In the pivoting variant 1 we will restrict the pivoting search within diagonal-blocks of the supernodes as shown in Figure 4.1 (a) and (b).
We now present the results of the numerical experiments that demonstrate the robustness and the effectiveness of our pivoting approach.
In order to evaluate the quality of the different pivoting methods for symmetric indefinite linear systems we used performance profiles as a tool for benchmarking and for comparing the algorithms.
The profiles are generated by running the set of pivoting methods [mu] on the set of sparse matrices [sigma] and recording information of interest, e.g.
The performance profile [p.sub.m]([alpha]) of the pivoting method m is then defined by
[p.sub.m] ([alpha]), gives the fraction of which the pivoting method m is the most effective method and [lim.sub.[alpha][right arrow][infinity]] indicates the fraction for which the algorithm succeeded.
pivoting strategy, then [PAP.sup.T] = [LDL.sup.T], where L is a unit lower triangular matrix strictly diagonally dominant by columns.
The following example shows that applying symmetric complete pivoting to a Stieljes matrix does not guarantee that the triangular factors are diagonally dominant.
then the permutation matrix associated to the symmetric complete pivoting is the identity and A = [LDL.sup.T], where
pivoting interchanges the first and second rows and columns in the first step of Gauss elimination and it still produces another row and column exchange in the second step.
pivoting also can be applied to an n x n M-matrix A of rank r < n diagonally dominant by columns in order to obtain an upper triangular matrix [A.sup.(r+1)] = DU, whose r first diagonal entries are nonzero.
pivoting also provides an LDU-decomposition with L and U well conditioned as the following result shows.