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Convergence Rate

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.



next up previous
Next: Comparing Communications Paradigms Up: Empirical Results for Previous: Analyzing Algorithm Component



David P. Koester
Sun Oct 22 17:27:14 EDT 1995