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Dulmage-Mendelsohn decomposition
p = dmperm(A)
[p,q,r,s,cc,rr] = dmperm(A)
p = dmperm(A) finds a vector p such that p(j) = i if column j is matched to row i, or zero if column j is unmatched. If A is a square matrix with full structural rank, p is a maximum matching 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) where A need not be square or full structural rank, finds the Dulmage-Mendelsohn decomposition of A. p and q are row and column permutation vectors, respectively, such that A(p,q) has a block upper triangular form. r and s are index 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). For 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.
[1] Pothen, Alex and Chin-Ju Fan "Computing the Block Triangular Form of a Sparse Matrix" ACM Transactions on Mathematical Software Vol 16, No. 4 Dec. 1990, pp. 303-324.