Summary: Solving linear systems through nested dissection
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.
Key words. Gaussian elimination, linear system, nested dissection.
AMS subject classifications. 68W30, 15A15, 05C50
Solving a linear system is the most basic, and perhaps the most important problem in computational
linear algebra. Considerable effort has been devoted to obtaining algorithms that solve a linear
system faster than the naive cubic implementation of Gaussian elimination.
For the rest of this introduction we assume that the system is given by Ax = b, where A is a
non-singular n × n matrix over a field, b is an n-vector over that field, and xT = (x1, . . . , xn) is the
vector of variables.
The fastest general algorithm for solving Ax = b was obtained by Bunch and Hopcroft , and by