Accurate matrix exponential computation to solve coupled differential models in engineering

NOTICE: this is the author’s version of a work that was accepted for publication in Mathematical and Computer Modelling. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Mathematical and Computer Modelling [Volume 54, Issues 7–8, October 2011] DOI: 10.1016/j.mcm.2010.12.049The matrix exponential plays a fundamental role in linear systems arising in engineering, mechanics and control theory. This work presents a new scaling-squaring algorithm for matrix exponential computation. It uses forward and backward error analysis with improved bounds for normal and nonnormal matrices. Applied to the Taylor method, it has presented a lower or similar cost compared to the state-of-the-art Padé algorithms with better accuracy results in the majority of test matrices, avoiding Padé's denominator condition problems. © 2011 Elsevier Ltd.This work has been supported by Universidad Politecnica de Valencia grants PAID-05-09-4338, PAID-06-08-3307 and Spanish Ministerio de Educacion grant MTM2009-08587.Sastre, J.; Ibáñez González, JJ.; Defez Candel, E.; Ruiz Martínez, PA. (2011). Accurate matrix exponential computation to solve coupled differential models in engineering. Mathematical and Computer Modelling. 54(7-8):1835-1840. https://doi.org/10.1016/j.mcm.2010.12.049S18351840547-

Dense packing crystal structures of physical tetrahedra

We present a method for discovering dense packings of general convex hard particles and apply it to study the dense packing behavior of a one-parameter family of particles with tetrahedral symmetry representing a deformation of the ideal mathematical tetrahedron into a less ideal, physical, tetrahedron and all the way to the sphere. Thus, we also connect the two well studied problems of sphere packing and tetrahedron packing on a single axis. Our numerical results uncover a rich optimal-packing behavior, compared to that of other continuous families of particles previously studied. We present four structures as candidates for the optimal packing at different values of the parameter, providing an atlas of crystal structures which might be observed in systems of nano-particles with tetrahedral symmetry

High performance computing of the matrix exponential

This work presents a new algorithm for matrix exponential computation that significantly simplifies a Taylor scaling and squaring algorithm presented previously by the authors, preserving accuracy. A Matlab version of the new simplified algorithm has been compared with the original algorithm, providing similar results in terms of accuracy, but reducing processing time. It has also been compared with two state-of-the-art implementations based on Fade approximations, one commercial and the other implemented in Matlab, getting better accuracy and processing time results in the majority of cases. (C) 2015 Elsevier B.V. All rights reserved.Ruíz Martínez, PA.; Sastre Martinez, J.; Ibáñez González, JJ.; Defez Candel, E. (2016). High performance computing of the matrix exponential. Journal of Computational and Applied Mathematics. 291:370-379. doi:10.1016/j.cam.2015.04.001S37037929

CONTEST : a Controllable Test Matrix Toolbox for MATLAB

Large, sparse networks that describe complex interactions are a common feature across a number of disciplines, giving rise to many challenging matrix computational tasks. Several random graph models have been proposed that capture key properties of real-life networks. These models provide realistic, parametrized matrices for testing linear system and eigenvalue solvers. CONTEST (CONtrollable TEST matrices) is a random network toolbox for MATLAB that implements nine models. The models produce unweighted directed or undirected graphs; that is, symmetric or unsymmetric matrices with elements equal to zero or one. They have one or more parameters that affect features such as sparsity and characteristic pathlength and all can be of arbitrary dimension. Utility functions are supplied for rewiring, adding extra shortcuts and subsampling in order to create further classes of networks. Other utilities convert the adjacency matrices into real-valued coefficient matrices for naturally arising computational tasks that reduce to sparse linear system and eigenvalue problems

Population Genetic Structure and Species Delimitation of a Widespread, Neotropical Dwarf Gecko

Amazonia harbors the greatest biological diversity on Earth. One trend that spans Amazonian taxa is that most taxonomic groups either exhibit broad geographic ranges or small restricted ranges. This is likely because many traits that determine a species range size, such as dispersal ability or body size, are autocorrelated. As such, it is rare to find groups that exhibit both large and small ranges. Once identified, however, these groups provide a powerful system for isolating specific traits that influence species distributions. One group of terrestrial vertebrates, gecko lizards, tends to exhibit small geographic ranges. Despite one exception, this applies to the Neotropical dwarf geckos of the genus Gonatodes. This exception, Gonatodes humeralis, has a geographic distribution almost 1,000,000 km2 larger than the combined ranges of its 30 congeners. As the smallest member of its genus and a gecko lizard more generally, G. humeralis is an unlikely candidate to be a wide-ranged Amazonian taxon. To test whether or not G. humeralis is one or more species, we generated molecular genetic data using restriction-site associated sequencing (RADseq) and traditional Sanger methods for samples from across its range and conducted a phylogeographic study. We conclude that G. humeralis is, in fact, a single species across its contiguous range in South America. Thus, Gonatodes is a unique clade among Neotropical taxa, containing both wide-ranged and range-restricted taxa, which provides empiricists with a powerful model system to correlate complex species traits and distributions. Additionally, we provide evidence to support species-level divergence of the allopatric population from Trinidad and we resurrect the name Gonatodes ferrugineus from synonymy for this population

SHCal04 Southern Hemisphere Calibration, 0–11.0 cal kyr BP

Recent measurements on dendrochronologically-dated wood from the Southern Hemisphere have shown that there are differences between the structural form of the radiocarbon calibration curves from each hemisphere. Thus, it is desirable, when possible, to use calibration data obtained from secure dendrochronologically-dated wood from the corresponding hemisphere. In this paper, we outline the recent work and point the reader to the internationally recommended data set that should be used for future calibration of Southern Hemisphere ¹⁴C dates

Deep learning: an introduction for applied mathematicians

Multilayered artificial neural networks are becoming a pervasive tool in a host of application fields. At the heart of this deep learning revolution are familiar concepts from applied and computational mathematics; notably, in calculus, approximation theory, optimization and linear algebra. This article provides a very brief introduction to the basic ideas that underlie deep learning from an applied mathematics perspective. Our target audience includes postgraduate and final year undergraduate students in mathematics who are keen to learn about the area. The article may also be useful for instructors in mathematics who wish to enliven their classes with references to the application of deep learning techniques. We focus on three fundamental questions: what is a deep neural network? how is a network trained? what is the stochastic gradient method? We illustrate the ideas with a short MATLAB code that sets up and trains a network. We also show the use of state-of-the art software on a large scale image classification problem. We finish with references to the current literature

Communicability across evolving networks

Many natural and technological applications generate time ordered sequences of networks, defined over a fixed set of nodes; for example time-stamped information about ‘who phoned who’ or ‘who came into contact with who’ arise naturally in studies of communication and the spread of disease. Concepts and algorithms for static networks do not immediately carry through to this dynamic setting. For example, suppose A and B interact in the morning, and then B and C interact in the afternoon. Information, or disease, may then pass from A to C, but not vice versa. This subtlety is lost if we simply summarize using the daily aggregate network given by the chain A-B-C. However, using a natural definition of a walk on an evolving network, we show that classic centrality measures from the static setting can be extended in a computationally convenient manner. In particular, communicability indices can be computed to summarize the ability of each node to broadcast and receive information. The computations involve basic operations in linear algebra, and the asymmetry caused by time’s arrow is captured naturally through the non-mutativity of matrix-matrix multiplication. Illustrative examples are given for both synthetic and real-world communication data sets. We also discuss the use of the new centrality measures for real-time monitoring and prediction

Efficient orthogonal matrix polynomial based method for computing matrix exponential

The matrix exponential plays a fundamental role in the solution of differential systems which appear in different science fields. This paper presents an efficient method for computing matrix exponentials based on Hermite matrix polynomial expansions. Hermite series truncation together with scaling and squaring and the application of floating point arithmetic bounds to the intermediate results provide excellent accuracy results compared with the best acknowledged computational methods. A backward-error analysis of the approximation in exact arithmetic is given. This analysis is used to provide a theoretical estimate for the optimal scaling of matrices. Two algorithms based on this method have been implemented as MATLAB functions. They have been compared with MATLAB functions funm and expm obtaining greater accuracy in the majority of tests. A careful cost comparison analysis with expm is provided showing that the proposed algorithms have lower maximum cost for some matrix norm intervals. Numerical tests show that the application of floating point arithmetic bounds to the intermediate results may reduce considerably computational costs, reaching in numerical tests relative higher average costs than expm of only 4.43% for the final Hermite selected order, and obtaining better accuracy results in the 77.36% of the test matrices. The MATLAB implementation of the best Hermite matrix polynomial based algorithm has been made available online. © 2011 Elsevier Inc. All rights reserved.This work has been supported by the Programa de Apoyo a la Investigacion y el Desarrollo of the Universidad Politecnica de Valencia PAID-05-09-4338, 2009.Sastre, J.; Ibáñez González, JJ.; Defez Candel, E.; Ruiz Martínez, PA. (2011). Efficient orthogonal matrix polynomial based method for computing matrix exponential. Applied Mathematics and Computation. 217(14):6451-6463. https://doi.org/10.1016/j.amc.2011.01.004S645164632171