A generalization of the ABS algorithms and its application to some special real and integer matrix factorizations

Document Type : Research Article

Authors

• E. Golpar Raboky ¹
• N. Mahdavi-Amiri ²

¹ Department of Mathematics, University of Qom. Qom, Iran. Tel: [phone]

² Faculty of Mathematical Sciences, Sharif University of Technology, Tehran, Iran.

10.22067/ijnao.2021.70974.1043

Abstract

In 1984, Abaffy, Broyden, and Spediacto (ABS) introduced a class of the so-called ABS algorithms to solve systems of real linear equations. Later, the scaled ABS, the extended ABS, the block ABS, and the integer ABS algorithms were introduced leading to various well-known matrix factorizations. Here, we present a generalization of ABS algorithms containing all matrix factorizations such as triangular, W Z, and ZW . We discuss the octant interlocking factorization and make use of the generalized ABS algorithm as a more general approach for producing the octant interlocking factorization.

Keywords

• ABS algorithms
• Quadrant interlocking factorization
• Octant interlocking factorization

20.1001.1.24236977.2022.12.2.3.9

Main Subjects

• Numerical linear algebra and its applications

---

References

1. Abaffy, J., Broyden, C.G. and Spedicato, E. A class of direct methods for linear systems, Numer. Math. 45 (1984) 361–376.
2. Abaffy, J. and Galantai, A. Conjugate direction methods for linear and nonlinear systems of algebraic equations, Colloquia Mathematica Societatis Janos Bolyai 50 (1986) 481–502.
3. Abaffy, J. and Spedicato, E. ABS projection algorithms, mathematical techniques for linear and nonlinear equations, Halsted Press, Chichester, 1989.
4. Adib, M., Mahdavi-Amiri, N. and Spedicato, E. Broyden method as an ABS algorithm, Publ. Univ. Miskolc Ser. D Nat. Sci. Math. 40 (1999), 3–13.
5. Chen, Y. and Zhou, B. On g-inverses and nonsingularity of a bordered matrix (A B C O ) , Linear Algebra Appl. 133 (1990) 133–151.
6. Esmaeili, H., Mahdavi-Amiri, N. and Spedicato, E. Generating the integer null space and conditions for determination of an integer basis using the ABS algorithms, Bull. Iran. Math. Soc. 27(1) (2001) 1–18.
7. Esmaeili, H., Mahdavi-Amiri, N. and Spedicato, E. A class of ABS algorithms for linear Diophantine systems, Numer. Math. 90 (2001) 101–115.
8. Evans, D.J., Hadjidimos, A. and Noutsos, D. The parallel solution of banded linear equation by the new quadrant interlocking factorization (Q.I.F.) method, Internat. J. Comput. Math. 9(2) (1981) 151–161.
9. Evans, D.J. and Hatzopoulos, M. A parallel linear system solver, Int. J. Comput. Math. 7(3) (1979) 227–238.
10. Golpar-Raboky, E.and Mahdavi-Amiri, N. Diophantine quadratic equation and Smith normal form using scaled extended integer ABS algorithms, J. Optim. Theory Appl. 152(1) (2012) 75–96.
11. Golpar-Raboky, E. and Mahdavi-Amiri, N. WZ factorization via AbaffyBroyden-Spedicato algorithms, Bull. Iran. Math. Soc. 40(2) (2014) 1–13.
12. Golpar-Raboky, E. and Mahdavi-Amiri, N. A new interpretation of the integer and real WZ factorization using block scaled ABS algorithms, Stat. Optim. Inf. Comput. 2 (2014) 243–256.
13. Khorramizadeh, M. and Mahdavi-Amiri, N. Integer extended ABS algorithms and possible control of intermediate results for linear Diophantine systems, 4OR 7 (2009) 145–167.

14. Mahdavi-Amiri, N. and Golpar-Raboky, E. Real and integer Wedder burn rank reduction formulas for matrix decompositions, Optim. Methods Softw. 30(4) (2015) 864–879.
15. Rao, S.C.S. Existence and uniqueness of WZ factorization, Parallel Comput. 23 (1997) 1129–1139.
16. Spedicato, E., Bodon, E. Del Popolo, A. and Mahdavi-Amiri, N. ABS methods and ABSPACK for linear systems and optimization: A review, 4OR 1 (2003) 51–66.
17. Spedicato, E., Bodon, E., Del Popolo, A. and Xia, Z. ABS algorithms for linear systems and optimization: A review and a bibliography, Ricerca Operativa 29 (2000) 39–88.
18. Spedicato, E., Bodon, E., Zunquan, X. and Mahdavi-Amiri, N. ABS methods for continuous and integer linear equations and optimization, CEJOR Cent. Eur. J. Oper. Res. 18 (2010) 73–95.
19. Spedicato, E., Xia, Z. and Zhang, L. The implicit LX method of the ABS class, Optim. Methods Softw. 8 (1997) 99–110.