Revisiting Cramer's rule for solving dense linear systems

Abstract

The utility industry relies heavily on load flow modeling to design and manage public power systems. The speed and computational complexity of these models depends largely on the technique used to solve large scale linear systems. State-of-the-art software packages are predominantly founded on Gaussian elimination techniques (e.g. LU decomposition). Although such solutions lend themselves to parallel implementation, they require extensive communication overhead and non-uniform load distribution, often limiting their scalability. This paper presents a highly-parallelizable framework for solving large-scale linear systems by means of a novel utilization of Cramer's rule. While the latter is often perceived to be impractical when considered for large systems, it is shown that the algorithm proposed has an O(N3) complexity with pragmatic forward and backward stability. Moreover, minimal communication overhead between parallel processors is imposed. Most importantly, from the perspective of load-balancing across parallel nodes, the workload is inherently uniform throughout the process, thereby overcoming the drawbacks of existing schemes. Empirical results are provided to substantiate the stated accuracy and computational complexity claims.