**ABS methods**, where the acronym contains the initials of Jozsef Abaffy, Charles G. Broyden and Emilio Spedicato, have been developed since 1981 to generate a large class of algorithms for the following applications:

At the beginning of 2007 ABS literature consisted of over 400 papers and reports and two monographs, one due to Abaffy and Spedicato and published in 1989, one due to Xia and Zhang and published, in Chinese, in 1998. Moreover three conferences had been organized in China.

Research on ABS methods has been the outcome of an international collaboration coordinated by Spedicato of University of Bergamo, Italy. It has involved over forty mathematicians from Hungary, UK, China, Iran and other countries.

The central element in such methods is the use of a special matrix transformation due essentially to the Hungarian mathematician Jenő Egerváry, who investigated its main properties in some papers that went unnoticed. For the basic problem of solving a linear system of *m equations in*

**n**variables, where

- Given an arbitrary initial estimate of the solution, find one of the infinite solutions, defining a linear variety of dimension
*n*− 1, of the first equation. - Find a solution of the second equation that is also a solution of the first, i.e. find a solution lying in the intersection of the linear varieties of the solutions of the first two equations considered separately.
- By iteration of the above approach after
*m'*steps one gets a solution of the last equation that is also a solution of the previous equations, hence of the full system. Moreover it is possible to detect equations that are either redundant or incompatible.

Among the main results obtained so far:

Knowledge of ABS methods is still quite limited among mathematicians, but they have great potential for improving the methods currently in use.