RCM Numberer

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This command is used to construct an RCM degree-of-freedom numbering object to provide the mapping between the degrees-of-freedom at the nodes and the equation numbers. An RCM numberer uses the reverse Cuthill-McKee scheme to order the matrix equations. The command to construct an RCM numberer is a follows:

numberer RCM

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REFERENCES:

E. Cuthill and J. McKee. Reducing the bandwidth of sparse symmetric matrices In Proc. 24th Nat. Conf. ACM, pages 157-172, 1969.

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THEORY:

Given a symmetric (in structure) n-by-n matrix we create a graph of the matrix, where the vertices correspond to the equation numbers and the edges of the graph correspond to non-zero (in structure, not necessarily value) <math> a_{ij} </math> matrix terms. The reverse Cuthill-McKee algorithm is a relabeling of the vertices of the graph to reduce the bandwidth of the matrix.

The algorithm produces an ordered n-tuple R of vertices which is the new order of the vertices.

First we choose a peripheral vertex x and set R:= ({x}).

Then for i=1,2,... we iterate the following steps while |R| < n

• Construct the adjacency set <math>A_i</math> of <math>R_i</math> (with <math>R_i</math> the i-th component of R) and exclude the vertices we already have in R

<math>A_i := \operatorname{Adj}(R_i) \setminus R</math>

• Sort <math>A_i</math> with ascending vertex order.
• Append <math>A_i</math> to the Result set R.

In other words, number the vertices according to a particular breadth-first search, where neighboring vertices are visited in order from lowest to highest vertex order.

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Code Developed by: fmk