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Time series of EIT chest images using singular value decomposition and Fourier transform
Get PDFThe aim of this study is to propose a useful method for exploring regional ventilation and perfusion in the chest. The paper describes two methods based on singular value decomposition (SVD) and Fourier transform (FT) respectively. This work shows that power spectral density (PSD) and phase images (derived from the Fourier transform) are easier to interpret and more useful tools for exploiting in vivo EIT data in healthy volunteers in order to explore the cardiovascular and respiratory system
Approximating the logarithm of a matrix to specified accuracy
Get PDFThe standard inverse scaling and squaring algorithm for computing the matrix logarithm begins by transforming the matrix to Schur triangular form in order to facilitate subsequent matrix square root and Padé approximation computations. A transformation-free form of this method that exploits incomplete Denman--Beavers square root iterations and aims for a specified accuracy (ignoring roundoff) is presented. The error introduced by using approximate square roots is accounted for by a novel splitting lemma for logarithms of matrix products. The number of square root stages and the degree of the final Padé approximation are chosen to minimize the computational work. This new method is attractive for high-performance computation since it uses only the basic building blocks of matrix multiplication, LU factorization and matrix inversion
Structured pseudospectra for polynomial eigenvalue problems, with applications
Get PDFPseudospectra associated with the standard and generalized eigenvalue problems have been widely investigated in recent years. We extend the usual definitions in two respects, by treating the polynomial eigenvalue problem and by allowing structured perturbations of a type arising in control theory. We explore connections between structured pseudospectra, structured backward errors, and structured stability radii. Two main approaches for computing pseudospectra are described. One is based on a transfer function and employs a generalized Schur decomposition of the companion form pencil. The other, specific to quadratic polynomials, finds a solvent of the associated quadratic matrix equation and thereby factorizes the quadratic -matrix. Possible approaches for large, sparse problems are also outlined. A collection of examples from vibrating systems, control theory, acoustics, and fluid mechanics is given to illustrate the techniques
Serial Rings
No full textThis book presents an exhaustive and up-to-date overview of the structure theory of serial rings, and the various methods of treating them. Results have been scattered throughout the literature, and the achievements of some schools, such as the Kiev school, seem little-known. This volume endeavours to unify the wide spectrum of tools used in this area and state the theory of serial rings based on two constructions: firstly, localisation with respect to a semi-prime Goldie ideal; and, secondly, a hidden 'blow-up' construction in a serial ring. Part of the work deals with the theory of modules over a serial ring, especially with finitely presented and pure injective modules. Other topics include noetherian serial rings and Artinian serial rings. Audience: This volume can be used as a textbook in ring theory and in the model theory of modules, and will also be of interest to postgraduates and researchers whose work involves rings and algebras
Characteristic relations for a model for the flow of granular materials
No full textA model for the flow of granular materials is considered which is based upon the physical and kinematic concepts of yield on a slip surface, a shearing motion across the slip surface, dilatation or contraction normal to the slip surface and rotation of the slip surface. The equations governing the model are presented in both tensorial and Cartesian equation form. For planar deformations a full analysis of the characteristic directions is carried out. The characteristic equation is a sextic polynomial with five distinct real roots defining a pair of velocity characteristic directions, a pair of stress characteristic directions and a direction associated with both the continuity equation and the slip rotation. Using the characteristic directions to define characteristic coordinates, the equations governing the model relative to these characteristic coordinates are presented. An application of the model to chute flow is considered
Picard and Chazy solutions to the Painlevé VI equation
No full textWe study the solutions of a particular family of Painlevé VI equations with parameters b = g = 0, d = \frac12 and 2a = (2m-1)2, for 2m Î \mathbb Z. We show that in the case of half-integer m, all solutions can be written in terms of known functions and they are of two types: a two-parameter family of solutions found by Picard and a new one-parameter family of classical solutions which we call Chazy solutions. We give explicit formulae for them and completely determine their asymptotic behaviour near the singular points 0,1,¥ and their nonlinear monodromy. We study the structure of analytic continuation of the solutions to the PVIm equation for any m such that 2m Î \mathbb Z. As an application, we classify all the algebraic solutions. For m half-integer, we show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For m integer, we show that all algebraic solutions belong to a one-parameter family of rational solutions
The reliability of local error estimators for convection diffusion equations
No full textWe assess the reliability of a simple a posteriori error estimator for steady-state convection–diffusion equations in cases where convection dominates. Our estimator is computed by solving a local Poisson problem with Neumann boundary conditions. It gives global upper and local lower bounds on the error measured in the H1 semi-norm. However, the error may be overestimated locally within boundary layers if these are not resolved by the mesh, that is, when the local mesh Péclet number is significantly greater than unity. We discuss the implications of this overestimation in a practical context where the estimator is used as a local error indicator within a self-adaptive mesh refinement process
Stability of Structured Hamiltonian Eigensolvers
Get PDFVarious applications give rise to eigenvalue problems for which the matrices are Hamiltonian or skew-Hamiltonian and also symmetric or skew-symmetric. We define structured backward errors that are useful for testing the stability of numerical methods for the solution of these four classes of structured eigenproblems. We introduce the symplectic quasi-QR factorization and show that for three of the classes it enables the structured backward error to be efficiently computed. We also give a detailed rounding error analysis of some recently developed Jacobi-like algorithms of Fassbender, Mackey, and Mackey [Linear Algebra Appl., to appear] for these eigenproblems. Based on the direct solution of 4 × 4, and in one case 8 × 8, structured subproblems these algorithms produce a complete basis of symplectic orthogonal eigenvectors for the two symmetric cases and a symplectic orthogonal basis for all the real invariant subspaces for the two skew-symmetric cases. We prove that, when the rotations are implemented using suitable formulae, the algorithms are strongly backward stable and we show that the QR algorithm does not have this desirable property
A block algorithm for matrix 1-norm estimation, with an application to 1-norm pseudospectra
Get PDFThe matrix 1-norm estimation algorithm used in LAPACK and various other software libraries and packages has proved to be a valuable tool. However, it has the limitations that it offers the user no control over the accuracy and reliability of the estimate and that it is based on level 2 BLAS operations. A block generalization of the 1-norm power method underlying the estimator is derived here and developed into a practical algorithm applicable to both real and complex matrices. The algorithm works with n × t matrices, where t is a parameter. For t=1 the original algorithm is recovered, but with two improvements (one for real matrices and one for complex matrices). The accuracy and reliability of the estimates generally increase with t and the computational kernels are level 3 BLAS operations for t > 1. The last t-1 columns of the starting matrix are randomly chosen, giving the algorithm a statistical flavor. As a by-product of our investigations we identify a matrix for which the 1-norm power method takes the maximum number of iterations. As an application of the new estimator we show how it can be used to efficiently approximate 1-norm pseudospectra
Surface tension-driven convection patterns in two liquid layers
Get PDFTwo superposed liquid layers display a variety of convective phenomena that are inaccessible in the traditional system where the upper layer is a gas. We consider several pairs of immiscible liquids. Once the liquids have been selected, the applied temperature difference and the depths of the layers are the only independent control parameters. Using a perfluorinated hydrocarbon and silicone oil system, we have made the first experimental observation of convection with the top plate hotter than the bottom plate. Since the system is stably stratified, this convective flow is solely due to thermocapillary forces. We also have found oscillatory convection at onset in an acetonitrile and n-hexane system heated from below. The experimental observations are in reasonable agreement with linear stability analyses