An Efficient Algorithm for Nontrivial Eigenvectors in MaxPlus Algebra
Abstract
:1. Introduction
2. MaxPlus Algebra
Algorithm 1 Eigenproblems in MaxPlus Algebra 

3. Latin Square
Algorithm 2 Eigenproblems for Latin Square 

Algorithm 3 Eigenproblems for MaxPlus Algebra 

 1.
 By Algorithm 2, the maximal number in M is the eigenvalue, $i.e.$,$$\lambda =max\left(M\right)=3.$$Consequently,$${M}_{\lambda}=\left[\begin{array}{cccc}1& 2& 0& \u03f5\\ 2& \u03f5& 1& 0\\ 0& 1& \u03f5& 2\\ \u03f5& 0& 2& 1\end{array}\right].$$Next, consider the initial vector$${u}^{*}\left(0\right)=\left(\begin{array}{c}0\\ \u03f5\\ \u03f5\\ \u03f5\end{array}\right).$$After iterating $\left(3\right)$, the following sequence is obtained:$$\begin{array}{ccc}\hfill {u}^{*}\left(1\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\to \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(2\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\to \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(3\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}& \to \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(4\right)& ={u}^{*}\left(2\right)\hfill \\ \hfill \left(\begin{array}{c}1\\ 2\\ 0\\ \u03f5\end{array}\right)\to \left(\begin{array}{c}0\\ 1\\ 1\\ 2\end{array}\right)\to \left(\begin{array}{c}1\\ 2\\ 0\\ 1\end{array}\right)& \to \left(\begin{array}{c}0\\ 1\\ 1\\ 2\end{array}\right)& ={u}^{*}\left(2\right).\hfill \end{array}$$It follows that $s=2,r=4$. To compute a corresponding eigenvector we proceed with the computation of the vector v.$$\begin{array}{ccc}\hfill v& =& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(s\right)\oplus \dots \oplus {u}^{*}(r1)\hfill \\ & =& \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(2\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\oplus \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}{u}^{*}\left(3\right)\hfill \\ & =& \left(\begin{array}{c}0\\ 1\\ 1\\ 2\end{array}\right)\oplus \left(\begin{array}{c}1\\ 2\\ 0\\ 1\end{array}\right)=\left(\begin{array}{c}0\\ 1\\ 0\\ 1\end{array}\right).\hfill \end{array}$$It is easy to check whether or not $M\otimes v=\lambda \otimes v$. It follows that$$M\otimes v=\left(\begin{array}{c}3\\ 2\\ 3\\ 2\end{array}\right)=\lambda \otimes v.$$Hence for the eigenvalue $\lambda =3$, the corresponding eigenvector v of the matrix M resulting from the above algorithm is a correct eigenvector.
 2.
 For Algorithm 3, consider the initial vector$$u\left(0\right)=\left(\begin{array}{c}0\\ \u03f5\\ \u03f5\\ \u03f5\end{array}\right).$$Iterating $\left(1\right)$, we obtain$$\begin{array}{ccc}\hfill u\left(1\right)\to \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\left(2\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\to \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\left(3\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}& \to u\left(4\right)& =u\left(2\right)\hfill \\ \hfill \left(\begin{array}{c}2\\ 1\\ 3\\ \u03f5\end{array}\right)\to \left(\begin{array}{c}6\\ 5\\ 5\\ 4\end{array}\right)\to \left(\begin{array}{c}8\\ 7\\ 9\\ 8\end{array}\right)& \to \left(\begin{array}{c}12\\ 11\\ 11\\ 12\end{array}\right)& =6\otimes u\left(2\right).\hfill \end{array}$$It follows that $q=2,p=4$. The vector v resulting from Algorithm 3, is given as$$\begin{array}{ccc}\hfill v& =& {\oplus}_{j=1}^{pq}({\lambda}^{\otimes (pqj)}\otimes u(q+j1))\hfill \\ & =& {\oplus}_{j=1}^{2}({\lambda}^{\otimes (2j)}\otimes u(2+j1))\hfill \\ & =& \lambda \otimes u\left(2\right)\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\oplus \phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4.pt}{0ex}}u\left(3\right)\hfill \\ & =& 3\otimes \left(\begin{array}{c}6\\ 5\\ 5\\ 4\end{array}\right)\oplus \left(\begin{array}{c}8\\ 7\\ 9\\ 8\end{array}\right)\hfill \\ & =& \left(\begin{array}{c}9\\ 8\\ 8\\ 7\end{array}\right)\oplus \left(\begin{array}{c}8\\ 7\\ 9\\ 8\end{array}\right)=\left(\begin{array}{c}9\\ 8\\ 9\\ 8\end{array}\right).\hfill \end{array}$$
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
 Halburd, R.G.; Southall, N.J. Tropical nevanlinna theory and ultradiscrete equations. Int. Math. Res. Not. 2009, 5, 887–911. [Google Scholar]
 CuninghameGreen, R.A. Lecture notes in economics and mathematical systems. In Minimax Algebra; SpringerVerlag: New York, NY, USA, 1979. [Google Scholar]
 De Shutter, B. On the ultimate behavior of the sequence of consecutive powers of a matrix in the maxplus algebra. Linear Algebra Its Appl. 2000, 307, 103–117. [Google Scholar] [CrossRef] [Green Version]
 Gaubert, S. Methods and applications of (max,+) linear algebra. In Annual Symposium on Theoretical Aspects of Computer Science; SpringerVerlag: Berlin/Heidelberg, Germany, 1997; pp. 261–282. [Google Scholar] [Green Version]
 Subiono on Classes of MinMaxPlus Systems and Their Application. Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 2000.
 Kubo, S.; Nishinari, K. Applications of maxplus algebra to flow shop scheduling problems. Discret. Appl. Math. 2018, 247, 278–293. [Google Scholar] [CrossRef]
 Santoso, K.A.; Suprajitno, H. On MaxPlus Algebra and Its Application on Image Steganography. Sci. World J. 2018, 2018, 6718653. [Google Scholar] [CrossRef] [PubMed]
 Braker, J.G.; Olsder, G.J. The power algorithm in max algebra. Linear Algebra Its Appl. 1993, 182, 67–89. [Google Scholar] [CrossRef] [Green Version]
 Subiono; van der Woude, J. Power algorithms for (max,+) and bipartite (min,max,+)systems. Discret. Event Dyn. Syst. 2000, 10, 369–389. [Google Scholar] [CrossRef]
 GarcaPlanas, M.I.; Magret, M.D. Eigenvectors of permutation matrices. Adv. Pure Math. 2015, 5, 390–394. [Google Scholar] [CrossRef]
 Mufid, M.S.U. Subiono Eigenvalues and eigenvectors of latin squares in maxplus algebra. J. Indones. Math. Soc. 2014, 20, 37–45. [Google Scholar] [CrossRef]
 Karp, R.M. A characterization of the minimum cycle mean in a digraph. Discret. Math. 1978, 23, 309–311. [Google Scholar] [CrossRef] [Green Version]
 Akian, M.; Gaubert, S.; Nitica, V.; Singer, I. Best approximation in maxplus semimodules. Linear Algebra Its Appl. 2011, 435, 3261–3296. [Google Scholar] [CrossRef]
 Gunawardena, J. Minmax functions. Discret. Event Dyn. Syst. 1994, 4, 377–407. [Google Scholar] [CrossRef]
 Brualdi, R.A.; Ryser, H.J. Combinatorial Matrix Theory; Cambridge University Press: Cambridge, UK, 1991. [Google Scholar]
 Subiono, A.M. Aljabar maxplus dan terapannya. Buku Ajar Mata Kuliah Pilihan Pasca Sarjana Matematika. In Proceedings of the National Graduate Seminar XII—ITS, Surabaya, Indonesia, 11–12 October 2012. [Google Scholar]
 McKay, B.D.; Wanless, I.M. On the number of Latin squares. Ann. Comb. 2005, 9, 334–344. [Google Scholar] [CrossRef]
 Tomaskova, H. Eigenproblem for circulant matrices in maxplus algebra. In Proceedings of the 12th WSEAS International Conference on Mathematical Methods, Computational Techniques and Intelligent Systems, Kantaoui, Sousse, Tunisia, 3–6 May 2010; World Scientific and Engineering Academy and Society (WSEAS): Cambridge, UK, 2010; pp. 158–161. [Google Scholar]
 Ulfa, M. Analysis on Eigenvector of Circulant Matrix in MaxPlus Algebra. Master’s Thesis, Mathematics and Natural Science Faculty, Institut Teknologi Sepuluh Nopember, Surabaya, Indonesia, 2009. (In Indonesian). [Google Scholar]
 Gavalec, M.; Plavka, J. Structure of the eigenspace of a Monge matrix in maxplus algebra. Discret. Appl. Math. 2008, 10, 596–606. [Google Scholar] [CrossRef]
 Imaev, A.A.; Judd, R.P. Computing an eigenvector of an inverse Monge matrix in maxplus algebra. Discret. Appl. Math. 2010, 158, 1701–1707. [Google Scholar] [CrossRef]
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Umer, M.; Hayat, U.; Abbas, F. An Efficient Algorithm for Nontrivial Eigenvectors in MaxPlus Algebra. Symmetry 2019, 11, 738. https://doi.org/10.3390/sym11060738
Umer M, Hayat U, Abbas F. An Efficient Algorithm for Nontrivial Eigenvectors in MaxPlus Algebra. Symmetry. 2019; 11(6):738. https://doi.org/10.3390/sym11060738
Chicago/Turabian StyleUmer, Mubasher, Umar Hayat, and Fazal Abbas. 2019. "An Efficient Algorithm for Nontrivial Eigenvectors in MaxPlus Algebra" Symmetry 11, no. 6: 738. https://doi.org/10.3390/sym11060738