NAG FL Interface
f11yef (sym_​rcm)

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1 Purpose

f11yef reduces the bandwidth of a sparse symmetric matrix stored in compressed column storage format using the Reverse Cuthill–McKee algorithm.

2 Specification

Fortran Interface
Subroutine f11yef ( n, nnz, icolzp, irowix, lopts, mask, perm, info, ifail)
Integer, Intent (In) :: n, nnz, icolzp(n+1), irowix(nnz), mask(*)
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: perm(n), info(4)
Logical, Intent (In) :: lopts(5)
C Header Interface
#include <nag.h>
void  f11yef_ (const Integer *n, const Integer *nnz, const Integer icolzp[], const Integer irowix[], const logical lopts[], const Integer mask[], Integer perm[], Integer info[], Integer *ifail)
The routine may be called by the names f11yef or nagf_sparse_sym_rcm.

3 Description

f11yef takes the compressed column storage (CCS) representation (see Section 2.1.3 in the F11 Chapter Introduction) of an n×n symmetric matrix A and applies the Reverse Cuthill–McKee (RCM) algorithm which aims to minimize the bandwidth of the matrix A by reordering the rows and columns symmetrically. This also results in a lower profile of the matrix (see Section 9).
f11yef can be useful for solving systems of equations Ax=b, as the permuted system PAPT(Px)=Pb (where P is the permutation matrix described by the vector perm returned by f11yef) may require less storage space and/or less computational steps when solving (see Wai-Hung and Sherman (1976)).
f11yef may be used prior to f11jaf and f11jbf (see Section 10 in f11jbf).

4 References

Pissanetsky S (1984) Sparse Matrix Technology Academic Press
Wai-Hung L and Sherman A H (1976) Comparative analysis of the Cuthill–McKee and the reverse Cuthill–McKee ordering algorithms for sparse matrices SIAM J. Numer. Anal. 13(2) 198–213

5 Arguments

1: n Integer Input
On entry: n, the order of the matrix A.
Constraint: n1.
2: nnz Integer Input
On entry: the number of nonzero elements in the matrix A.
Constraint: nnz0.
3: icolzp(n+1) Integer array Input
On entry: icolzp records the index into irowix which starts each new column.
  • 1icolzp(i)nnz+1, for i=2,3,,n;
  • icolzp(1)=1;
  • icolzp(n+1)=nnz+1, where icolzp(i) holds the position integer for the starts of the columns in irowix.
4: irowix(nnz) Integer array Input
On entry: the row indices corresponding to the nonzero elements in the matrix A.
Constraint: 1irowix(i)n, for i=1,2,,nnz.
5: lopts(5) Logical array Input
On entry: the options to be used by f11yef.
Row/column i of the matrix A will only be referenced if mask(i)0, otherwise mask will be ignored.
The final permutation will not be reversed, that is, the Cuthill–McKee ordering will be returned. The bandwidth of the non-reversed matrix will be the same but the profile will be the same or larger (see Wai-Hung and Sherman (1976)).
The matrix A will be checked for symmetrical sparsity pattern, otherwise not.
The bandwidth and profile of the unpermuted matrix will be calculated, otherwise not.
The bandwidth and profile of the permuted matrix will be calculated, otherwise not.
6: mask(*) Integer array Input
Note: the dimension of the array mask must be at least n if lopts(1)=.TRUE., and at least 0 otherwise.
On entry: mask is only referenced if lopts(1) is .TRUE.. A value of mask(i)=0 indicates that the node corresponding to row or column i is not to be referenced. A value of mask(i)0 indicates that the node corresponding to row or column i is to be referenced. In particular, rows and columns not referenced will not be permuted.
7: perm(n) Integer array Output
On exit: this will contain the permutation vector that describes the permutation matrix P for the reordering of the matrix A. The elements of the permutation matrix P are zero except for the unit elements in row i and column perm(i), i=1,2,n.
8: info(4) Integer array Output
On exit: statistics about the matrix A and the permuted matrix. The quantities below are calculated using any masking in effect otherwise the value zero is returned.
The bandwidth of the matrix A, if lopts(4)=.TRUE..
The profile of the matrix A, if lopts(4)=.TRUE..
The bandwidth of the permuted matrix PAPT, if lopts(5)=.TRUE..
The profile of the permuted matrix PAPT, if lopts(5)=.TRUE..
9: ifail Integer Input/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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
On entry, n=value.
Constraint: n1.
On entry, nnz=value.
Constraint: nnz1.
On entry, irowix(value)=value and n=value.
Constraint: 1irowix(i)n for all i.
On entry, icolzp(value)=value and nnz=value.
Constraint: 1icolzp(i)nnz for all i.
On entry, icolzp(1)=value.
Constraint: icolzp(1)=1.
On entry, icolzp(n+1)=value and nnz=value.
Constraint: icolzp(n+1)=nnz+1.
On entry, the matrix A is not symmetric.
Element (value,value) has no symmetric element.
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.
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.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
f11yef is not threaded in any implementation.

9 Further Comments

The bandwidth for a matrix A=(aij) is defined as
b = maxij |i-j| ,   i,j=1,2,,n​ s.t. ​aij0 .  
The profile is defined as
p = j=1 n bj ,  where ​ bj = max i |i-j| ,   i=1,2,n ​ s.t. ​ aij0 .  

10 Example

This example reads the CCS representation of a real sparse matrix A and calls f11yef to reorder the rows and columns and displays the results.

10.1 Program Text

Program Text (f11yefe.f90)

10.2 Program Data

Program Data (f11yefe.d)

10.3 Program Results

Program Results (f11yefe.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 "f11yefe.r" ind 0 Example Program Figure 1 : Original Matrix Ordering
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 "f11yefe.r" ind 1 Figure 2 : Reverse Cuthill-McKee Reordering