# The SUNLinearSolver API¶

The SUNLinSol API defines several linear solver operations that enable SUNDIALS packages to utilize any SUNLinSol implementation that provides the required functions. These functions can be divided into three categories. The first are the core linear solver functions. The second group of functions consists of set routines to supply the linear solver with functions provided by the SUNDIALS time integrators and to modify solver parameters. The final group consists of get routines for retrieving linear solver statistics. All of these functions are defined in the header file sundials/sundials_linearsolver.h.

## SUNLinearSolver core functions¶

The core linear solver functions consist of two required functions to get the linear solver type (SUNLinSolGetType()) and solve the linear system $$Ax=b$$ (SUNLinSolSolve()). The remaining functions are for getting the solver ID (SUNLinSolGetID()), initializing the linear solver object once all solver-specific options have been set (SUNLinSolInitialize()), setting up the linear solver object to utilize an updated matrix $$A$$ (SUNLinSolSetup()), and for destroying the linear solver object (SUNLinSolFree()) are optional.

SUNLinearSolver_Type SUNLinSolGetType(SUNLinearSolver LS)

Returns the type identifier for the linear solver LS. It is used to determine the solver type (direct, iterative, or matrix-iterative) from the abstract SUNLinearSolver interface. Returned values are one of the following:

• SUNLINEARSOLVER_DIRECT0, the SUNLinSol module requires a matrix, and computes an ‘exact’ solution to the linear system defined by that matrix.
• SUNLINEARSOLVER_ITERATIVE1, the SUNLinSol module does not require a matrix (though one may be provided), and computes an inexact solution to the linear system using a matrix-free iterative algorithm. That is it solves the linear system defined by the package-supplied ATimes routine (see SUNLinSolSetATimes() below), even if that linear system differs from the one encoded in the matrix object (if one is provided). As the solver computes the solution only inexactly (or may diverge), the linear solver should check for solution convergence/accuracy as appropriate.
• SUNLINEARSOLVER_MATRIX_ITERATIVE2, the SUNLinSol module requires a matrix, and computes an inexact solution to the linear system defined by that matrix using an iterative algorithm. That is it solves the linear system defined by the matrix object even if that linear system differs from that encoded by the package-supplied ATimes routine. As the solver computes the solution only inexactly (or may diverge), the linear solver should check for solution convergence/accuracy as appropriate.

Usage:

type = SUNLinSolGetType(LS);


Notes: See section Intended use cases for more information on intended use cases corresponding to the linear solver type.

SUNLinearSolver_ID SUNLinSolGetID(SUNLinearSolver LS)

Returns the identifier for the linear solver LS. It is recommended that a user-supplied SUNLinearSolver implementation return the SUNLINEARSOLVER_CUSTOM identifier.

Usage:

id = SUNLinSolGetID(LS);

int SUNLinSolInitialize(SUNLinearSolver LS)

Performs linear solver initialization (assuming that all solver-specific options have been set). This should return zero for a successful call, and a negative value for a failure, ideally returning one of the generic error codes listed in section SUNLinearSolver return codes.

Usage:

retval = SUNLinSolInitialize(LS);

int SUNLinSolSetup(SUNLinearSolver LS, SUNMatrix A)

Performs any linear solver setup needed, based on an updated system SUNMatrix A. This may be called frequently (e.g., with a full Newton method) or infrequently (for a modified Newton method), based on the type of integrator and/or nonlinear solver requesting the solves. This should return zero for a successful call, a positive value for a recoverable failure and a negative value for an unrecoverable failure, ideally returning one of the generic error codes listed in section SUNLinearSolver return codes.

Usage:

retval = SUNLinSolSetup(LS, A);

int SUNLinSolSolve(SUNLinearSolver LS, SUNMatrix A, N_Vector x, N_Vector b, realtype tol)

This required function Solves a linear system $$Ax = b$$.

Arguments:
• LS – a SUNLinSol object.
• A – a SUNMatrix object.
• x – a N_Vector object containing the initial guess for the solution of the linear system, and the solution to the linear system upon return.
• b – a N_Vector object containing the linear system right-hand side.
• tol – the desired linear solver tolerance.

Return value: This should return zero for a successful call, a positive value for a recoverable failure and a negative value for an unrecoverable failure, ideally returning one of the generic error codes listed in section SUNLinearSolver return codes.

Direct solvers: can ignore the tol argument.

Matrix-free solvers: (those that identify as SUNLINEARSOLVER_ITERATIVE) can ignore the SUNMatrix input A, and should rely on the matrix-vector product function supplied through the routine SUNLinSolSetATimes().

Iterative solvers: (those that identify as SUNLINEARSOLVER_ITERATIVE or SUNLINEARSOLVER_MATRIX_ITERATIVE) should attempt to solve to the specified tolerance tol in a weighted 2-norm. If the solver does not support scaling then it should just use a 2-norm.

Usage:

retval = SUNLinSolSolve(LS, A, x, b, tol);

int SUNLinSolFree(SUNLinearSolver LS)

Frees memory allocated by the linear solver. This should return zero for a successful call, and a negative value for a failure.

Usage:

retval = SUNLinSolFree(LS);


## SUNLinearSolver set functions¶

The following set functions are used to supply linear solver modules with functions defined by the SUNDIALS packages and to modify solver parameters. Only the routine for setting the matrix-vector product routine is required, and that is only for matrix-free linear solver modules. Otherwise, all other set functions are optional. SUNLinSol implementations that do not provide the functionality for any optional routine should leave the corresponding function pointer NULL instead of supplying a dummy routine.

int SUNLinSolSetATimes(SUNLinearSolver LS, void* A_data, ATimesFn ATimes)

This function is required for matrix-free linear solvers; otherwise it is optional.

Provides a ATimesFn function pointer, as well as a void* pointer to a data structure used by this routine, to a linear solver object. SUNDIALS packages will call this function to set the matrix-vector product function to either a solver-provided difference-quotient via vector operations or a user-supplied solver-specific routine. This routine should return zero for a successful call, and a negative value for a failure, ideally returning one of the generic error codes listed in section SUNLinearSolver return codes.

Usage:

retval = SUNLinSolSetATimes(LS, A_data, ATimes);

int SUNLinSolSetPreconditioner(SUNLinearSolver LS, void* P_data, PSetupFn Pset, PSolveFn Psol)

This optional routine provides PSetupFn and PSolveFn function pointers that implement the preconditioner solves $$P_1^{-1}$$ and $$P_2^{-1}$$. This routine will be called by a SUNDIALS package, which will provide translation between the generic Pset and Psol calls and the package- or user-supplied routines. This routine should return zero for a successful call, and a negative value for a failure, ideally returning one of the generic error codes listed in section SUNLinearSolver return codes.

Usage:

retval = SUNLinSolSetPreconditioner(LS, Pdata, Pset, Psol);

int SUNLinSolSetScalingVectors(SUNLinearSolver LS, N_Vector s1, N_Vector s2)

This optional routine provides left/right scaling vectors for the linear system solve. Here, s1 and s2 are N_Vectors of positive scale factors containing the diagonal of the matrices $$S_1$$ and $$S_2$$, respectively. Neither of these vectors need to be tested for positivity, and a NULL argument for either indicates that the corresponding scaling matrix is the identity. This routine should return zero for a successful call, and a negative value for a failure, ideally returning one of the generic error codes listed in section SUNLinearSolver return codes.

Usage:

retval = SUNLinSolSetScalingVectors(LS, s1, s2);


## SUNLinearSolver get functions¶

The following get functions allow SUNDIALS packages to retrieve results from a linear solve. All routines are optional.

int SUNLinSolNumIters(SUNLinearSolver LS)

This optional routine should return the number of linear iterations performed in the last “solve” call.

Usage:

its = SUNLinSolNumIters(LS);

realtype SUNLinSolResNorm(SUNLinearSolver LS)

This optional routine should return the final residual norm from the last “solve” call.

Usage:

rnorm = SUNLinSolResNorm(LS);

N_Vector SUNLinSolResid(SUNLinearSolver LS)

If an iterative method computes the preconditioned initial residual and returns with a successful solve without performing any iterations (i.e., either the initial guess or the preconditioner is sufficiently accurate), then this optional routine may be called by the SUNDIALS package. This routine should return the N_Vector containing the preconditioned initial residual vector.

Usage:

rvec = SUNLinSolResid(LS);


Note: since N_Vector is actually a pointer, and the results are not modified, this routine should not require additional memory allocation. If the SUNLinSol object does not retain a vector for this purpose, then this function pointer should be set to NULL in the implementation.

sunindextype SUNLinSolLastFlag(SUNLinearSolver LS)

This optional routine should return the last error flag encountered within the linear solver. This is not called by the SUNDIALS packages directly; it allows the user to investigate linear solver issues after a failed solve.

Usage:

lflag = SUNLinLastFlag(LS);

int SUNLinSolSpace(SUNLinearSolver LS, long int *lenrwLS, long int *leniwLS)

This optional routine should return the storage requirements for the linear solver LS. lrw is a long int containing the number of realtype words and liw is a long int containing the number of integer words. The return value is an integer flag denoting success/failure of the operation.

This function is advisory only, for use by users to help determine their total space requirements.

Usage:

retval = SUNLinSolSpace(LS, &lrw, &liw);


## Functions provided by SUNDIALS packages¶

To interface with SUNLinSol modules, the SUNDIALS packages supply a variety of routines for evaluating the matrix-vector product, and setting up and applying the preconditioniner. These package-provided routines translate between the user-supplied ODE, DAE, or nonlinear systems and the generic interfaces to the linear systems of equations that result in their solution. The types for functions provided to a SUNLinSol module are defined in the header file sundials/sundials_iterative.h, and are described below.

typedef int (*ATimesFn)(void *A_data, N_Vector v, N_Vector z)

These functions compute the action of a matrix on a vector, performing the operation $$z = Av$$. Memory for z will already be allocated prior to calling this function. The parameter A_data is a pointer to any information about $$A$$ which the function needs in order to do its job. The vector $$v$$ should be left unchanged. This routine should return 0 if successful and a non-zero value if unsuccessful.

typedef int (*PSetupFn)(void *P_data)

These functions set up any requisite problem data in preparation for calls to the corresponding PSolveFn. This routine should return 0 if successful and a non-zero value if unsuccessful.

typedef int (*PSolveFn)(void *P_data, N_Vector r, N_Vector z, realtype tol, int lr)

These functions solve the preconditioner equation $$Pz = r$$ for the vector $$z$$. Memory for z will already be allocated prior to calling this function. The parameter P_data is a pointer to any information about $$P$$ which the function needs in order to do its job (set up by the corresponding PSetupFn). The parameter lr is input, and indicates whether $$P$$ is to be taken as the left or right preconditioner: lr = 1 for left and lr = 2 for right. If preconditioning is on one side only, lr can be ignored. If the preconditioner is iterative, then it should strive to solve the preconditioner equation so that

$\| Pz - r \|_{\text{wrms}} < tol$

where the error weight vector for the WRMS norm may be accessed from the main package memory structure. The vector r should not be modified by the PSolveFn. This routine should return 0 if successful and a non-zero value if unsuccessful. On a failure, a negative return value indicates an unrecoverable condition, while a positive value indicates a recoverable one, in which the calling routine may reattempt the solution after updating preconditioner data.

## SUNLinearSolver return codes¶

The functions provided to SUNLinSol modules by each SUNDIALS package, and functions within the SUNDIALS-provided SUNLinSol implementations utilize a common set of return codes, listed below. These adhere to a common pattern: 0 indicates success, a postitive value corresponds to a recoverable failure, and a negative value indicates a non-recoverable failure. Aside from this pattern, the actual values of each error code are primarily to provide additional information to the user in case of a linear solver failure.

• SUNLS_SUCCESS (0) – successful call or converged solve
• SUNLS_MEM_NULL (-801) – the memory argument to the function is NULL
• SUNLS_ILL_INPUT (-802) – an illegal input has been provided to the function
• SUNLS_MEM_FAIL (-803) – failed memory access or allocation
• SUNLS_ATIMES_NULL (-804) – the Atimes function is NULL
• SUNLS_ATIMES_FAIL_UNREC (-805) – an unrecoverable failure occurred in the ATimes routine
• SUNLS_PSET_FAIL_UNREC (-806) – an unrecoverable failure occurred in the Pset routine
• SUNLS_PSOLVE_NULL (-807) – the preconditioner solve function is NULL
• SUNLS_PSOLVE_FAIL_UNREC (-808) – an unrecoverable failure occurred in the Psolve routine
• SUNLS_PACKAGE_FAIL_UNREC (-809) – an unrecoverable failure occurred in an external linear solver package
• SUNLS_GS_FAIL (-810) – a failure occurred during Gram-Schmidt orthogonalization (SPGMR/SPFGMR)
• SUNLS_QRSOL_FAIL (-811) – a singular $R$ matrix was encountered in a QR factorization (SPGMR/SPFGMR)
• SUNLS_VECTOROP_ERR (-812) – a vector operation error occurred
• SUNLS_RES_REDUCED (801) – an iterative solver reduced the residual, but did not converge to the desired tolerance
• SUNLS_CONV_FAIL (802) – an iterative solver did not converge (80and the residual was not reduced)
• SUNLS_ATIMES_FAIL_REC (803) – a recoverable failure occurred in the ATimes routine
• SUNLS_PSET_FAIL_REC (804) – a recoverable failure occurred in the Pset routine
• SUNLS_PSOLVE_FAIL_REC (805) – a recoverable failure occurred in the Psolve routine
• SUNLS_PACKAGE_FAIL_REC (806) – a recoverable failure occurred in an external linear solver package
• SUNLS_QRFACT_FAIL (807) – a singular matrix was encountered during a QR factorization (SPGMR/SPFGMR)
• SUNLS_LUFACT_FAIL (808) – a singular matrix was encountered during a LU factorization

## The generic SUNLinearSolver module¶

SUNDIALS packages interact with specific SUNLinSol implementations through the generic SUNLinSol module on which all other SUNLinSol iplementations are built. The SUNLinearSolver type is a pointer to a structure containing an implementation-dependent content field, and an ops field. The type SUNLinearSolver is defined as

typedef struct _generic_SUNLinearSolver *SUNLinearSolver;

struct _generic_SUNLinearSolver {
void *content;
struct _generic_SUNLinearSolver_Ops *ops;
};


where the _generic_SUNLinearSolver_Ops structure is a list of pointers to the various actual linear solver operations provided by a specific implementation. The _generic_SUNLinearSolver_Ops structure is defined as

struct _generic_SUNLinearSolver_Ops {
SUNLinearSolver_Type (*gettype)(SUNLinearSolver);
SUNLinearSolver_ID   (*getid)(SUNLinearSolver);
int                  (*setatimes)(SUNLinearSolver, void*, ATimesFn);
int                  (*setpreconditioner)(SUNLinearSolver, void*,
PSetupFn, PSolveFn);
int                  (*setscalingvectors)(SUNLinearSolver,
N_Vector, N_Vector);
int                  (*initialize)(SUNLinearSolver);
int                  (*setup)(SUNLinearSolver, SUNMatrix);
int                  (*solve)(SUNLinearSolver, SUNMatrix, N_Vector,
N_Vector, realtype);
int                  (*numiters)(SUNLinearSolver);
realtype             (*resnorm)(SUNLinearSolver);
sunindextype         (*lastflag)(SUNLinearSolver);
int                  (*space)(SUNLinearSolver, long int*, long int*);
N_Vector             (*resid)(SUNLinearSolver);
int                  (*free)(SUNLinearSolver);
};


The generic SUNLinSol module defines and implements the linear solver operations defined in Sections SUNLinearSolver core functions through SUNLinearSolver get functions. These routines are in fact only wrappers to the linear solver operations defined by a particular SUNLinSol implementation, which are accessed through the ops field of the SUNLinearSolver structure. To illustrate this point we show below the implementation of a typical linear solver operation from the generic SUNLinearSolver module, namely SUNLinSolInitialize, which initializes a SUNLinearSolver object for use after it has been created and configured, and returns a flag denoting a successful or failed operation:

int SUNLinSolInitialize(SUNLinearSolver S)
{
return ((int) S->ops->initialize(S));
}


## Compatibility of SUNLinearSolver modules¶

We note that not all SUNLinearSolver types are compatible with all SUNMatrix and N_Vector types provided with SUNDIALS. In Table Compatible SUNLinearSolver and SUNMatrix implementations we show the matrix-based linear solvers available as SUNLinearSolver modules, and the compatible matrix implementations. Recall that Table SUNDIALS linear solver interfaces and vector implementations that can be used for each shows the compatibility between all SUNLinearSolver modules and vector implementations.

### Compatible SUNLinearSolver and SUNMatrix implementations¶

Linear Solver Dense Banded Sparse User Supplied
Dense X     X
LapackDense X     X
Band   X   X
LapackBand   X   X
KLU     X X
SuperLU_MT     X X
User supplied X X X X

## Implementing a custom SUNLinearSolver module¶

A particular implementation of the SUNLinearSolver module must:

• Specify the content field of the SUNLinSol module.

• Define and implement the required linear solver operations. See the section ARKode SUNLinearSolver interface to determine which SUNLinSol operations are required for this SUNDIALS package.

Note that the names of these routines should be unique to that implementation in order to permit using more than one SUNLinSol module (each with different SUNLinearSolver internal data representations) in the same code.

• Define and implement user-callable constructor and destructor routines to create and free a SUNLinearSolver with the new content field and with ops pointing to the new linear solver operations.

We note that the function pointers for all unsupported optional routines should be set to NULL in the ops structure. This allows the SUNDIALS package that is using the SUNLinSol object to know that the associated functionality is not supported.

To aid in the creation of custom SUNLinearSolver modules the generic SUNLinearSolver module provides the utility function SUNLinSolNewEmpty(). When used in custom SUNLinearSolver constructors this function will ease the introduction of any new optional linear solver operations to the SUNLinearSolver API by ensuring only required operations need to be set.

SUNLinearSolver SUNLinSolNewEmpty()

This function allocates a new generic SUNLinearSolver object and initializes its content pointer and the function pointers in the operations structure to NULL.

Return value: If successful, this function returns a SUNLinearSolver object. If an error occurs when allocating the object, then this routine will return NULL.

void SUNLinSolFreeEmpty(SUNLinearSolver LS)

This routine frees the generic SUNLinearSolver object, under the assumption that any implementation-specific data that was allocated within the underlying content structure has already been freed. It will additionally test whether the ops pointer is NULL, and, if it is not, it will free it as well.

Arguments:
• LS – a SUNLinearSolver object

Additionally, a SUNLinearSolver implementation may do the following:

• Define and implement additional user-callable “set” routines acting on the SUNLinearSolver, e.g., for setting various configuration options to tune the linear solver to a particular problem.
• Provide additional user-callable “get” routines acting on the SUNLinearSolver object, e.g., for returning various solve statistics.

### Intended use cases¶

The SUNLinSol (and SUNMATRIX) APIs are designed to require a minimal set of routines to ease interfacing with custom or third-party linear solver libraries. External solvers provide similar routines with the necessary functionality and thus will require minimal effort to wrap within custom SUNMATRIX and SUNLinSol implementations. Sections SUNMATRIX functions required by ARKode and ARKode SUNLinearSolver interface include a list of the required set of routines that compatible SUNMATRIX and SUNLinSol implementations must provide. As SUNDIALS packages utilize generic SUNLinSol modules allowing for user-supplied SUNLinearSolver implementations, there exists a wide range of possible linear solver combinations. Some intended use cases for both the SUNDIALS-provided and user-supplied SUNLinSol modules are discussd in the following sections.

#### Direct linear solvers¶

Direct linear solver modules require a matrix and compute an ‘exact’ solution to the linear system defined by the matrix. Multiple matrix formats and associated direct linear solvers are supplied with SUNDIALS through different SUNMATRIX and SUNLinSol implementations. SUNDIALS packages strive to amortize the high cost of matrix construction by reusing matrix information for multiple nonlinear iterations. As a result, each package’s linear solver interface recomputes Jacobian information as infrequently as possible.

Alternative matrix storage formats and compatible linear solvers that are not currently provided by or interfaced with SUNDIALS can leverage this infrastructure with minimal effort. To do so, a user must implement custom SUNMATRIX and SUNLinSol wrappers for the desired matrix format and/or linear solver following the APIs described in the sections Matrix Data Structures and Description of the SUNLinearSolver module. This user-supplied SUNLinSol module must then self-identify as having SUNLINEARSOLVER_DIRECT type.

#### Matrix-free iterative linear solvers¶

Matrix-free iterative linear solver modules do not require a matrix and compute an inexact solution to the linear system defined by the package-supplied ATimes routine. SUNDIALS supplies multiple scaled, preconditioned iterative linear solver (spils) SUNLinSol modules that support scaling to allow users to handle non-dimensionalization (as best as possible) within each SUNDIALS package and retain variables and define equations as desired in their applications. For linear solvers that do not support left/right scaling, the tolerance supplied to the linear solver is adjusted to compensate (see section Iterative linear solver tolerance for more details); however, this use case may be non-optimal and cannot handle situations where the magnitudes of different solution components or equations vary dramatically within a single problem.

To utilize alternative linear solvers that are not currently provided by or interfaced with SUNDIALS a user must implement a custom SUNLinSol wrapper for the linear solver following the API described in the section Description of the SUNLinearSolver module. This user-supplied SUNLinSol module must then self-identify as having SUNLINEARSOLVER_ITERATIVE type.

#### Matrix-based iterative linear solvers (reusing $$A$$)¶

Matrix-based iterative linear solver modules require a matrix and compute an inexact solution to the linear system defined by the matrix. This matrix will be updated infrequently and resued across multiple solves to amortize cost of matrix construction. As in the direct linear solver case, only wrappers for the matrix and linear solver in SUNMATRIX and SUNLinSol implementations need to be created to utilize a new linear solver. This user-supplied SUNLinSol module must then self-identify as having SUNLINEARSOLVER_MATRIX_ITERATIVE type.

At present, SUNDIALS has one example problem that uses this approach for wrapping a structured-grid matrix, linear solver, and preconditioner from the hypre library that may be used as a template for other customized implementations (see examples/arkode/CXX_parhyp/ark_heat2D_hypre.cpp).

#### Matrix-based iterative linear solvers (current $$A$$)¶

For users who wish to utilize a matrix-based iterative linear solver module where the matrix is purely for preconditioning and the linear system is defined by the package-supplied ATimes routine, we envision two current possibilities.

The preferred approach is for users to employ one of the SUNDIALS scaled, preconditioned iterative linear solver (spils) implementations (SUNLinSol_SPGMR(), SUNLinSol_SPFGMR(), SUNLinSol_SPBCGS(), SUNLinSol_SPTFQMR(), or SUNLinSol_PCG()) as the outer solver. The creation and storage of the preconditioner matrix, and interfacing with the corresponding linear solver, can be handled through a package’s preconditioner ‘setup’ and ‘solve’ functionality (see the sections Preconditioner setup (iterative linear solvers) and Preconditioner solve (iterative linear solvers), respectively) without creating SUNMATRIX and SUNLinSol implementations. This usage mode is recommended primarily because the SUNDIALS-provided spils modules support the scaling as described above.

A second approach supported by the linear solver APIs is as follows. If the SUNLinSol implementation is matrix-based, self-identifies as having SUNLINEARSOLVER_ITERATIVE type, and also provides a non-NULL :c:func:SUNLinSolSetATimes() routine, then each SUNDIALS package will call that routine to attach its package-specific matrix-vector product routine to the SUNLinSol object. The SUNDIALS package will then call the SUNLinSol-provided SUNLinSolSetup() routine (infrequently) to update matrix information, but will provide current matrix-vector products to the SUNLinSol implementation through the package-supplied ATimesFn routine.