|MATLAB Function Reference|
Sparse reverse Cuthill-McKee ordering
r = symrcm(S)
returns the symmetric reverse Cuthill-McKee ordering of
S. This is a permutation
r such that
S(r,r) tends to have its nonzero elements closer to the diagonal. This is a good preordering for LU or Cholesky factorization of matrices that come from long, skinny problems. The ordering works for both symmetric and nonsymmetric
For a real, symmetric sparse matrix,
S, the eigenvalues of
S(r,r) are the same as those of
eig(S(r,r)) probably takes less time to compute than
The algorithm first finds a pseudoperipheral vertex of the graph of the matrix. It then generates a level structure by breadth-first search and orders the vertices by decreasing distance from the pseudoperipheral vertex. The implementation is based closely on the SPARSPAK implementation described by George and Liu.
uses an M-file in the
demos toolbox to generate the adjacency graph of a truncated icosahedron. This is better known as a soccer ball, a Buckminster Fuller geodesic dome (hence the name
bucky), or, more recently, as a 60-atom carbon molecule. There are 60 vertices. The vertices have been ordered by numbering half of them from one hemisphere, pentagon by pentagon; then reflecting into the other hemisphere and gluing the two halves together. With this numbering, the matrix does not have a particularly narrow bandwidth, as the first spy plot shows
The reverse Cuthill-McKee ordering is obtained with
spy plot shows a much narrower bandwidth.
This example is continued in the reference pages for
The bandwidth can also be computed with
The bandwidths of
R are 35 and 12, respectively.
 George, Alan and Joseph Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, 1981.
 Gilbert, John R., Cleve Moler, and Robert Schreiber, "Sparse Matrices in MATLAB: Design and Implementation," to appear in SIAM Journal on Matrix Analysis, 1992. A slightly expanded version is also available as a technical report from the Xerox Palo Alto Research Center.