jac, KINLS provides the function KINSetJacFn(). The KINLS interface passes the pointer user_data to the Jacobian function. This allows the user to create an arbitrary structure with relevant problem data and access it during the execution of the user-supplied Jacobian function, without using global data in the program. The pointer user_data may be specified through KINSetUserData().

int KINSetJacFn(void *kin_mem, KINLsJacFn jac)

The function KINSetJacFn() specifies the Jacobian approximation function to be used for a matrix-based solver within the KINLS interface.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• jac – user-defined Jacobian approximation function. See KINLsJacFn for more details.

Return value:
• KINLS_SUCCESS – The optional value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver interface has not been initialized.

Notes:

This function must be called after the KINLS linear solver interface has been initialized through a call to KINSetLinearSolver(). By default, KINLS uses an internal difference quotient function for the SUNMATRIX_DENSE and SUNMATRIX_BAND modules. If NULL is passed to jac, this default function is used. An error will occur if no jac is supplied when using other matrix types.

Warning

The previous routine KINDlsSetJacFn() is now a wrapper for this routine, and may still be used for backward-compatibility. However, this will be deprecated in future releases, so we recommend that users transition to the new routine name soon.

When using matrix-free linear solver modules, the KINLS linear solver interface requires a function to compute an approximation to the product between the Jacobian matrix $$J(u)$$ and a vector $$v$$. The user can supply his/her own Jacobian-times-vector approximation function, or use the internal difference quotient approximation that comes with the KINLS solver interface.

A user-defined Jacobian-vector function must be of type KINLsJacTimesVecFn and can be specified through a call to KINLsSetJacTimesVecFn() (see §8.4.6.5 for specification details). The pointer user_data received through KINSetUserData() (or a pointer to NULL if user_data was not specified) is passed to the Jacobian-times-vector function jtimes each time it is called. This allows the user to create an arbitrary structure with relevant problem data and access it during the execution of the user-supplied functions without using global data in the program.

int KINSetJacTimesVecFn(void *kin_mem, KINLsJacTimesVecFn jtimes)

The function KINSetJacTimesVecFn() specifies the Jacobian-vector product function.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• jtimes – user-defined Jacobian-vector product function.

Return value:
• KINLS_SUCCESS – The optional value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

• KINLS_SUNLS_FAIL – An error occurred when setting up the system matrix-times-vector routines in the SUNLINSOL object used by the KINLS interface.

Notes:

The default is to use an internal difference quotient for jtimes. If NULL is passed as jtimes, this default is used. This function must be called after the KINLS linear solver interface has been initialized through a call to KINSetLinearSolver(). The function type KINLsJacTimesVecFn is described in §8.4.6.5. The previous routine KINSpilsSetJacTimesVecFn() is now a wrapper for this routine, and may still be used for backward-compatibility. However, this will be deprecated in future releases, so we recommend that users transition to the new routine name soon.

When using the internal difference quotient the user may optionally supply an alternative system function for use in the Jacobian-vector product approximation by calling KINSetJacTimesVecSysFn(). The alternative system function should compute a suitable (and differentiable) approximation of the system function provided to KINInit(). For example, as done in [44] when solving the nonlinear systems that arise in the implicit integration of ordinary differential equations, the alternative function may use lagged values when evaluating a nonlinearity to avoid differencing a potentially non-differentiable factor.

int KINSetJacTimesVecSysFn(void *kin_mem, KINSysFn jtimesSysFn)

The function KINSetJacTimesVecSysFn() specifies an alternative system function for use in the internal Jacobian-vector product difference quotient approximation.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• jtimesSysFn – is the CC function which computes the alternative system function to use in Jacobian-vector product difference quotient approximations. This function has the form func(u, fval, user_data). (For full details see §8.4.6.1.)

Return value:
• KINLS_SUCCESS – The optional value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

• KINLS_ILL_INPUT – The internal difference quotient approximation is disabled.

Notes:

The default is to use the system function provided to KINInit() in the internal difference quotient. If the input system function is NULL, the default is used. This function must be called after the KINLS linear solver interface has been initialized through a call to KINSetLinearSolver().

When using an iterative linear solver, the user may supply a preconditioning operator to aid in solution of the system. This operator consists of two user-supplied functions, psetup and psolve, that are supplied to KINLS using the function KINSetPreconditioner(). The psetup function supplied to this routine should handle evaluation and preprocessing of any Jacobian data needed by the user’s preconditioner solve function, psolve. Both of these functions are fully specified in §8.4.6. The user data pointer received through KINSetUserData() (or a pointer to NULL if user data was not specified) is passed to the psetup and psolve functions. This allows the user to create an arbitrary structure with relevant problem data and access it during the execution of the user-supplied preconditioner functions without using global data in the program.

int KINSetPreconditioner(void *kin_mem, KINLsPrecSetupFn psetup, KINLsPrecSolveFn psolve)

The function KINSetPreconditioner() specifies the preconditioner setup and solve functions.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• psetup – user-defined function to set up the preconditioner. See KINLsPrecSetupFn for more details. Pass NULL if no setup is necessary.

• psolve – user-defined preconditioner solve function. See KINLsPrecSolveFn for more details.

Return value:
• KINLS_SUCCESS – The optional values have been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

• KINLS_SUNLS_FAIL – An error occurred when setting up preconditioning in the SUNLinearSolver object used by the KINLS interface.

Notes:

The default is NULL for both arguments (i.e., no preconditioning). This function must be called after the KINLS linear solver interface has been initialized through a call to KINSetLinearSolver().

Warning

The previous routine KINSpilsSetPreconditioner() is now a wrapper for this routine, and may still be used for backward-compatibility. However, this will be removed in future releases, so we recommend that users transition to the new routine name soon.

### 8.4.5.5. Optional output functions

KINSOL provides an extensive list of functions that can be used to obtain solver performance information. Table 8.2 lists all optional output functions in KINSOL, which are then described in detail in the remainder of this section, beginning with those for the main KINSOL solver and continuing with those for the KINLS linear solver interface. Where the name of an output from a linear solver module would otherwise conflict with the name of an optional output from the main solver, a suffix LS (for Linear Solver) has been added here (e.g., lenrwLS).

Table 8.2 Optional outputs from KINSOL and KINLS

Optional output

Function name

KINSOL main solver

Size of KINSOL real and integer workspaces

KINGetWorkSpace()

Number of function evaluations

KINGetNumFuncEvals()

Number of nonlinear iterations

KINGetNumNonlinSolvIters()

Number of $$\beta$$-condition failures

KINGetNumBetaCondFails()

Number of backtrack operations

KINGetNumBacktrackOps()

Scaled norm of $$F$$

KINGetFuncNorm()

Scaled norm of the step

KINGetStepLength()

User data pointer

KINGetUserData()

Print all statistics

KINPrintAllStats()

Name of constant associated with a return flag

KINGetReturnFlagName()

KINLS linear solver interface

Stored Jacobian of the nonlinear system

KINGetJac()

Nonlinear iteration number at which the Jacobian was evaluated

KINGetJacNumIters()

Size of real and integer workspaces

KINGetLinWorkSpace()

No. of Jacobian evaluations

KINGetNumJacEvals()

No. of $$F$$ calls for D.Q. Jacobian[-vector] evals.

KINGetNumLinFuncEvals()

No. of linear iterations

KINGetNumLinIters()

No. of linear convergence failures

KINGetNumLinConvFails()

No. of preconditioner evaluations

KINGetNumPrecEvals()

No. of preconditioner solves

KINGetNumPrecSolves()

No. of Jacobian-vector product evaluations

KINGetNumJtimesEvals()

Last return from a KINLS function

KINGetLastLinFlag()

Name of constant associated with a return flag

KINGetLinReturnFlagName()

#### 8.4.5.5.1. Main solver optional output functions

KINSOL provides several user-callable functions that can be used to obtain different quantities that may be of interest to the user, such as solver workspace requirements and solver performance statistics. These optional output functions are described next.

int KINGetWorkSpace(void *kin_mem, long int lenrw, long int leniw)

The function KINGetWorkSpace() returns the KINSOL integer and real workspace sizes.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• lenrw – the number of realtype values in the KINSOL workspace.

• leniw – the number of integer values in the KINSOL workspace.

Return value:
• KIN_SUCCESS – The optional output values have been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

Notes:

KINSOL solver In terms of the problem size $$N$$, the actual size of the real workspace is $$17 + 5 N$$ realtype words. The real workspace is increased by an additional $$N$$ words if constraint checking is enabled (see KINSetConstraints()).

The actual size of the integer workspace (without distinction between int and long int) is $$22 + 5 N$$ (increased by $$N$$ if constraint checking is enabled).

int KINGetNumFuncEvals(void *kin_mem, long int nfevals)

The function KINGetNumFuncEvals() returns the number of evaluations of the system function.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• nfevals – number of calls to the user-supplied function that evaluates $$F(u)$$.

Return value:
• KIN_SUCCESS – The optional output value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

int KINGetNumNonlinSolvIters(void *kin_mem, long int nniters)

The function KINGetNumNonlinSolvIters() returns the number of nonlinear iterations.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• nniters – number of nonlinear iterations.

Return value:
• KIN_SUCCESS – The optional output value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

int KINGetNumBetaCondFails(void *kin_mem, long int nbcfails)

The function KINGetNumBetaCondFails() returns the number of $$\beta$$-condition failures.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• nbcfails – number of $$\beta$$ -condition failures.

Return value:
• KIN_SUCCESS – The optional output value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

int KINGetNumBacktrackOps(void *kin_mem, long int nbacktr)

The function KINGetNumBacktrackOps() returns the number of backtrack operations (step length adjustments) performed by the line search algorithm.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• nbacktr – number of backtrack operations.

Return value:
• KIN_SUCCESS – The optional output value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

int KINGetFuncNorm(void *kin_mem, realtype fnorm)

The function KINGetFuncNorm() returns the scaled Euclidean $$\ell_2$$ norm of the nonlinear system function $$F(u)$$ evaluated at the current iterate.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• fnorm – current scaled norm of $$F(u)$$.

Return value:
• KIN_SUCCESS – The optional output value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

int KINGetStepLength(void *kin_mem, realtype steplength)

The function KINGetStepLength() returns the scaled Euclidean $$\ell_2$$ norm of the step used during the previous iteration.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• steplength – scaled norm of the Newton step.

Return value:
• KIN_SUCCESS – The optional output value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

int KINGetUserData(void *kin_mem, void **user_data)

The function KINGetUserData() returns the user data pointer provided to KINSetUserData().

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• user_data – memory reference to a user data pointer.

Return value:
• KIN_SUCCESS – The optional output value has been successfully set.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

New in version 6.3.0.

int KINPrintAllStats(void *cvode_mem, FILE *outfile, SUNOutputFormat fmt)

The function KINPrintAllStats() outputs all of the nonlinear solver, linear solver, and other statistics.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• outfile – pointer to output file.

• fmt – the output format:

Return value:
• KIN_SUCCESS – The output was successfully.

• KIN_MEM_NULL – The kin_mem pointer is NULL.

• KIN_ILL_INPUT – An invalid formatting option was provided.

Note

The file scripts/sundials_csv.py provides python utility functions to read and output the data from a SUNDIALS CSV output file using the key and value pair format.

New in version 6.2.0.

char *KINGetReturnFlagName(int flag)

The function KINGetReturnFlagName() returns the name of the KINSOL constant corresponding to flag.

Arguments:
• flag – return flag from a KINSOL function.

Return value:
• A string containing the name of the corresponding constant

#### 8.4.5.5.2. KINLS linear solver interface optional output functions

The following optional outputs are available from the KINLS modules:

int KINGetJac(void *kin_mem, SUNMatrix *J)

Returns the internally stored copy of the Jacobian matrix of the nonlinear system function.

Parameters
• kin_mem – the KINSOL solver object

• J – the Jacobian matrix

Return values
• KINLS_SUCCESS – the output value has been successfully set

• KINLS_MEM_NULLkin_mem was NULL

• KINLS_LMEM_NULL – the linear solver interface has not been initialized

Warning

With linear solvers that overwrite the input Jacobian matrix as part of the linear solver setup (e.g., performing an in-place LU factorization) the matrix returned by KINGetJac() may differ from the matrix returned by the last Jacobian evaluation.

Warning

This function is provided for debugging purposes and the values in the returned matrix should not be altered.

int KINGetJacNumIters(void *kin_mem, sunrealtype *nni_J)

Returns the nonlinear iteration number at which the Jacobian was evaluated.

Parameters
• kin_mem – the KINSOL memory structure

• nni_J – the nonlinear iteration number

Return values
• KINLS_SUCCESS – the output value has been successfully set

• KINLS_MEM_NULLkin_mem was NULL

• KINLS_LMEM_NULL – the linear solver interface has not been initialized

int KINGetLinWorkSpace(void *kin_mem, long int *lenrwLS, long int *leniwLS)

The function KINGetLinWorkSpace() returns the sizes of the real and integer workspaces used by the KINLS linear solver interface.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• lenrwLS – the number of real values in the KINLS workspace.

• leniwLS – the number of integer values in the KINLS workspace.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

Notes:

The workspace requirements reported by this routine correspond only to memory allocated within this interface and to memory allocated by the SUNLinearSolver object attached to it. The template Jacobian matrix allocated by the user outside of KINLS is not included in this report.

Warning

The previous routines KINDlsGetWorkspace() and KINSpilsGetWorkspace() are now deprecated.

int KINGetNumJacEvals(void *kin_mem, long int *njevals)

The function KINGetNumJacEvals() returns the cumulative number of calls to the KINLS Jacobian approximation function.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• njevals – the cumulative number of calls to the Jacobian function total so far.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

Warning

The previous routine KINDlsGetNumJacEvals() is now deprecated,

int KINGetNumLinFuncEvals(void *kin_mem, long int *nrevalsLS)

The function KINGetNumLinResEvals() returns the cumulative number of calls to the user residual function due to the finite difference Jacobian approximation or finite difference Jacobian-vector product approximation.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• nrevalsLS – the cumulative number of calls to the user residual function.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

Notes:

The value nrevalsLS is incremented only if one of the default internal difference quotient functions is used.

Warning

The previous routines KINDlsGetNumRhsEvals() and KINSpilsGetNumRhsEvals() are now deprecated.

int KINGetNumLinIters(void *kin_mem, long int *nliters)

The function KINGetNumLinIters() returns the cumulative number of linear iterations.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• nliters – the current number of linear iterations.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

Warning

The previous routine KINSpilsGetNumLinIters() is now deprecated.

int KINGetNumLinConvFails(void *kin_mem, long int *nlcfails)

The function KINGetNumLinConvFails() returns the cumulative number of linear convergence failures.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• nlcfails – the current number of linear convergence failures.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

Warning

The previous routine KINSpilsGetNumConvFails() is now deprecated.

int KINGetNumPrecEvals(void *kin_mem, long int *npevals)

The function KINGetNumPrecEvals() returns the cumulative number of preconditioner evaluations, i.e., the number of calls made to psetup.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• npevals – the cumulative number of calls to psetup.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

Warning

The previous routine KINSpilsGetNumPrecEvals() is now deprecated.

int KINGetNumPrecSolves(void *kin_mem, long int *npsolves)

The function KINGetNumPrecSolves() returns the cumulative number of calls made to the preconditioner solve function, psolve.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• npsolves – the cumulative number of calls to psolve.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

Warning

The previous routine KINSpilsGetNumPrecSolves() is now deprecated.

int KINGetNumJtimesEvals(void *kin_mem, long int *njvevals)

The function KINGetNumJtimesEvals() returns the cumulative number of calls made to the Jacobian-vector product function, jtimes.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• njvevals – the cumulative number of calls to jtimes.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

Warning

The previous routine KINSpilsGetNumJtimesEvals() is now deprecated.

int KINGetLastLinFlag(void *kin_mem, long int *lsflag)

The function KINGetLastLinFlag() returns the last return value from an KINLS routine.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• lsflag – the value of the last return flag from an KINLS function.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer is NULL.

• KINLS_LMEM_NULL – The KINLS linear solver has not been initialized.

Notes:

If the KINLS setup function failed (i.e., KINSolve() returned KIN_LSETUP_FAIL) when using the SUNLINSOL_DENSE or SUNLINSOL_BAND modules, then the value of lsflag is equal to the column index (numbered from one) at which a zero diagonal element was encountered during the LU factorization of the (dense or banded) Jacobian matrix.

If the KINLS setup function failed when using another SUNLinearSolver object, then lsflag will be SUNLS_PSET_FAIL_UNREC, SUNLS_ASET_FAIL_UNREC, or SUNLS_PACKAGE_FAIL_UNREC.

If the KINLS solve function failed (KINSolve() returned KIN_LSOLVE_FAIL), lsflag contains the error return flag from the SUNLinearSolver object, which will be one of: SUNLS_MEM_NULL, indicating that the SUNLinearSolver memory is NULL; SUNLS_ATIMES_FAIL_UNREC, indicating an unrecoverable failure in the $$J*v$$ function; SUNLS_PSOLVE_FAIL_UNREC, indicating that the preconditioner solve function psolve failed unrecoverably; SUNLS_GS_FAIL, indicating a failure in the Gram-Schmidt procedure (generated only in SPGMR or SPFGMR); SUNLS_QRSOL_FAIL, indicating that the matrix $$R$$ was found to be singular during the QR solve phase (SPGMR and SPFGMR only); or SUNLS_PACKAGE_FAIL_UNREC, indicating an unrecoverable failure in an external iterative linear solver package.

Warning

The previous routines KINDlsGetLastFlag() and KINSpilsGetLastFlag() are now deprecated.

char *KINGetLinReturnFlagName(long int lsflag)

The function KINGetLinReturnFlagName() returns the name of the KINLS constant corresponding to lsflag.

Arguments:
• flag – the flag returned by a call to an KINSOL function

Return value:
• char* – the flag name string or if $$1 \leq \mathtt{lsflag} \leq N$$ (LU factorization failed), this function returns “NONE”.

Warning

The previous routines KINDlsGetReturnFlagName() and KINSpilsGetReturnFlagName() are now deprecated.

## 8.4.6. User-supplied functions

The user-supplied functions consist of one function defining the nonlinear system, (optionally) a function that handles error and warning messages, (optionally) a function that handles informational messages, (optionally) one or two functions that provides Jacobian-related information for the linear solver, and (optionally) one or two functions that define the preconditioner for use in any of the Krylov iterative algorithms.

### 8.4.6.1. Problem defining function

The user must provide a function of type KINSysFn defined as follows:

typedef int (*KINSysFn)(N_Vector u, N_Vector fval, void *user_data)

This function computes the $$F(u)$$ (or $$G(u)$$ for fixed-point iteration and Anderson acceleration) for a given value of the vector $$u$$.

Arguments:
• u – is the current value of the dependent variable vector, $$u$$

• fval – is the output vector $$F(u)$$

• user_data – is a pointer to user data, the same as the user_data pointer parameter passed to KINSetUserData()

Return value:

An KINSysFn function type should return a value of $$0$$ if successful, a positive value if a recoverable error occurred (in which case KINSOL will attempt to correct), or a negative value if a nonrecoverable error occurred. In the last case, the integrator halts. If a recoverable error occurred, the integrator will attempt to correct and retry.

Notes:

Allocation of memory for fval is handled within KINSOL.

### 8.4.6.2. 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 KINSetErrFile()), the user may provide a function of type KINErrHandlerFn to process any such messages. The function type KINErrHandlerFn is defined as follows:

typedef void (*KINErrHandlerFn)(int error_code, const char *module, const char *function, char *msg, void *user_data)

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

Arguments:
• error_code – is the error code

• module – is the name of the KINSOL module reporting the error

• function – is the name of the function in which the error occurred

• eH_data – is a pointer to user data, the same as the eh_data parameter passed to KINSetErrHandlerFn()

Return value:

This function has no return value.

Notes:

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

### 8.4.6.3. Informational message handler function

As an alternative to the default behavior of directing informational (meaning non-error) messages to the file pointed to by infofp (see KINSetInfoFile()), the user may provide a function of type KINInfoHandlerFn to process any such messages. The function type KINInfoHandlerFn is defined as follows:

typedef void (*KINInfoHandlerFn)(const char *module, const char *function, char *msg, void *ih_data)

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

Arguments:
• error_code – is the error code

• module – is the name of the KINSOL module reporting the error

• function – is the name of the function in which the error occurred

• ih_data – is a pointer to user data, the same as the ih_data parameter passed to KINSetInfoHandlerFn()

Return value:

This function has no return value.

### 8.4.6.4. Jacobian construction (matrix-based linear solvers)

If a matrix-based linear solver module is used (i.e. a non-NULL SUNMatrix object was supplied to KINSetLinearSolver()), the user may provide a function of type KINLsJacFn defined as follows:

typedef int (*KINLsJacFn)(N_Vector u, N_Vector fu, SUNMatrix J, void *user_data, N_Vector tmp1, N_Vector tmp2)

This function computes the Jacobian matrix $$J(u)$$ (or an approximation to it).

Arguments:
• u – is the current (unscaled) iterate.

• fu – is the current value of the vector, $$F(u)$$.

• J – is the output (approximate) Jacobian matrix (of type SUNMatrix), $$F'(u)$$.

• user_data - is a pointer to user data, the same as the user_data parameter passed to KINSetUserData().

• tmp1, tmp2, – are pointers to memory allocated for variables of type N_Vector which can be used by KINLsJacFn function as temporary storage or work space.

Return value:

An KINLsJacFn should return $$0$$ if successful, or a non-zero value otherwise.

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 Chapter §10 for details).

With direct linear solvers (i.e., linear solvers with type SUNLINEARSOLVER_DIRECT), the Jacobian matrix $$J(u)$$ is zeroed out prior to calling the user-supplied Jacobian function so only nonzero elements need to be loaded into J.

If the user’s KINLsJacFn function uses difference quotient approximations, it may need to access quantities not in the call list. These quantities may include the scale vectors and the unit roundoff. To obtain the scale vectors, the user will need to add to user_data pointers to u_scale and/or f_scale as needed. The unit roundoff can be accessed as UNIT_ROUNDOFF defined in sundials_types.h.

dense:

A user-supplied dense Jacobian function must load the N $$\times$$ N dense matrix J with an approximation to the Jacobian matrix $$J(u)$$ at the point (u). The accessor macros SM_ELEMENT_D and SM_COLUMN_D allow the user to read and write dense matrix elements without making explicit references to the underlying representation of the SUNMATRIX_DENSE type. SM_ELEMENT_D(J, i, j) references the (i, j)-th element of the dense matrix J (with i, j$$= 0\ldots \texttt{N}-1$$). This macro is meant for small problems for which efficiency of access is not a major concern. Thus, in terms of the indices $$m$$ and $$n$$ ranging from $$1$$ to $$N$$, the Jacobian element $$J_{m,n}$$ can be set using the statement SM_ELEMENT_D(J, m-1, n-1) = $$J_{m,n}$$. Alternatively, SM_COLUMN_D(J, j) returns a pointer to the first element of the j-th column of J (with j$$= 0\ldots \texttt{N}-1$$), and the elements of the j-th column can then be accessed using ordinary array indexing. Consequently, $$J_{m,n}$$ can be loaded using the statements col_n = SM_COLUMN_D(J, n-1); col_n[m-1] = $$J_{m,n}$$. For large problems, it is more efficient to use SM_COLUMN_D than to use SM_ELEMENT_D. Note that both of these macros number rows and columns starting from $$0$$. The SUNMATRIX_DENSE type and accessor macros are documented in §10.9.

banded:

A user-supplied banded Jacobian function must load the N $$\times$$ N banded matrix J with an approximation to the Jacobian matrix $$J(u)$$ at the point (u). The accessor macros SM_ELEMENT_B, SM_COLUMN_B, and SM_COLUMN_ELEMENT_B allow the user to read and write banded matrix elements without making specific references to the underlying representation of the SUNMATRIX_BAND type. SM_ELEMENT_B(J, i, j) references the (i, j)-th element of the banded matrix J, counting from $$0$$. This macro is meant for use in small problems for which efficiency of access is not a major concern. Thus, in terms of the indices $$m$$ and $$n$$ ranging from $$1$$ to $$\texttt{N}$$ with $$(m,n)$$ within the band defined by mupper and mlower, the Jacobian element $$J_{m,n}$$ can be loaded using the statement SM_ELEMENT_B(J, m-1, n-1) = $$J_{m,n}$$. The elements within the band are those with -mupper $$\le$$ m-n $$\le$$ mlower. Alternatively, SM_COLUMN_B(J, j) returns a pointer to the diagonal element of the j-th column of J, and if we assign this address to realtype *col_j, then the i-th element of the j-th column is given by SM_COLUMN_ELEMENT_B(col_j, i, j), counting from $$0$$. Thus, for $$(m,n)$$ within the band, $$J_{m,n}$$ can be loaded by setting col_n = SM_COLUMN_B(J, n-1); and SM_COLUMN_ELEMENT_B(col_n, m-1, n-1) = $$J_{m,n}$$. The elements of the j-th column can also be accessed via ordinary array indexing, but this approach requires knowledge of the underlying storage for a band matrix of type SUNMATRIX_BAND. The array col_n can be indexed from $$-$$mupper to mlower. For large problems, it is more efficient to use SM_COLUMN_B and SM_COLUMN_ELEMENT_B than to use the SM_ELEMENT_B macro. As in the dense case, these macros all number rows and columns starting from $$0$$. The SUNMATRIX_BAND type and accessor macros are documented in §10.12.

sparse:

A user-supplied sparse Jacobian function must load the N $$\times$$ N compressed-sparse-column or compressed-sparse-row matrix J with an approximation to the Jacobian matrix $$J(u)$$ at the point (u). Storage for J already exists on entry to this function, although the user should ensure that sufficient space is allocated in J 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. The amount of allocated space in a SUNMATRIX_SPARSE object may be accessed using the macro SM_NNZ_S or the routine SUNSparseMatrix_NNZ. The SUNMATRIX_SPARSE type and accessor macros are documented in §10.14.

Warning

The previous function type KINDlsJacFn() is identical to KINLsJacFn, and may still be used for backward-compatibility. However, this will be deprecated in future releases, so we recommend that users transition to the new function type name soon.

### 8.4.6.5. Jacobian-vector product (matrix-free linear solvers)

If a matrix-free linear solver is to be used (i.e., a NULL-valued SUNMatrix was supplied to KINSetLinearSolver()), the user may provide a function of type KINLsJacTimesVecFn in the following form, to compute matrix-vector products $$Jv$$. If such a function is not supplied, the default is a difference quotient approximation to these products.

typedef int (*KINLsJacTimesVecFn)(N_Vector v, N_Vector Jv, N_Vector u, booleantype *new_u, void *user_data)

This function computes the product $$J v$$ (or an approximation to it).

Arguments:
• v – is the vector by which the Jacobian must be multplied to the right.

• Jv – is the computed output vector.

• u – is the current value of the dependent variable vector.

• user_data – is a pointer to user data, the same as the user_data parameter passed to KINSetUserData().

Return value:

The value returned by the Jacobian-times-vector function should be 0 if successful. If a recoverable failure occurred, the return value should be positive. In this case, KINSOL will attempt to correct by calling the preconditioner setup function. If this information is current, KINSOL halts. If the Jacobian-times-vector function encounters an unrecoverable error, it should return a negative value, prompting KINSOL to halt.

Notes:

If a user-defined routine is not given, then an internal jtimes function, using a difference quotient approximation, is used.

This function must return a value of $$J*v$$ that uses the current value of $$J$$, i.e. as evaluated at the current $$u$$.

If the user’s KINLsJacTimesVecFn function uses difference quotient approximations, it may need to access quantities not in the call list. These might include the scale vectors and the unit roundoff. To obtain the scale vectors, the user will need to add to user_data pointers to u_scale and/or f_scale as needed. The unit roundoff can be accessed as UNIT_ROUNDOFF defined in sundials_types.h.

Warning

The previous function type KINSpilsJacTimesVecFn is identical to KINLsJacTimesVecFn, and may still be used for backward-compatibility. However, this will be removed in future releases, so we recommend that users transition to the new function type name soon.

### 8.4.6.6. Preconditioner solve (iterative linear solvers)

If a user-supplied preconditioner is to be used with a SUNLinearSolver solver module, then the user must provide a function to solve the linear system $$Pz = r$$ where $$P$$ is the preconditioner matrix which approximates (at least crudely) the Jacobian matrix $$J = F'(u)$$. This function must be of type KINLsPrecSolveFn, defined as follows:

typedef int (*KINLsPrecSolveFn)(N_Vector u, N_Vector uscale, N_Vector fval, N_Vector fscale, N_Vector v, void *user_data)

This function solves the preconditioning system $$Pz = r$$.

Arguments:
• u – is the current (unscaled) value of the iterate.

• uscale – is a vector containing diagonal elements of the scaling matrix u

• fval – is the vector $$F(u)$$ evaluated at u

• fscale – is a vector containing diagonal elements of the scaling matrix for fval

• v – on inpuut, v is set to the right-hand side vector of the linear system, r. On output, v must contain the solution z of the linear system $$Pz=r$$

• user_data – is a pointer to user data, the same as the user_data parameter passed to KINSetUserData().

Return value:

The value returned by the preconditioner solve function should be 0 if successful, positive for a recoverable error, or negative for an unrecoverable error.

Notes:

If the preconditioner solve function fails recoverably and if the preconditioner information (set by the preconditioner setup function) is out of date, KINSOL attempts to correct by calling the setup function. If the preconditioner data is current, KINSOL halts.

### 8.4.6.7. Preconditioner setup (iterative linear solvers)

If the user’s preconditioner requires that any Jacobian-related data be evaluated or preprocessed, then this needs to be done in a user-supplied function of type KINLsPrecSetupFn, defined as follows:

typedef int (*KINLsPrecSetupFn)(N_Vector u, N_Vector uscale, N_Vector fval, N_Vector fscale, void *user_data)

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

Arguments:
• u – is the current (unscaled) value of the iterate.

• uscale – is a vector containing diagonal elements of the scaling matrix u

• fval – is the vector $$F(u)$$ evaluated at u

• fscale – is a vector containing diagonal elements of the scaling matrix for fval

• user_data – is a pointer to user data, the same as the user_data parameter passed to KINSetUserData().

Return value:

The value returned by the preconditioner 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:

The user-supplied preconditioner setup subroutine should compute the right preconditioner matrix $$P$$ (stored in the memory block referenced by the user_data pointer) used to form the scaled preconditioned linear system

$(D_F J(u) P^{-1} D_u^{-1}) (D_u P x) = - D_F F(u) \, ,$

where $$D_u$$ and $$D_F$$ denote the diagonal scaling matrices whose diagonal elements are stored in the vectors uscale and fscale, respectively.

The preconditioner setup routine will not be called prior to every call made to the preconditioner solve function, but will instead be called only as often as necessary to achieve convergence of the Newton iteration.

If the user’s KINLsPrecSetupFn function uses difference quotient approximations, it may need to access quantities not in the call list. These might include the scale vectors and the unit roundoff. To obtain the scale vectors, the user will need to add to user_data pointers to u_scale and/or f_scale as needed. The unit roundoff can be accessed as UNIT_ROUNDOFF defined in sundials_types.h.

If the preconditioner solve routine requires no preparation, then a preconditioner setup function need not be given.

## 8.4.7. A parallel band-block-diagonal preconditioner module

The efficiency of Krylov iterative methods for the solution of linear systems can be greatly enhanced through preconditioning. For problems in which the user cannot define a more effective, problem-specific preconditioner, KINSOL provides a band-block-diagonal preconditioner module KINBBDPRE, to be used with the parallel N_Vector module described in §9.10.

This module provides a preconditioner matrix for KINSOL that is block-diagonal with banded blocks. The blocking corresponds to the distribution of the dependent variable vector $$u$$ amongst the processes. Each preconditioner block is generated from the Jacobian of the local part (associated with the current process) of a given function $$G(u)$$ approximating $$F(u)$$ ($$G = F$$ is allowed). The blocks are generated by each process via a difference quotient scheme, utilizing a specified band structure. This structure is given by upper and lower half-bandwidths, mudq and mldq, defined as the number of non-zero diagonals above and below the main diagonal, respectively. However, from the resulting approximate Jacobain blocks, only a matrix of bandwidth mukeep $$+$$ mlkeep $$+ 1$$ is retained.

Neither pair of parameters need be the true half-bandwidths of the Jacobian of the local block of $$G$$, if smaller values provide a more efficient preconditioner. Such an efficiency gain may occur if the couplings in the system outside a certain bandwidth are considerably weaker than those within the band. Reducing mukeep and mlkeep while keeping mudq and mldq at their true values, discards the elements outside the narrower band. Reducing both pairs has the additional effect of lumping the outer Jacobian elements into the computed elements within the band, and requires more caution and experimentation to see whether the lower cost of narrower band matrices offsets the loss of accuracy in the blocks.

The KINBBDPRE module calls two user-provided functions to construct $$P$$: a required function Gloc (of type KINBBDLocalFn) which approximates the nonlinear system function $$G(u) \approx F(u)$$ and which is computed locally, and an optional function Gcomm (of type KINBBDCommFn) which performs all interprocess communication necessary to evaluate the approximate function $$G$$. These are in addition to the user-supplied nonlinear system function that evaluates $$F(u)$$. Both functions take as input the same pointer user_data as that passed by the user to KINSetUserData() and passed to the user’s function func, and neither function has a return value. The user is responsible for providing space (presumably within user_data) for components of u that are communicated by Gcomm from the other processes, and that are then used by Gloc, which should not do any communication.

typedef int (*KINBBDLocalFn)(sunindextype Nlocal, N_Vector u, N_Vector gval, void *user_data)

This Gloc function computes $$G(u)$$, and outputs the resulting vector as gval.

Arguments:
• Nlocal – is the local vector length.

• u – is the current value of the iterate.

• gval – is the output vector.

• user_data – is a pointer to user data, the same as the user_data parameter passed to KINSetUserData().

Return value:

An KINBBDLocalFn function type should return 0 to indicate success, or non-zero if an error occured.

Notes:

This function must assume that all inter-processor communication of data needed to calculate gval has already been done, and this data is accessible within user_data.

The case where $$G$$ is mathematically identical to $$F$$ is allowed.

typedef int (*KINBBDCommFn)(sunindextype Nlocal, N_Vector u, void *user_data)

This Gcomm function performs all inter-processor communications necessary for the execution of the Gloc function above, using the input vectors u.

Arguments:
• Nlocal – is the local vector length.

• u – is the current value of the iterate.

• user_data – is a pointer to user data, the same as the user_data parameter passed to KINSetUserData().

Return value:

An KINBBDLocalFn function type should return 0 to indicate success, or non-zero if an error occured.

Notes:

The Gcomm function is expected to save communicated data in space defined within the structure user_data.

Each call to the Gcomm function is preceded by a call to the residual function func with the same u argument. Thus Gcomm can omit any communications done by func if relevant to the evaluation of Gloc. If all necessary communication was done in func, then Gcomm = NULL can be passed in the call to KINBBDPrecInit().

Besides the header files required for the integration of the DAE problem (see §8.4.3), to use the KINBBDPRE module, the main program must include the header file kin_bbdpre.h which declares the needed function prototypes.

The following is a summary of the usage of this module and describes the sequence of calls in the user main program. Steps that are unchanged from the user main program presented in §8.4.4 are not bold.

1. Initialize parallel or multi-threaded environment (if appropriate)

2. Create the SUNDIALS context object

3. Set the problem dimensions etc.

4. Create the vector with the initial guess

5. Create matrix object (if appropriate)

6. Create linear solver object (if appropriate)

When creating the iterative linear solver object, specify the use of right preconditioning (SUN_PREC_RIGHT) as KINSOL only supports right preconditioning.

7. Create nonlinear solver object (if appropriate)

8. Create KINSOL object

9. Initialize KINSOL solver

10. Attach the linear solver (if appropriate)

11. Set linear solver optional inputs (if appropriate)

Note that the user should not overwrite the preconditioner setup function or solve function through calls to KINSetPreconditioner() optional input function.

12. Initialize the KINBBDPRE preconditioner module

Call KINBBDPrecInit() to allocate memory and initialize the internal preconditioner data. The last two arguments of KINBBDPrecInit() are the two user-supplied functions described above.

13. Set optional inputs

14. Solve problem

15. Get optional outputs

Additional optional outputs associated with KINBBDPRE are available by way of two routines described below, KINBBDPrecGetWorkSpace() and KINBBDPrecGetNumGfnEvals().

16. Deallocate memory

17. Finalize MPI, if used

The user-callable functions that initialize or re-initialize the KINBBDPRE preconditioner module are described next.

int KINBBDPrecInit(void *kin_mem, sunindextype Nlocal, sunindextype mudq, sunindexype mldq, sunindextype mukeep, sunindextype mlkeep, realtype dq_rel_u, KINBBDLocalFn Gloc, KINBBDCommFn Gcomm)

The function KINBBDPrecInit() initializes and allocates memory for the KINBBDPRE preconditioner.

Arguments:
• kin_mem – pointer to the KINSOL memory block.

• Nlocal – local vector length.

• mudq – upper half-bandwidth to be used in the difference-quotient Jacobian approximation.

• mldq – lower half-bandwidth to be used in the difference-quotient Jacobian approximation.

• mukeep – upper half-bandwidth of the retained banded approximate Jacobian block.

• mlkeep – lower half-bandwidth of the retained banded approximate Jacobian block.

• dq_rel_u – the relative increment in components of u used in the difference quotient approximations. The default is $$\texttt{dq\_rel\_u} = \sqrt{\text{unit roundoff}}$$ , which can be specified by passing dq_rel_u= 0.0.

• Gloc – the CC function which computes the approximation $$G(u) \approx F(u)$$.

• Gcomm – the optional CC function which performs all interprocess communication required for the computation of $$G(u)$$.

Return value:
• KINLS_SUCCESS – The call to KINBBDPrecInit() was successful.

• KINLS_MEM_NULL – The kin_mem pointer was NULL.

• KINLS_MEM_FAIL – A memory allocation request has failed.

• KINLS_LMEM_NULL – The KINLS linear solver interface has not been initialized.

• KINLS_ILL_INPUT – The supplied vector implementation was not compatible with the block band preconditioner.

Notes:

If one of the half-bandwidths mudq or mldq to be used in the difference-quotient calculation of the approximate Jacobian is negative or exceeds the value Nlocal-1, it is replaced with 0 or Nlocal-1 accordingly.

The half-bandwidths mudq and mldq need not be the true half-bandwidths of the Jacobian of the local block of $$G$$, when smaller values may provide greater efficiency.

Also, the half-bandwidths mukeep and mlkeep of the retained banded approximate Jacobian block may be even smaller, to reduce storage and computation costs further.

For all four half-bandwidths, the values need not be the same for every process.

The following two optional output functions are available for use with the KINBBDPRE module:

int KINBBDPrecGetWorkSpace(void *kin_mem, long int *lenrwBBDP, long int *leniwBBDP)

The function KINBBDPrecGetWorkSpace() returns the local sizes of the KINBBDPRE real and integer workspaces.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• lenrwBBDP – local number of real values in the KINBBDPRE workspace.

• leniwBBDP – local number of integer values in the KINBBDPRE workspace.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer was NULL.

• KINLS_PMEM_NULL – The KINBBDPRE preconditioner has not been initialized.

Notes:

The workspace requirements reported by this routine correspond only to memory allocated within the KINBBDPRE module (the banded matrix approximation, banded SUNLinearSolver object, temporary vectors). These values are local to each process.

The workspaces referred to here exist in addition to those given by the corresponding KINGetLinWorkSpace() function.

int KINBBDPrecGetNumGfnEvals(void *kin_mem, long int *ngevalsBBDP)

The function KINBBDPrecGetNumGfnEvals() returns the cumulative number of calls to the user Gres function due to the finite difference approximation of the Jacobian blocks used within KINBBDPRE’s preconditioner setup function.

Arguments:
• kin_mem – pointer to the KINSOL solver object.

• ngevalsBBDP – the cumulative number of calls to the user Gres function.

Return value:
• KINLS_SUCCESS – The optional output value has been successfully set.

• KINLS_MEM_NULL – The kin_mem pointer was NULL.

• KINLS_PMEM_NULL – The KINBBDPRE preconditioner has not been initialized.

In addition to the ngevalsBBDP evaluations of Gres, the costs associated with KINBBDPRE also includes nlinsetups LU factorizations, nlinsetups calls to Gcomm, npsolves banded backsolve calls, and nrevalsLS residual function evaluations, where nlinsetups is an optional KINSOL output (see §8.4.5.5.1), and npsolves and nrevalsLS are linear solver optional outputs (see §8.4.5.5.2).

## 8.4.8. Alternative to KINSOL for difficult systems

A nonlinear system $$F(u) = 0$$ may be difficult to solve with KINSOL (or any other nonlinear system solver) for a variety of reasons. The possible reasons include high nonlinearity, small region of convergence, and lack of a good initial guess. For systems with such difficulties, there is an alternative approach that may be more successful. This is an old idea, but deserves some new attention.

If the nonlinear system is $$F(u) = 0$$, consider instead the ODE system $$du/dt = - M^{-1} F(u)$$, where $$M$$ is a nonsingular matrix that is an approximation (even a crude approximation) to the system Jacobian $$F_u = dF/du$$. Whatever $$M$$ is, if this ODE is solved until it reaches a steady state $$u^*$$, then $$u^*$$ is a zero of the right-hand side of the ODE, and hence a solution to $$F(u) = 0$$. There is no issue of having a close enough initial guess.

A further basis for this choice of ODE is the following: If $$M$$ approximates $$F_u$$, then the Jacobian of the ODE system, $$-M^{-1}F_u$$, is approximately equal to $$-I$$ where $$I$$ is the identity matrix. This means that (in a local approximation sense) the solution modes of the ODE behave like $$\exp(-t)$$, and that asymptotically the approach to the steady state goes as $$\exp(-t)$$. Of course, the closer $$M$$ is to $$F_u$$, the better this basis applies.

Using (say) CVODE to solve the above ODE system requires, in addition to the objective function $$F(u)$$, the calculation of a suitable matrix $$M$$ and its inverse, or at least a routine that solves linear systems $$Mx = b$$. This is similar to the KINSOL requirement of supplying the system Jacobian $$J$$ (or solutions to $$Jx = b$$), but differs in that $$M$$ may be simpler than $$J$$ and hence easier to deal with. Depending on the nature of $$M$$, this may be handled best with a direct solver, or with a preconditioned Krylov solver. The latter calls for the use of a preconditioner $$P$$ that may be a crude approximation to $$M$$, hence even easier to solve. Note if using ARKODE, the ODE system may be posed in the linearly implicit from $$M du/dt = -F(u)$$ where $$M$$ is the “mass matrix” for the system. This use case requires supplying ARKODE with a function to evaluate $$M$$ or to compute its action on a vector ($$Mv = w$$) and attaching a linear solver (direct or iterative) to solve the linear systems $$Mx = b$$.

The solution of the ODE may be made easier by solving instead the equivalent DAE, $$M du/dt + F(u) = 0$$. Applying IDA to this system requires solutions to linear systems whose matrix is the DAE system Jacobian, $$J = F_u + \alpha M$$, where $$\alpha$$ is the scalar coefficient $$c_j$$ supplied to the user’s Jacobian or preconditioner routine. Selecting a preconditioned Krylov method requires an approximation to this Jacobian as preconditioner $$P$$. Given that $$M$$ approximates $$F_u$$ (possibly crudely), the appropriate approximation to $$J$$ is $$P = M + \alpha M = (1 + \alpha)M$$. Again the user must supply a routine that solves linear systems $$Px = b$$, or $$M x = b/(1 + \alpha)$$. If M is too difficult to solve, than an approximation $$M'$$ that is easier can be substituted, as long as it achieves convergence. As always, there is a trade-off between the expense of solving $$M$$ and the difficulty of achieving convergence in the linear solver.

For the solution of either the ODE or DAE system above, the chances for convergence can be improved with a piecewise constant choice for $$M$$. Specifically, starting from an initial guess $$u_0$$, an initial choice for $$M$$ might be $$M_0 = F_u(u_0)$$, or some approximation to this Jacobian. Then one could integrate $$M_0 du/dt + F(u) = 0$$ from $$t = 0$$ to $$t = T$$ for some sizable value $$T$$, evaluate $$F_u(u(T))$$, and take $$M_1$$ to be an approximation to that Jacobian. Then integrate using $$M_1$$ from $$t = T$$ to $$t = 2T$$, and repeat the process until it converges to a steady state.