ABS methods
Encyclopedia
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 algorithm
s 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
, ABS methods use the following
simple geometric idea:
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.
Charles George Broyden
Charles George Broyden was a mathematician who specialized in optimization problems and numerical linear algebra...
and Emilio Spedicato
Emilio Spedicato
Emilio Spedicato is full professor of operations research at the University of Bergamo in Italy. He attended the Liceo Classico Manzoni, obtaining the highest score in northern Italy at the final exams...
, have been developed since 1981 to generate a large class of algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...
s for the following applications:
- solution of general linear algebraic systems, determined or underdetermined,
- full or deficient rank;
- solution of linear Diophantine systems, i.e. equation systems where the coefficient matrix and the right hand side are integer valued and an integer solution is sought; this is a special but important case of Hilbert's tenth problemHilbert's tenth problemHilbert's tenth problem is the tenth on the list of Hilbert's problems of 1900. Its statement is as follows:Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite...
, the only one in practice soluble; - solution of nonlinear algebraic equations;
- solution of continuous unconstrained or constrained optimization.
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
Bergamo
Bergamo is a town and comune in Lombardy, Italy, about 40 km northeast of Milan. The comune is home to over 120,000 inhabitants. It is served by the Orio al Serio Airport, which also serves the Province of Bergamo, and to a lesser extent the metropolitan area of Milan...
, 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
Jenő Egerváry
Jenő Egerváry was a Hungarian mathematician.-Biography:Egerváry was born in Debrecen in 1891. In 1914, he received his doctorate at the Pázmány Péter University in Budapest, where he studied under the supervision of Lipót Fejér...
, 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
, ABS methods use the followingsimple geometric idea:
- 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:
- unification of algorithms for linear, nonlinear algebraic equations and for linearly constrained nonlinear optimization, including the LP problem as a special case;
- the method of Gauss has been improved by reducing the required memory and eliminating the need for pivoting;
- new methods for nonlinear systems with convergence properties better than for Newton method;
- derivation of a general algorithm for Hilbert tenth problem, linear case, with the extension of a classic Euler theorem from one equation to a system;
- solvers have been obtained that are more stable than classical ones, especially for the problem arising in primal-dual interior point method;
- ABS methods are usually faster on vector or parallel machines;
- ABS methods provide a simpler approach for teaching for a variety of classes of problems, since particular methods are obtained just by specific parameter choices.
Knowledge of ABS methods is still quite limited among mathematicians, but they have great potential for improving the methods currently in use.