On rank-diminishing operations and their applications to the solution of linear equations

Egerváry, E.

doi:10.1007/bf01604497

Cited by 29 publications

(21 citation statements)

References 8 publications

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“…A stability analysis for decent methods is given in [7]. See also [3,4,10]. As Corollary 2.7, we conclude that the pair (P, I) is A T -conjugate for Algorithm 1, considering nonsingular matrix of system (2).…”

Section: Algorithmmentioning

confidence: 65%

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

et al. 2017

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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.

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Section: Algorithmmentioning

confidence: 65%

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

et al. 2017

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“…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.…”

Section: The Emas Algorithmsmentioning

confidence: 99%

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

Khorramizadeh

Mahdavi–Amiri

2008

4OR-Q J Oper Res

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We present a new class of integer extended ABS algorithms for solving linear Diophantine systems. The proposed class contains the integer ABS (the so-called EMAS and our proposed MEMAS) algorithms and the generalized Rosser's algorithm as its members. After an application of each member of the class a particular solution of the system and an integer basis for the null space of the coefficient matrix are at hand. We show that effective algorithms exist within this class by appropriately setting the parameters of the members of the new class to control the growth of intermediate results. Finally, we propose two effective heuristic rules for selecting certain parameters in the new class of integer extended ABS algorithms.

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“…We explore in this paper how to apply them to Diophantine equations, deriving a class of ABS algorithms for this problem and establishing at the same time a new ABS criterion for deciding whether the system is solvable. Several years ago, Egervary (1955Egervary ( , 1960) gave a method, which is a special case of our class, for solving the homogeneous Diophantine equation (b = 0). His method starts with H 1 the identity matrix of order n and generates a sequence of n × n integer matrices {H i+1 }, i = 1, .…”

Section: Introductionmentioning

confidence: 99%

“…Egervary's (1955) paper may have been the first paper giving a solution for a system (albeit only the homogeneous one). Egervary's (1960) paper is mainly concerned with applying his rank-one reduction process also to a non integer linear system, essentially developing the ABS method ante litteram.…”

Section: Introductionmentioning

confidence: 99%

A class of ABS algorithms for Diophantine linear systems

Esmaeili

Mahdavi–Amiri

Spedicato

2001

Numerische Mathematik

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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.

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On rank-diminishing operations and their applications to the solution of linear equations

Cited by 29 publications

References 8 publications

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

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

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

A class of ABS algorithms for Diophantine linear systems

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