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symrcm - Sparse reverse Cuthill-McKee ordering

Syntax

r = symrcm(S)

Description

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 S.

For a real, symmetric sparse matrix, S, the eigenvalues of S(r,r) are the same as those of S, but eig(S(r,r)) probably takes less time to compute than eig(S).

Algorithm

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.

Examples

The statement

B = bucky;

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

subplot(1,2,1), spy(B), title('B')

The reverse Cuthill-McKee ordering is obtained with

p = symrcm(B);
R = B(p,p);

The spy plot shows a much narrower bandwidth.

subplot(1,2,2), spy(R), title('B(p,p)')

This example is continued in the reference pages for symamd.

The bandwidth can also be computed with

[i,j] = find(B);
bw = max(i-j) + 1;

The bandwidths of B and R are 35 and 12, respectively.

References

[1] George, Alan and Joseph Liu, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, 1981.

[2] Gilbert, John R., Cleve Moler, and Robert Schreiber, "Sparse Matrices in MATLAB: Design and Implementation," SIAM Journal on Matrix Analysis, 1992. A slightly expanded version is also available as a technical report from the Xerox Palo Alto Research Center.