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The key step is to remove some inessential moves from the James-Peel tree for D(w), namely those which only permute rows or columns.
Otherwise, SCPRE permutes the blocks and builds a block upper-triangular preconditioner.
H] permutes the edges of H with respect to descending edge weights.
Such a matrix is usually easy to factorize if we initially permute it to BTF, so that a direct solver like KLU [9] only needs to factorize the diagonal blocks.
One motivation for our approach is the fact that the singular vectors of B are obtained from those of A by the same permutation that permutes A to B.
1), that permutes B into a block diagonally-dominant matrix of the form (4.
2 Sort it in increasing order and use the resulting permutation P to permute the matrix B.
Once we know how to permute one large block that contains at least two subblocks into two blocks, we can employ this procedure recursively.
5 explains how to permute a matrix with SSS blocks into a single SSS matrix.
5 to permute it into a single SSS matrix by using a permutation matrix [PSI].
Recall from Section 2 that we can permute the saddle-point system with MSSS blocks into a global MSSS system.