NAG C Library Function Document

nag_sparse_sym_rcm (f11yec)


nag_sparse_sym_rcm (f11yec) reduces the bandwidth of a sparse symmetric matrix stored in compressed column storage format using the Reverse Cuthill–McKee algorithm.


#include <nag.h>
#include <nagf11.h>
void  nag_sparse_sym_rcm (Integer n, Integer nnz, const Integer icolzp[], const Integer irowix[], const Nag_Boolean lopts[], const Integer mask[], Integer perm[], Integer info[], NagError *fail)


nag_sparse_sym_rcm (f11yec) takes the compressed column storage (CCS) representation (see Section 2.1.3 in the f11 Chapter Introduction) of an n by 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).
nag_sparse_sym_rcm (f11yec) can be useful for solving systems of equations Ax=b, as the permuted system PAPTPx=Pb (where P is the permutation matrix described by the vector perm returned by nag_sparse_sym_rcm (f11yec)) may require less storage space and/or less computational steps when solving (see Wai-Hung and Sherman (1976)).
nag_sparse_sym_rcm (f11yec) may be used prior to nag_sparse_sym_chol_fac (f11jac) and nag_sparse_sym_precon_ichol_solve (f11jbc) (see Section 10 in nag_sparse_sym_precon_ichol_solve (f11jbc)).


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


1:     n IntegerInput
On entry: n, the order of the matrix A.
Constraint: n1.
2:     nnz IntegerInput
On entry: the number of nonzero elements in the matrix A.
Constraint: nnz0.
3:     icolzp[n+1] const IntegerInput
On entry: icolzp records the index into irowix which starts each new column.
  • 1icolzp[i-1]nnz+1, for i=2,3,,n;
  • icolzp[0]=1;
  • icolzp[n]=nnz+1, where icolzp[i-1] holds the position integer for the starts of the columns in irowix.
4:     irowix[nnz] const IntegerInput
On entry: the row indices corresponding to the nonzero elements in the matrix A.
Constraint: 1irowix[i-1]n, for i=1,2,,nnz.
5:     lopts[5] const Nag_BooleanInput
On entry: the options to be used by nag_sparse_sym_rcm (f11yec).
Row/column i of the matrix A will only be referenced if mask[i-1]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[dim] const IntegerInput
Note: the dimension, dim, of the array mask must be at least
  • n when lopts[0]=Nag_TRUE;
  • otherwise mask may be NULL.
On entry: mask is only referenced if lopts[0] is Nag_TRUE otherwise mask may be set to NULL. A value of mask[i-1]=0 indicates that the node corresponding to row or column i is not to be referenced. A value of mask[i-1]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] IntegerOutput
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-1], i=1,2,n.
8:     info[4] IntegerOutput
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[3]=Nag_TRUE.
The profile of the matrix A, if lopts[3]=Nag_TRUE.
The bandwidth of the permuted matrix PAPT, if lopts[4]=Nag_TRUE.
The profile of the permuted matrix PAPT, if lopts[4]=Nag_TRUE.
9:     fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, n=value.
Constraint: n1.
On entry, nnz=value.
Constraint: nnz1.
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
On entry, icolzp[0]=value.
Constraint: icolzp[0]=1.
On entry, icolzp[n]=value and nnz=value.
Constraint: icolzp[n]=nnz+1.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
On entry, the matrix A is not symmetric.
Element value,value has no symmetric element.
On entry, icolzp[value]=value and nnz=value.
Constraint: 1icolzp[i-1]nnz for all i.
On entry, irowix[value]=value and n=value.
Constraint: 1irowix[i-1]n for all i.


Not applicable.

Parallelism and Performance

nag_sparse_sym_rcm (f11yec) is not threaded in any implementation.

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 .  


This example reads the CCS representation of a real sparse matrix A and calls nag_sparse_sym_rcm (f11yec) to reorder the rows and columns and displays the results.

Program Text

Program Text (f11yece.c)

Program Data

Program Data (f11yece.d)

Program Results

Program Results (f11yece.r)

GnuplotProduced by GNUPLOT 4.6 patchlevel 3 Example Program Figure 1 : Original Matrix Ordering gnuplot_plot_1
GnuplotProduced by GNUPLOT 4.6 patchlevel 3 Figure 2 : Reverse Cuthill-McKee Reordering gnuplot_plot_1