F11 Chapter Contents
F11 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF11MDF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F11MDF computes a column permutation suitable for $LU$ factorization (by F11MEF) of a real sparse matrix in compressed column (Harwell–Boeing) format and applies it to the matrix. This routine must be called prior to F11MEF.

## 2  Specification

 SUBROUTINE F11MDF ( SPEC, N, ICOLZP, IROWIX, IPRM, IFAIL)
 INTEGER N, ICOLZP(*), IROWIX(*), IPRM(7*N), IFAIL CHARACTER(1) SPEC

## 3  Description

Given a sparse matrix in compressed column (Harwell–Boeing) format $A$ and a choice of column permutation schemes, the routine computes those data structures that will be needed by the $LU$ factorization routine F11MEF and associated routines F11MMF, F11MFF and F11MHF. The column permutation choices are:
• original order (that is, no permutation);
• user-supplied permutation;
• a permutation, computed by the routine, designed to minimize fill-in during the $LU$ factorization.
The algorithm for this computed permutation is based on the approximate minimum degree column ordering algorithm COLAMD. The computed permutation is not sensitive to the magnitude of the nonzero values of $A$.

## 4  References

Amestoy P R, Davis T A and Duff I S (1996) An approximate minimum degree ordering algorithm SIAM J. Matrix Anal. Appl. 17 886–905
Gilbert J R and Larimore S I (2004) A column approximate minimum degree ordering algorithm ACM Trans. Math. Software 30,3 353–376
Gilbert J R, Larimore S I and Ng E G (2004) Algorithm 836: COLAMD, an approximate minimum degree ordering algorithm ACM Trans. Math. Software 30, 3 377–380

## 5  Parameters

1:     SPEC – CHARACTER(1)Input
On entry: indicates the permutation to be applied.
${\mathbf{SPEC}}=\text{'N'}$
The identity permutation is used (i.e., the columns are not permuted).
${\mathbf{SPEC}}=\text{'U'}$
The permutation in the IPRM array is used, as supplied by you.
${\mathbf{SPEC}}=\text{'M'}$
The permutation computed by the COLAMD algorithm is used
Constraint: ${\mathbf{SPEC}}=\text{'N'}$, $\text{'U'}$ or $\text{'M'}$.
2:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     ICOLZP($*$) – INTEGER arrayInput
Note: the dimension of the array ICOLZP must be at least ${\mathbf{N}}+1$.
On entry: ${\mathbf{ICOLZP}}\left(i\right)$ contains the index in $A$ of the start of a new column. See Section 2.1.3 in the F11 Chapter Introduction.
4:     IROWIX($*$) – INTEGER arrayInput
Note: the dimension of the array IROWIX must be at least ${\mathbf{ICOLZP}}\left({\mathbf{N}}+1\right)-1$, the number of nonzeros of the sparse matrix $A$.
On entry: ${\mathbf{IROWIX}}\left(i\right)$ contains the row index in $A$ for element $A\left(i\right)$. See Section 2.1.3 in the F11 Chapter Introduction.
5:     IPRM($7×{\mathbf{N}}$) – INTEGER arrayInput/Output
On entry: the first ${\mathbf{N}}$ entries contain the column permutation supplied by you. This will be used if ${\mathbf{SPEC}}=\text{'U'}$, and ignored otherwise. If used, it must consist of a permutation of all the integers in the range $\left[0,\left({\mathbf{N}}-1\right)\right]$, the leftmost column of the matrix $A$ denoted by $0$ and the rightmost by ${\mathbf{N}}-1$. Labelling columns in this way, ${\mathbf{IPRM}}\left(i\right)=j$ means that column $i-1$ of $A$ is in position $j$ in $A{P}_{c}$, where ${P}_{r}A{P}_{c}=LU$ expresses the factorization to be performed.
On exit: a new permutation is returned in the first ${\mathbf{N}}$ entries. The rest of the array contains data structures that will be used by other routines. The routine computes the column elimination tree for $A$ and a post-order permutation on the tree. It then compounds the IPRM permutation given or computed by the COLAMD algorthm with the post-order permutation. This array is needed by the $LU$ factorization routine F11MEF and associated routines F11MFF, F11MHF and F11MMF and should be passed to them unchanged.
6:     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

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{SPEC}}\ne \text{'N'}$, $\text{'U'}$ or $\text{'M'}$, or ${\mathbf{N}}<0$.
${\mathbf{IFAIL}}=2$
On entry, ${\mathbf{SPEC}}=\text{'U'}$, but IPRM does not represent a valid permutation of the integers in $\left[0,\left({\mathbf{N}}-1\right)\right]$. An input value of IPRM is either out of range or repeated.
${\mathbf{IFAIL}}=3$
Unspecified failure of the COLAMD algorithm. This should not happen and should be reported to NAG.
${\mathbf{IFAIL}}=4$
On entry, the array ICOLZP failed to satisfy the following constraints:
• ${\mathbf{ICOLZP}}\left(1\right)=1$;
• ${\mathbf{ICOLZP}}\left(\mathit{i}-1\right)\le {\mathbf{ICOLZP}}\left(\mathit{i}\right)$, for $\mathit{i}=2,3,\dots ,{\mathbf{N}}+1$;
• ${\mathbf{ICOLZP}}\left(\mathit{i}\right)\le {\mathbf{N}}×{\mathbf{N}}+1$, for $\mathit{i}=1,2,\dots ,{\mathbf{N}}+1$.
${\mathbf{IFAIL}}=5$
On entry, the array IROWIX failed to satisfy the following constraints:
• $1\le {\mathbf{IROWIX}}\left(\mathit{i}\right)\le {\mathbf{N}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{ICOLZP}}\left({\mathbf{N}}+1\right)-1$;
• for each column $i=1\dots {\mathbf{N}}$, the row indices ${\mathbf{IROWIX}}\left(j\right)$, where $j={\mathbf{ICOLZP}}\left(i\right)\dots {\mathbf{ICOLZP}}\left(i+1\right)-1$, do not repeat.
${\mathbf{IFAIL}}=301$
Unable to allocate required internal workspace.

## 7  Accuracy

Not applicable. This computation does not use floating point numbers.

We recommend calling this routine with ${\mathbf{SPEC}}=\text{'M'}$ before calling F11MEF. The COLAMD algorithm computes a sparsity-preserving permutation ${P}_{c}$ solely from the pattern of $A$ such that the $LU$ factorization ${P}_{r}A{P}_{c}=LU$ remains as sparse as possible, regardless of the subsequent choice of ${P}_{r}$. The algorithm takes advantage of the existence of super-columns (columns with the same sparsity pattern) to reduce running time.

## 9  Example

This example computes a sparsity preserving column permutation for the $LU$ factorization of the matrix $A$, where
 $A= 2.00 1.00 0 0 0 0 0 1.00 -1.00 0 4.00 0 1.00 0 1.00 0 0 0 1.00 2.00 0 -2.00 0 0 3.00 .$

### 9.1  Program Text

Program Text (f11mdfe.f90)

### 9.2  Program Data

Program Data (f11mdfe.d)

### 9.3  Program Results

Program Results (f11mdfe.r)