NAG Library Routine Document

1Purpose

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.

2Specification

Fortran Interface
 Subroutine f11mdf ( spec, n, iprm,
 Integer, Intent (In) :: n, icolzp(max(1,(n+1))), irowix(max(1,(ICOLZP(max(1,(n+1)))-1))) Integer, Intent (Inout) :: iprm(max(1,(7*n))), ifail Character (1), Intent (In) :: spec
#include nagmk26.h
 void f11mdf_ ( const char *spec, const Integer *n, const Integer icolzp[], const Integer irowix[], Integer iprm[], Integer *ifail, const Charlen length_spec)

3Description

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 $A$.

4References

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

5Arguments

1:     $\mathbf{spec}$ – 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:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{icolzp}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\left({\mathbf{n}}+1\right)\right)\right)$ – Integer arrayInput
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:     $\mathbf{irowix}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\left({\mathbf{icolzp}}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\left({\mathbf{n}}+1\right)\right)\right)-1\right)\right)\right)$ – Integer arrayInput
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:     $\mathbf{iprm}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\left(7×{\mathbf{n}}\right)\right)\right)$ – 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:     $\mathbf{ifail}$ – 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).

6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, 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, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{spec}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{spec}}=\text{'N'}$, $\text{'U'}$ or $\text{'M'}$.
${\mathbf{ifail}}=2$
Incorrect column permutations in array iprm.
${\mathbf{ifail}}=3$
COLAMD algorithm failed.
${\mathbf{ifail}}=4$
Incorrect specification of argument icolzp.
${\mathbf{ifail}}=5$
Incorrect specification of argument irowix.
${\mathbf{ifail}}=-99$
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.

7Accuracy

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

8Parallelism and Performance

f11mdf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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.

10Example

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

10.1Program Text

Program Text (f11mdfe.f90)

10.2Program Data

Program Data (f11mdfe.d)

10.3Program Results

Program Results (f11mdfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017