Matrix Sparsification and Nested Dissection over Arbitrary Fields
The generalized nested dissection method, developed by Lipton et al. [ 1979], is a seminal method for solving a linear system Ax = b where A is a symmetric positive definite matrix. The method runs extremely fast whenever A is a well-separable matrix (such as matrices whose underlying support is planar or avoids a fixed minor). In this work, we extend the nested dissection method to apply to any nonsingular well-separable matrix over any field. The running times we obtain essentially match those of the nested dissection method. An important tool is a novel method for matrix sparsification that preserves determinants and minors, and that guarantees that constant powers of the sparsified matrix remain sparse.
matrix sparsification nested dissection linear system algorithms theory gaussian elimination