S. Furtado | Searcharxiv

arXiv2024

Analysis of natural cardinal ranking vectors for pairwise comparisons and the universal efficiency of the Perron geometric mean

In models using pair-wise (ratio) comparisons among alternatives, a cardinal ranking vector should be deduced from a reciprocal matrix. The right Perron eigenvector (RP) was traditionally used, though several other options have emerged. We consider some alternatives, mostly new, namely the entry-wise reciprocal of the left Perron vector (LP), the left singular vector (LS), the entry-wise reciprocal of the right singular vector (RS), the arithmetic and geometric means of RP and LP (AP and GP), and of LS and RS (AS and GS). (The ranking vector AS was proposed by Gass and Rapcs\'ak (2004)). All 8 of these vectors produce the natural vector in the consistent case. We compare them empirically, in terms of efficiency, for random matrices, as a function of the number of alternatives. It turns out that the vector GP is universily efficient, and this fact is proven. The vector GS performs better that the remaining 6 vectors. We show that, for reciprocal matrices obtained from consistent matrices by modifying one column and the corresponding row, all 8 vectors are efficient. Moreover, the cone generated by the columns is efficient.

math.CO↗

arXiv2017

Conditioning and backward error of block-symmetric block-tridiagonal linearizations of matrix polynomials

For each square matrix polynomial $P(\lambda)$ of odd degree, a block-symmetric block-tridiagonal pencil $\mathcal{T}_{P}(\lambda)$ was introduced by Antoniou and Vologiannidis in 2004, and a variation $\mathcal{R}_P(\lambda)$ was introduced by Mackey et al. in 2010. These two pencils have several appealing properties, namely they are always strong linearizations of $P(\lambda)$, they are easy to construct from the coefficients of $P(\lambda)$, the eigenvectors of $P(\lambda)$ can be recovered easily from those of $\mathcal{T}_P(\lambda)$ and $\mathcal{R}_P(\lambda)$, the two pencils are symmetric (resp. Hermitian) when $P(\lambda)$ is, and they preserve the sign characteristic of $P(\lambda)$ when $P(\lambda)$ is Hermitian. In this paper we study the numerical behavior of $\mathcal{T}_{P}(\lambda)$ and $\mathcal{R}_P(\lambda)$. We compare the conditioning of a finite, nonzero, simple eigenvalue $\delta$ of $P(\lambda)$, when considered an eigenvalue of $P(\lambda)$ and an eigenvalue of $\mathcal{T}_{P}(\lambda)$. We also compare the backward error of an approximate eigenpair $(z,\delta)$ of $\mathcal{T}_{P}(\lambda)$ with the backward error of an approximate eigenpair $(x,\delta)$ of $P(\lambda)$, where $x$ was recovered from $z$ in an appropriate way. When the matrix coefficients of $P(\lambda)$ have similar norms and $P(\lambda)$ is scaled so that the largest norm of the matrix coefficients of $P(\lambda)$ is one, we conclude that $\mathcal{T}_{P}(\lambda)$ and $\mathcal{R}_P(\lambda)$ have good numerical properties in terms of eigenvalue conditioning and backward error. Moreover, we compare the numerical behavior of $\mathcal{T}_{P}(\lambda)$ with that of other well-studied linearizations in the literature, and conclude that $\mathcal{T}_{P}(\lambda)$ performs better than these linearizations when $P(\lambda)$ has odd degree and has been scaled.

math.NA↗