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Symmetric and Non-Symmetric ABS Methods for Solving Diophantine Systems of Equations

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  • Szabina Fodor

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. Copyright Kluwer Academic Publishers 2001

Suggested Citation

  • Szabina Fodor, 2001. "Symmetric and Non-Symmetric ABS Methods for Solving Diophantine Systems of Equations," Annals of Operations Research, Springer, vol. 103(1), pages 291-314, March.
    • Handle: RePEc:spr:annopr:v:103:y:2001:i:1:p:291-314:10.1023/a:1012971509934
    DOI: 10.1023/A:1012971509934
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    Cited by:

    1. Szabina Fodor & Zoltán Németh, 2019. "Numerical analysis of parallel implementation of the reorthogonalized ABS methods," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 27(2), pages 437-454, June.
    2. József Abaffy & Szabina Fodor, 2021. "ABS-Based Direct Method for Solving Complex Systems of Linear Equations," Mathematics, MDPI, vol. 9(19), pages 1-17, October.
    3. Spedicato Emilio & Bodon Elena & Xia Zunquan & Mahdavi-Amiri Nezam, 2010. "ABS methods for continuous and integer linear equations and optimization," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 18(1), pages 73-95, March.

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