If s is symmetric, then colperm generates a permutation so that

both the rows and columns of S(p,p) are ordered in nondecreasing

order of nonzero count. If S is positive definite, this is

sometimes useful as a preordering for Cholesky factorization:

chol(S(p,p)).

COLPERM is not the best ordering in the world, but it's fast to

compute, and it does a pretty good job.

See also colamd, symamd, symrcm.

Reference page in Help browser

doc colperm

<dmperm> - Dulmage-Mendelsohn permutation.

DMPERM Dulmage-Mendelsohn permutation.

p = DMPERM(A) finds a maximum matching p such that p(j) = i if column j

is matched to row i, or 0 if column j is unmatched. If A is a square

matrix with full structural rank, p is a row permutation and A(p,:) has a

zero-free diagonal. The structural rank of A is sprank(A) = sum(p>0).

[p,q,r,s,cc,rr] = DMPERM(A) finds the Dulmage-Mendelsohn decomposition

of A. p and q are row and column permutation vectors, respectively

such that A(p,q) has block upper triangular form. r and s are vectors

indicating the block boundaries for the fine decomposition. cc and rr

are vectors of length five indicating the block boundaries of the

coarse decomposition.

C = A(p,q) is split into a 4-by-4 set of coarse blocks:

A11 A12 A13 A14

0 0 A23 A24

0 0 0 A34

0 0 0 A44

where A12, A23, and A34 are square with zero-free diagonals. The

columns of A11 are the unmatched columns, and the rows of A44 are the

unmatched rows. Any of these blocks can be empty. In the coarse

decomposition, the (i,j)th block is C(rr(i):rr(i+1)-1,cc(j):cc(j+1)-1).

In terms of a linear system, [A11 A12] is the underdetermined part of

the system (it is always rectangular and with more columns and rows, or

0-by-0), A23 is the well-determined part of the system (it is always

square), and [A34 ; A44] is the overdetermined part of the system (it

is always rectangular with more rows than columns, or 0-by-0).

The structural rank of A is sprank(A) = rr(4)-1, which is an upper

bound on the numerical rank of A. sprank(A) = rank(full(sprand(A)))

with probability 1 in exact arithmetic.

The A23 submatrix is further subdivided into block upper triangular

form via the fine decomposition (the strongly-connected components of

A23). If A is square and structurally nonsingular, A23 is the entire

matrix.

C(r(i):r(i+1)-1,s(j):s(j+1)-1) is the (i,j)th block of the fine

decomposition. The (1,1) block is the rectangular block [A11 A12],

unless this block is 0-by-0. The (b,b) block is the rectangular block

[A34 ; A44], unless this block is 0-by-0, where b = length(r)-1. All

other blocks of the form C(r(i):r(i+1)-1,s(i):s(i+1)-1) are diagonal

blocks of A23, and are square with a zero-free diagonal.

DMPERM uses CSparse.

See also sprank.

Reference page in Help browser

doc dmperm