Solving Linear Systems through Nested Dissection

Abstract

The generalized nested dissection method, developed by Lipton, Rose, and Tarjan, 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 non-singular well-separable matrix over any field. The running times we obtain essentially match those of the nested dissection method.