Approximate minimum degree permutation
P = amd(A)
P = amd(A,opts)
P = amd(A) returns the
approximate minimum degree permutation vector for the sparse matrix
= A + A'. The Cholesky factorization of
to be sparser than that of
amd function tends to be faster than
symamd, and also tends to return better
be square. If
A is a full matrix, then
P = amd(A,opts) allows additional
options for the reordering. The
opts input is a
structure with the two fields shown below. You only need to set the
fields of interest:
dense — A nonnegative scalar value that indicates what is considered to be dense. If A is n-by-n, then rows and columns with more than
A + A'are considered to be "dense" and are ignored during the ordering. MATLAB® software places these rows and columns last in the output permutation. The default value for this field is 10.0 if this option is not present.
aggressive — A scalar value controlling aggressive absorption. If this field is set to a nonzero value, then aggressive absorption is performed. This is the default if this option is not present.
MATLAB software performs an assembly tree post-ordering,
which is typically the same as an elimination tree post-ordering.
It is not always identical because of the approximate degree update
used, and because “dense” rows and columns do not take
part in the post-order. It well-suited for a subsequent
chol operation, however, If you require
a precise elimination tree post-ordering, you can use the following
P = amd(S); C = spones(S)+spones(S'); [ignore, Q] = etree(C(P,P)); P = P(Q);
S is already symmetric, omit the second
C = spones(S)+spones(S').
Effect of Preordering on Sparsity of Cholesky Factors
Compute the Cholesky factor of a matrix before and after it is ordered using
amd to examine the effect on sparsity.
Load the barbell graph sparse matrix and add a sparse identity matrix to it to ensure it is positive definite. Compute two Cholesky factors: one of the original matrix and one of the original matrix preordered with
load barbellgraph.mat A = A+speye(size(A)); p = amd(A); L = chol(A,'lower'); Lp = chol(A(p,p),'lower');
Plot the sparsity patterns of all four matrices. The Cholesky factor obtained from the preordered matrix is much sparser compared to the factor of the matrix in its natural ordering.
figure subplot(2,2,1) spy(A) title('Original Matrix A') subplot(2,2,2) spy(A(p,p)) title('AMD ordered A') subplot(2,2,3) spy(L) title('Cholesky factor of A') subplot(2,2,4) spy(Lp) title('Cholesky factor of AMD ordered A')
 Amestoy, Patrick R., Timothy A. Davis, and Iain S. Duff. “Algorithm 837: AMD, an Approximate Minimum Degree Ordering Algorithm.” ACM Transactions on Mathematical Software 30, no. 3 (September 2004): 381–388. https://doi.org/10.1145/1024074.1024081.
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