Donald J. Rose

Triangulated graphs and the elimination process

Abstract A triangulated graph is a graph in which for every cycle of length l > 3, there is an edge joining two nonconsecutive vertices. In this paper we study triangulated graphs and show that they play an important role in the elimination process. The results have application in the study of the numerical solution of sparse positive definite systems of linear equations by Gaussian elimination.

On simple characterizations of k-trees

k-trees are a special class of perfect elimination graphs which arise in the study of sparse linear systems. We present four simple characterizations of k-trees involving cliques, paths, and separators.

Partitioning, Tearing, and Modification of Sparse Linear Systems

Abstract The computational complexity of partitioning sparse matrices is developed graph-theoretically. The results are used to study tearing and modification, and to show that single-element tearing of symmetric systems is rarely advantageous when the torn system is solved by elimination.

An algorithm for solving a special class of tridiagonal systems of linear equations

An algorithm is presented for solving a system of linear equations <italic>Bu</italic> = <italic>k</italic> where <italic>B</italic> is tridiagonal and of a special form. This form arises when discretizing the equation - d/d<italic>x</italic> (<italic>p</italic>(<italic>x</italic>) <italic>du</italic>/<italic>dx</italic>) = <italic>k</italic>(<italic>x</italic>) (with appropriate boundary conditions) using central differences. It is shown that this algorithm is almost twice as fast as the Gaussian elimination method usually suggested for solving such systems. In addition, explicit formulas for the inverse and determinant of the matrix <italic>B</italic> are given.

A Recursive Analysis of Dissection Strategies.

Publisher Summary An idea used frequently in the top level design and analysis of computer algorithms is to partition a problem into smaller subproblems whose individual solutions are combined to give the solution to the original problem. If the subproblems have a structure essentially identical to the original problem, this process is described recursively, and the partitioning proceeds until the subproblems can be solved trivially. The chapter presents a simple formalism for analyzing structurally recursive algorithms and presents an application of this analysis to study the strategy known as nested dissection for ordering sparse matrices whose graphs are n × m grids. It also discusses structural recursion formalism and presents the derivation of asymptotic complexity (order of magnitude) of several numerical algorithms including n × n nested dissection.

Automatic nested dissection

Nested dissection is an ordering technique used to order the sparse symmetric positive definite systems of linear equations arising from discretizations to elliptic boundary value problems yielding regular n × n grids. By taking a recursive view of nested dissection we develop an ordering strategy which is particularly simple and efficient when n &equil; 2@−1. A FORTRAN IV subroutine of our algorithm is included and some experiments are presented.