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Item type:Publication, Access status: with files , On Sinkhorn's DAD theorem and the self-consistency equation in COSMO-based activity coefficient modelsIn a 1966 paper, Sinkhorn proved that for any real square matrix A which has only positive entries there exists a uniquely determined real diagonal matrix D with positive diagonal entries such that 𝐵:=DAD is stochastic, i.e. all row sums of B are equal to 1. Moreover, Sinkhorn stated an iterative method for computing D. Nowadays, Sinkhorn's result and its variants are often referred to as DAD theorems. The purpose of this article is twofold. On the one hand, we give the link between Sinkhorn's DAD theorem and the self-consistency equation in COSMO-based activity coefficient models in chemical engineering. On the other hand, we give a new constructive proof of Sinkhorn's DAD theorem by using classical fixed-point theory. Hereby, the larger class of nonnegative matrices with positive diagonal is considered. Our proof uniformly provides convergence for a number of iterative methods for computing D. Some of them are used in practice although, to the best of our knowledge, a formal proof of convergence is missing.Publicationtype: Journal Article46 58 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: with files , Perturbation properties of the generalized spectral radiusLet n∈N, K∈{R,C} and Mn(K) be the set of all n-by-n matrices with entries in K. We investigate the sensitivity of the generalized spectral radius ρK(A):=max{|λ|:λ∈Kand|Ax|=|λx|foranx∈Kn∖{0}}of A∈Mn(K) under perturbations, i.e. we ask how ρK(A) is related to ρK(A+E) for perturbations E∈Mn(K). For example and somewhat surprisingly, Elsner's famous bound on the spectral variation of two complex square matrices fully translates to |ρC(A)−ρC(B)|⩽∥A+B∥1−1n2∥A−B∥1n2for all A,B∈Mn(C), where ∥⋅∥2 is the matrix 2-norm. For ρR this result holds true locally.Publicationtype: Journal Article39 101 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , When does ‖f(A)‖ = f(‖A‖) hold true?(2022); Let a strongly stable norm (Formula presented.) on the set (Formula presented.) of complex n-by-n matrices be given, which means that (Formula presented.) for all (Formula presented.) and all (Formula presented.). Furthermore, let (Formula presented.) be a power series with nonnegative coefficients (Formula presented.) and radius of convergence R>0. If (Formula presented.), we additionally suppose that (Formula presented.). We aim to characterize those A with (Formula presented.), which fulfil (Formula presented.). We first show how to reduce the discussion of f to Neumann series. For matrix norms induced by uniformly convex vector norms, like the (Formula presented.) -norms, (Formula presented.), it follows from known results of the Daugavet equation (Formula presented.) that (Formula presented.) holds true if, and only if, (Formula presented.) is an eigenvalue of A, provided that (Formula presented.) for some (Formula presented.). Under adapted assumptions on the (Formula presented.) we prove that this equivalence remains true for the (Formula presented.) - and the (Formula presented.) -norm, for unitarily invariant matrix norms and for the numerical radius. We conjecture this equivalence to be valid for all strongly stable norms if (Formula presented.) for all (Formula presented.).Publicationtype: Journal Article101 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Almost sharp bounds for the componentwise distance to the nearest singular matrixThe normwise distance of a regular matrix A ∈ M (ℝ) to the nearest singular matrix is well known to be ∥A ∥ . Such a normwise distance neglects small entries in the matrix, and it does not allow for weights in a perturbation. The reciprocal ∥ |A |·E∥ of the Bauer-Skeel condition number is known to be a lower bound for the componentwise distance of A to the nearest singular matrix weighted by the nonnegative matrix E. In this paper we derive an upper bound for this componentwise distance involving the Bauer-Skeel condition number. We show that this upper bound is sharp up to a constant factor less than 3 + 2√2, independent of A and E. For finite values of n, improved constants are given as well. © 1997 OPA (Overseas Publishers Association) Amsterdam B.V. Published in The Netherlands under license by Gordon and Breach Science Publishers. n -1 -1 -1 -1Publicationtype: Journal Article28