NAG Library Routine Document

f11bef  (real_gen_basic_solver)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{irevcm}$ – IntegerInput/Output

On initial entry: $\mathbf{irevcm}=0$, otherwise an error condition will be raised.

On intermediate re-entry: 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. f11bef will then return with the termination code $\mathbf{irevcm}=4$. Note that before calling f11bef with $\mathbf{irevcm}=5$ the calling program must have performed the tasks required by the value of irevcm returned by the previous call to f11bef, otherwise subsequently returned values may be invalid.

$\mathbf{irevcm}=6$

Immediate termination: f11bef 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 matrix-vector 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.

$\mathbf{irevcm}=-1$

The calling program must compute the matrix-vector product $v=A^{\mathrm{T}}u$, where $u$ and $v$ are stored in u and v, respectively; RGMRES, CGS and Bi-CGSTAB($\ell$) methods return $\mathbf{irevcm}=-1$ only if the matrix norm ${‖A‖}_1$ or ${‖A‖}_{\infty }$ is estimated internally using Higham's method. This can only happen if $\mathbf{iterm}=1$ in f11bdf.

$\mathbf{irevcm}=1$

The calling program must compute the matrix-vector 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. f11bff can be called at this step to return additional information.

On final exit: $\mathbf{irevcm}=4$: f11bef has completed its tasks. The value of ifail 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 re-entry, either irevcm must remain unchanged or be reset to $5$ or $6$.

Note: any values you return to f11bef as part of the reverse communication procedure should not include floating-point NaN (Not a Number) or infinity values, since these are not handled by f11bef. If your code inadvertently does return any NaNs or infinities, f11bef is likely to produce unexpected results.

2:     $\mathbf{u}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension 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 re-entry: must remain unchanged.

On intermediate exit: the returned value of irevcm determines the contents of u in the following way:

• if $\mathbf{irevcm}=-1$, $1$ 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 $\mathbf{ifail}=\mathbf{3}$ or $\mathbf{ifail}<\mathbf{0}$, the array u is unchanged from the initial entry to f11bef.

If $\mathbf{ifail}=\mathbf{1}$, the array u is unchanged from the last entry to f11bef.

Otherwise, u holds the last available iterate of the solution of the system of equations, for all returned values of ifail.

3:     $\mathbf{v}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output

Note: the dimension of the array v must be at least $\mathit{n}$.

On initial entry: the right-hand side $b$ of the system of equations $Ax=b$.

On intermediate re-entry: the returned value of irevcm determines the contents of v in the following way:

• if $\mathbf{irevcm}=-1$, $1$ 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 f11bef;
• 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 $\mathbf{ifail}=\mathbf{3}$ or $\mathbf{ifail}<\mathbf{0}$, the array v is unchanged from the initial entry to f11bef.

If $\mathbf{ifail}=\mathbf{1}$, the array v is unchanged from the last entry to f11bef.

If $\mathbf{ifail}=\mathbf{0}$ or $\mathbf{2}$, 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 $\mathbf{ifail}=\mathbf{8}$ is returned on last entry, in which case v is set to $0.0$.

4:     $\mathbf{wgt}\left(*\right)$ – Real (Kind=nag_wp) arrayInput

Note: the dimension of the array wgt must be at least $\max \left(1,\mathit{n}\right)$.

On entry: the user-supplied weights, if these are to be used in the computation of the vector norms in the termination criterion (see Sections 3 and 5 in f11bdf).

Constraint: if weights are to be used, at least one element of wgt must be nonzero.

5:     $\mathbf{work}\left(\mathbf{lwork}\right)$ – Real (Kind=nag_wp) arrayCommunication Array

On initial entry: the array work as returned by f11bdf (see also Section 5 in f11bdf).

On intermediate re-entry: must remain unchanged.

6:     $\mathbf{lwork}$ – IntegerInput

On initial entry: the dimension of the array work as declared in the (sub)program from which f11bef is called (see also Sections 3 and 5 in f11bdf). The required amount of workspace is as follows:

Method	Requirements
RGMRES	$\mathbf{lwork}=100+\mathit{n}\left(m+3\right)+m\left(m+5\right)+1$, where $m$ is the dimension of the basis.
CGS	$\mathbf{lwork}=100+7\mathit{n}$.
Bi-CGSTAB($\ell$)	$\mathbf{lwork}=100+\left(2\mathit{n}+\ell \right)\left(\ell +2\right)+p$, 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 f11bdf.

7:     $\mathbf{ifail}$ – IntegerInput/Output

On initial entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output arguments may be useful even if $\mathbf{ifail}\ne \mathbf{0}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of ifail on exit.

On intermediate exit: the value of ifail is meaningless and should be ignored.

On final exit: (i.e., when $\mathbf{irevcm}=4$) $\mathbf{ifail}=\mathbf{0}$, unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

$\mathbf{ifail}=-i$

On entry, the $i$th argument had an illegal value.

$\mathbf{ifail}=1$

f11bef 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 f11bff have remained unchanged.

$\mathbf{ifail}=2$

The required accuracy could not be obtained. However, f11bef has terminated with reasonable accuracy: the last iterate of the residual satisfied the termination criterion but the exact residual $r=b-Ax$, did not. After the first occurrence of this situation, the iteration was restarted once, but f11bef 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 f11bff to check the values of the left- and right-hand sides of the termination condition.

$\mathbf{ifail}=3$

f11bdf was either not called before calling f11bef or it returned an error. The arguments u and v remain unchanged.

$\mathbf{ifail}=4$

The calling program requested a tidy termination before the solution had converged. The arrays u and v return the last iterates available of the solution and of the residual vector, respectively.

$\mathbf{ifail}=5$

The solution did not converge within the maximum number of iterations allowed. The arrays u and v return the last iterates available of the solution and of the residual vector, respectively.

$\mathbf{ifail}=6$

Algorithmic breakdown. The last available iterates of the solution and residuals are returned, although it is possible that they are completely inaccurate.

$\mathbf{ifail}=8$

The calling program requested an immediate termination. However, 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.

$\mathbf{ifail}=10$

User-supplied weights are to be used, but all elements of the array wgt are zero.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

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Accuracy

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Parallelism and Performance

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Further Comments

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Example