NAG Library Routine Document

F11MEF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     N – INTEGERInput

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

Constraint: $\mathbf{N}\ge 0$.

2:     IROWIX($*$) – INTEGER arrayInput

Note: the dimension of the array IROWIX must be at least $\mathit{nnz}$, the number of nonzeros of the sparse matrix $A$.

On entry: the row index array of sparse matrix $A$.

3:     A($*$) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array A must be at least $\mathit{nnz}$, the number of nonzeros of the sparse matrix $A$.

On entry: the array of nonzero values in the sparse matrix $A$.

4:     IPRM($7\times \mathbf{N}$) – INTEGER arrayInput/Output

On entry: contains the column permutation which defines the permutation $P_c$ and associated data structures as computed by routine F11MDF.

On exit: part of the array is modified to record the row permutation $P_r$ determined by pivoting.

5:     THRESH – REAL (KIND=nag_wp)Input

On entry: the diagonal pivoting threshold, $t$. At step $j$ of the Gaussian elimination, if $\left|A_{jj}\right|\ge t\left({\displaystyle \underset{i\ge j}{\max }}\left|A_{ij}\right|\right)$, use $A_{jj}$ as a pivot, otherwise use ${\displaystyle \underset{i\ge j}{\max }}\left|A_{ij}\right|$. A value of $t=1$ corresponds to partial pivoting, a value of $t=0$ corresponds to always choosing the pivot on the diagonal (unless it is zero).

Suggested value: $\mathbf{THRESH}=1.0$. Smaller values may result in a faster factorization, but the benefits are likely to be small in most cases. It might be possible to use $\mathbf{THRESH}=0.0$ if you are confident about the stability of the factorization, for example, if $A$ is diagonally dominant.

Constraint: $0.0\le \mathbf{THRESH}\le 1.0$.

6:     NZLMX – INTEGERInput

On entry: indicates the available size of array IL. The dimension of IL should be at least $7\times \mathbf{N}+\mathbf{NZLMX}+4$. A good range for NZLMX that works for many problems is $\mathit{nnz}$ to $8\times \mathit{nnz}$, where $\mathit{nnz}$ is the number of nonzeros in the sparse matrix $A$. If, on exit, $\mathbf{IFAIL}=\mathbf{2}$, the given NZLMX was too small and you should attempt to provide more storage and call the routine again.

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

7:     NZLUMX – INTEGERInput/Output

On entry: indicates the available size of array LVAL. The dimension of LVAL should be at least NZLUMX.

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

On exit: if $\mathbf{IFAIL}=\mathbf{4}$, the given NZLUMX was too small and is reset to a value that will be sufficient. You should then provide the indicated storage and call the routine again.

8:     NZUMX – INTEGERInput

On entry: indicates the available sizes of arrays IU and UVAL. The dimension of IU should be at least $2\times \mathbf{N}+\mathbf{NZUMX}+1$ and the dimension of UVAL should be at least NZUMX. A good range for NZUMX that works for many problems is $\mathit{nnz}$ to $8\times \mathit{nnz}$, where $\mathit{nnz}$ is the number of nonzeros in the sparse matrix $A$. If, on exit, $\mathbf{IFAIL}=\mathbf{3}$, the given NZUMX was too small and you should attempt to provide more storage and call the routine again.

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

9:     IL($7\times \mathbf{N}+\mathbf{NZLMX}+4$) – INTEGER arrayOutput

On exit: encapsulates the sparsity pattern of matrix $L$.

10:   LVAL(NZLUMX) – REAL (KIND=nag_wp) arrayOutput

On exit: records the nonzero values of matrix $L$ and some of the nonzero values of matrix $U$.

11:   IU($2\times \mathbf{N}+\mathbf{NZUMX}+1$) – INTEGER arrayOutput

On exit: encapsulates the sparsity pattern of matrix $U$.

12:   UVAL($\mathbf{NZUMX}$) – REAL (KIND=nag_wp) arrayOutput

On exit: records some of the nonzero values of matrix $U$.

13:   NNZL – INTEGEROutput

On exit: the number of nonzero values in the matrix $L$.

14:   NNZU – INTEGEROutput

On exit: the number of nonzero values in the matrix $U$.

15:   FLOP – REAL (KIND=nag_wp)Output

On exit: the number of floating point operations performed.

16:   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,	$\mathbf{N}<0$,
or	$\mathbf{NZLMX}<1$,
or	$\mathbf{NZLUMX}<1$,
or	$\mathbf{NZUMX}<1$,
or	$\mathbf{THRESH}<0.0$,
or	$\mathbf{THRESH}>1.0$.

$\mathbf{IFAIL}=2$

$\mathbf{NZLMX}$ was not large enough. You should repeat the call with a larger value of $\mathbf{NZLMX}$, providing more storage for the output array $\mathbf{IL}$.

$\mathbf{IFAIL}=3$

$\mathbf{NZUMX}$ was not large enough. You should repeat the call with a larger value of $\mathbf{NZUMX}$, providing more storage for the output arrays $\mathbf{IU}$ and $\mathbf{UVAL}$.

$\mathbf{IFAIL}=4$

$\mathbf{NZLUMX}$ was not large enough. You should repeat the call with the value of $\mathbf{NZLUMX}$ returned on exit, providing more storage for the output array $\mathbf{LVAL}$.

$\mathbf{IFAIL}=5$

The matrix $A$ is singular and no factorization will be attempted.

$\mathbf{IFAIL}=301$

Unable to allocate required internal workspace.

7  Accuracy

8  Further Comments

9  Example

9.1  Program Text

Program Text (f11mefe.f90)

9.2  Program Data

Program Data (f11mefe.d)

9.3  Program Results

Program Results (f11mefe.r)