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.