NAG Library Routine Document

F11DEF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     METHOD – CHARACTER(*)Input

On entry: 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:     PRECON – CHARACTER(1)Input

On entry: specifies the type of preconditioning to be used.

$\mathbf{PRECON}=\text{'N'}$

No preconditioning.

$\mathbf{PRECON}=\text{'J'}$

Jacobi.

$\mathbf{PRECON}=\text{'S'}$

Symmetric successive-over-relaxation.

Constraint: $\mathbf{PRECON}=\text{'N'}$, $\text{'J'}$ or $\text{'S'}$.

3:     N – INTEGERInput

On entry: $n$, the order of the matrix $A$.

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

4:     NNZ – INTEGERInput

On entry: the number of nonzero elements in the matrix $A$.

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

5:     A(NNZ) – REAL (KIND=nag_wp) arrayInput

On entry: the nonzero elements of the matrix $A$, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The routine F11ZAF may be used to order the elements in this way.

6:     IROW(NNZ) – INTEGER arrayInput

7:     ICOL(NNZ) – INTEGER arrayInput

On entry: the row and column indices of the nonzero elements supplied in A.

Constraints:

IROW and ICOL must satisfy the following constraints (which may be imposed by a call to F11ZAF):

• $1\le \mathbf{IROW}\left(\mathit{i}\right)\le \mathbf{N}$ and $1\le \mathbf{ICOL}\left(\mathit{i}\right)\le \mathbf{N}$, for $\mathit{i}=1,2,\dots ,\mathbf{NNZ}$;
• $\mathbf{IROW}\left(\mathit{i}-1\right)<\mathbf{IROW}\left(\mathit{i}\right)$ or $\mathbf{IROW}\left(\mathit{i}-1\right)=\mathbf{IROW}\left(\mathit{i}\right)$ and $\mathbf{ICOL}\left(\mathit{i}-1\right)<\mathbf{ICOL}\left(\mathit{i}\right)$, for $\mathit{i}=2,3,\dots ,\mathbf{NNZ}$.

8:     OMEGA – REAL (KIND=nag_wp)Input

On entry: if $\mathbf{PRECON}=\text{'S'}$, OMEGA is the relaxation parameter $\omega$ to be used in the SSOR method. Otherwise OMEGA need not be initialized and is not referenced.

Constraint: $0.0<\mathbf{OMEGA}<2.0$.

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

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

10:   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 $\mathbf{METHOD}=\text{'BICGSTAB'}$, $0<\mathbf{M}\le \min \left(\mathbf{N},10\right)$.

11:   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: $\mathbf{TOL}<1.0$.

12:   MAXITN – INTEGERInput

On entry: the maximum number of iterations allowed.

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

13:   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 $x$.

14:   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.

15:   ITN – INTEGEROutput

On exit: the number of iterations carried out.

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

17:   LWORK – INTEGERInput

On entry: the dimension of the array WORK as declared in the (sub)program from which F11DEF is called.

Constraints:

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

where $\mathit{\nu }=\mathbf{N}$ for $\mathbf{PRECON}=\text{'J'}$ or $\text{'S'}$, and $0$ otherwise

18:   IWORK($2\times \mathbf{N}+1$) – INTEGER arrayWorkspace

19:   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	$\mathbf{PRECON}\ne \text{'N'}$, $\text{'J'}$ or $\text{'S'}$,
or	$\mathbf{N}<1$,
or	$\mathbf{NNZ}<1$,
or	$\mathbf{NNZ}>{\mathbf{N}}^2$,
or	$\mathbf{PRECON}=\text{'S'}$ and OMEGA lies outside the interval $\left(0.0,2.0\right)$,
or	$\mathbf{M}<1$,
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 arrays IROW and ICOL fail to satisfy the following constraints:

• $1\le \mathbf{IROW}\left(i\right)\le \mathbf{N}$ and $1\le \mathbf{ICOL}\left(i\right)\le \mathbf{N}$, for $i=1,2,\dots ,\mathbf{NNZ}$;
• $\mathbf{IROW}\left(i-1\right)<\mathbf{IROW}\left(i\right)$, or $\mathbf{IROW}\left(i-1\right)=\mathbf{IROW}\left(i\right)$ and $\mathbf{ICOL}\left(i-1\right)<\mathbf{ICOL}\left(i\right)$, for $i=2,3,\dots ,\mathbf{NNZ}$.

Therefore a nonzero element has been supplied which does not lie within the matrix $A$, is out of order, or has duplicate row and column indices. Call F11ZAF to reorder and sum or remove duplicates.

$\mathbf{IFAIL}=3$

On entry, the matrix $A$ has a zero diagonal element. Jacobi and SSOR preconditioners are not appropriate for this problem.

$\mathbf{IFAIL}=4$

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 (f11defe.f90)

9.2  Program Data

Program Data (f11defe.d)

9.3  Program Results

Program Results (f11defe.r)