Geometric Algebra and Covariant Methods in Physics and Cosmology
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Improving the Accuracy of Computed Eigenvalues and Eigenvectors*
SIAM J. NUMER. ANAL. 1983 Society for Industrial and Applied Mathematics Vol. 20, No. 1, February 1983 0036-1429/83/2001-0002 $01.25/0 IMPROVING THE ACCURACY OF COMPUTED EIGENVALUES AND EIGENVECTORS* J. J. DONGARRA,t C. B. MOLER AND J. H. WILKINSON Abstract. This paper describes and analyzes several variants of a computational method for improving the numerical accuracy of, and for obtaining numerical bounds on, matrix eigenvalues and eigenvectors. The method, which is essentially a numerically stable implementation of Newton's method, may be used to "fine tune" the results obtained from standard subroutines such as those in EISPACK [Lecture Notes in Computer Science 6, 51, Springer-Verlag, Berlin, 1976, 1977]. Extended precision arithmetic is required in the computation of certain residuals. Introduction. The calculation of an eigenvalue , and the corresponding eigenvec- tor x (here after referred to as an eigenpair) of a matrix A involves the solution of the nonlinear system of equations (A AI)x O. Starting from an approximation h and , a sequence of iterates may be determined using Newton's method or variants of it. The conditions on and guaranteeing convergence have been treated extensively in the literature. For a particularly lucid account the reader is referred to the book by Rail [3]. In a recent paper Wilkinson [7] describes an algorithm for determining error bounds for a computed eigenpair based on these mathematical concepts. Considerations of numerical stability were an essential feature of that paper and indeed were its main raison d'etre. In general this algorithm provides an improved eigenpair and error bounds for it; unless the eigenpair is very ill conditioned the improved eigenpair is usually correct to the precision of the computation used in the main body of the algorithm. -
Abstract Tensor Systems and Diagrammatic Representations
Abstract tensor systems and diagrammatic representations J¯anisLazovskis September 28, 2012 Abstract The diagrammatic tensor calculus used by Roger Penrose (most notably in [7]) is introduced without a solid mathematical grounding. We will attempt to derive the tools of such a system, but in a broader setting. We show that Penrose's work comes from the diagrammisation of the symmetric algebra. Lie algebra representations and their extensions to knot theory are also discussed. Contents 1 Abstract tensors and derived structures 2 1.1 Abstract tensor notation . 2 1.2 Some basic operations . 3 1.3 Tensor diagrams . 3 2 A diagrammised abstract tensor system 6 2.1 Generation . 6 2.2 Tensor concepts . 9 3 Representations of algebras 11 3.1 The symmetric algebra . 12 3.2 Lie algebras . 13 3.3 The tensor algebra T(g)....................................... 16 3.4 The symmetric Lie algebra S(g)................................... 17 3.5 The universal enveloping algebra U(g) ............................... 18 3.6 The metrized Lie algebra . 20 3.6.1 Diagrammisation with a bilinear form . 20 3.6.2 Diagrammisation with a symmetric bilinear form . 24 3.6.3 Diagrammisation with a symmetric bilinear form and an orthonormal basis . 24 3.6.4 Diagrammisation under ad-invariance . 29 3.7 The universal enveloping algebra U(g) for a metrized Lie algebra g . 30 4 Ensuing connections 32 A Appendix 35 Note: This work relies heavily upon the text of Chapter 12 of a draft of \An Introduction to Quantum and Vassiliev Invariants of Knots," by David M.R. Jackson and Iain Moffatt, a yet-unpublished book at the time of writing. -
Numerical Analysis Notes for Math 575A
Numerical Analysis Notes for Math 575A William G. Faris Program in Applied Mathematics University of Arizona Fall 1992 Contents 1 Nonlinear equations 5 1.1 Introduction . 5 1.2 Bisection . 5 1.3 Iteration . 9 1.3.1 First order convergence . 9 1.3.2 Second order convergence . 11 1.4 Some C notations . 12 1.4.1 Introduction . 12 1.4.2 Types . 13 1.4.3 Declarations . 14 1.4.4 Expressions . 14 1.4.5 Statements . 16 1.4.6 Function definitions . 17 2 Linear Systems 19 2.1 Shears . 19 2.2 Reflections . 24 2.3 Vectors and matrices in C . 27 2.3.1 Pointers in C . 27 2.3.2 Pointer Expressions . 28 3 Eigenvalues 31 3.1 Introduction . 31 3.2 Similarity . 31 3.3 Orthogonal similarity . 34 3.3.1 Symmetric matrices . 34 3.3.2 Singular values . 34 3.3.3 The Schur decomposition . 35 3.4 Vector and matrix norms . 37 3.4.1 Vector norms . 37 1 2 CONTENTS 3.4.2 Associated matrix norms . 37 3.4.3 Singular value norms . 38 3.4.4 Eigenvalues and norms . 39 3.4.5 Condition number . 39 3.5 Stability . 40 3.5.1 Inverses . 40 3.5.2 Iteration . 40 3.5.3 Eigenvalue location . 41 3.6 Power method . 43 3.7 Inverse power method . 44 3.8 Power method for subspaces . 44 3.9 QR method . 46 3.10 Finding eigenvalues . 47 4 Nonlinear systems 49 4.1 Introduction . 49 4.2 Degree . 51 4.2.1 Brouwer fixed point theorem . -
Numerically Stable Generation of Correlation Matrices and Their Factors ∗
BIT 0006-3835/00/4004-0640 $15.00 2000, Vol. 40, No. 4, pp. 640–651 c Swets & Zeitlinger NUMERICALLY STABLE GENERATION OF CORRELATION MATRICES AND THEIR FACTORS ∗ ‡ PHILIP I. DAVIES† and NICHOLAS J. HIGHAM Department of Mathematics, University of Manchester, Manchester, M13 9PL, England email: [email protected], [email protected] Abstract. Correlation matrices—symmetric positive semidefinite matrices with unit diagonal— are important in statistics and in numerical linear algebra. For simulation and testing it is desirable to be able to generate random correlation matrices with specified eigen- values (which must be nonnegative and sum to the dimension of the matrix). A popular algorithm of Bendel and Mickey takes a matrix having the specified eigenvalues and uses a finite sequence of Givens rotations to introduce 1s on the diagonal. We give im- proved formulae for computing the rotations and prove that the resulting algorithm is numerically stable. We show by example that the formulae originally proposed, which are used in certain existing Fortran implementations, can lead to serious instability. We also show how to modify the algorithm to generate a rectangular matrix with columns of unit 2-norm. Such a matrix represents a correlation matrix in factored form, which can be preferable to representing the matrix itself, for example when the correlation matrix is nearly singular to working precision. Key words: Random correlation matrix, Bendel–Mickey algorithm, eigenvalues, sin- gular value decomposition, test matrices, forward error bounds, relative error bounds, IMSL, NAG Library, Jacobi method. AMS subject classification: 65F15. 1 Introduction. An important class of symmetric positive semidefinite matrices is those with unit diagonal. -
Model Problems in Numerical Stability Theory for Initial Value Problems*
SIAM REVIEW () 1994 Society for Industrial and Applied Mathematics Vol. 36, No. 2, pp. 226-257, June 1994 004 MODEL PROBLEMS IN NUMERICAL STABILITY THEORY FOR INITIAL VALUE PROBLEMS* A.M. STUART AND A.R. HUMPHRIES Abstract. In the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with simple dynamics; this has been motivated by the need to study error propagation mechanisms in stiff problems, a question modeled effectively by contractive linear or nonlinear problems. While this has resulted in a coherent and self-contained body of knowledge, it has never been entirely clear to what extent this theory is relevant for problems exhibiting more complicated dynamics. Recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics. This on-going work is unified and, in particular, striking similarities between this new developing stability theory and the classical linear and nonlinear stability theories are emphasized. The classical theories of A, B and algebraic stability for Runge-Kutta methods are briefly reviewed; the dynamics of solutions within the classes of equations to which these theories apply--linear decay and contractive problemsw are studied. Four other categories of equationsmgradient, dissipative, conservative and Hamiltonian systemsmare considered. Relationships and differences between the possible dynamics in each category, which range from multiple competing equilibria to chaotic solutions, are highlighted. Runge-Kutta schemes that preserve the dynamical structure of the underlying problem are sought, and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given. -
Optimising Cosmic Shear Surveys to Measure Modifications to Gravity on Cosmic Scales
Mon. Not. R. Astron. Soc. 000, 1–13 (2009) Printed 11 November 2019 (MN LATEX style file v2.2) Optimising cosmic shear surveys to measure modifications to gravity on cosmic scales Donnacha Kirk1,Istvan Laszlo2, Sarah Bridle1, Rachel Bean2 1Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK 2Department of Astronomy, Cornell University, Ithaca, NY 14853, USA 11 November 2019 ABSTRACT We consider how upcoming photometric large scale structure surveys can be opti- mized to measure the properties of dark energy and possible cosmic scale modifications to General Relativity in light of realistic astrophysical and instrumental systematic uncertainities. In particular we include flexible descriptions of intrinsic alignments, galaxy bias and photometric redshift uncertainties in a Fisher Matrix analysis of shear, position and position-shear correlations, including complementary cosmolog- ical constraints from the CMB. We study the impact of survey tradeoffs in depth versus breadth, and redshift quality. We parameterise the results in terms of the Dark Energy Task Force figure of merit, and deviations from General Relativity through an analagous Modified Gravity figure of merit. We find that intrinsic alignments weaken the dependence of figure of merit on area and that, for a fixed observing time, halving the area of a Stage IV reduces the figure of merit by 20% when IAs are not included and by only 10% when IAs are included. While reducing photometric redshift scatter improves constraining power, the dependence is shallow. The variation in constrain- ing power is stronger once IAs are included and is slightly more pronounced for MG constraints than for DE. -
Stability Analysis for Systems of Differential Equations
Stability Analysis for Systems of Differential Equations David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/ or send a letter to Creative Commons, PO Box 1866, Mountain View, CA 94042, USA. Created: February 8, 2003 Last Modified: March 2, 2008 Contents 1 Introduction 2 2 Physical Stability 2 3 Numerical Stability 4 3.1 Stability for Single-Step Methods..................................4 3.2 Stability for Multistep Methods...................................5 3.3 Choosing a Stable Step Size.....................................7 4 The Simple Pendulum 7 4.1 Numerical Solution of the ODE...................................8 4.2 Physical Stability for the Pendulum................................ 13 4.3 Numerical Stability of the ODE Solvers.............................. 13 1 1 Introduction In setting up a physical simulation involving objects, a primary step is to establish the equations of motion for the objects. These equations are formulated as a system of second-order ordinary differential equations that may be converted to a system of first-order equations whose dependent variables are the positions and velocities of the objects. Such a system is of the generic form x_ = f(t; x); t ≥ 0; x(0) = x0 (1) where x0 is a specified initial condition for the system. The components of x are the positions and velocities of the objects. The function f(t; x) includes the external forces and torques of the system. A computer implementation of the physical simulation amounts to selecting a numerical method to approximate the solution to the system of differential equations. -
Algorithmic Structure for Geometric Algebra Operators and Application to Quadric Surfaces Stephane Breuils
Algorithmic structure for geometric algebra operators and application to quadric surfaces Stephane Breuils To cite this version: Stephane Breuils. Algorithmic structure for geometric algebra operators and application to quadric surfaces. Operator Algebras [math.OA]. Université Paris-Est, 2018. English. NNT : 2018PESC1142. tel-02085820 HAL Id: tel-02085820 https://pastel.archives-ouvertes.fr/tel-02085820 Submitted on 31 Mar 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ÉCOLE DOCTORALE MSTIC THÈSE DE DOCTORAT EN INFORMATIQUE Structure algorithmique pour les opérateurs d’Algèbres Géométriques et application aux surfaces quadriques Auteur: Encadrants: Stéphane Breuils Dr. Vincent NOZICK Dr. Laurent FUCHS Rapporteurs: Prof. Dominique MICHELUCCI Prof. Pascal SCHRECK Examinateurs: Dr. Leo DORST Prof. Joan LASENBY Prof. Raphaëlle CHAINE Soutenue le 17 Décembre 2018 iii Thèse effectuée au Laboratoire d’Informatique Gaspard-Monge, équipe A3SI, dans les locaux d’ESIEE Paris LIGM, UMR 8049 École doctorale Paris-Est Cité Descartes, Cité Descartes, Bâtiment Copernic-5, bd Descartes 6-8 av. Blaise Pascal Champs-sur-Marne Champs-sur-Marne, 77 454 Marne-la-Vallée Cedex 2 77 455 Marne-la-Vallée Cedex 2 v Abstract Geometric Algebra is considered as a very intuitive tool to deal with geometric problems and it appears to be increasingly efficient and useful to deal with computer graphics problems. -
The Making of a Geometric Algebra Package in Matlab Computer Science Department University of Waterloo Research Report CS-99-27
The Making of a Geometric Algebra Package in Matlab Computer Science Department University of Waterloo Research Report CS-99-27 Stephen Mann∗, Leo Dorsty, and Tim Boumay [email protected], [email protected], [email protected] Abstract In this paper, we describe our development of GABLE, a Matlab implementation of the Geometric Algebra based on `p;q (where p + q = 3) and intended for tutorial purposes. Of particular note are the C matrix representation of geometric objects, effective algorithms for this geometry (inversion, meet and join), and issues in efficiency and numerics. 1 Introduction Geometric algebra extends Clifford algebra with geometrically meaningful operators, and its purpose is to facilitate geometrical computations. Present textbooks and implementation do not always convey this geometrical flavor or the computational and representational convenience of geometric algebra, so we felt a need for a computer tutorial in which representation, computation and visualization are combined to convey both the intuition and the techniques of geometric algebra. Current software packages are either Clifford algebra only (such as CLICAL [9] and CLIFFORD [1]) or do not include graphics [6], so we decide to build our own. The result is GABLE (Geometric Algebra Learning Environment) a hands-on tutorial on geometric algebra that should be accessible to the second year student in college [3]. The GABLE tutorial explains the basics of Geometric Algebra in an accessible manner. It starts with the outer product (as a constructor of subspaces), then treats the inner product (for perpendilarity), and moves via the geometric product (for invertibility) to the more geometrical operators such as projection, rotors, meet and join, and end with to the homogeneous model of Euclidean space. -
Two-Spinors, Oscillator Algebras, and Qubits: Aspects of Manifestly Covariant Approach to Relativistic Quantum Information
Quantum Information Processing manuscript No. (will be inserted by the editor) Two-spinors, oscillator algebras, and qubits: Aspects of manifestly covariant approach to relativistic quantum information Marek Czachor Received: date / Accepted: date Abstract The first part of the paper reviews applications of 2-spinor methods to rela- tivistic qubits (analogies between tetrads in Minkowski space and 2-qubit states, qubits defined by means of null directions and their role for elimination of the Peres-Scudo- Terno phenomenon, advantages and disadvantages of relativistic polarization operators defined by the Pauli-Lubanski vector, manifestly covariant approach to unitary rep- resentations of inhomogeneous SL(2,C)). The second part deals with electromagnetic fields quantized by means of harmonic oscillator Lie algebras (not necessarily taken in irreducible representations). As opposed to non-relativistic singlets one has to dis- tinguish between maximally symmetric and EPR states. The distinction is one of the sources of ‘strange’ relativistic properties of EPR correlations. As an example, EPR averages are explicitly computed for linear polarizations in states that are antisymmet- ric in both helicities and momenta. The result takes the familiar form p cos 2(α β) ± − independently of the choice of representation of harmonic oscillator algebra. Parameter p is determined by spectral properties of detectors and the choice of EPR state, but is unrelated to detector efficiencies. Brief analysis of entanglement with vacuum and vacuum violation of Bell’s inequality is given. The effects are related to inequivalent notions of vacuum states. Technical appendices discuss details of the representation I employ in field quantization. In particular, M-shaped delta-sequences are used to define Dirac deltas regular at zero. -
Keiichi Umetsu — CURRICULUM VITAE (Updated on July 8, 2021)
Keiichi Umetsu —CURRICULUM VITAE (Updated on September 8, 2021) Contact and Personal Information Work Address: Institute of Astronomy and Astrophysics, Academia Sinica (ASIAA), 11F of Astronomy-Mathematics Building, National Taiwan University (NTU), No. 1, Section 4, Roosevelt Road, Taipei 10617, Taiwan Email: [email protected] Web: http://idv.sinica.edu.tw/keiichi/index.php ORCID: 0000-0002-7196-4822 WOS ResearcherID: AAZ-7589-2020 Academic Appointments Full Research Fellow [rank equivalent to Full Professor], ASIAA (01/2014 – present) Kavli Visiting Scholar, Kavli Institute for Astronomy and Astrophysics, Peking University, China (2016) Associate Research Fellow [tenured], ASIAA (01/2010 – 12/2013) Adjunct Research Fellow, Leung Center for Cosmology and Particle Astrophysics, NTU (01/2008 – 12/2012) Assistant Research Fellow [tenure track], ASIAA (06/2006 – 12/2009) Science Lead for the Yuan Tseh Lee Array: AMiBA, Mauna Loa, Hawaii, USA (08/2005 – 07/2011) Faculty Staff Scientist, ASIAA (07/2005 – 05/2006) Postdoctoral Research Fellow, ASIAA (06/2001 – 06/2005) Education Ph.D. in Astronomy, Tohoku University, Japan (04/1998 – 03/2001) M.Sc. in Astronomy, Tohoku University, Japan (04/1996 – 03/1998) B.Sc. in Physics, Tohoku University, Japan (04/1992 – 03/1996) Publication Summary According to the NASA Astrophysics Data System, I have • 181 research papers published or in press in peer-reviewed journals, with a total h-index of 53 • 21 lead (first or corresponding*) author publications with a total of over 1400 citations • Lead author publications: 1 paper cited at least 200 times (Umetsu* et al. 2014), 6 cited at least 100 times and +1 cited 99 times (Umetsu* et al. -
Annual Review 2011/12
UCL DEPARTMENT OF PHYSICS AND ASTRONOMY PHYSICS AND ASTRONOMY 2011–12 ANNUAL REVIEW PHYSICS AND ASTRONOMY ANNUAL REVIEW 2011–12 Contents Welcome It is an honour to write this WELCOME 1 introduction standing, to misquote Newton, on the shoulders of giants. COMMUNITY FOCUS 3 Jonathan Tennyson finished his Teaching Lowdown 4 tenure as Head of Department in September 2011 and so the majority Student Accolades 4 of the material contained in this Outstanding PhD Theses Published 5 Review records achievements under his leadership. In addition, Tony Career Profiles 6 Harker is currently acting as Head of Science in Action 8 Department in many matters and will continue to do so until October 2012. Alumni Matters 9 This is due to my on-going commitments with the ATLAS experiment on the ACADEMIC SHOWCASE 11 Large Hadron Collider (LHC) at CERN. Staff Accolades 12 I currently spend a large amount of my time in Geneva and I am very grateful Academic Appointments 14 to both Tony and Jonathan, as well as to other members of staff for helping Doctor of Philosophy (PhD) 15 to make this transition a success. In Portrait of Dr Phil Jones 16 particular I would also like to thank Hilary Wigmore as Departmental Manager and Raman Prinja as the new Director of RESEARCH SPOTLIGHT 17 Teaching for their continued support. Atomic, Molecular, Optical and Position Physics (AMOPP) 19 High Energy Physics (HEP) 21 “Success in such long- Condensed Matter and term, high-impact projects Materials Physics (CMMP) 25 requires sustained vision Astrophysics (Astro) 29 and dedicated work by Biological Physics (BioP) 33 excellent scientists over Research Statistics 35 many years.” Staff Snapshot 38 Underpinning this success is the outstanding quality of scientific research and education within the Department.