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.
The routine may be called by the names f11mdf or nagf_sparse_direct_real_gen_setup.
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. Software30,3 353–376
Gilbert J R, Larimore S I and Ng E G (2004) Algorithm 836: COLAMD, an approximate minimum degree ordering algorithm ACM Trans. Math. Software30, 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 $\text{'M'}$.
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.
On entry: the first ${\mathbf{n}}$ entries contain the column permutation if supplied by the user. 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 $[0,({\mathbf{n}}-1)]$, 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: The column permutation given or computed is returned in the second ${\mathbf{n}}$ entries. The rest of the array contains data structures that will be used by other routines in the suite. 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 and this permutation is returned in the first ${\mathbf{n}}$ entries. This whole array is needed by the $LU$ factorization routine f11mef and associated routines f11mff,f11mhfandf11mmf and should be passed to them unchanged.
6: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{spec}}=\u27e8\mathit{\text{value}}\u27e9$. Constraint: ${\mathbf{spec}}=\text{'N'}$, $\text{'U'}$ or $\text{'M'}$.
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface 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.
9Further Comments
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