1,720,975 research outputs found
Iterative and doubling algorithms for Riccati‐type matrix equations: A comparative introduction
No full textWe review a family of algorithms for Lyapunov- and Riccati-type equations which are all related to each other by the idea of doubling: they construct the iterate (Formula presented.) of another naturally-arising fixed-point iteration (Xh) via a sort of repeated squaring. The equations we consider are Stein equations X − A∗ X A = Q, Lyapunov equations A∗ X + X A + Q = 0, discrete-time algebraic Riccati equations X = Q + A∗ X(I + G X)−1A, continuous-time algebraic Riccati equations Q + A∗ X + X A − X G X = 0, palindromic quadratic matrix equations A + Q Y + A∗Y2 = 0, and nonlinear matrix equations X + A∗ X−1A = Q. We draw comparisons among these algorithms, highlight the connections between them and to other algorithms such as subspace iteration, and discuss open issues in their theory
Comparison theorems for splittings of M-matrices in (block) Hessenberg form
No full textSome variants of the (block) Gauss–Seidel iteration for the solution of linear systems with M-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix ρ(PGS) ≥ ρ(PS) ≥ ρ(PAGS) , where PGS, PS, PAGS are the iteration matrices of the Gauss–Seidel, staircase, and anti-Gauss–Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation
Nearest Ω -stable matrix via Riemannian optimization
Full text linkWe study the problem of finding the nearest Ω-stable matrix to a certain matrix A, i.e., the nearest matrix with all its eigenvalues in a prescribed closed set Ω. Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity
Deflating subspaces of T-palindromic pencils and algebraic T-Riccati equations
No full textBy exploiting the connection between solving algebraic (Formula presented.) -Riccati equations and computing certain deflating subspaces of (Formula presented.) -palindromic matrix pencils, we obtain theoretical and computational results on both problems. Theoretically, we introduce conditions to avoid the presence of modulus-one eigenvalues in a (Formula presented.) -palindromic matrix pencil and conditions for the existence of solutions of a (Formula presented.) -Riccati equation. Computationally, we improve the palindromic QZ algorithm with a new ordering procedure and introduce new algorithms for computing deflating subspaces of the (Formula presented.) -palindromic pencil, based on quadraticizations of the pencil or on an integral representation of the projector on the sought deflating subspace
Markov-Chain based centralities and Space Syntax’ Angular Analysis: an initial overview and application
No full textCentrality measures of Integration and Choice have performed a crucial role for Space Syntax in depicting complex relations among form, function, and movement within cities. However, while still relevant, those measures are unable to address certain innate network properties regarding the relative importance of certain road elements, essential for urban analyses focused on road-network resilience. The overreliance on Integration and Choice metrics to explain urban phenomena left several configurational patterns derived from connectivity rather unaddressed by Space Syntax and currently constitutes the methodology’s main limitation. With those points in consideration, this paper proposes an initial overview regarding the adaptation of Markov-based centrality measures to the Space Syntax framework and graph representation. These measures, often computable only in primal graphs, are adapted to the Road-Centre Line representation, in a first approach that aims to further integrate them into the Angular Analysis’ framework. We use the measures of Normalized PageRank Centrality and Normalized Kemeny-based centrality that estimate the connective relative importance of individual road-elements within the system. These measures are based on the notions of strong-ties and weak-ties, both well-known concepts in social networks; Weak-ties are important to establish bridges among interconnected communities of strong-tied individuals. In the urban configuration, weak-ties give information about crucial bridges among spaces characterized by strong-ties, areas that possess a high number of interconnected road elements – common pattern found in urban settlements. Results indicate that adapting Markov-based centralities to Space Syntax is feasible and maintains a configurational and spatial sense, hence it introduces new dimensions to be evaluated in urban-regional analysis
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
A general framework for the rigorous computation of invariant densities and the coarse-fine strategy
Full text linkIn this paper we present a general, axiomatical framework for the rigorous approximation of invariant densities and other important statistical features of dynamics. We approximate the system through a finite element reduction, by composing the associated transfer operator with a suitable finite dimensional projection (a discretization scheme) as in the well-known Ulam method.
We introduce a general framework based on a list of properties (of the system and of the projection) that need to be verified so that we can take advantage of a so-called “coarse-fine” strategy. This strategy is a novel method in which we exploit information coming from a coarser approximation of the system to get useful information on a finer approximation, speeding up the computation. This coarse-fine strategy allows a precise estimation of invariant densities and also allows to estimate rigorously the speed of mixing of the system by the speed of mixing of a coarse approximation of it, which can easily be estimated by the computer.
The estimates obtained her e are rigorous, i.e., they come with exact error bounds that are guaranteed to hold and take into account both the discretization and the approximations induced by finite-precision arithmetic.
We apply this framework to several discretization schemes and examples of invariant density computation from previous works, obtaining a remarkable reduction in computation time.
We have implemented the numerical methods described here in the Julia programming language, and released our implementation publicly as a Julia package
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