Lawrence E. Garey

A parallel algorithm for solving Toeplitz linear systems

Numerical methods of solution are considered for systems which are Toeplitz and symmetric. In our case, the coefficient matrix is essentially tridiagonal and sparse. There are two distinct approaches to be considered each of which is efficient in its own way. Here we will combine the two approaches which will allow application of the cyclic reduction method to coefficient matrices of more general forms. The convergence of the approximations to the exact solution will also be examined. Solving linear systems by the adapted cyclic reduction method can be parallel processed.

A fast method for solving second Order boundary value volterra Integro-differential equations

Using finite difference methods to solve nonlinear Volterra integro-differential equations with two point boundary conditions give rise to a symmetric banded coefficient matrix. A typical method for solving systems of this form involves the LU method. In this paper the original system is modified to allow the implementation of a special fast algorithm for solving tridiagonal systems. Numerical examples are given to compare an efficient form of the LU method with the new approach.

A split-correct parallel algorithm for solving tridiagonal symmetric toeplitz systems

In 1994, Yan and Chung produced a fast algorithm for solving a diagonally dominant symmetric Toeplitz tridiagonal system of linear equations Ax = b. In this work a method will be presented which will allow for problems of the above nature to be split into two separate systems which can be solved in parallel, and then combined and corrected to obtain a solution to the original system. An error analysis will be provided along with example cases and time comparison results.

Direct numerical methods for first-order fredholm integro-differential equations

Numerical methods for Volterra integro-differential equations are adapted for the direct solution of a Fredholm integro-differential equation. A convergence analysis is presented and it is shown for certain test equations, instabilities may arise which may be eliminated with the introduction of a parameter (see Brunner [3]) to modify the methods.

An Efficient Method for Second Order Boundary Value Problems with Two Point Boundary Conditions

The discrete approximation to a second order boundary value problem by finite difference methods gives rise to a system with a coefficient matrix which is typically banded. Several articles discuss the solution of such systems. Here the efficient method is based on the idea of a system perturbation followed by corrections and is competitive with standard methods

A parallel method for linear equations with tridiagonal Toeplitz coefficient matrices

Abstract There are many articles on symmetric tridiagonal Toeplitz and circulant systems. Such systems in areas including numerical methods for solving boundary value differential equations and in graph theory. These matrices can often be written as the product of bidiagonal matrices. In this article, nonsymmetric Toepliz systems and nonsymmetric circulant systems are examined. The coefficient matrix is split into two bidiagonal matrices and the efficient solution of the resulting systems is considered.

Solving banded and near symmetric systems

In an earlier work by the authors [8], finite difference methods applied to obtain numerical approximations for the solution of nonlinear Volterra integro-differential equations with two point boundary conditions were considered. In [8], the auxiliary conditions required to complete the system, were chosen to preserve the symmetry of the banded system. In general, applying natural auxiliary conditions (NAC) would give rise to a nonsymmetric banded system. In this article, we consider the efficient solving of a linear banded system with a near-symmetric coefficient matrix.

Parallel algorithms for solving tridiagonal and near-circulant systems

Many problems in mathematics and applied science lead to the solution of linear systems having circulant coefficient matrices. This paper presents a new stable method for the exact solution of non-symmetric tridiagonal circulant linear systems of equations. The method presented in this paper is quite competitive with Gaussian elimination both in terms of arithmetic operations and storage requirements. It is also competitive with the modified double sweep method. This method can be applied to solve the near-circulant tridiagonal system. In addition, the method is modified to allow for parallel processing.

A parallel numerical algorithm for fredholm integro-differential two-point boundary value problems

Numerical methods for second order differential equations with two-point boundary conditions are incorporated into a three part method for the solution of a second order nonlinear Fredholm integro-differential equation. The interest in this paper is the development of an algorithm for parallel processing the discrete nonlinear system.Numerical examples are given.

An efficient algorithm for a model with a bidiagonal coefficient matrix

Bidiagonal matrices arise directly in the discretization of models using one step methods and indirectly through the factoring of tridiagonal systems with special characteristics. We first look at the solution of a bidiagonal system and then relate it to an experimental method that has been used to determine the overall heat transfer coefficient in a concentric tube, parallel flow heat exchanger. The discretization of the governing partial differential equation with its boundary conditions gives rise to a near Toeplitz bidiagonal system. Here we consider a method for the parallel processing of the resulting systems to provide measurements at fixed spatial and time increments.