In this paper a fast algorithm for solving a large system with an essentially Toeplitz .veband coe.cient matrix is presented. The first two and last two rows are influenced by boundary conditions. The five band core of this matrix is factored as the product of tridiagonal matrices for the purpose of obtaining a solution of a linear system more e.ciently. An error term for the approximate solution is presented following the work by Yan and Chung [12]. An algorithm is developed for solving the two systems and is tested using two multiprocessor machines with different architectures.