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Sparse column permutation based on nonzero count
j = colperm(S)
j = colperm(S)
generates a permutation vector j such that the columns of S(:,j) are ordered according to increasing count of nonzero entries. This is sometimes useful as a preordering for LU factorization; in this case use lu(S(:,j)).
If S is symmetric, then j = colperm(S) generates a permutation j so that both the rows and columns of S(j,j) are ordered according to increasing count of nonzero entries. If S is positive definite, this is sometimes useful as a preordering for Cholesky factorization; in this case use chol(S(j,j)).
The algorithm involves a sort on the counts of nonzeros in each column.
The n-by-n arrowhead matrix
A = [ones(1,n); ones(n-1,1) speye(n-1,n-1)]has a full first row and column. Its LU factorization,
lu(A), is almost completely full. The statement
j = colperm(A)returns
j = [2:n 1]. So A(j,j) sends the full row and column to the bottom and the rear, and lu(A(j,j)) has the same nonzero structure as A itself.
On the other hand, the Bucky ball example, B = bucky,
has exactly three nonzero elements in each row and column, so j = colperm(B) is the identity permutation and is no help at all for reducing fill-in with subsequent factorizations.
chol Cholesky factorization
colmmd Sparse minimum degree ordering
lu LU matrix factorization
symrcm Sparse reverse Cuthill-McKee ordering