Efficient Alternative Method for Computing Multivariate Resultant Formation

Authors

  • Surajo Sulaiman Department of Mathematics, Faculty of Science, Northwest University Kano 700221, Kano
  • Noraini Aris Department of Mathematical Sciences, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor,
  • Shamsatun Nahar Ahmad Department of Computer and Mathematical Sciences, Universiti Teknologi Mara Segamat, Jalan Universiti Off Km. 12 Jalan Muar, 85000 Segamat, Johor Darul Ta'zim

DOI:

https://doi.org/10.5614/j.math.fund.sci.2019.51.1.2

Keywords:

Dixon resultant, hybrid resultant, Jouanolou??s resultant, resultant matrix

Abstract

In elimination theory, the matrix method of computing resultant remains the most popular due to its less computational complexity compared to Groebner basis and set characteristics approaches. However, for a matrix method to be effective, the size and the nature of elements of the matrix play an important role, since if the resultant is not an exact resultant, it has to be extracted from the determinant of the corresponding resultant matrix.. In this paper, a new resultant matrix is proposed. The proposed construction consists of four blocks, one of the blocks uses an entry formula of computing a Dixon matrix, while, two of the blocks use a mapping from the Jouanolou's method and the last block consists of only zero elements. The new formulation is computed without intermediate cancelling terms which reduces the complexity of the construction and enhances its effectiveness.

References

Yang, L., Fu, H. & Zeng, Z., A Practical Symbolic Algorithm for the Inverse Kinematics of 6R Manipulators with Simple Geometry, McCune W. (Eds) Automated Deduction - CADE-14, Lecture Notes in Computer Science (Lecture Notes in Artificial Intelligence), Springer, Berlin, Heidelberg, 1249, pp. 73-86, 1997. DOI: 10.1007/3-540-63104-6_11

Lewis, R.H. & Stiller, P.F., Solving the Recognition Problem for Six Lines using the Dixon Resultant, Mathematics and Computers in Simulation, 49(3), pp. 205-219, 1999.

Awange, J.L., Grafarend, E.W., Palancz, B. & Zaletnyik, P., Algebraic Geodesy and Geoinformatics, Springer Science & Business Media, 2010.

Sylvester, J.J., On a Theory of the Syzygetic Relations of Two Rational Integral Functions, Comprising an Application to the Theory of Sturm's Functions, and that of the Greatest Algebraical Common Measure. Philosophical Transactions of the Royal Society of London, 143, pp. 407-548, 1853.

Bezout, E., General Theory of Algebraic Equations, PhD Dissertation, Royal Academy of Sciences, Eleves & Alpirans at the Royal Corps of Artillery Paris, Paris, 1779. (Text in French)

Macaulay, F., Some Formulae in Elimination, Proceedings of the London Mathematical Society, 1(1), pp. 3-27, 1902.

Sulaiman, S. & Aris, N., Comparison of Some Multivariable Hybrid Resultant Matrix Formulations, Indian Journal of Science and Technology, 9(46), pp. 1-8, 2016.

Wang, W. & Lian, X. Computations of Multi-resultant with Mechanization. Applied Mathematics and Computation, 170(1), pp. 237-257, 2005.

Kapur, D. & Saxena, T., Comparison of Various Multivariate Resultant Formulations, in Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation, Association for Computing Machinery, 1995.

Jouanolou, J.P., Forms of Inertia and Resultant: A Formula, Advances in Mathematics, 126(2), pp. 119-250, 1997. (Text in French)

D'Andrea, C. & Dickenstein, A., Explicit Formulas for the Multivariate Resultant, Journal of Pure and Applied Algebra, 164(1), pp. 59-86, 2001.

Szanto, A., Multivariate Subresultants using Jouanolou's Resultant Matrices, Preprint, 2001.

Chionh, E.W., Concise Parallel Dixon Determinant, Computer Aided Geometric Design, 14(6), pp. 561-570, 1997.

Chionh, E.W., Zhang, M. & Goldman, R.N., Fast Computation of the Bezout and Dixon Resultant Matrices, Journal of Symbolic Computation, 33(1), pp. 13-29, 2002.

Zhao, S. & Fu, H., An Extended Fast Algorithm for Constructing the Dixon Resultant Matrix, Science in China Series A: Mathematics, 48(1), pp. 131-143, 2005.

Qin, X., Wu, D., Tang, L. & Ji, Z., Complexity of Constructing Dixon Resultant Matrix, International Journal of Computer Mathematics, 94(10), pp. 2074-2088, 2017.

Khetan, A., The Resultant of An Unmixed Bivariate System, Journal of Symbolic Computation, 36(3-4), pp. 425-442, 2003.

Zhao, S. & Fu, H., Multivariate Sylvester Resultant and Extraneous Factors , Science in China Series A: Mathematics, 40(7), pp. 649-660, 2010. (Text in Chinese)

Zhao, S. & Fu, H., Three Kinds of Extraneous Factors in Dixon Resultants"? Science in China Series A: Mathematics, 52(1), pp. 160-172, 2009.

Downloads

Published

2019-04-30

Issue

Section

Articles