3.3 Computing the eigenvector by cyclic coordinates
3.3.2 Principal eigenvector of positive matrices by cyclic co-
The details of this algorithm can be found in section 5.2 in the dissertation, as well as in manuscript [4]. An iterative algorithm is given for computing the principal eigenvector and eigenvalue of positive matrices. This method works in very general cases, but one application is the calculation of the Eigenvector Method for pairwise comparison matrices.
The algorithm uses form (2.13) to approxiamte λmax, but this is an arbi-trary choice: the algorithm can be easily adapted to use form (2.12). Later however, both forms will be used to give the stopping condition.
The method of cyclic coordinates will be used in this case as well. The variables are the elements of the right principal eigenvector, w: w1, . . . , wn. As discussed earlier, cyclic coordinates considers only one variable as a proper variable in each step. Let the index of this variable be denoted by k, thus in every step wk will be the actual variable, while the values of all other variables are xed at their values calculated in the previous step.
Thus, as described earlier, in each step we are looking for the value ofwk for which the following is true:
wk = arg min
wk
i=1,...,nmax
(Aw)i wi
. (3.1)
Because all other wj,j 6=k values are xed, for all i(3.1) is only dependent on wk. Thus the following notation can be introduced:
fi(wk) = (Aw)i
wi , i= 1, . . . , n. (3.2) Therefore, what we are searching for is the wk > 0 value, for which wk = arg minw
kmaxi=1,...,nfi(wk), or in other words where the maximum function of fi has the minimum point. The fi(wk) value will be the approximation (upper bound) of λmax. It can be shown, that the fi functions for i 6=k are linear, while fori=k,fk(wk) is a hyperbolic function. It can also be shown,
20 CHAPTER 3. NEW RESULTS OF THE DISSERTATION that it is sucient to calculate only the intersection points of fk with each fi, i6=k. Because of the strict monotonic descent of fk, the value of wk >0 which satises (3.1), will be the smallest wk, which is in the intersection of the hyperbolic and a linear function. The intersection point itself can be calculated by the quadratic formula.
An opportunity for faster running arises if we consider that the calculation of all intersection points is unnecessary. Those linear functions that have no common points with the maximum of the linear functions can be ignored.
The stopping condition is when the estimation of the expression minw>0maxi=1,...,n (Aw)w i
i in the CollatzWielandt formula (the minimum point of the maximum function), and the estimation of the expression maxw>0mini=1,...,n (Aw)w i
i (the maximum point of the minimum function) are closer to each other than a predened threshold, the algorithm stops.
For starting values, any positive vector is acceptable. A possible sim-ple starting value is 1 for all variables, wi(0) = 1, i = 1, . . . , n. In case of PCMs though, the principal eigenvector (the weight vector of the eigenvec-tor method) is close to the row-wise geometric mean (the weight veceigenvec-tor for the logarithmic least squares method) [16]. Therefore, the starting values in this case should be
w(0)i =
n
Y
j=1
√n
aij. (3.3)
This starting value set can also be used in case of general positive matrices.
The algorithm presented above is a new method for calculating the princi-pal eigenvector and eigenvalue, which is tailored for speed on large matrices, and its simplicity comes from the method of cyclic coordinates and the arith-metically simple calculations.
Chapter 4
List of publications
Scientic articles in English
1. K. Ábele-Nagy. Minimization of the Perron eigenvalue of incomplete pairwise comparison matrices by Newton iteration. Acta Universitatis Sapientiae, Informatica, 7(1):5871, 2015.
2. K. Ábele-Nagy and S. Bozóki. Eciency analysis of simple perturbed pairwise comparison matrices. Fundamenta Informaticae, 144:279289, 2016.
3. K. Ábele-Nagy, S. Bozóki, and Ö. Rebák. Eciency analysis of double perturbed pairwise comparison matrices. Journal of the Operational Research Society, 69(5):707713, 2018.
Manuscript in English
4. K. Ábele-Nagy and J. Fülöp. On computing the principal eigenvector of positive matrices by the method of cyclic coordinates. kézirat, 2019.
21
22 CHAPTER 4. LIST OF PUBLICATIONS
Master's theses in Hungarian
5. K. Ábele-Nagy. Nem teljesen kitöltött páros összehasonlítás mátrixok a többszempontú döntésekben. Diplomamunka, Eötvös Loránd Tu-dományegyetem, 2010.
6. K. Ábele-Nagy. Nem teljesen kitöltött páros összehasonlítás mátrixok aggregálása. MA Szakdolgozat, Budapesti Corvinus Egyetem, 2012.
Bibliography
[1] K. Ábele-Nagy. Minimization of the Perron eigenvalue of incomplete pairwise comparison matrices by Newton iteration. Acta Universitatis Sapientiae, Informatica, 7(1):5871, 2015.
[2] K. Ábele-Nagy and S. Bozóki. Eciency analysis of simple perturbed pairwise comparison matrices. Fundamenta Informaticae, 144:279289, 2016.
[3] K. Ábele-Nagy, S. Bozóki, and Ö. Rebák. Eciency analysis of double perturbed pairwise comparison matrices. Journal of the Operational Research Society, 69(5):707713, 2018.
[4] K. Ábele-Nagy and J. Fülöp. On computing the principal eigenvector of positive matrices by the method of cyclic coordinates. kézirat, 2019.
[5] R. Blanquero, E. Carrizosa, and E. Conde. Inferring ecient weights from pairwise comparison matrices. Mathematical Methods of Operations Research, 64(2):271284, 2006.
[6] S. Bozóki. Inecient weights from pairwise comparison matrices with arbitrarily small inconsistency. Optimization, 63(12):18931901, 2014.
[7] S. Bozóki, J. Fülöp, and L. Rónyai. On optimal completion of incomplete pairwise comparison matrices. Mathematical and Computer Modelling, 52(1-2):318333, 2010.
23
24 BIBLIOGRAPHY [8] A. T. W. Chu, R. E. Kalaba, and K. Spingarn. A comparison of two methods for determining the weights of belonging to fuzzy sets. Journal of Optimization Theory and Applications, 27(4):531538, 1979.
[9] G. Crawford and C. Williams. A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology, 29(4):387405, 1985.
[10] J. G. de Graan. Extensions of the multiple criteria analysis method of T.L. Saaty. Technical report, National Institute for Water Supply, Leidschendam, The Netherlands, 1980.
[11] P. de Jong. A statistical approach to Saaty's scaling method for priori-ties. Journal of Mathematical Psychology, 28(4):467478, 1984.
[12] A. Farkas. The analysis of the principal eigenvector of pairwise compar-ison matrices. Acta Polytechnica Hungarica, 4(2):99115, 2007.
[13] J. Fülöp. A sajátvektor módszer egy optimalizálási megközelítése. XXX.
Magyar Operációkutatási Konferencia, 2013.
[14] P. Harker. Incomplete pairwise comparisons in the Analytic Hierarchy Process. Mathematical Modelling, 9(11):837848, 1987.
[15] P. T. Harker. Derivatives of the Perron root of a positive reciprocal matrix: With application to the Analytic Hierarchy Process. Applied Mathematics and Computation, 22(2-3):217232, 1987.
[16] M. Kwiesielewicz. The logarithmic least squares and the generalized pseudoinverse in estimating ratios. European Journal of Operational Research, 93(3):611619, 1996.
[17] D. G. Luenberger and Y. Ye. Linear and Nonlinear Programming, vol-ume 116 of International Series in Operations Research & Management Science. Springer, 3rd edition, 2008.
BIBLIOGRAPHY 25 [18] G. Rabinowitz. Some comments on measuring world inuence. Journal
of Peace Science, 2(1):4955, feb 1976.
[19] T. L. Saaty. A scaling method for priorities in hierarchical structures.
Journal of Mathematical Psychology, 15(3):234281, 1977.
[20] T. L. Saaty. The Analytic Hierarchy Process. McGraw-Hill, 1980.
[21] S. Shiraishi, T. Obata, and M. Daigo. Properties of a positive reciprocal matrix and their application to AHP. Journal of the Operations Research Society of Japan, 41(3):404414, 1998.
[22] W. C. Wedley. Consistency prediction for incomplete AHP matrices.
Mathematical and Computer Modelling, 17(4-5):151161, 1993.