11.2. ARKODE SUNLinearSolver interface
In Table 11.3, we list the SUNLinSol module functions used within the ARKLS interface. As with the SUNMATRIX module, we emphasize that the ARKODE user does not need to know detailed usage of linear solver functions by the ARKODE code modules in order to use ARKODE. The information is presented as an implementation detail for the interested reader.
Routine 
DIRECT 
ITERATIVE 
MATRIX ITERATIVE 
MATRIX EMBEDDED 

X 
X 
X 
X 

O 
X 
O 

O 
O 
O 

O 
O 
O 

X 
X 
X 

X 
X 
X 

X 
X 
X 
X 

O 
O 

O 
O 



O 
O 
O 
O 
Notes:
SUNLinSolNumIters()
is only used to accumulate overall iterative linear solver statistics. If it is not implemented by theSUNLinearSolver
module, then ARKLS will consider all solves as requiring zero iterations.Although
SUNLinSolResNorm()
is optional, if it is not implemented by theSUNLinearSolver
then ARKLS will consider all solves a being exact.Although ARKLS does not call
SUNLinSolLastFlag()
directly, this routine is available for users to query linear solver failure modes.Although ARKLS does not call
SUNLinSolFree()
directly, this routine should be available for users to call when cleaning up from a simulation.
Since there are a wide range of potential SUNLinSol use cases, the following subsections describe some details of the ARKLS interface, in the case that interested users wish to develop custom SUNLinSol modules.
11.2.1. Lagged matrix information
If the SUNLinSol module identifies as having type
SUNLINEARSOLVER_DIRECT
or SUNLINEARSOLVER_MATRIX_ITERATIVE
,
then it solves a linear system defined by a SUNMATRIX object. ARKLS
will update the matrix information infrequently according to the strategies
outlined in §3.2.10.2.3. To this end, we
differentiate between the desired linear system
\(\mathcal A x = b\) with \(\mathcal A = (M\gamma J)\)
and the actual linear system
Since ARKLS updates the SUNMATRIX object infrequently, it is likely
that \(\gamma\ne\tilde{\gamma}\), and in turn \(\mathcal
A\ne\tilde{\mathcal A}\). Therefore, after calling the
SUNLinSolprovided SUNLinSolSolve()
routine, we test whether
\(\gamma / \tilde{\gamma} \ne 1\), and if this is the case we
scale the solution \(\tilde{x}\) to obtain the desired linear
system solution \(x\) via
The motivation for this selection of the scaling factor \(c = 2/(1 + \gamma/\tilde{\gamma})\) follows the derivation in [21, 63]. In short, if we consider a stationary iteration for the linear system as consisting of a solve with \(\tilde{\mathcal A}\) followed with a scaling by \(c\), then for a linear constantcoefficient problem, the error in the solution vector will be reduced at each iteration by the error matrix \(E = I  c \tilde{\mathcal A}^{1} \mathcal A\), with a convergence rate given by the spectral radius of \(E\). Assuming that stiff systems have a spectrum spread widely over the left halfplane, \(c\) is chosen to minimize the magnitude of the eigenvalues of \(E\).
11.2.2. Iterative linear solver tolerance
If the SUNLinSol object selfidentifies as having type
SUNLINEARSOLVER_ITERATIVE
or SUNLINEARSOLVER_MATRIX_ITERATIVE
,
then ARKLS will set the input tolerance delta
as described in
§3.2.10.3.2. However, if the iterative linear
solver does not support scaling matrices (i.e., the
SUNLinSolSetScalingVectors()
routine is NULL
), then
ARKLS will attempt to adjust the linear solver tolerance to account
for this lack of functionality. To this end, the following
assumptions are made:
All solution components have similar magnitude; hence the residual weight vector \(w\) used in the WRMS norm (see §3.2.6), corresponding to the left scaling matrix \(S_1\), should satisfy the assumption
\[w_i \approx w_{mean},\quad \text{for}\quad i=0,\ldots,n1.\]The SUNLinSol object uses a standard 2norm to measure convergence.
Under these assumptions, ARKLS adjusts the linear solver convergence requirement as follows (using the notation from (11.2)):
Therefore we compute the tolerance scaling factor
and supply the scaled tolerance delta
\(= \text{tol} / w_{mean}\) to the SUNLinSol object.
11.2.3. Providing a custom SUNLinearSolver
In certain instances, users may wish to provide a custom SUNLinSol implementation to ARKODE in order to leverage the structure of a problem. While the “standard” API for these routines is typically sufficient for most users, others may need additional ARKODEspecific information on top of what is provided. For these purposes, we note the following advanced ouptut functions available in ARKStep and MRIStep:
ARKStep advanced outputs: when solving the Newton nonlinear system of equations in predictorcorrector form,
ARKStepGetCurrentTime()
– when called within the computation of a step (i.e., within a solve) this returns \(t^I_{n,i}\). Otherwise the current internal solution time is returned.ARKStepGetCurrentState()
– when called within the computation of a step (i.e., within a solve) this returns the current stage vector \(z_{i} = z_{cor} + z_{pred}\). Otherwise the current internal solution is returned.ARKStepGetCurrentGamma()
– returns \(\gamma\).ARKStepGetCurrentMassMatrix()
– returns \(M(t)\).ARKStepGetNonlinearSystemData()
– returns \(z_{i}\), \(z_{pred}\), \(f^I(t^I_{n,i}, y_{cur})\), \(\tilde{a}_i\), and \(\gamma\).
MRIStep advanced outputs: when solving the Newton nonlinear system of equations in predictorcorrector form,
MRIStepGetCurrentTime()
– when called within the computation of a step (i.e., within a solve) this returns \(t^S_{n,i}\). Otherwise the current internal solution time is returned.MRIStepGetCurrentState()
– when called within the computation of a step (i.e., within a solve) this returns the current stage vector \(z_{i} = z_{cor} + z_{pred}\). Otherwise the current internal solution is returned.MRIStepGetCurrentGamma()
– returns \(\gamma\).MRIStepGetNonlinearSystemData()
– returns \(z_{i}\), \(z_{pred}\), \(f^I(t^I_{n,i}, y_{cur})\), \(\tilde{a}_i\), and \(\gamma\).
11.3. CVODE SUNLinearSolver interface
Table 11.4 below lists the SUNLinearSolver
module linear solver
functions used within the CVLS interface. As with the SUNMatrix
module, we
emphasize that the CVODE user does not need to know detailed usage of linear
solver functions by the CVODE code modules in order to use CVODE. The
information is presented as an implementation detail for the interested reader.
The linear solver functions listed below are marked with ‘x’ to
indicate that they are required, or with \(\dagger\) to indicate that
they are only called if they are nonNULL
in the SUNLinearSolver
implementation that is being used. Note:
SUNLinSolNumIters
is only used to accumulate overall iterative linear solver statistics. If it is not implemented by theSUNLinearSolver
module, then CVLS will consider all solves as requiring zero iterations.Although CVLS does not call
SUNLinSolLastFlag
directly, this routine is available for users to query linear solver issues directly.Although CVLS does not call
SUNLinSolFree
directly, this routine should be available for users to call when cleaning up from a simulation.
DIRECT 
ITERATIVE 
MATRIX_ITERATIVE 


x 
x 
x 

\(\dagger\) 
x 
\(\dagger\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 

x 
x 
x 

x 
x 
x 

x 
x 
x 

\(^1\) 
\(\dagger\) 
\(\dagger\) 

\(^2\) 

\(^3\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 
Since there are a wide range of potential SUNLinearSolver
use cases, the following
subsections describe some details of the CVLS interface, in the case that
interested users wish to develop custom SUNLinearSolver
modules.
11.3.1. Lagged matrix information
If the SUNLinearSolver
object selfidentifies as having type
SUNLINEARSOLVER_DIRECT
or SUNLINEARSOLVER_MATRIX_ITERATIVE
, then the
SUNLinearSolver
object solves a linear system defined by a SUNMatrix
object. CVLS will update the matrix information infrequently according to the
strategies outlined in §4.2. To this end, we
differentiate between the desired linear system \(Mx=b\) with \(M =
(I\gamma J)\), and the actual linear system
Since CVLS updates the SUNMatrix
object infrequently, it is likely that
\(\gamma\ne\bar{\gamma}\), and in turn \(M\ne\bar{M}\). When using a
BDF method, after calling the SUNLinearSolver
provided SUNLinSolSolve
routine, we test whether \(\gamma / \bar{\gamma} \ne 1\), and if this is
the case we scale the solution \(\bar{x}\) to correct the linear system
solution \(x\) via
The motivation for this selection of the scaling factor \(c = 2/(1 +\gamma/\bar{\gamma})\) is discussed in detail in [21, 63]. In short, if we consider a stationary iteration for the linear system as consisting of a solve with \(\bar{M}\) followed by scaling by \(c\), then for a linear constantcoefficient problem, the error in the solution vector will be reduced at each iteration by the error matrix \(E = I  c \bar{M}^{1} M\), with a convergence rate given by the spectral radius of \(E\). Assuming that stiff systems have a spectrum spread widely over the left halfplane, \(c\) is chosen to minimize the magnitude of the eigenvalues of \(E\).
11.3.2. Iterative linear solver tolerance
If the SUNLinearSolver
object selfidentifies as having type
SUNLINEARSOLVER_ITERATIVE
or
SUNLINEARSOLVER_MATRIX_ITERATIVE
then CVLS will set the input
tolerance delta
as described in §4.2.1. However, if the
iterative linear solver does not support scaling matrices (i.e., the
SUNLinSolSetScalingVectors
routine is NULL
), then CVLS will attempt
to adjust the linear solver tolerance to account for this lack of functionality.
To this end, the following assumptions are made:
All solution components have similar magnitude; hence the error weight vector \(W\) used in the WRMS norm (see §4.2.1) should satisfy the assumption
\[W_i \approx W_{mean},\quad \text{for}\quad i=0,\ldots,n1.\]The
SUNLinearSolver
object uses a standard 2norm to measure convergence.
Since CVODE uses identical left and right scaling matrices, \(S_1 = S_2 = S = \operatorname{diag}(W)\), then the linear solver convergence requirement is converted as follows (using the notation from equations (11.1) – (11.2)):
Therefore the tolerance scaling factor
is computed and the scaled tolerance delta
\(= \text{tol} / W_{mean}\) is
supplied to the SUNLinearSolver
object.
11.4. CVODES SUNLinearSolver interface
Table 11.5 below lists the SUNLinearSolver
module linear solver
functions used within the CVLS interface. As with the SUNMatrix
module, we
emphasize that the CVODES user does not need to know detailed usage of linear
solver functions by the CVODES code modules in order to use CVODES. The
information is presented as an implementation detail for the interested reader.
The linear solver functions listed below are marked with “x” to
indicate that they are required, or with “\(\dagger\)” to indicate that
they are only called if they are nonNULL
in the SUNLinearSolver
implementation that is being used. Note:
SUNLinSolNumIters
is only used to accumulate overall iterative linear solver statistics. If it is not implemented by theSUNLinearSolver
module, then CVLS will consider all solves as requiring zero iterations.Although CVLS does not call
SUNLinSolLastFlag
directly, this routine is available for users to query linear solver issues directly.Although CVLS does not call
SUNLinSolFree
directly, this routine should be available for users to call when cleaning up from a simulation.
DIRECT 
ITERATIVE 
MATRIX_ITERATIVE 


x 
x 
x 

\(\dagger\) 
x 
\(\dagger\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 

x 
x 
x 

x 
x 
x 

x 
x 
x 

\(^1\) 
\(\dagger\) 
\(\dagger\) 

\(^2\) 

\(^3\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 
Since there are a wide range of potential SUNLinearSolver
use cases, the following
subsections describe some details of the CVLS interface, in the case that
interested users wish to develop custom SUNLinearSolver
modules.
11.4.1. Lagged matrix information
If the SUNLinearSolver
object selfidentifies as having type
SUNLINEARSOLVER_DIRECT
or SUNLINEARSOLVER_MATRIX_ITERATIVE
, then the
SUNLinearSolver
object solves a linear system defined by a SUNMatrix
object. CVLS will update the matrix information infrequently according to the
strategies outlined in §5.2. To this end, we
differentiate between the desired linear system \(Mx=b\) with \(M =
(I\gamma J)\), and the actual linear system
Since CVLS updates the SUNMatrix
object infrequently, it is likely that
\(\gamma\ne\bar{\gamma}\), and in turn \(M\ne\bar{M}\). When using a
BDF method, after calling the SUNLinearSolver
provided SUNLinSolSolve
routine, we test whether \(\gamma / \bar{\gamma} \ne 1\), and if this is
the case we scale the solution \(\bar{x}\) to correct the linear system
solution \(x\) via
The motivation for this selection of the scaling factor \(c = 2/(1 +\gamma/\bar{\gamma})\) is discussed in detail in [21, 63]. In short, if we consider a stationary iteration for the linear system as consisting of a solve with \(\bar{M}\) followed by scaling by \(c\), then for a linear constantcoefficient problem, the error in the solution vector will be reduced at each iteration by the error matrix \(E = I  c \bar{M}^{1} M\), with a convergence rate given by the spectral radius of \(E\). Assuming that stiff systems have a spectrum spread widely over the left halfplane, \(c\) is chosen to minimize the magnitude of the eigenvalues of \(E\).
11.4.2. Iterative linear solver tolerance
If the SUNLinearSolver
object selfidentifies as having type
SUNLINEARSOLVER_ITERATIVE
or
SUNLINEARSOLVER_MATRIX_ITERATIVE
then CVLS will set the input
tolerance delta
as described in §5.2.1. However, if the
iterative linear solver does not support scaling matrices (i.e., the
SUNLinSolSetScalingVectors
routine is NULL
), then CVLS will attempt
to adjust the linear solver tolerance to account for this lack of functionality.
To this end, the following assumptions are made:
All solution components have similar magnitude; hence the error weight vector \(W\) used in the WRMS norm (see §5.2.1) should satisfy the assumption
\[W_i \approx W_{mean},\quad \text{for}\quad i=0,\ldots,n1.\]The
SUNLinearSolver
object uses a standard 2norm to measure convergence.
Since CVODES uses identical left and right scaling matrices, \(S_1 = S_2 = S = \operatorname{diag}(W)\), then the linear solver convergence requirement is converted as follows (using the notation from equations (11.1) – (11.2)):
Therefore the tolerance scaling factor
is computed and the scaled tolerance delta
\(= \text{tol} / W_{mean}\) is
supplied to the SUNLinearSolver
object.
11.5. IDA SUNLinearSolver interface
Table 11.6 below lists the SUNLinearSolver
module linear solver
functions used within the IDALS interface. As with the SUNMatrix
module, we
emphasize that the IDA user does not need to know detailed usage of linear
solver functions by the IDA code modules in order to use IDA. The
information is presented as an implementation detail for the interested reader.
The linear solver functions listed below are marked with ‘x’ to
indicate that they are required, or with \(\dagger\) to indicate that
they are only called if they are nonNULL
in the SUNLinearSolver
implementation that is being used. Note:
Although IDALS does not call
SUNLinSolLastFlag
directly, this routine is available for users to query linear solver issues directly.Although IDALS does not call
SUNLinSolFree
directly, this routine should be available for users to call when cleaning up from a simulation.
DIRECT 
ITERATIVE 
MATRIX_ITERATIVE 


x 
x 
x 

\(\dagger\) 
x 
\(\dagger\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 

x 
x 
x 

x 
x 
x 

x 
x 
x 

x 
x 

x 
x 

\(^1\) 

\(^2\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 
Since there are a wide range of potential SUNLinearSolver
use cases, the following
subsections describe some details of the IDALS interface, in the case that
interested users wish to develop custom SUNLinearSolver
modules.
11.5.1. Lagged matrix information
If the SUNLinearSolver
object selfidentifies as having type
SUNLINEARSOLVER_DIRECT
or SUNLINEARSOLVER_MATRIX_ITERATIVE
, then the
SUNLinearSolver
object solves a linear system defined by a SUNMatrix
object. IDALS will update the matrix information infrequently according to the
strategies outlined in §6.2. To this end, we
differentiate between the desired linear system \(Jx=b\) with
\(J = \left(\dfrac{\partial F}{\partial y}c_j \dfrac{\partial F}{\partial\dot{y}}\right)\),
and the actual linear system \(\bar{J}\bar{x}=b\) with
where the overlines indicate the lagged versions of these numbers and matrices.
Since IDALS updates the SUNMatrix
objects infrequently and it is likely that
\(c_j\ne\bar{c}_j\), then typically \(J\ne\bar{J}\). Thus after calling
the SUNLinearSolver
provided SUNLinSolSolve
routine, we test whether
\(\dfrac{c_j}{\bar{c}_j} \ne 1\), and if this is
the case we scale the solution \(\bar{x}\) to correct the linear system
solution \(x\) via
The motivation for this selection of the scaling factor \(c = 2/(1 + c_j/\bar{c}_j)\) is discussed in detail in [21, 63]. In short, if we consider a stationary iteration for the linear system as consisting of a solve with \(\bar{J}\) followed by scaling by \(c\), then for a linear constantcoefficient problem, the error in the solution vector will be reduced at each iteration by the error matrix \(E = I  c \bar{J}^{1} J\), with a convergence rate given by the spectral radius of \(E\). Assuming that stiff systems have a spectrum spread widely over the left halfplane, \(c\) is chosen to minimize the magnitude of the eigenvalues of \(E\).
11.5.2. Iterative linear solver tolerance
If the SUNLinearSolver
object selfidentifies as having type
SUNLINEARSOLVER_ITERATIVE
or
SUNLINEARSOLVER_MATRIX_ITERATIVE
then IDALS will set the input
tolerance delta
as described in §6.2.2. However, if the
iterative linear solver does not support scaling matrices (i.e., the
SUNLinSolSetScalingVectors
routine is NULL
), then IDALS will attempt
to adjust the linear solver tolerance to account for this lack of functionality.
To this end, the following assumptions are made:
All solution components have similar magnitude; hence the error weight vector \(W\) used in the WRMS norm (see §6.2.2) should satisfy the assumption
\[W_i \approx W_{mean},\quad \text{for}\quad i=0,\ldots,n1.\]The
SUNLinearSolver
object uses a standard 2norm to measure convergence.
Since IDA uses identical left and right scaling matrices, \(S_1 = S_2 = S = \operatorname{diag}(W)\), then the linear solver convergence requirement is converted as follows (using the notation from equations (11.1) – (11.2)):
Therefore the tolerance scaling factor
is computed and the scaled tolerance delta
\(= \text{tol} / W_{mean}\) is
supplied to the SUNLinearSolver
object.
11.6. IDAS SUNLinearSolver interface
Table 11.7 below lists the SUNLinearSolver
module linear solver
functions used within the IDALS interface. As with the SUNMatrix
module, we
emphasize that the IDA user does not need to know detailed usage of linear
solver functions by the IDA code modules in order to use IDA. The
information is presented as an implementation detail for the interested reader.
The linear solver functions listed below are marked with ‘x’ to
indicate that they are required, or with \(\dagger\) to indicate that
they are only called if they are nonNULL
in the SUNLinearSolver
implementation that is being used. Note:
Although IDALS does not call
SUNLinSolLastFlag
directly, this routine is available for users to query linear solver issues directly.Although IDALS does not call
SUNLinSolFree
directly, this routine should be available for users to call when cleaning up from a simulation.
DIRECT 
ITERATIVE 
MATRIX_ITERATIVE 


x 
x 
x 

\(\dagger\) 
x 
\(\dagger\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 

x 
x 
x 

x 
x 
x 

x 
x 
x 

x 
x 

x 
x 

\(^1\) 

\(^2\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 
Since there are a wide range of potential SUNLinearSolver
use cases, the following
subsections describe some details of the IDALS interface, in the case that
interested users wish to develop custom SUNLinearSolver
modules.
11.6.1. Lagged matrix information
If the SUNLinearSolver
object selfidentifies as having type
SUNLINEARSOLVER_DIRECT
or SUNLINEARSOLVER_MATRIX_ITERATIVE
, then the
SUNLinearSolver
object solves a linear system defined by a SUNMatrix
object. IDALS will update the matrix information infrequently according to the
strategies outlined in §7.2. To this end, we
differentiate between the desired linear system \(Jx=b\) with
\(J = \left(\dfrac{\partial F}{\partial y}c_j \dfrac{\partial F}{\partial\dot{y}}\right)\),
and the actual linear system \(\bar{J}\bar{x}=b\) with
where the overlines indicate the lagged versions of these numbers and matrices.
Since IDALS updates the SUNMatrix
objects infrequently and it is likely that
\(c_j\ne\bar{c}_j\), then typically \(J\ne\bar{J}\). Thus after calling
the SUNLinearSolver
provided SUNLinSolSolve
routine, we test whether
\(\dfrac{c_j}{\bar{c}_j} \ne 1\), and if this is
the case we scale the solution \(\bar{x}\) to correct the linear system
solution \(x\) via
The motivation for this selection of the scaling factor \(c = 2/(1 + c_j/\bar{c}_j)\) is discussed in detail in [21, 63]. In short, if we consider a stationary iteration for the linear system as consisting of a solve with \(\bar{J}\) followed by scaling by \(c\), then for a linear constantcoefficient problem, the error in the solution vector will be reduced at each iteration by the error matrix \(E = I  c \bar{J}^{1} J\), with a convergence rate given by the spectral radius of \(E\). Assuming that stiff systems have a spectrum spread widely over the left halfplane, \(c\) is chosen to minimize the magnitude of the eigenvalues of \(E\).
11.6.2. Iterative linear solver tolerance
If the SUNLinearSolver
object selfidentifies as having type
SUNLINEARSOLVER_ITERATIVE
or
SUNLINEARSOLVER_MATRIX_ITERATIVE
then IDALS will set the input
tolerance delta
as described in §7.2.2. However, if the
iterative linear solver does not support scaling matrices (i.e., the
SUNLinSolSetScalingVectors
routine is NULL
), then IDALS will attempt
to adjust the linear solver tolerance to account for this lack of functionality.
To this end, the following assumptions are made:
All solution components have similar magnitude; hence the error weight vector \(W\) used in the WRMS norm (see §7.2.2) should satisfy the assumption
\[W_i \approx W_{mean},\quad \text{for}\quad i=0,\ldots,n1.\]The
SUNLinearSolver
object uses a standard 2norm to measure convergence.
Since IDA uses identical left and right scaling matrices, \(S_1 = S_2 = S = \operatorname{diag}(W)\), then the linear solver convergence requirement is converted as follows (using the notation from equations (11.1) – (11.2)):
Therefore the tolerance scaling factor
is computed and the scaled tolerance delta
\(= \text{tol} / W_{mean}\) is
supplied to the SUNLinearSolver
object.
11.7. KINSOL SUNLinearSolver interface
Table 11.8 below lists the SUNLinearSolver
module linear
solver functions used within the KINLS interface. As with the SUNMatrix
module, we emphasize that the KINSOL user does not need to know detailed usage
of linear solver functions by the KINSOL code modules in order to use KINSOL.
The information is presented as an implementation detail for the interested
reader.
The linear solver functions listed below are marked with “x” to indicate that they
are required, or with “\(\dagger\)” to indicate that they are only called if
they are nonNULL
in the SUNLinearSolver
implementation that is being
used. Note:
SUNLinSolNumIters()
is only used to accumulate overall iterative linear solver statistics. If it is not implemented by theSUNLinearSolver
module, then KINLS will consider all solves as requiring zero iterations.Although
SUNLinSolResNorm()
is optional, if it is not implemented by theSUNLinearSolver
then KINLS will consider all solves a being exact.Although KINLS does not call
SUNLinSolLastFlag()
directly, this routine is available for users to query linear solver issues directly.Although KINLS does not call
SUNLinSolFree()
directly, this routine should be available for users to call when cleaning up from a simulation.

x 
x 
x 

\(\dagger\) 
x 
\(\dagger\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 

\(\dagger\) 
\(\dagger\) 
\(\dagger\) 

x 
x 
x 

x 
x 
x 

x 
x 
x 
\(^1\) 
\(\dagger\) 
\(\dagger\) 

\(^2\) 
\(\dagger\) 
\(\dagger\) 

\(^3\) 

\(^4\) 


\(\dagger\) 
\(\dagger\) 
\(\dagger\) 
Since there are a wide range of potential SUNLinearSolver
use cases, the following subsections describe some details of
the KINLS interface, in the case that interested users wish to develop custom SUNLinearSolver
modules.
11.7.1. Lagged matrix information
If the SUNLinearSolver
object selfidentifies as having type
SUNLINEARSOLVER_DIRECT
or SUNLINEARSOLVER_MATRIX_ITERATIVE
, then the
SUNLinearSolver
object solves a linear system defined by a
SUNMatrix
object. As a result, KINSOL can perform its optional residual
monitoring scheme, described in §8.2.8.
11.7.2. Iterative linear solver tolerance
If the SUNLinearSolver
object selfidentifies as having type
SUNLINEARSOLVER_ITERATIVE
or SUNLINEARSOLVER_MATRIX_ITERATIVE
then KINLS
will adjust the linear solver tolerance delta
as described in
§8.2.9 during the course of
the nonlinear solve process. However, if the iterative linear solver does not
support scaling matrices (i.e., the SUNLinSolSetScalingVectors
routine is
NULL
), then KINLS will be unable to fully handle illconditioning in the
nonlinear solve process through the solution and residual scaling operators
described in §8.2.4. In this case, KINLS will attempt
to adjust the linear solver tolerance to account for this lack of functionality.
To this end, the following assumptions are made:
All residual components have similar magnitude; hence the scaling matrix \(D_F\) used in computing the linear residual norm (see §8.2.4) should satisfy the assumption
\[(D_F)_{i,i} \approx D_{F,mean},\quad \text{for}\quad i=0,\ldots,n1.\]The
SUNLinearSolver
object uses a standard 2norm to measure convergence.
Since KINSOL uses \(D_F\) as the leftscaling matrix, \(S_1 = D_F\), then the linear solver convergence requirement is converted as follows (using the notation from equations (11.1) – (11.2):
Therefore the tolerance scaling factor
is computed and the scaled tolerance delta
\(= \text{tol} / D_{F,mean}\) is supplied to the SUNLinearSolver
object.
11.7.3. Matrixembedded solver incompatibility
At present, KINLS is incompatible with SUNLinearSolver
objects that
selfidentify as having type SUNLINEARSOLVER_MATRIX_EMBEDDED
. Support for
such usersupplied linear solvers may be added in a future release. Users
interested in such support are recommended to contact the SUNDIALS development
team.