Abstract
In the present paper we describe a new class of algorithms for solving Diophantine systems of equations in integer arithmetic. This algorithm, designated as the integer ABS (iABS) algorithm, is based on the ABS methods in the real space, with extensive modifications to ensure that all calculations remain in the integer space. Importantly, the iABS solves Diophantine systems of equations without determining the Hermite normal form. The algorithm is suitable for solving determined, over- or underdetermined, full rank or rank deficient linear integer equations. We also present a scaled integer ABS system and two special cases for solving general Diophantine systems of equations. In the scaled symmetric iABS (ssiABS), the Abaffian matrix H i is symmetric, allowing that only half of its elements need to be calculated and stored. The scaled non-symmetric iABS system (snsiABS) provides more freedom in selecting the arbitrary parameters and thus the maximal values of H i can be maintained at a certainly lower level. In addition to the above theoretical results, we also provide numerical experiments to test the performance of the ssiABS and the snsiABS algorithms. These experiments have confirmed the suitability of the iABS system for practical applications.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. Abaffy, A lineáris egyenletrendszerek általános megoldásának egy miódszerosztálya, Alkalmaz. Mat. Lapok 5 (1979) 223–240.
J. Abaffy, Generalization of the ABS class, in: Proceeding of the Second Conference on Automata Languages and Mathematical Systems (University of Economics, 1984) pp. 7-11.
J. Abaffy, C.G. Broyden and E. Spedicato, A class of direct methods for linear equations, Numer. Math. 45 (1994) 361–376.
J. Abaffy and A. Galántai, Conjugate direction methods for linear and nonlinear systems of algebraic equations, Colloq. Math. Soc. János Bolyai 50 (1986) 481–502.
J. Abaffy and E. Spedicato, ABS Projection Algorithms Mathematical Techniques for Linear and Nonlinear Equations (Ellis Horwood, Chichester, 1989).
S. Cabay and T.P.L. Lam, Exact solution of systems of linear equations, ACM Transactions on Mathematical Software 3 (1977) 404–410.
P.D. Domich, R. Kannan and L.E. Trotter, Jr., Hermite normal form computation using modulo determinant arithmetic, Mathematics of Operations Research 12 (1987) 50–59.
E. Egerváry, On rank-diminishing operations and their applications to the solution of linear equations, ZAMP 9 (1960) 376–386.
H. Esmaeili, N. Mahdavi-Amiri and E. Spedicato, Solution of Diophantine linear systems via the ABS methods, Preprint, University of Bergamo, Bergamo, Italy (1999).
H.Y. Huang, A Direct method for the general solution of a system of linear equations, J. Optim. Theory Appl. 16 (1975) 429–445.
R. Kannan and A. Bachem, Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix, SIAM J. Comput. 8 (1979) 499–507.
T.M. Rao, K. Subramanian and E.V. Krishnamurthy, Residue arithmetic algorithmetic algorithms for exact compution of g-inverses of matrices, SIAM J. Numer. Anal. 13 (1976) 155–171.
K.H. Rosen, Elementary Number Theory and Its Applications (Addison-Wesley, 1986).
A. Schrijver, Theory of Linear and Integer Programming (Wiley, Chichester, 1986).
J. Springer, Exact solution of general integer systems of linear equations, ACM Transactions on Mathematical Software 12 (1986) 51–61.
G.W. Stewart, Introduction to Matrix Computations (Academic Press, London, 1973).
G. Zielke, Die Auflösung beliebiger linearer algebraischer Gleichungssysteme durch Blockzerlegung, Beitr. Numer. Math. 8 (1980) 181–199.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Fodor, S. Symmetric and Non-Symmetric ABS Methods for Solving Diophantine Systems of Equations. Annals of Operations Research 103, 291–314 (2001). https://doi.org/10.1023/A:1012971509934
Issue Date:
DOI: https://doi.org/10.1023/A:1012971509934