NAG Library Routine Document

F11DXF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     STORE – CHARACTER(1)Input

On entry: specifies whether the matrix $A$ is stored using symmetric coordinate storage (SCS) (applicable only to a Hermitian matrix $A$) or coordinate storage (CS) (applicable to both Hermitian and non-Hermitian matrices).

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

The complete matrix $A$ is stored in CS format.

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

The lower triangle of the Hermitian matrix $A$ is stored in SCS format.

Constraint: $\mathbf{STORE}=\text{'N'}$ or $\text{'S'}$.

2:     TRANS – CHARACTER(1)Input

On entry: if $\mathbf{STORE}=\text{'N'}$, specifies whether the approximate solution of $Ax=b$ or of $A^{\mathrm{T}}x=b$ is required.

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

The approximate solution of $Ax=b$ is calculated.

$\mathbf{TRANS}=\text{'T'}$

The approximate solution of $A^{\mathrm{T}}x=b$ is calculated.

Suggested value: if the matrix $A$ is Hermitian and stored in CS format, it is recommended that $\mathbf{TRANS}=\text{'N'}$ for reasons of efficiency.

Constraint: $\mathbf{TRANS}=\text{'N'}$ or $\text{'T'}$.

3:     INIT – CHARACTER(1)Input

On entry: on first entry, INIT should be set to 'I', unless the diagonal elements of $A$ are already stored in the array DIAG. If DIAG already contains the diagonal of $A$, it must be set to 'N'.

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

DIAG must contain the diagonal of $A$.

$\mathbf{INIT}=\text{'I'}$

DIAG will store the diagonal of $A$ on exit.

Suggested value: $\mathbf{INIT}=\text{'I'}$ on first entry; $\mathbf{INIT}=\text{'N'}$, subsequently, unless DIAG has been overwritten.

Constraint: $\mathbf{INIT}=\text{'N'}$ or $\text{'I'}$.

4:     NITER – INTEGERInput

On entry: the number of Jacobi iterations requested.

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

5:     N – INTEGERInput

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

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

6:     NNZ – INTEGERInput

On entry: if $\mathbf{STORE}=\text{'N'}$, the number of nonzero elements in the matrix $A$.

If $\mathbf{STORE}=\text{'S'}$, the number of nonzero elements in the lower triangle of the matrix $A$.

Constraints:

• if $\mathbf{STORE}=\text{'N'}$, $1\le \mathbf{NNZ}\le {\mathbf{N}}^2$;
• if $\mathbf{STORE}=\text{'S'}$, $1\le \mathbf{NNZ}\le \mathbf{N}\times \left(\mathbf{N}+1\right)/2$.

7:     A(NNZ) – COMPLEX (KIND=nag_wp) arrayInput

On entry: if $\mathbf{STORE}=\text{'N'}$, the nonzero elements in the matrix $A$ (CS format).

If $\mathbf{STORE}=\text{'S'}$, the nonzero elements in the lower triangle of the matrix $A$ (SCS format).

In both cases, the elements of either $A$ or of its lower triangle must be ordered by increasing row index and by increasing column index within each row. Multiple entries for the same row and columns indices are not permitted. The routine F11ZNF or F11ZPF may be used to reorder the elements in this way for CS and SCS storage, respectively.

8:     IROW(NNZ) – INTEGER arrayInput

9:     ICOL(NNZ) – INTEGER arrayInput

On entry: if $\mathbf{STORE}=\text{'N'}$, the row and column indices of the nonzero elements supplied in A.

If $\mathbf{STORE}=\text{'S'}$, the row and column indices of the nonzero elements of the lower triangle of the matrix $A$ supplied in A.

Constraints:

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

10:   CHECK – CHARACTER(1)Input

On entry: specifies whether or not the CS or SCS representation of the matrix $A$ should be checked.

$\mathbf{CHECK}=\text{'C'}$

Checks are carried out on the values of N, NNZ, IROW, ICOL; if $\mathbf{INIT}=\text{'N'}$, DIAG is also checked.

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

None of these checks are carried out.

See also Section 8.2.

Constraint: $\mathbf{CHECK}=\text{'C'}$ or $\text{'N'}$.

11:   B(N) – COMPLEX (KIND=nag_wp) arrayInput

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

12:   X(N) – COMPLEX (KIND=nag_wp) arrayOutput

On exit: the approximate solution vector $x_{\mathbf{NITER}}$.

13:   DIAG(N) – COMPLEX (KIND=nag_wp) arrayInput/Output

On entry: if $\mathbf{INIT}=\text{'N'}$, the diagonal elements of $A$.

On exit: if $\mathbf{INIT}=\text{'N'}$, unchanged on exit.

If $\mathbf{INIT}=\text{'I'}$, the diagonal elements of $A$.

14:   WORK(N) – COMPLEX (KIND=nag_wp) arrayWorkspace

15:   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,	$\mathbf{STORE}\ne \text{'N'}$ or $\text{'S'}$,
or	$\mathbf{TRANS}\ne \text{'N'}$ or $\text{'T'}$,
or	$\mathbf{INIT}\ne \text{'N'}$ or $\text{'I'}$,
or	$\mathbf{CHECK}\ne \text{'C'}$ or $\text{'N'}$,
or	$\mathbf{NITER}\le 0$.

$\mathbf{IFAIL}=2$

On entry,	$\mathbf{N}<1$,
or	$\mathbf{NNZ}<1$,
or	$\mathbf{NNZ}>{\mathbf{N}}^2$, if $\mathbf{STORE}=\text{'N'}$,
or	$1\le \mathbf{NNZ}\le \left[\mathbf{N}\left(\mathbf{N}+1\right)\right]/2$, if $\mathbf{STORE}=\text{'S'}$.

$\mathbf{IFAIL}=3$

On entry, the arrays IROW and ICOL fail to satisfy the following constraints:

• $1\le \mathbf{IROW}\left(i\right)\le \mathbf{N}$ and
  • if $\mathbf{STORE}=\text{'N'}$ then $1\le \mathbf{ICOL}\left(i\right)\le \mathbf{N}$, or
  • if $\mathbf{STORE}=\text{'S'}$ then $1\le \mathbf{ICOL}\left(\mathit{i}\right)\le \mathbf{IROW}\left(\mathit{i}\right)$, 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 $\mathit{i}=2,3,\dots ,\mathbf{NNZ}$.

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 either F11ZAF or F11ZBF to reorder and sum or remove duplicates when $\mathbf{STORE}=\text{'N'}$ or $\mathbf{STORE}=\text{'S'}$, respectively.

$\mathbf{IFAIL}=4$

On entry, $\mathbf{INIT}=\text{'N'}$ and some diagonal elements of $A$ stored in DIAG are zero.

$\mathbf{IFAIL}=5$

On entry, $\mathbf{INIT}=\text{'I'}$ and some diagonal elements of $A$ are zero.

7  Accuracy

8  Further Comments

8.1  Timing

8.2  Use of CHECK

9  Example

9.1  Program Text

Program Text (f11dxfe.f90)

9.2  Program Data

Program Data (f11dxfe.d)

9.3  Program Results

Program Results (f11dxfe.r)