Critical to the performance of an iterative linear solver is the
convergence of the technique for a given data set. We have applied
our sparse Gauss-Seidel solver to sample positive definite matrices
with the sparsity pattern from actual power systems networks and
random values for the entries. We have examined convergence for
various matrices and various matrix orderings. Samples of the
measured convergence data are presented in tables 7.2
and 7.3 for the BCSPWR09 and BCSPWR10 power systems
networks respectively. These tables present the total error for an
iteration, and the minimum and maximum values encountered that
iteration. All initial values, , have been defined to
equal zero.
Table 7.2: Convergence for the BCSPWR09 Power Systems Network
Table 7.3: Convergence for the BCSPWR10 Power Systems Network
In both tables 7.2 and 7.3, convergence is
rather rapid, and after twelve iterations, total error is less than . Consequently, only eight iterations are required
for six decimal place accuracy with these data sets. In a positive
definite matrix, the maximum values in the matrix fall on the
diagonal. In this generated data, the magnitude of the diagonals were
set equal to the number of non-zeros in the row plus a uniformly
distributed random number between zero and one while the
off-diagonal values were set equal to a uniformly distributed
randomly number between zero and one. The values of
were set
equal to one plus a uniformly distributed random number between zero
and one. If the relative magnitude of the diagonals with respect to
the off-diagonals is larger, convergence will be even faster
We hypothesize that this good convergence rate is in part due to having good estimates of the initial starting vector. For actual solutions of power systems networks, this solver would be used within an iterative non-linear solver, so even better estimates of starting points for each solution will be readily available, especially for transient stability simulations where differential-algebraic equations are solved for small time increments.