4.10 Scaling the Inner Linear Equation System

Scaling is the final step before the inner linear equation system is passed to the solver module in order to obtain its solution. Since preconditioners like the Incomplete-LU factorization compare the entries per row, a normalized representation of the matrix has to be provided. Such a normalization is not required when external modules include their own capabilities or when different kind of preconditioners are used (see Section 5.2.5). In those cases, the scaling should be switched off in order to save the computational effort.

The standard algorithm used by default works with a two-stage strategy [66]: In the first stage, the matrix is scaled such that the diagonal elements equal unity. The second stage attempts to suppress the off-diagonals while keeping the diagonals at unity. The resulting scaling matrices $\ensuremath{\mathbf{S}}_{\mathrm{r}}$ and $\ensuremath{\mathbf{S}}_{\mathrm{c}}$ are diagonal matrices. With $\ensuremath{\mathbf{A}}_i \ensuremath{\mathbf{x}}_i = \ensuremath{\mathbf{b}}_i$ as the inner system, the effect of sorting and scaling is given as:

$$({\ensuremath{\ensuremath{\mathbf{S}}_{\mathrm{r}}}} ({\ensuremat... ...uremath{\mathbf{R}}_{\mathrm{s}}^{\mathrm{T}}}} {\ensuremath{\mathbf{b}}_i})\ .$$ (4.25)

In Figure 4.3 a cut-out of the scaled inner system matrix is shown. Since the values are modified while keeping the structure constant, only the colors are changed. Note the red color of the diagonal entries indicating the unity entries.

Figure 4.3: In comparison to the pre-eliminated structure, the reordering algorithm significantly reduces the bandwidth from 2,867 to 102 in order to reduce the factorization fill-in (left). The circle indicates the range of the cut-out of the scaled matrix shown in the right figure. The scaled inner system matrix has diagonal entries equal unity, which is demonstrated by the red color. Since only the values are changed, no structural difference can be seen in comparison to the sorted matrix.