A class of ABS algorithms for Diophantine linear systems

Abstract

Systems of integer linear (Diophantine) equations arise from various applications. In this paper we present an approach, based upon the ABS methods, to solve a general system of linear Diophantine equations. This approach determines if the system has a solution, generalizing the classical fundamental theorem of the single linear Diophantine equation. If so, a solution is found along with an integer Abaffian (rank deficient) matrix such that the integer combinations of its rows span the integer null space of the cofficient matrix, implying that every integer solution is obtained by the sum of a single solution and an integer combination of the rows of the Abaffian. We show by a counterexample that, in general, it is not true that any set of linearly independent rows of the Abaffian forms an integer basis for the null space, contrary to a statement by Egervary. Finally we show how to compute the Hermite normal form for an integer matrix in the ABS framework.

Integer extended ABS algorithms and possible control of intermediate results for linear Diophantine systems

“…The ABS class of algorithms is a generalization of Egervary's rank reducing algebraic process (Egervary 1956(Egervary , 1960, first introduced by Abaffy, Broyden and Spedicato (see Abaffy et al 1984) for solving linear systems, later extended to solve nonlinear equations and optimization problems (see Abaffy and Spedicato 1989) and recently specialized to solve systems of linear Diophantine equations (Esmaeili et al 2001a); for a recent review, see Spedicato et al (2003). An ABS method starts with an arbitrary initial vector x 1 ∈ R n and a nonsingular matrix H 1 ∈ R n,n , Spedicato's parameter.…”

“…After performing a finite number of iterations, an algorithm in this class finds an integer matrix spanning the integer null space of A, and a particular solution of (1.1). The new proposed class contains the class of EMAS algorithms (see Esmaeili et al 2001a), and some other well known efficient algorithms, such as the LDSSBR of Chou and Collins (1982). Choosing the parameters of the new class appropriately, we can design effective and competitive algorithms for solving linear Diophantine systems.…”

“…They also developed an algorithm based on Rosser's idea (see Rosser 1941), the so-called LDSSBR, and showed numerically that the LDSSBR was more successful than the LDSMKB in controlling the growth of intermediate results. Esmaeili et al (2001a), presented a class of algorithms, the so-called EMAS algorithms, for solving systems of linear Diophantine equations based on the ABS algorithms (see Abaffy et al 1984;Abaffy and Spedicato 1989).…”

Can string theory predict the Weinberg angle?

“…[18]. (vii) The solution to the system of linear diophantine equations is given by the sum of a particular solution and a linear combination of (not necessarily linearly independent, see Ref.…”

“…(vii) The solution to the system of linear diophantine equations is given by the sum of a particular solution and a linear combination of (not necessarily linearly independent, see Ref. [18]) homogeneous solutions, 8 where the coefficients are all integers. In most cases, there are no more than 2 homogeneous solutions, so that we can simply iterate over the integer coefficients to obtain candidates for hypercharge.…”