NAG Library Routine Document

f11dgf  (real_gen_solve_bdilu)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{method}$ – 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:     $\mathbf{n}$ – IntegerInput

3:     $\mathbf{nnz}$ – IntegerInput

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

5:     $\mathbf{la}$ – IntegerInput

6:     $\mathbf{irow}\left(\mathbf{la}\right)$ – Integer arrayInput

7:     $\mathbf{icol}\left(\mathbf{la}\right)$ – Integer arrayInput

8:     $\mathbf{nb}$ – IntegerInput

9:     $\mathbf{istb}\left(\mathbf{nb}+1\right)$ – Integer arrayInput

10:   $\mathbf{indb}\left(\mathbf{lindb}\right)$ – Integer arrayInput

11:   $\mathbf{lindb}$ – IntegerInput

12:   $\mathbf{ipivp}\left(\mathbf{lindb}\right)$ – Integer arrayInput

13:   $\mathbf{ipivq}\left(\mathbf{lindb}\right)$ – Integer arrayInput

14:   $\mathbf{istr}\left(\mathbf{lindb}+1\right)$ – Integer arrayInput

15:   $\mathbf{idiag}\left(\mathbf{lindb}\right)$ – Integer arrayInput

On entry: the values returned in arrays irow, icol, ipivp, ipivq, istr and idiag by a previous call to f11dff.

The arrays istb, indb and a together with the the scalars n, nnz, la, nb and lindb must be the same values that were supplied in the preceding call to f11dff.

16:   $\mathbf{b}\left(\mathbf{n}\right)$ – Real (Kind=nag_wp) arrayInput

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

17:   $\mathbf{m}$ – 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 $\mathbf{method}=\text{'BICGSTAB'}$, $0<\mathbf{m}\le \min \left(\mathbf{n},10\right)$.

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

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

19:   $\mathbf{maxitn}$ – IntegerInput

On entry: the maximum number of iterations allowed.

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

20:   $\mathbf{x}\left(\mathbf{n}\right)$ – 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 $x$.

21:   $\mathbf{rnorm}$ – 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.

22:   $\mathbf{itn}$ – IntegerOutput

On exit: the number of iterations carried out.

23:   $\mathbf{work}\left(\mathbf{lwork}\right)$ – Real (Kind=nag_wp) arrayWorkspace

24:   $\mathbf{lwork}$ – IntegerInput

On entry: the dimension of the array work as declared in the (sub)program from which f11dgf is called.

Constraints:

• if $\mathbf{method}=\text{'RGMRES'}$, $\mathbf{lwork}\ge 6\times \mathbf{n}+\mathbf{m}\times \left(\mathbf{m}+\mathbf{n}+5\right)+101$;
• if $\mathbf{method}=\text{'CGS'}$, $\mathbf{lwork}\ge 10\times \mathbf{n}+100$;
• if $\mathbf{method}=\text{'BICGSTAB'}$, $\mathbf{lwork}\ge 2\times \mathbf{n}\times \mathbf{m}+8\times \mathbf{n}+\mathbf{m}\times \left(\mathbf{m}+2\right)+100$;
• if $\mathbf{method}=\text{'TFQMR'}$, $\mathbf{lwork}\ge 13\times \mathbf{n}+100$.

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

On 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, if you are not familiar with this argument, 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, for $b=〈\mathit{\text{value}}〉$, $\mathbf{istb}\left(b+1\right)=〈\mathit{\text{value}}〉$ and $\mathbf{istb}\left(b\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{istb}\left(\mathit{b}+1\right)>\mathbf{istb}\left(\mathit{b}\right)$, for $\mathit{b}=1,2,\dots ,\mathbf{nb}$.

On entry, $\mathbf{indb}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $1\le \mathbf{indb}\left(\mathit{m}\right)\le \mathbf{n}$, for $\mathit{m}=1,2,\dots ,\mathbf{istb}\left(\mathbf{nb}+1\right)-1$ 

On entry, $\mathbf{istb}\left(1\right)=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{istb}\left(1\right)\ge 1$.

On entry, $\mathbf{la}=〈\mathit{\text{value}}〉$ and $\mathbf{nnz}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{la}\ge 2\times \mathbf{nnz}$.

On entry, $\mathbf{lindb}=〈\mathit{\text{value}}〉$, $\mathbf{istb}\left(\mathbf{nb}+1\right)-1=〈\mathit{\text{value}}〉$ and $\mathbf{nb}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lindb}\ge \mathbf{istb}\left(\mathbf{nb}+1\right)-1$.

On entry, $\mathbf{lwork}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lwork}\ge 〈\mathit{\text{value}}〉$.

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: if $\mathbf{method}=\text{'RGMRES'}$, $1\le \mathbf{m}\le \min \left(\mathbf{n},〈\mathit{\text{value}}〉\right)$.
If $\mathbf{method}=\text{'BICGSTAB'}$, $1\le \mathbf{m}\le \min \left(\mathbf{n},〈\mathit{\text{value}}〉\right)$.

On entry, $\mathbf{maxitn}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{maxitn}\ge 1$.

On entry, $\mathbf{method}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{method}=\text{'RGMRES'}$, $\text{'CGS'}$ or $\text{'BICGSTAB'}$.

On entry, $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{n}\ge 1$.

On entry, $\mathbf{nb}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $1\le \mathbf{nb}\le \mathbf{n}$.

On entry, $\mathbf{nnz}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nnz}\ge 1$.

On entry, $\mathbf{nnz}=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nnz}\le {\mathbf{n}}^2$.

On entry, $\mathbf{tol}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{tol}<1.0$.

$\mathbf{ifail}=2$

On entry, element $〈\mathit{\text{value}}〉$ of a was out of order.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dff and f11dgf.

On entry, $\mathbf{icol}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $1\le \mathbf{icol}\left(\mathit{i}\right)\le \mathbf{n}$, for $\mathit{i}=1,2,\dots ,\mathbf{nnz}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dff and f11dgf.

On entry, $\mathbf{irow}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$ and $\mathbf{n}=〈\mathit{\text{value}}〉$.
Constraint: $1\le \mathbf{irow}\left(\mathit{i}\right)\le \mathbf{n}$, for $\mathit{i}=1,2,\dots ,\mathbf{nnz}$.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dff and f11dgf.

On entry, location $〈\mathit{\text{value}}〉$ of $\left(\mathbf{irow},\mathbf{icol}\right)$ was a duplicate.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dff and f11dgf.

$\mathbf{ifail}=3$

The CS representation of the preconditioner is invalid.
Check that a, irow, icol, ipivp, ipivq, istr and idiag have not been corrupted between calls to f11dff and f11dgf.

$\mathbf{ifail}=4$

The required accuracy could not be obtained. However a reasonable accuracy may have been achieved. 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.

$\mathbf{ifail}=5$

The solution has not converged after $〈\mathit{\text{value}}〉$ iterations.

$\mathbf{ifail}=6$

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

$\mathbf{ifail}=-99$

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.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (f11dgfe.f90)

10.2

Program Data

Program Data (f11dgfe.d)

10.3

Program Results

Program Results (f11dgfe.r)