NAG CL Interface
f11bec (real_gen_basic_solver)
1
Purpose
f11bec is an iterative solver for a real general (nonsymmetric) system of simultaneous linear equations;
f11bec is the second in a suite of three functions, where the first function,
f11bdc, must be called prior to
f11bec to set up the suite, and the third function in the suite,
f11bfc, can be used to return additional information about the computation.
These functions are suitable for the solution of large sparse general (nonsymmetric) systems of equations.
2
Specification
The function may be called by the names: f11bec, nag_sparse_real_gen_basic_solver or nag_sparse_nsym_basic_solver.
3
Description
f11bec solves the general (nonsymmetric) system of linear simultaneous equations
$Ax=b$ of order
$\mathit{n}$, where
$\mathit{n}$ is large and the coefficient matrix
$A$ is sparse, using one of four available methods: RGMRES, the preconditioned restarted generalized minimum residual method (see
Saad and Schultz (1986)); CGS, the preconditioned conjugate gradient squared method (see
Sonneveld (1989)); BiCGSTAB(
$\ell $), the biconjugate gradient stabilized method of order
$\ell $ (see
Van der Vorst (1989) and
Sleijpen and Fokkema (1993)); or TFQMR, the transposefree quasiminimal residual method (see
Freund and Nachtigal (1991) and
Freund (1993)).
For a general description of the methods employed you are referred to
Section 3 in
f11bdc.
f11bec can solve the system after the first function in the suite,
f11bdc, has been called to initialize the computation and specify the method of solution. The third function in the suite,
f11bfc, can be used to return additional information generated by the computation, during monitoring steps and after
f11bec has completed its tasks.
f11bec uses
reverse communication, i.e., it returns repeatedly to the calling program with the argument
irevcm (see
Section 5) set to specified values which require the calling program to carry out one of the following tasks:

–compute the matrixvector product $v=Au$ or $v={A}^{\mathrm{T}}u$ (the four methods require the matrix transposevector product only if ${\Vert A\Vert}_{1}$ or ${\Vert A\Vert}_{\infty}$ is estimated internally by Higham's method (see Higham (1988)));

–solve the preconditioning equation $Mv=u$;

–notify the completion of the computation;

–allow the calling program to monitor the solution.
Through the argument
irevcm the calling program can cause immediate or tidy termination of the execution. On final exit, the last iterates of the solution and of the residual vectors of the original system of equations are returned.
Reverse communication has the following advantages.

1.Maximum flexibility in the representation and storage of sparse matrices: all matrix operations are performed outside the solver function, thereby avoiding the need for a complicated interface with enough flexibility to cope with all types of storage schemes and sparsity patterns. This applies also to preconditioners.

2.Enhanced user interaction: you can closely monitor the progress of the solution and tidy or immediate termination can be requested. This is useful, for example, when alternative termination criteria are to be employed or in case of failure of the external functions used to perform matrix operations.
4
References
Freund R W (1993) A transposefree quasiminimal residual algorithm for nonHermitian linear systems SIAM J. Sci. Comput. 14 470–482
Freund R W and Nachtigal N (1991) QMR: a QuasiMinimal Residual Method for NonHermitian Linear Systems Numer. Math. 60 315–339
Higham N J (1988) FORTRAN codes for estimating the onenorm of a real or complex matrix, with applications to condition estimation ACM Trans. Math. Software 14 381–396
Saad Y and Schultz M (1986) GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 7 856–869
Sleijpen G L G and Fokkema D R (1993) BiCGSTAB$\left(\ell \right)$ for linear equations involving matrices with complex spectrum ETNA 1 11–32
Sonneveld P (1989) CGS, a fast Lanczostype solver for nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 10 36–52
Van der Vorst H (1989) BiCGSTAB, a fast and smoothly converging variant of BiCG for the solution of nonsymmetric linear systems SIAM J. Sci. Statist. Comput. 13 631–644
5
Arguments
Note: this function uses
reverse communication. Its use involves an initial entry, intermediate exits and reentries, and a final exit, as indicated by the argument
irevcm. Between intermediate exits and reentries,
all arguments other than irevcm and v must remain unchanged.

1:
$\mathbf{irevcm}$ – Integer *
Input/Output

On initial entry: ${\mathbf{irevcm}}=0$, otherwise an error condition will be raised.
On intermediate reentry: must either be unchanged from its previous exit value, or can have one of the following values.
 ${\mathbf{irevcm}}=5$
 Tidy termination: the computation will terminate at the end of the current iteration. Further reverse communication exits may occur depending on when the termination request is issued. f11bec will then return with the termination code ${\mathbf{irevcm}}=4$. Note that before calling f11bec with ${\mathbf{irevcm}}=5$ the calling program must have performed the tasks required by the value of irevcm returned by the previous call to f11bec, otherwise subsequently returned values may be invalid.
 ${\mathbf{irevcm}}=6$
 Immediate termination: f11bec will return immediately with termination code ${\mathbf{irevcm}}=4$ and with any useful information available. This includes the last iterate of the solution. The residual vector is generally not available.
Immediate termination may be useful, for example, when errors are detected during matrixvector multiplication or during the solution of the preconditioning equation.
Changing
irevcm to any other value between calls will result in an error.
On intermediate exit:
has the following meanings.
 ${\mathbf{irevcm}}=\mathrm{1}$
 The calling program must compute the matrixvector product $v={A}^{\mathrm{T}}u$, where $u$ and $v$ are stored in u and v, respectively; RGMRES, CGS and BiCGSTAB($\ell $) methods return ${\mathbf{irevcm}}=\mathrm{1}$ only if the matrix norm ${\Vert A\Vert}_{1}$ or ${\Vert A\Vert}_{\infty}$ is estimated internally using Higham's method. This can only happen if ${\mathbf{iterm}}=1$ in f11bdc.
 ${\mathbf{irevcm}}=1$
 The calling program must compute the matrixvector product $v=Au$, where $u$ and $v$ are stored in u and v, respectively.
 ${\mathbf{irevcm}}=2$
 The calling program must solve the preconditioning equation $Mv=u$, where $u$ and $v$ are stored in u and v, respectively.
 ${\mathbf{irevcm}}=3$
 Monitoring step: the solution and residual at the current iteration are returned in the arrays u and v, respectively. No action by the calling program is required. f11bfc can be called at this step to return additional information.
On final exit:
${\mathbf{irevcm}}=4$:
f11bec has completed its tasks. The value of
fail determines whether the iteration has been successfully completed, errors have been detected or the calling program has requested termination.
Constraint:
on initial entry,
${\mathbf{irevcm}}=0$; on reentry, either
irevcm must remain unchanged or be reset to
$5$ or
$6$.
Note: any values you return to f11bec as part of the reverse communication procedure should not include floatingpoint NaN (Not a Number) or infinity values, since these are not handled by f11bec. If your code inadvertently does return any NaNs or infinities, f11bec is likely to produce unexpected results.

2:
$\mathbf{u}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
u
must be at least
$\mathit{n}$.
On initial entry: an initial estimate, ${x}_{0}$, of the solution of the system of equations $Ax=b$.
On intermediate reentry: must remain unchanged.
On intermediate exit:
the returned value of
irevcm determines the contents of
u in the following way:
 if ${\mathbf{irevcm}}=\mathrm{1}$, $1$ or $2$, u holds the vector $u$ on which the operation specified by irevcm is to be carried out;
 if ${\mathbf{irevcm}}=3$, u holds the current iterate of the solution vector.
On final exit: if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_OUT_OF_SEQUENCE or
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT, the array
u is unchanged from the last entry to
f11bec.
Otherwise,
u holds the last available iterate of the solution of the system of equations, for all returned values of
fail.

3:
$\mathbf{v}\left[\mathit{dim}\right]$ – double
Input/Output

Note: the dimension,
dim, of the array
v
must be at least
$\mathit{n}$.
On initial entry: the righthand side $b$ of the system of equations $Ax=b$.
On intermediate reentry: the returned value of
irevcm determines the contents of
v in the following way:
 if ${\mathbf{irevcm}}=\mathrm{1}$, $1$ or $2$, v must store the vector $v$, the result of the operation specified by the value of irevcm returned by the previous call to f11bec;
 if ${\mathbf{irevcm}}=3$, v must remain unchanged.
On intermediate exit:
if
${\mathbf{irevcm}}=3$,
v holds the current iterate of the residual vector. Note that this is an approximation to the true residual vector. Otherwise, it does not contain any useful information.
On final exit: if
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_OUT_OF_SEQUENCE or
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_INT, the array
v is unchanged from the last entry to
f11bec.
If
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR or
NE_ACCURACY, the array
v contains the true residual vector of the system of equations (see also
Section 6).
Otherwise,
v stores the last available iterate of the residual vector unless
${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_USER_STOP is returned on last entry, in which case
v is set to
$0.0$.

4:
$\mathbf{wgt}\left[\mathit{dim}\right]$ – const double
Input

Note: the dimension,
dim, of the array
wgt
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,\mathit{n})$.
On entry: the usersupplied weights, if these are to be used in the computation of the vector norms in the termination criterion (see
Sections 3 and
5 in
f11bdc).
Constraint:
if weights are to be used, at least one element of
wgt must be nonzero.

5:
$\mathbf{work}\left[{\mathbf{lwork}}\right]$ – double
Communication Array

On initial entry: the array
work as returned by
f11bdc (see also
Section 5 in
f11bdc).
On intermediate reentry: must remain unchanged.

6:
$\mathbf{lwork}$ – Integer
Input

On initial entry: the dimension of the array
work (see also
Sections 3 and
5 in
f11bdc). The required amount of workspace is as follows:
Method 
Requirements 
RGMRES 
${\mathbf{lwork}}=100+\mathit{n}(m+3)+m(m+5)+1$, where $m$ is the dimension of the basis. 
CGS 
${\mathbf{lwork}}=100+7\mathit{n}$. 
BiCGSTAB($\ell $) 
${\mathbf{lwork}}=100+(2\mathit{n}+\ell )(\ell +2)+p$, where $\ell $ is the order of the method. 
TFQMR 
${\mathbf{lwork}}=100+10\mathit{n}$, 
where 
$p=2\mathit{n}$ 
if $\ell >1$ and ${\mathbf{iterm}}=2$ was supplied. 

$p=\mathit{n}$ 
if $\ell >1$ and a preconditioner is used or ${\mathbf{iterm}}=2$ was supplied. 

$p=0$ 
otherwise. 
Constraint:
${\mathbf{lwork}}\ge {\mathbf{lwreq}}$, where
lwreq is returned by
f11bdc.

7:
$\mathbf{fail}$ – NagError *
Input/Output

The NAG error argument (see
Section 7 in the Introduction to the NAG Library CL Interface).
6
Error Indicators and Warnings
 NE_ACCURACY

The required accuracy could not be obtained. However, a reasonable accuracy may have been achieved.
Userrequested tidy termination. The required accuracy has not been achieved. However, a reasonable accuracy may have been achieved.
f11bec has terminated with reasonable accuracy: the last iterate of the residual satisfied the termination criterion but the exact residual
$r=bAx$, did not. After the first occurrence of this situation, the iteration was restarted once, but
f11bec could not improve on the accuracy. This error code usually implies that your problem has been fully and satisfactorily solved to within or close to the accuracy available on your system. Further iterations are unlikely to improve on this situation. You should call
f11bfc to check the values of the left and righthand sides of the termination condition.
 NE_ALG_FAIL

Algorithm breakdown at iteration no. $\u27e8\mathit{\text{value}}\u27e9$.
The last available iterates of the solution and residuals are returned, although it is possible that they are completely inaccurate.
 NE_ALLOC_FAIL

Dynamic memory allocation failed.
See
Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
 NE_BAD_PARAM

On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
 NE_CONVERGENCE

The solution has not converged after $\u27e8\mathit{\text{value}}\u27e9$ iterations.
Userrequested tidy termination. The solution has not converged after $\u27e8\mathit{\text{value}}\u27e9$ iterations.
 NE_INT

On entry,
${\mathbf{lwork}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint:
${\mathbf{lwork}}\ge {\mathbf{lwreq}}$, where
lwreq is returned by
f11bdc.
On initial entry, ${\mathbf{irevcm}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{irevcm}}=0$.
On intermediate reentry,
${\mathbf{irevcm}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: either
irevcm must be unchanged from its previous exit value or
${\mathbf{irevcm}}=5$ or
$6$.
 NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact
NAG for assistance.
See
Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
 NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library CL Interface for further information.
 NE_OUT_OF_SEQUENCE

Either
f11bdc was not called before calling
f11bec or it has returned an error.
f11bec has already completed its tasks. You need to set a new problem.
f11bec has been called again after returning the termination code
${\mathbf{irevcm}}=4$. No further computation has been carried out and all input data and data stored for access by
f11bfc have remained unchanged.
 NE_USER_STOP

Userrequested immediate termination.
The array
u returns the last iterate of the solution, the array
v returns the last iterate of the residual vector, for the CGS and TFQMR methods only.
 NE_WEIGHT_ZERO

The weights in array
wgt are all zero.
7
Accuracy
On completion, i.e.,
${\mathbf{irevcm}}=4$ on exit, the arrays
u and
v will return the solution and residual vectors,
${x}_{k}$ and
${r}_{k}=bA{x}_{k}$, respectively, at the
$k$th iteration, the last iteration performed, unless an immediate termination was requested.
On successful completion, the termination criterion is satisfied to within the userspecified tolerance, as described in
Section 3 in
f11bdc. The computed values of the left and righthand sides of the termination criterion selected can be obtained by a call to
f11bfc.
8
Parallelism and Performance
Background information to multithreading can be found in the
Multithreading documentation.
f11bec is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f11bec makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The number of operations carried out by f11bec for each iteration is likely to be principally determined by the computation of the matrixvector products $v=Au$ and by the solution of the preconditioning equation $Mv=u$ in the calling program. Each of these operations is carried out once every iteration.
The number of the remaining operations in f11bec for each iteration is approximately proportional to $\mathit{n}$.
The number of iterations required to achieve a prescribed accuracy cannot be easily determined at the onset, as it can depend dramatically on the conditioning and spectrum of the preconditioned matrix of the coefficients $\overline{A}={M}^{1}A$ (RGMRES, CGS and TFQMR methods) or $\overline{A}=A{M}^{1}$ (BiCGSTAB($\ell $) method).
Additional matrixvector products are required for the computation of
${\Vert A\Vert}_{1}$ or
${\Vert A\Vert}_{\infty}$, when this has not been supplied to
f11bdc and is required by the termination criterion employed.
If the termination criterion
${\Vert {r}_{k}\Vert}_{p}\le \tau ({\Vert b\Vert}_{p}+{\Vert A\Vert}_{p}\times {\Vert {x}_{k}\Vert}_{p})$ is used (see
Section 3 in
f11bdc) and
$\Vert {x}_{0}\Vert \gg \Vert {x}_{k}\Vert $, then the required accuracy cannot be obtained due to loss of significant digits. The iteration is restarted automatically at some suitable point:
f11bec sets
${x}_{0}={x}_{k}$ and the computation begins again. For particularly badly scaled problems, more than one restart may be necessary. This does not apply to the RGMRES method which, by its own nature, selfrestarts every superiteration. Naturally, restarting adds to computational costs: it is recommended that the iteration should start from a value
${x}_{0}$ which is as close to the true solution
$\stackrel{~}{x}$ as can be estimated. Otherwise, the iteration should start from
${x}_{0}=0$.
10
Example