Showing papers on "Tridiagonal matrix algorithm published in 1975"
TL;DR: The problem of solving tridmgonal linear systems on vector computers is considered and implementations of several direct and lterative methods are given for the Control Data Corporatlon STAR-100 computer.
Abstract: The problem of solving tridmgonal linear systems on vector computers is considered. In particular, implementations of several direct and lterative methods are given for the Control Data Corporatlon STAR-100 computer. The direct methods considered are Gaussian elimination, a parallel method due to Stone, and cyclic reduction; the iteratlve methods considered are Jacobi's method, successive overrelaxaUon, and a parallel method due to Traub. In addition, timing formulas for the methods based on current information are included to provide a basis for comparison In general, the direct methods are found to be superior to the iterative methods. The choice of direct methods depends on the size of the system, but for more than 125 equations, cyclic reduction is the fastest algorithm.
140 citations
1 Nov 1975
TL;DR: Graph-theoretic ideas which apply to the problem of solving a sparse system of linear equations by Gaussian elimination are surveyed and algorithms based on these ideas are presented.
Abstract: This paper surveys graph-theoretic ideas which apply to the problem of solving a sparse system of linear equations by Gaussian elimination Included are a discussion of bandwidth, profile, and general sparse elimination schemes, and of two graph-theoretic partitioning methods Algorithms based on these ideas are presented
75 citations
TL;DR: It is shown that in many respects suspicions are unfounded, and in general the absolute error in the solution is strictly comparable with that corresponding to Gaussian elimination with partial pivoting plus back substitution.
Abstract: The stability of the Gauss-Jordan algorithm with partial pivoting for the solution of general systems of linear equations is commonly regarded as suspect. It is shown that in many respects suspicions are unfounded, and in general the absolute error in the solution is strictly comparable with that corresponding to Gaussian elimination with partial pivoting plus back substitution. However, when A is ill conditioned, the residual corresponding to the Gauss-Jordan solution will often be much greater than that corresponding to the Gaussian elimination solution.
61 citations
1 Nov 1975
TL;DR: Computer procedures are described for error-free matrix computations, using thep-adic arithmetic, and the exact solution of a highly ill-conditioned linear system of equations is obtained by using the Gaussian elimination method.
Abstract: Computer procedures are described for error-free matrix computations, using thep-adic arithmetic. As an example, the exact solution of a highly ill-conditioned linear system of equations is obtained, by using the Gaussian elimination method.
32 citations
TL;DR: In this article, the Thomas algorithm is used to solve a system of linear finite difference equations, where the coefficients are determined by simple quadrature schemes applied to each increment, and an expression is derived for the roundoff error associated with the final Thomas iteration.
Abstract: The definite integral is generally interpreted geometrically as an “area”. An alternate interpretation as a steady-state “flux” through a unit slab is derived, which leads to a new method of numerical integration. The usual sum of a large number of approximate areas is replaced by the flux through a “single” increment. The method involves the solution of a system of linear finite difference equations. The coefficient matrix is tri-diagonal and is solved efficiently by the Thomas algorithm. During the iterative process the coefficients are determined by simple quadrature schemes applied to each increment. Error analysis revealed that an expression could be derived for the roundoff error associated with the final Thomas iteration. It is shown that the roundoff error is smallest when the matrix coefficients a k \S>1. The method is shown to be superior to the classical methods due to its simplicity and tolerance for variable increment size. In addition, a new function is determined which is useful in diffusion studies. Numerical data are presented confirming these results.
1 citations
TL;DR: This work considers a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse systems of linear eauations, and gives efficient algorithms to calculate the fill-in produced by any elimination ordering and find a minimal elimination ordering.
Abstract: We consider a graph-theoretic elimination process which is related to performing Gaussian elimination on sparse systems of linear eauations. We give efficient algorithms to: (1) calculate the fill-in produced by any elimination ordering; (2) find a perfect elimination ordering if one exists; and (3) find a minimal elimination ordering. We also show that problems (1) and (2) are at least as time-consuming as testing whether a directed graph is transitive, and that the problem of finding a minimum ordering is NP-complete.