Research and Implementation of Fast Decomposition Method for Symmetric Indefinite Sparse Linear Systems

The fast solution of symmetric indefinite sparse ma-trices is of great significance in the fields of scientific computing and engineering design. However, the traditional solution methods are limited by the sparsity and indeterminacy of these matrices and fail to fully utilize their symmetry, leading to a large amount of storage redundancy and invalid computation. To address the aforementioned problems, a new method is designed to first reorder the matrices using the Reverse Cuthill-McKee (RCM) algorithm to efficiently compress the bandwidth of the matrices, and then an improved circuit implementation of the Aasen decomposition algorithm is proposed to accelerate the solution of indeterminate sparse matrices. The parallelism of the algorithm is fully exploited by constructing the chunked column compression storage mechanism and dynamic configuration control rules. Experimental results show that the method proposed in this paper can significantly improve the computational efficiency and stability of solving symmetric indeterminate sparse matrices, especially in specific applications, achieving a performance improvement of about 30x.