Published in Volume XXXIV, Issue 1, 2024, pages 1-22, doi: 10.47743/SACS.2024.1.1
Authors: M. Dehghani Darmian
Abstract
In this paper, we study and compute the inverse of matrices with parametric entries. We demonstrate that the Gauss-Jordan method can be extended to compute the inverse of parametric matrices, offering a powerful tool for solving systems of linear equations and analyzing parametric systems. Using this new expansion (so-called Gauss-Jordan systems) and also utilizing linearly dependency systems for linear systems involving parameters [4, 5], we introduce the notion of an inverse matrix system for a parametric matrix. In doing so, we decompose the space of parameters into a finite partition and for each partition, we give the corresponding inverse matrix without applying Gröbner systems. We also present an algorithm for computing an inverse system for a given parametric matrix. All mentioned algorithms have been implemented in Maple, and their efficiency and behavior have been experimented on a set of benchmark matrices.
References
[1] Bruno Buchberger. Bruno Buchberger’s PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal. Journal of Symbolic Computation, 41(3-4):475–511, 2006. doi:10.1016/J.JSC.2005.09.007.
[2] David A. Cox, John Little, and Donal O’Shea. Ideals, Varieties, and Algorithms – An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics. Springer, 4th edition, 2015. doi:10.1007/978-3-319-16721-3.
[3] Mahdi Dehghani Darmian, Amir Hashemi, and Antonio Montes. Erratum to ”A new algorithm for discussing Gröbner bases with parameters” Journal of Symbolic Computation 33(1-2) (2002) 183-208. Journal of Symbolic Computation, 46(10):1187–1188, 2011. doi:10.1016/J.JSC.2011.05.002.
[4] Mahdi Dehghani Darmian and Amir Hashemi. Parametric FGLM algorithm. Journal of Symbolic Computation, 82:38–56, 2017. doi:10.1016/j.jsc.2016.12.006.
[5] Mahdi Dehghani Darmian and Amir Hashemi. Generalization of the F4 algorithm to parametric polynomial ideals. In 7th International Conference on Combinatorics, Cryptography, Computer Science and Computation, I4C 2022, pages 1014–1021. 2022. URL: https://i4c.iust.ac.ir/UPL/Paper2022/accpapers/i4c2022-1028.pdf.
[6] Amir Hashemi, Mahdi Dehghani Darmian, and Marzieh Barkhordar. Gröbner systems conversion. Mathematics in Computer Science, 11(1):61–77, 2017. doi:10.1007/S11786-017-0295-3.
[7] Amir Hashemi, Benyamin M.-Alizadeh, and Mahdi Dehghani Darmian. Computing comprehensive Gröbner systems: A comparison of two methods. The Computer Science Journal of Moldova, 25(3):278–302, 2017. URL: http://www.math.md/publications/csjm/issues/v25-n3/12511/.
[8] Deepak Kapur, Yao Sun, and Dingkang Wang. A new algorithm for computing comprehensive Gröbner systems. In Wolfram Koepf, editor, 23rd International Symposium on Symbolic and Algebraic Computation, ISSAC 2010, pages 29–36. ACM, 2010. doi:10.1145/1837934.1837946.
[9] Antonio Montes. A new algorithm for discussing Gröbner bases with parameters. Journal of Symbolic Computation, 33(2):183–208, 2002. doi:10.1006/JSCO.2001.0504.
[10] Antonio Montes and Michael Wibmer. Gröbner bases for polynomial systems with parameters. Journal of Symbolic Computation, 45(12):1391–1425, 2010. doi:10.1016/J.JSC.2010.06.017.
[11] Katsusuke Nabeshima. A speed-up of the algorithm for computing comprehensive Gröbner systems. In Dongming Wang, editor, 20th Symbolic and Algebraic Computation, International Symposium, ISSAC 2007, pages 299–306. ACM, 2007. doi:10.1145/1277548.1277589.
[12] Akira Suzuki and Yosuke Sato. A simple algorithm to compute comprehensive Gröbner bases using Gröbner bases. In Barry M. Trager, editor, 19th International Symposium on Symbolic and Algebraic Computation, ISSAC 2006, pages 326–331. ACM, 2006. doi:10.1145/1145768.1145821.
[13] Volker Weispfenning. Comprehensive Gröbner bases. Journal of Symbolic Computation, 14(1):1–30, 1992. doi:10.1016/0747-7171(92)90023-W.
Bibtex
@article{sacscuza:dehghani2024eacipm, title={Efficient Algorithm for Computing Inverse of Parametric Matrices}, author={M. Dehghani Darmian}, journal={Scientific Annals of Computer Science}, volume={34}, number={1}, organization={Alexandru Ioan Cuza University, Ia\c si, Rom\^ania}, year={2024}, pages={1-22}, publisher={Alexandru Ioan Cuza University Press, Ia\c si}, doi={10.7561/SACS.2024.1.1} }