The filled boxes indicate diagonal supernodes of different sizes [n.sub.s] where Supernode-Bunch-Kaufman pivoting is performed and the dotted boxes indicate the remaining part of the supernodes.
An interchange among the rows and columns of a supernode of diagonal size [n.sub.s], referred to as Supernode-Bunch-Kaufman pivoting, has no effect on the overall fill-in and this is the mechanism for finding a suitable pivot in our SBK method.
Algorithm 2.1 describes the usual 1 x 1 and 2 x 2 Bunch-Kaufman pivoting strategy within the diagonal block corresponding to a supernode of size [n.sub.s].
Symmetric reorderings to improve the results of pivoting on restricted subsets.
Pivoting Variant 1: Fill-in reduction [P.sub.Fill] based on compressed subgraphs.
Pivoting Variant 2: Use All Preselected 2 x 2 Pivots.
As we have commented above, the triangular factors of the LDU-decomposition appearing when we apply complete pivoting to an n x n nonsingular matrix have off-diagonal elements with absolute value bounded above by 1 and then their condition numbers are bounded in terms of n.
Let us start by showing in the next result a first example, where pivoting is not necessary.
Gaussian elimination with a given pivoting strategy, for nonsingular matrices A = [([alpha].sub.ij]).sub.1[less than or equal to] i,j [less than or equal to] n], consists of a succession of at most n - 1 major steps resulting in a sequence of matrices as follows:
The matrix [[??].sup.(t)] = [([[??].sup.(t).sub.ij]).sub.1 [less than or equal to] i, j [less than or equal to] n] is obtained from the matrix [A.sup.(t)] by reordering the rows and/or columns t, t + 1, ..., n of [A.sup.(t)] according to the given pivoting strategy and satisfying [[??].sup.(t).sub.tt] [not equal to] 0.
In [12] we defined a symmetric maximal absolute diagonal dominance (m.a.d.d.) pivoting as a symmetric pivoting which chooses as pivot at the tth step (1 [less than or equal to] t [less than or equal to] n - 1) a row [i.sub.t] ([greater than or equal to] t) satisfying
Although Gauss elimination without pivoting of a positive definite symmetric matrix is stable, it does not guarantee the well conditioning of the triangular factors.
