Solving linear systems through nested dissection Raphael Yuster Summary: Solving linear systems through nested dissection Noga Alon Raphael Yuster 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. Key words. Gaussian elimination, linear system, nested dissection. AMS subject classifications. 68W30, 15A15, 05C50 1 Introduction 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 [2], and by Collections: Mathematics