We show that Muir's law of extensible minors, Cayley's law of complementaries and Jacobi's identity for minors of the adjugate [Determinantal identities Linear Algebra and its Applications 52/53 (1983) pp. 769-791] are equivalent. We also show our generalization of Mühlbach/Muir's extension principle [A generalization of Mühlbach's extension principle for determinantal identities. Linear and Multilinear Algebra 61 (10) (2013) pp. 1363-1376] is equivalent to its previous form derived by Mühlbach. As a corollary, we show that Mühlbach-Gasca-(Lopez-Carmona)-Ramirez identity [A generalization of Sylvester's identity on determinants and some applications. Linear Algebra and its Applications 66 (1985) pp. 221-234/On extending determinantal identities. Linear Algebra and its Applications 132 (1990) pp. 145-162] is equivalent to its generalization found by Beckermann and Mühlbach [A general determinantal identity of Sylvester type and some applications. Linear Algebra and its Applications 197,198 (1994) pp. 93-112].

We study Compositional Models based on Dirichlet Regression where, given a (vector) covariate x, one considers the response variable y = (y 1 ,. .. , y D) to be a positive vector with a conditional Dirichlet distribution, y|x ∼ D(α 1 (x). .. α D (x)). We introduce a new method for estimating the parameters of the Dirichlet Covariate Model when α j (x) is a linear model on x, and also propose a Bayesian model selection approach. We present some numerical results which suggest that our proposals are more stable and robust than traditional approaches.

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We present a new analysis of the stability of extended Floater-Hormann interpolants, in which both noisy data and rounding errors are considered. Contrary to what is claimed in the current literature, we show that the Lebesgue constant of these interpolants can grow exponentially with the parameters that define them, and we emphasize the importance of using the proper interpretation of the Lebesgue constant in order to estimate correctly the effects of noise and rounding errors. We also present a simple condition that implies the backward instability of the barycentric formula used to implement extended interpolants. Our experiments show that extended interpolants mentioned in the literature satisfy this condition and, therefore, the formula used to implement them is not backward stable. Finally, we explain that the extrapolation step is a significant source of numerical instability for extended interpolants based on extrapolation.

We analyze the effects of rounding errors in the nodes on polynomial barycentric interpolation. These errors are particularly relevant for the first barycentric formula with the Chebyshev points of the second kind. Here, we propose a method for reducing them.Mathematics Subject Classification (2010) 65D05, 65G50