General solution of full row rank linear systems of equations using a new compression ABS model

Abstract

This paper describes two new ABS algorithms based on two-step ABS methods for solving general solution of full row rank linear systems of equations. For both of our works, the ith iteration solves the first 2i equations, but for the second algorithm, we compress the space. We investigate the numerical stability of our models and a class of methods having optimal stability is defined. The condition for the residual perturbation to be minimal is also given. Computational complexity and numerical results demonstrate that, for our new methods, we need less work than corresponding two-step ABS algorithms and Huang's method. The number of multiplications for these new schemes is half of the storage needed by the Gaussian elimination method. Our new version ABS algorithms are computationally better than the classical Gaussian elimination method, having the same arithmetic cost, but using less memory and no pivoting is necessary. The compression ABS model economizes the space more than the first algorithm, via deleting the zero rows of the Abaffian matrix by two in every iteration.