# symmmd

## Purpose

Symmetric minimum degree ordering for elimination sparsity.

## Synopsis

    p = symmmd(S)

## Description

`p = symmmd(S)` returns a symmetric minimum degree ordering of `S`. For a symmetric positive definite matrix `S`, this is a permutation `p` such that `S(p,p)` tends to have a sparser Cholesky factor than `S`. Sometimes `symmmd` works well for symmetric indefinite matrices too.

The minimum degree ordering is automatically used by `\` and `/` for the solution of symmetric, positive definite, sparse linear systems.

Some options and parameters associated with heuristics in the algorithm can be changed with `spparms`.

## Algorithm

The symmetric minimum degree algorithm is based on the column minimum degree algorithm. In fact, `symmmd(A)` just creates a nonzero structure `K` such that `K'`*`K` has the same nonzero structure as `A` and then calls the column minimum degree code for `K`.

## Examples

Here is a comparison of reverse Cuthill-McKee and minimum degree on the Bucky ball example mentioned in the `symrcm` reference page.

    B = bucky+4*speye(60);
    r = symrcm(B);
    p = symmmd(B);
    R = B(r,r);
    S = B(p,p);
    subplot(2,2,1), spy(R), title('B(r,r)')
    subplot(2,2,2), spy(S), title('B(s,s)')
    subplot(2,2,3), spy(chol(R)), title('chol(B(r,r))')
    subplot(2,2,4), spy(chol(S)), title('chol(B(s,s))')

``Even though this is a very small problem, the behavior of both orderings is typical. RCM produces a matrix with a narrow bandwidth which fills in almost completely during the Cholesky factorization. Minimum degree produces a structure with large blocks of contiguous zeros which do not fill in during the factorization. Consequently, the minimum degree ordering requires less time and storage for the factorization.

## See Also

    \, chol, colmmd, colperm, spparms, symrcm

## References

[1] John R. Gilbert, Cleve Moler, and Robert Schreiber, "Sparse Matrices in MATLAB: Design and Implementation," SIAM Journal on Matrix Analysis and Applications 13, pp. 333-356, 1992.

(c) Copyright 1994 by The MathWorks, Inc.