1,720,957 research outputs found
Block rational Krylov methods for matrix equations and matrix functions
Full text linkIn this thesis, we describe block rational Krylov subspaces, highlighting their correspondence with rational matrices and generalizing important properties that apply to the non-block case. We develop procedures based on block rational Krylov subspaces for solving Sylvester and tensor Sylvester equations, computing functions of Hermitian hierarchically semiseparable (HSS) matrices, and applying functions of Hermitian matrices to block vectors.In the context of Sylvester equations with low-rank right-hand sides, we devise a new formulation of the residual obtained when projection methods based on block rational Krylov subspaces are utilized. This formulation allows the development of various adaptive pole selection strategies, offering a theoretical analysis of established heuristics as well as effective novel techniques.A natural extension is represented by tensor Sylvester equations, involving summands and -dimensional tensors for both the unknown and the right-hand side. Methods based on projection onto Krylov subspaces typically assume a right-hand side of rank one. In this thesis, we demonstrate how to apply block rational Krylov methods to handle right-hand sides with low multilinear or tensor train rank. By extending the results established for Sylvester equations, we present a new formulation of the residual and several adaptive pole selection strategies for the tensor case as well.In the context of matrix functions, we utilize block rational Krylov methods in an unconventional manner. Our aim is to leverage the rapid convergence of block rational Krylov subspaces while circumventing costly operations such as solving large linear systems.We present a fast algorithm designed to approximate , where represents a Hermitian HSS matrix. We use a telescopic decomposition of , inspired by the recent work of Levitt and Martinsson, allowing us to approximate by recursively performing low-rank updates with block rational Krylov subspaces while keeping the size of the matrices involved in the rational Krylov subspaces small. In particular, no large-scale linear system needs to be solved, yielding favorable complexity estimates and reduced execution times compared to existing methods.Finally, we develop a memory-efficient algorithm for computing , where is a Hermitian matrix (not necessarily HSS), and is a block vector. Our approach combines a block Lanczos algorithm with a basis compression technique based on block rational Krylov subspaces involving only small matrices, enabling us to avoid storing the entire Lanczos basis, resulting in significant reductions in memory usage. Theoretical results demonstrate that, for a wide variety of functions, the proposed algorithm differs from block Lanczos by an error term that is typically negligible
An Efficient Block Rational Krylov Solver for Sylvester Equations with Adaptive Pole Selection
Full text linkWe present an algorithm for the solution of Sylvester equations with right-hand side of low rank. The method is based on projection onto a block rational Krylov subspace, with two key contributions with respect to the state of the art. First, we show how to maintain the last pole equal to infinity throughout the iteration, by means of pole reordering, which allows for a cheap evaluation of the true residual at every step. Second, we extend the convergence analysis in [B. Beckermann, SIAM J. Numer. Anal., 49 (2011), pp. 2430--2450] to the block case. This extension allows us to link the convergence with the problem of minimizing the norm of a small rational matrix over the spectra or field -of -values of the involved matrices. This is in contrast with the nonblock case, where the minimum problem is scalar, instead of matrix valued. Replacing the norm of the objective function with a more easily evaluated function yields several adaptive pole selection strategies, providing a theoretical analysis for known heuristics, as well as effective novel techniques
Solving large-scale symmetric Lyapunov equations via a Lanczos method with compression
No full textThis work considers large-scale Lyapunov matrix equations of the form AX + XA = cc^T,
where A is a symmetric positive definite matrix and c is a vector. Motivated by the need to solve such equations in a wide range of applications, various numerical methods have been developed to compute low-rank approximations of the solution matrix X. In this work, we focus on the Lanczos method, which has the distinct advantage of requiring only matrix-vector products with A, making it broadly applicable. However, the Lanczos method may suffer from slow convergence when A is ill-conditioned, leading to excessive memory requirements for storing the Krylov subspace basis generated by the algorithm. To address this issue, we propose a novel compression strategy for the Krylov subspace basis that significantly reduces memory usage without hindering convergence. This is supported by both numerical experiments and a convergence analysis. Our analysis also accounts for the loss of orthogonality due to round-off errors in the Lanczos process
Tensorized block rational Krylov methods for tensor Sylvester equations
Full text linkWe introduce the definition of tensorized block rational Krylov subspaces and
its relation with multivariate rational functions, extending the formulation of
tensorized Krylov subspaces introduced in [Kressner D., Tobler C., Krylov
subspace methods for linear systems with tensor product structure, SIMAX,
2010]. Moreover, we develop methods for the solution of tensor Sylvester
equations with low multilinear or Tensor Train rank, based on projection onto a
tensor block rational Krylov subspace. We provide a convergence analysis, some
strategies for pole selection, and techniques to efficiently compute the
residual.Comment: 22 pages, 6 figures, 3 table
Fast computation of the roots of polynomials in the Chebyshev basis
No full textThe aim of the thesis is to study methods for computing roots of polynomials and matrix polynomials expressed in the Chebyshev basis, with lower computational cost than the available methods.
Chebyshev polynomials have a fundamental role in approximation theory, in numerical linear algebra and in related applications. A recent application, which is a representative example of their good numerical properties, is the Matlab package ”Chebfun”, which allows to numerically handle smooth functions. The approach used in Chebfun is to interpolate smooth functions at the Chebyshev points and to write the interpolant polynomial in the Chebyshev basis. This technique has good stability properties ad can be done in a fast way by using inverse cosine transforms.
In such a context, finding roots of a polynomial in the Chebyshev basis becomes fundamental.
The main idea used to find the roots of a polynomial is to build a linearization matrix, usually called companion matrix, and find the eigenvalues of that matrix. This is what the Chebfun command roots does for computing the roots of a polynomial expressed in Chebyshev basis. This method computes the roots of a polynomial using O(n^2) memory storage and O(n^3) operations, where n is the polynomial degree.
To reduce this cost, it can be exploited the structure of the companion matrix.
The approach we use in this thesis is to exploit the structure of the companion matrix, for polynomials expressed in the Chebyshev basis, viewed as a symmetric matrix plus a rank-one correction. This fact will allow to design a fast and backward stable rootfinding algorithm based on the QR iteration. In fact, the overall computation cost of this algorithm is O(n^2) arithmetic operations with O(n) memory storage, where n is the degree of the polynomial
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
Variations on the Author
Full text link“Variations on the Author” discusses two of Eduardo Coutinho’s recent films (Um Dia na Vida, from 2010, and Últimas Conversas, posthumously released in 2015) and their contribution to the general question of documentary authorship. The director’s filmography is characterized by a consistent yet self-effacing form of authorial self-inscription: Coutinho often features as an interviewer that rather than express opinions propels discourses; an interviewer that is good at listening. This mode of self-inscription characterizes him as an author who is not expressive but who is nonetheless markedly present on the screen. In Um Dia na Vida, however, Coutinho is completely absent form the image, while Últimas Conversas, on the contrary, includes a confessional prologue that moves the director from the margins to the center of his films. This article examines the ways in which these works stand out in the filmography of a director who offers new insights into the notion of cinematic authorship
Appropriate Similarity Measures for Author Cocitation Analysis
Full text linkWe provide a number of new insights into the methodological discussion about author cocitation analysis. We first argue that the use of the Pearson correlation for measuring the similarity between authors’ cocitation profiles is not very satisfactory. We then discuss what kind of similarity measures may be used as an alternative to the Pearson correlation. We consider three similarity measures in particular. One is the well-known cosine. The other two similarity measures have not been used before in the bibliometric literature. Finally, we show by means of an example that our findings have a high practical relevance.information science;Pearson correlation;cosine;similarity measure;author cocitation analysis
Dispelling the Myths Behind First-author Citation Counts
Full text linkWe conducted a full-scale evaluative citation analysis study of scholars in the XML research field to explore just how different from each other author rankings resulting from different citation counting methods actually are, and to demonstrate the capability of emerging data and tools on the Web in supporting more realistic citation counting methods. Our results contest some common arguments for the continued
use of first-author citation counts in the evaluation of scholars, such as high correlations between author rankings by first-author citation counts and other citation
counting methods, and high costs of using more realistic citation counting methods that are not well-supported by the ISI databases. It is argued that increasingly available digital full text research papers make it possible for citation analysis studies to go beyond what the ISI databases have directly supported and to employ more
sophisticated methods
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