NAG Library Routine Document

F11DCF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     METHOD – CHARACTER(*)Input

On entry: specifies the iterative method to be used.

$\mathbf{METHOD}=\text{'RGMRES'}$

Restarted generalized minimum residual method.

$\mathbf{METHOD}=\text{'CGS'}$

Conjugate gradient squared method.

$\mathbf{METHOD}=\text{'BICGSTAB'}$

Bi-conjugate gradient stabilized ($\ell$) method.

$\mathbf{METHOD}=\text{'TFQMR'}$

Transpose-free quasi-minimal residual method.

Constraint: $\mathbf{METHOD}=\text{'RGMRES'}$, $\text{'CGS'}$, $\text{'BICGSTAB'}$ or $\text{'TFQMR'}$.

2:     N – INTEGERInput

On entry: $n$, the order of the matrix $A$. This must be the same value as was supplied in the preceding call to F11DAF.

Constraint: $\mathbf{N}\ge 1$.

3:     NNZ – INTEGERInput

On entry: the number of nonzero elements in the matrix $A$. This must be the same value as was supplied in the preceding call to F11DAF.

Constraint: $1\le \mathbf{NNZ}\le {\mathbf{N}}^2$.

4:     A(LA) – REAL (KIND=nag_wp) arrayInput

On entry: the values returned in the array A by a previous call to F11DAF.

5:     LA – INTEGERInput

On entry: the dimension of the arrays A, IROW and ICOL as declared in the (sub)program from which F11DCF is called. This must be the same value as was supplied in the preceding call to F11DAF.

Constraint: $\mathbf{LA}\ge 2\times \mathbf{NNZ}$.

6:     IROW(LA) – INTEGER arrayInput

7:     ICOL(LA) – INTEGER arrayInput

8:     IPIVP(N) – INTEGER arrayInput

9:     IPIVQ(N) – INTEGER arrayInput

10:   ISTR($\mathbf{N}+1$) – INTEGER arrayInput

11:   IDIAG(N) – INTEGER arrayInput

On entry: the values returned in arrays IROW, ICOL, IPIVP, IPIVQ, ISTR and IDIAG by a previous call to F11DAF.

IPIVP and IPIVQ are restored on exit.

12:   B(N) – REAL (KIND=nag_wp) arrayInput

On entry: the right-hand side vector $b$.

13:   M – INTEGERInput

On entry: if $\mathbf{METHOD}=\text{'RGMRES'}$, M is the dimension of the restart subspace.

If $\mathbf{METHOD}=\text{'BICGSTAB'}$, M is the order $\ell$ of the polynomial Bi-CGSTAB method; otherwise, M is not referenced.

Constraints:

• if $\mathbf{METHOD}=\text{'RGMRES'}$, $0<\mathbf{M}\le \min \left(\mathbf{N},50\right)$;
• if $\mathbf{METHOD}=\text{'BICGSTAB'}$, $0<\mathbf{M}\le \min \left(\mathbf{N},10\right)$.

14:   TOL – REAL (KIND=nag_wp)Input

On entry: the required tolerance. Let $x_k$ denote the approximate solution at iteration $k$, and $r_k$ the corresponding residual. The algorithm is considered to have converged at iteration $k$ if

$${‖r_k‖}_{\infty }\le \tau \times \left({‖b‖}_{\infty }+{‖A‖}_{\infty }{‖x_k‖}_{\infty }\right)\text{.}$$

If $\mathbf{TOL}\le 0.0$, $\tau =\max \left(\sqrt{\epsilon },\sqrt{n}\epsilon \right)$ is used, where $\epsilon$ is the machine precision. Otherwise $\tau =\max \left(\mathbf{TOL},10\epsilon ,\sqrt{n}\epsilon \right)$ is used.

Constraint: $\mathbf{TOL}<1.0$.

15:   MAXITN – INTEGERInput

On entry: the maximum number of iterations allowed.

Constraint: $\mathbf{MAXITN}\ge 1$.

16:   X(N) – REAL (KIND=nag_wp) arrayInput/Output

On entry: an initial approximation to the solution vector $x$.

On exit: an improved approximation to the solution vector $x$.

17:   RNORM – REAL (KIND=nag_wp)Output

On exit: the final value of the residual norm ${‖r_k‖}_{\infty }$, where $k$ is the output value of ITN.

18:   ITN – INTEGEROutput

On exit: the number of iterations carried out.

19:   WORK(LWORK) – REAL (KIND=nag_wp) arrayWorkspace

20:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F11DCF is called.

Constraints:

• if $\mathbf{METHOD}=\text{'RGMRES'}$, $\mathbf{LWORK}\ge 4\times \mathbf{N}+\mathbf{M}\times \left(\mathbf{M}+\mathbf{N}+5\right)+101$;
• if $\mathbf{METHOD}=\text{'CGS'}$, $\mathbf{LWORK}\ge 8\times \mathbf{N}+100$;
• if $\mathbf{METHOD}=\text{'BICGSTAB'}$, $\mathbf{LWORK}\ge 2\times \mathbf{N}\times \left(\mathbf{M}+3\right)+\mathbf{M}\times \left(\mathbf{M}+2\right)+100$;
• if $\mathbf{METHOD}=\text{'TFQMR'}$, $\mathbf{LWORK}\ge 11\times \mathbf{N}+100$.

21:   IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{ or }\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.

On exit: $\mathbf{IFAIL}=\mathbf{0}$ unless the routine detects an error or a warning has been flagged (see Section 6).

6  Error Indicators and Warnings

$\mathbf{IFAIL}=1$

On entry,	$\mathbf{METHOD}\ne \text{'RGMRES'},\text{'CGS'},\text{'BICGSTAB'}$, or 'TFQMR',
or	$\mathbf{N}<1$,
or	$\mathbf{NNZ}<1$,
or	$\mathbf{NNZ}>{\mathbf{N}}^2$,
or	$\mathbf{LA}<2\times \mathbf{NNZ}$,
or	$\mathbf{M}<1$ and $\mathbf{METHOD}=\text{'RGMRES'}$ or $\mathbf{METHOD}=\text{'BICGSTAB'}$,
or	$\mathbf{M}>\min \left(\mathbf{N},50\right)$, with $\mathbf{METHOD}=\text{'RGMRES'}$,
or	$\mathbf{M}>\min \left(\mathbf{N},10\right)$, with $\mathbf{METHOD}=\text{'BICGSTAB'}$,
or	$\mathbf{TOL}\ge 1.0$,
or	$\mathbf{MAXITN}<1$,
or	LWORK too small.

$\mathbf{IFAIL}=2$

On entry, the CS representation of $A$ is invalid. Further details are given in the error message. Check that the call to F11DCF has been preceded by a valid call to F11DAF, and that the arrays A, IROW, and ICOL have not been corrupted between the two calls.

$\mathbf{IFAIL}=3$

On entry, the CS representation of the preconditioning matrix $M$ is invalid. Further details are given in the error message. Check that the call to F11DCF has been preceded by a valid call to F11DAF and that the arrays A, IROW, ICOL, IPIVP, IPIVQ, ISTR and IDIAG have not been corrupted between the two calls.

$\mathbf{IFAIL}=4$

The required accuracy could not be obtained. However, a reasonable accuracy may have been obtained, and further iterations could not improve the result. You should check the output value of RNORM for acceptability. 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.

$\mathbf{IFAIL}=5$

Required accuracy not obtained in MAXITN iterations.

$\mathbf{IFAIL}=6$

Algorithmic breakdown. A solution is returned, although it is possible that it is completely inaccurate.

$\mathbf{IFAIL}=7$ (F11BDF, F11BEF or F11BFF)

A serious error has occurred in an internal call to one of the specified routines. Check all subroutine calls and array sizes. Seek expert help.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f11dcfe.f90)

9.2  Program Data

Program Data (f11dcfe.d)

9.3  Program Results

Program Results (f11dcfe.r)