DOCSLIB.ORG

Algorithms and Perturbation Theory for Matrix Eigenvalue Problems and the Singular Value Decomposition by YUJI NAKATSUKASA B.S

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Algorithms and Perturbation Theory for Matrix Eigenvalue Problems and the Singular Value Decomposition by YUJI NAKATSUKASA B.S Algorithms and Perturbation Theory for Matrix Eigenvalue Problems and the Singular Value Decomposition By YUJI NAKATSUKASA B.S. (University of Tokyo) 2005 M.S. (University of Tokyo) 2007 DISSERTATION Submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in APPLIED MATHEMATICS in the OFFICE OF GRADUATE STUDIES of the UNIVERSITY OF CALIFORNIA DAVIS Approved: Roland Freund (Chair) Fran¸cois Gygi Naoki Saito Committee in Charge 2011 -i- Contents Abstract v Acknowledgments vi Chapter 1. Introduction 1 Chapter2. Overviewandsummaryofcontributions 4 2.1. Notations 4 2.2. The symmetric eigendecomposition and the singular valuedecomposition 5 2.3. The polar decomposition 6 2.4. Backward stability of an algorithm 7 2.5. The dqds algorithm 8 2.6. Aggressive early deflation 10 2.7. Hermitian eigenproblems 13 2.8. Generalized eigenvalue problems 14 2.9. Eigenvalue first-order expansion 16 2.10. Perturbation of eigenvectors 17 2.11. Gerschgorin’s theorem 21 Part 1. Algorithms for matrix decompositions 22 Chapter 3. Communication minimizing algorithm for the polardecomposition 23 3.1. Newton-type methods 25 3.2. Halley’smethodanddynamicallyweightedHalley 27 3.3. QR-based implementations 33 3.4. Backward stability proof 36 3.5. Numerical examples 48 3.6. Solving the max-min problem 50 3.7. Applicationinmoleculardynamicssimulations 56 Chapter 4. Efficient, communication minimizing algorithm for the symmetric eigenvalue problem 59 4.1. Algorithm QDWH-eig 60 4.2. Practical considerations 66 4.3. Numerical experiments 69 Chapter 5. Efficient, communication minimizing algorithm for the SVD 73 5.1. Algorithm QDWH-SVD 73 5.2. Practical considerations 75 -ii- 5.3. Numerical experiments 76 Chapter 6. dqds with aggressive early deflation for computing singular values of bidiagonal matrices 80 6.1. Aggressive early deflation for dqds - version 1: Aggdef(1) 81 6.2. Aggressive early deflation for dqds - version 2: Aggdef(2) 83 6.3. Convergence analysis 98 6.4. Numerical experiments 102 Part 2. Eigenvalue perturbation theory 108 Chapter 7. Eigenvalue perturbation bounds for Hermitian block tridiagonal matrices 109 7.1. Basic approach 110 7.2. 2-by-2 block case 111 7.3. Block tridiagonal case 114 7.4. Two case studies 118 7.5. Effectofthepresenceofmultipleeigenvalues 123 Chapter8. Perturbationofgeneralizedeigenvalues 126 8.1. Absolute Weyl theorem 127 8.2. Relative Weyl theorem 130 8.3. Quadratic perturbation bounds for Hermitian definite pairs 132 8.4. An extension to non-Hermitian pairs 146 Chapter 9. Perturbation and condition numbers of a multiple generalized eigenvalue 148 9.1. Perturbation bounds for multiple generalized eigenvalues 150 9.2. Another explanation 154 9.3. Implication in the Rayleigh-Ritz process 157 9.4. Condition numbers of a multiple generalized eigenvalue 161 9.5. Hermitian definite pairs 163 9.6. Non-Hermitian pairs 165 9.7. Multiple singular value 170 Chapter 10. Perturbation of eigenvectors 172 10.1. The tan θ theoremunderrelaxedconditions 173 10.2. The generalized tan θ theoremwithrelaxedconditions 177 10.3. RefinedRayleigh-Ritzapproximationbound 178 10.4. New bounds for the angles between Ritz vectors and exact eigenvectors 179 10.5. Singular vectors 183 10.6. The cos θ theorem 187 Chapter 11. Gerschgorin theory 192 11.1. AnewGerschgorin-typeeigenvalueinclusionset 194 11.2. Examples and applications 199 11.3. Gerschgorin theorems for generalized eigenproblems 202 11.4. Examples 213 -iii- 11.5. Applications 214 Chapter 12. Summary and future work 219 Bibliography 220 -iv- Yuji Nakatsukasa September 2011 Applied Mathematics Algorithms and Perturbation Theory for Matrix Eigenvalue Problems and the Singular Value Decomposition Abstract This dissertation is about algorithmic and theoretical developments for eigenvalue prob- lems in numerical linear algebra. The first part of this dissertation proposes algorithms for two important matrix decom- positions, the symmetric eigenvalue decomposition and the singular value decomposition. Recent advances and changes in computational architectures have made it necessary for basic linear algebra algorithms to be well-adapted for parallel computing. A few decades ago, an algorithm was considered faster if it required fewer arithmetic operations. This is not the case anymore, and now it is vital to minimize both arithmetic and communication when designing algorithms that are well-suited for high performance scientific computing. Unfortunately, for the above two matrix decompositions, no known algorithm minimizes communication without needing significantly more arithmetic. The development of such al- gorithms is the main theme of the first half of the dissertation. Our algorithms have great potential as the future approach to computing these matrix decompositions. The second part of this dissertation explores eigenvalue perturbation theory. Besides being of theoretical interest, perturbation theory is a useful tool that plays important roles in many applications. For example, it is frequently employed in the stability analysis of a numerical algorithm, for examining whether a given problem is well-conditioned under perturbation, or for bounding errors of a computed solution. However, there are a number of phenomena that still cannot be explained by existing theory. We make contributions by deriving refined eigenvalue perturbation bounds for Hermitian block tridiagonal matrices and generalized Hermitian eigenproblems, giving explanations for the perturbation behavior of a multiple generalized eigenvalue, presenting refined eigenvector perturbation bounds, and developing new Gerschgorin-type eigenvalue inclusion sets. -v- Acknowledgments Roland Freund advised me for this dissertation work and I am grateful to his support, encouragement and sharing mathematical insights. His view towards numerical linear algebra greatly influenced my perspective for research directions. A number of individuals have given me valuable suggestions and advices during my graduate studies. With Kensuke Aishima I had exciting communications on the dqds algorithm. I thank Zhaojun Bai for many discussions and introducing me to the polar decomposition. Fran¸cois Gygi introduced me to the computational aspects of molecular dynamics. Nick Higham always inspired me with his insights and knowledge, and gave me countless suggestions to a number of my manuscripts. Ren-Cang Li provided his expertise on eigenvalue perturbation theory and discussions with him has always been exciting and illuminating. Beresford Parlett shared his wisdom on tridiagonal and bidiagonal matrices. Efrem Rensi made our office an excellent place to work in and I enjoyed many discussions with him inside and outside the office. Naoki Saito introduced me to Laplacian eigenvalues, and provided help beyond mathematics, always being attentive despite his busy schedule. Fran¸coise Tisseur generously spared her time to go over my manuscripts and presentations and gave me invaluable suggestions. Ichitaro Yamazaki offered great help with coding, especially for the work on dqds. Shao-Liang Zhang introduced me to the field of numerical linear algebra, and has been understanding and welcoming whenever I visited Japan. Beyond my studies, I would like to extend my special acknowledgment to Celia Davis, who provided me with inestimable support and advices during difficult times. And last but not least, I would like to thank my family for always being incredibly patient, understanding and supportive. Without their support this work would not have been completed. This work was partially supported by NSF grant OCI-0749217 and DOE grant DE-FC02- 06ER25794. -vi- 1 CHAPTER 1 Introduction This dissertation is about design and perturbation analysis of numerical algorithms for matrix eigenvalue problems and the singular value decomposition. The most computation- ally challenging part of many problems in scientific computing is often to solve large-scale matrix equations, the two most frequently arising of which are linear systems of equations and eigenvalue problems. The rapid development of computer technology makes possible simulations and computations of ever larger scale. It is vital that the algorithms are well- suited for the computing architecture to make full use of the computational power, and that we understand the underlying theory to ensure the computed quantities are meaningful and reliable. These two issues are the fundamental motivation for this dissertation work. For a numerical algorithm to maximize the computational resources, not only does it need to do as few arithmetic operations as possible, it must be suitable for parallel or pipelined processing. In particular, on the emerging multicore and heterogeneous computing systems, communication costs have exceeded arithmetic costs by orders of magnitude, and the gap is growing exponentially over time. Hence it is vital to minimize both arithmetic and communication when designing algorithms that are well-suited for high performance scientific computing. The asymptotic communication lower bound is analyzed in [9], and algorithms that attain these asymptotic lower bounds are said to minimize communication. There has been much progress in the development of such algorithms in numerical linear algebra, and communication-minimizing implementations are now known for many basic matrix oper- ations such as matrix multiplications, Cholesky factorization and QR decomposition [9]. However, the reduction in communication cost sometimes comes at the expense of signifi- cantly more arithmetic. This includes the existing
Load more

Recommended publications
  • Notes on Statistical Field Theory
    Lecture Notes on Statistical Field Theory Kevin Zhou [email protected] These notes cover statistical field theory and the renormalization group. The primary sources were: • Kardar, Statistical Physics of Fields. A concise and logically tight presentation of the subject, with good problems. Possibly a bit too terse unless paired with the 8.334 video lectures. • David Tong's Statistical Field Theory lecture notes. A readable, easygoing introduction covering the core material of Kardar's book, written to seamlessly pair with a standard course in quantum field theory. • Goldenfeld, Lectures on Phase Transitions and the Renormalization Group. Covers similar material to Kardar's book with a conversational tone, focusing on the conceptual basis for phase transitions and motivation for the renormalization group. The notes are structured around the MIT course based on Kardar's textbook, and were revised to include material from Part III Statistical Field Theory as lectured in 2017. Sections containing this additional material are marked with stars. The most recent version is here; please report any errors found to [email protected]. 2 Contents Contents 1 Introduction 3 1.1 Phonons...........................................3 1.2 Phase Transitions......................................6 1.3 Critical Behavior......................................8 2 Landau Theory 12 2.1 Landau{Ginzburg Hamiltonian.............................. 12 2.2 Mean Field Theory..................................... 13 2.3 Symmetry Breaking.................................... 16 3 Fluctuations 19 3.1 Scattering and Fluctuations................................ 19 3.2 Position Space Fluctuations................................ 20 3.3 Saddle Point Fluctuations................................. 23 3.4 ∗ Path Integral Methods.................................. 24 4 The Scaling Hypothesis 29 4.1 The Homogeneity Assumption............................... 29 4.2 Correlation Lengths.................................... 30 4.3 Renormalization Group (Conceptual)..........................
    [Show full text]
  • Effective Field Theories, Reductionism and Scientific Explanation Stephan
    To appear in: Studies in History and Philosophy of Modern Physics Effective Field Theories, Reductionism and Scientific Explanation Stephan Hartmann∗ Abstract Effective field theories have been a very popular tool in quantum physics for almost two decades. And there are good reasons for this. I will argue that effec- tive field theories share many of the advantages of both fundamental theories and phenomenological models, while avoiding their respective shortcomings. They are, for example, flexible enough to cover a wide range of phenomena, and concrete enough to provide a detailed story of the specific mechanisms at work at a given energy scale. So will all of physics eventually converge on effective field theories? This paper argues that good scientific research can be characterised by a fruitful interaction between fundamental theories, phenomenological models and effective field theories. All of them have their appropriate functions in the research process, and all of them are indispens- able. They complement each other and hang together in a coherent way which I shall characterise in some detail. To illustrate all this I will present a case study from nuclear and particle physics. The resulting view about scientific theorising is inherently pluralistic, and has implications for the debates about reductionism and scientific explanation. Keywords: Effective Field Theory; Quantum Field Theory; Renormalisation; Reductionism; Explanation; Pluralism. ∗Center for Philosophy of Science, University of Pittsburgh, 817 Cathedral of Learning, Pitts- burgh, PA 15260, USA (e-mail: [email protected]) (correspondence address); and Sektion Physik, Universit¨at M¨unchen, Theresienstr. 37, 80333 M¨unchen, Germany. 1 1 Introduction There is little doubt that effective field theories are nowadays a very popular tool in quantum physics.
    [Show full text]
  • Hamiltonian Perturbation Theory on a Lie Algebra. Application to a Non-Autonomous Symmetric Top
    Hamiltonian Perturbation Theory on a Lie Algebra. Application to a non-autonomous Symmetric Top. Lorenzo Valvo∗ Michel Vittot Dipartimeno di Matematica Centre de Physique Théorique Università degli Studi di Roma “Tor Vergata” Aix-Marseille Université & CNRS Via della Ricerca Scientifica 1 163 Avenue de Luminy 00133 Roma, Italy 13288 Marseille Cedex 9, France January 6, 2021 Abstract We propose a perturbation algorithm for Hamiltonian systems on a Lie algebra V, so that it can be applied to non-canonical Hamiltonian systems. Given a Hamil- tonian system that preserves a subalgebra B of V, when we add a perturbation the subalgebra B will no longer be preserved. We show how to transform the perturbed dynamical system to preserve B up to terms quadratic in the perturbation. We apply this method to study the dynamics of a non-autonomous symmetric Rigid Body. In this example our algebraic transform plays the role of Iterative Lemma in the proof of a KAM-like statement. A dynamical system on some set V is a flow: a one-parameter group of mappings associating to a given element F 2 V (the initial condition) another element F (t) 2 V, for any value of the parameter t. A flow on V is determined by a linear mapping H from V to itself. However, the flow can be rarely computed explicitly. In perturbation theory we aim at computing the flow of H + V, where the flow of H is known, and V is another linear mapping from V to itself. In physics, the set V is often a Lie algebra: for instance in classical mechanics [4], fluid dynamics and plasma physics [20], quantum mechanics [24], kinetic theory [18], special and general relativity [17].
    [Show full text]
  • A Fast Method for Computing the Inverse of Symmetric Block Arrowhead Matrices
    Appl. Math. Inf. Sci. 9, No. 2L, 319-324 (2015) 319 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/092L06 A Fast Method for Computing the Inverse of Symmetric Block Arrowhead Matrices Waldemar Hołubowski1, Dariusz Kurzyk1,∗ and Tomasz Trawi´nski2 1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, Gliwice 44–100, Poland 2 Mechatronics Division, Silesian University of Technology, Akademicka 10a, Gliwice 44–100, Poland Received: 6 Jul. 2014, Revised: 7 Oct. 2014, Accepted: 8 Oct. 2014 Published online: 1 Apr. 2015 Abstract: We propose an effective method to find the inverse of symmetric block arrowhead matrices which often appear in areas of applied science and engineering such as head-positioning systems of hard disk drives or kinematic chains of industrial robots. Block arrowhead matrices can be considered as generalisation of arrowhead matrices occurring in physical problems and engineering. The proposed method is based on LDLT decomposition and we show that the inversion of the large block arrowhead matrices can be more effective when one uses our method. Numerical results are presented in the examples. Keywords: matrix inversion, block arrowhead matrices, LDLT decomposition, mechatronic systems 1 Introduction thermal and many others. Considered subsystems are highly differentiated, hence formulation of uniform and simple mathematical model describing their static and A square matrix which has entries equal zero except for dynamic states becomes problematic. The process of its main diagonal, a one row and a column, is called the preparing a proper mathematical model is often based on arrowhead matrix. Wide area of applications causes that the formulation of the equations associated with this type of matrices is popular subject of research related Lagrangian formalism [9], which is a convenient way to with mathematics, physics or engineering, such as describe the equations of mechanical, electromechanical computing spectral decomposition [1], solving inverse and other components.
    [Show full text]
  • 1 Introduction 2 Classical Perturbation Theory
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server Classical and Quantum Perturbation Theory for two Non–Resonant Oscillators with Quartic Interaction Luca Salasnich Dipartimento di Matematica Pura ed Applicata, Universit`a di Padova, Via Belzoni 7, I–35131 Padova, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Padova, Via Marzolo 8, I–35131 Padova, Italy Istituto Nazionale per la Fisica della Materia, Unit`a di Milano, Via Celoria 16, I–20133 Milano, Italy Abstract. We study the classical and quantum perturbation theory for two non–resonant oscillators coupled by a nonlinear quartic interaction. In particular we analyze the question of quantum corrections to the torus quantization of the classical perturbation theory (semiclassical mechanics). We obtain up to the second order of perturbation theory an explicit analytical formula for the quantum energy levels, which is the semiclassical one plus quantum corrections. We compare the ”exact” quantum levels obtained numerically to the semiclassical levels studying also the effects of quantum corrections. Key words: Classical Perturbation Theory; Quantum Mechanics; General Mechanics 1 Introduction Nowadays there is considerable renewed interest in the transition from classical mechanics to quantum mechanics, a powerful motivation behind that being the problem of the so–called quantum chaos [1–3]. An important aspect is represented by the semiclassical quantization formula of the (regular) energy levels for quasi–integrable systems [4–6], the so–called torus quantization, initiated by Einstein [7] and completed by Maslov [8]. It has been recently shown [9,10] that, for perturbed non–resonant harmonic oscillators, the algorithm of classical perturbation theory can be used to formulate the quantum mechanical perturbation theory as the semiclassically quantized classical perturbation theory equipped with the quantum corrections in powers of ¯h ”correcting” the classical Hamiltonian that appears in the classical algorithm.
    [Show full text]
  • Perturbation Theory and Exact Solutions
    PERTURBATION THEORY AND EXACT SOLUTIONS by J J. LODDER R|nhtdnn Report 76~96 DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS J J. LODOER ASSOCIATIE EURATOM-FOM Jun»»76 FOM-INST1TUUT VOOR PLASMAFYSICA RUNHUIZEN - JUTPHAAS - NEDERLAND DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS by JJ LODDER R^nhuizen Report 76-95 Thisworkwat performed at part of th«r«Mvchprogmmncof thcHMCiattofiafrccmentof EnratoniOTd th« Stichting voor FundtmenteelOiutereoek der Matctk" (FOM) wtihnnmcWMppoft from the Nederhmdie Organiutic voor Zuiver Wetemchap- pcigk Onderzoek (ZWO) and Evntom It it abo pabHtfMd w a the* of Ac Univenrty of Utrecht CONTENTS page SUMMARY iii I. INTRODUCTION 1 II. GENERALIZED FUNCTIONS DEFINED ON DISCONTINUOUS TEST FUNC­ TIONS AND THEIR FOURIER, LAPLACE, AND HILBERT TRANSFORMS 1. Introduction 4 2. Discontinuous test functions 5 3. Differentiation 7 4. Powers of x. The partie finie 10 5. Fourier transforms 16 6. Laplace transforms 20 7. Hubert transforms 20 8. Dispersion relations 21 III. PERTURBATION THEORY 1. Introduction 24 2. Arbitrary potential, momentum coupling 24 3. Dissipative equation of motion 31 4. Expectation values 32 5. Matrix elements, transition probabilities 33 6. Harmonic oscillator 36 7. Classical mechanics and quantum corrections 36 8. Discussion of the Pu strength function 38 IV. EXACTLY SOLVABLE MODELS FOR DISSIPATIVE MOTION 1. Introduction 40 2. General quadratic Kami1tonians 41 3. Differential equations 46 4. Classical mechanics and quantum corrections 49 5. Equation of motion for observables 51 V. SPECIAL QUADRATIC HAMILTONIANS 1. Introduction 53 2. Hamiltcnians with coordinate coupling 53 3. Double coordinate coupled Hamiltonians 62 4. Symmetric Hamiltonians 63 i page VI. DISCUSSION 1. Introduction 66 ?.
    [Show full text]
  • Conformal Invariance and Quantum Integrability of Sigma Models on Symmetric Superspaces
    Physics Letters B 648 (2007) 254–261 www.elsevier.com/locate/physletb Conformal invariance and quantum integrability of sigma models on symmetric superspaces A. Babichenko a,b a Racah Institute of Physics, the Hebrew University, Jerusalem 91904, Israel b Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel Received 7 December 2006; accepted 2 March 2007 Available online 7 March 2007 Editor: L. Alvarez-Gaumé Abstract We consider two dimensional nonlinear sigma models on few symmetric superspaces, which are supergroup manifolds of coset type. For those spaces where one loop beta function vanishes, two loop beta function is calculated and is shown to be zero. Vanishing of beta function in all orders of perturbation theory is shown for the principal chiral models on group supermanifolds with zero Killing form. Sigma models on symmetric (super) spaces on supergroup manifold G/H are known to be classically integrable. We investigate a possibility to extend an argument of absence of quantum anomalies in nonlocal current conservation from nonsuper case to the case of supergroup manifolds which are asymptotically free in one loop. © 2007 Elsevier B.V. All rights reserved. 1. Introduction super coset PSU(2, 2|4)/SO(1, 4) × SO(5). Hyperactivity in at- tempts to exploit integrability and methods of Bethe ansatz as a Two dimensional (2d) nonlinear sigma models (NLSM) on calculational tool in checks of ADS/CFT correspondence (see, supermanifolds, with and without WZ term, seemed to be ex- e.g., reviews [12] and references therein), also supports this in- otic objects, when they appeared in condensed matter physics terest, since both spin chains (on the gauge theory side) and 2d twenty years ago as an elegant calculational tool in problems NLSM (on the ADS side) appearing there, usually have a su- of self avoiding walks [1] and disordered metals [2].
    [Show full text]
  • Perturbation Theory in Celestial Mechanics
    Perturbation Theory in Celestial Mechanics Alessandra Celletti Dipartimento di Matematica Universit`adi Roma Tor Vergata Via della Ricerca Scientifica 1, I-00133 Roma (Italy) ([email protected]) December 8, 2007 Contents 1 Glossary 2 2 Definition 2 3 Introduction 2 4 Classical perturbation theory 4 4.1 The classical theory . 4 4.2 The precession of the perihelion of Mercury . 6 4.2.1 Delaunay action–angle variables . 6 4.2.2 The restricted, planar, circular, three–body problem . 7 4.2.3 Expansion of the perturbing function . 7 4.2.4 Computation of the precession of the perihelion . 8 5 Resonant perturbation theory 9 5.1 The resonant theory . 9 5.2 Three–body resonance . 10 5.3 Degenerate perturbation theory . 11 5.4 The precession of the equinoxes . 12 6 Invariant tori 14 6.1 Invariant KAM surfaces . 14 6.2 Rotational tori for the spin–orbit problem . 15 6.3 Librational tori for the spin–orbit problem . 16 6.4 Rotational tori for the restricted three–body problem . 17 6.5 Planetary problem . 18 7 Periodic orbits 18 7.1 Construction of periodic orbits . 18 7.2 The libration in longitude of the Moon . 20 1 8 Future directions 20 9 Bibliography 21 9.1 Books and Reviews . 21 9.2 Primary Literature . 22 1 Glossary KAM theory: it provides the persistence of quasi–periodic motions under a small perturbation of an integrable system. KAM theory can be applied under quite general assumptions, i.e. a non– degeneracy of the integrable system and a diophantine condition of the frequency of motion.
    [Show full text]
  • Accurate Eigenvalue Decomposition of Arrowhead Matrices and Applications
    Accurate eigenvalue decomposition of arrowhead matrices and applications Accurate eigenvalue decomposition of arrowhead matrices and applications Nevena Jakovˇcevi´cStor FESB, University of Split joint work with Ivan Slapniˇcar and Jesse Barlow IWASEP9 June 4th, 2012. 1/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. Under certain conditions, the aheig algorithm computes all eigenvalues and all components of corresponding eigenvectors with high rel. acc. in O(n2) operations. 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. Under certain conditions, the aheig algorithm computes all eigenvalues and all components of corresponding eigenvectors with high rel. acc. in O(n2) operations. 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. Under certain conditions, the aheig algorithm computes all eigenvalues and all components of corresponding eigenvectors with high rel. acc. in O(n2) operations. The algorithm is based on shift-and-invert technique and limited finite O(n) use of double precision arithmetic when necessary. 2/31 Accurate eigenvalue decomposition of arrowhead matrices and applications Introduction We present a new algorithm (aheig) for computing eigenvalue decomposition of real symmetric arrowhead matrix. Under certain conditions, the aheig algorithm computes all eigenvalues and all components of corresponding eigenvectors with high rel.
    [Show full text]
  • On the Spectrum of PP-Wave Matrix Theory
    Preprint typeset in JHEP style - HYPER VERSION hep-th/0207034 AEI-2002-047 On the Spectrum of PP-Wave Matrix Theory Nakwoo Kim and Jan Plefka Max-Planck-Institut f¨ur Gravitationsphysik Albert-Einstein-Institut Am M¨uhlenberg 1, D-14476 Golm, Germany kim,[email protected] Abstract: We study the spectrum of the recently proposed matrix model of DLCQ M-theory in a parallel plane (pp)-wave background. In contrast to matrix theory in a flat background this model contains mass terms, which lift the flat directions of the potential and renders its spectrum discrete. The supersymmetry algebra of the model groups the energy eigenstates into supermultiplets, whose members differ by fixed amounts of energy in great similarity to the representation of supersymmetry in AdS spaces. There is a unique and exact zero-energy groundstate along with a multitude of long and short multiplets of excited states. For large masses the arXiv:hep-th/0207034v3 10 Jul 2002 quantum mechanical model may be treated perturbatively and we study the leading order energy shifts of the first excited states up to level two. Most interestingly we uncover a protected short multiplet at level two, whose energies do not receive perturbative corrections. Moreover, we conjecture the existence of an infinite series of similar protected multiplets in the pp-wave matrix model. Contents 1. Introduction 1 2. The Model and its Quantization 2 3. Supersymmetry Algebra and Structure of the Spectrum 4 4. The Perturbative Energy Spectrum and Protected States 8 5. Conclusions and Outlook 12 A. The Supersymmetry Algebra 13 B.
    [Show full text]
  • High Performance Selected Inversion Methods for Sparse Matrices
    High performance selected inversion methods for sparse matrices Direct and stochastic approaches to selected inversion Doctoral Dissertation submitted to the Faculty of Informatics of the Università della Svizzera italiana in partial fulfillment of the requirements for the degree of Doctor of Philosophy presented by Fabio Verbosio under the supervision of Olaf Schenk February 2019 Dissertation Committee Illia Horenko Università della Svizzera italiana, Switzerland Igor Pivkin Università della Svizzera italiana, Switzerland Matthias Bollhöfer Technische Universität Braunschweig, Germany Laura Grigori INRIA Paris, France Dissertation accepted on 25 February 2019 Research Advisor PhD Program Director Olaf Schenk Walter Binder i I certify that except where due acknowledgement has been given, the work presented in this thesis is that of the author alone; the work has not been sub- mitted previously, in whole or in part, to qualify for any other academic award; and the content of the thesis is the result of work which has been carried out since the official commencement date of the approved research program. Fabio Verbosio Lugano, 25 February 2019 ii To my whole family. In its broadest sense. iii iv Le conoscenze matematiche sono proposizioni costruite dal nostro intelletto in modo da funzionare sempre come vere, o perché sono innate o perché la matematica è stata inventata prima delle altre scienze. E la biblioteca è stata costruita da una mente umana che pensa in modo matematico, perché senza matematica non fai labirinti. Umberto Eco, “Il nome della rosa” v vi Abstract The explicit evaluation of selected entries of the inverse of a given sparse ma- trix is an important process in various application fields and is gaining visibility in recent years.
    [Show full text]
  • Normalization of the Ground State of the Supersymmetric Harmonic
    Letters in High Energy Physics LHEP xx, xxx, 2018 Normalization of the Ground State of the Supersymmetric Harmonic Oscillator Ahmed Ayad School of Physics, University of the Witwatersrand, Private Bag 3, WITS-2050, Johannesburg, South Africa Abstract Supersymmetry plays a main role in all current thinking about superstring theory. Indeed, many remark- able properties of string theory have been explained using supersymmetry as a tool. So far, there has been no unbroken supersymmetry observed in nature, and if nature is described by supersymmetry, it must be broken. Supersymmetry may be broken spontaneously at any order of perturbation theory or dynamically due to nonperturbative effects. To examine the methods of supersymmetry breaking, special attention is given to discuss the normalization of the ground state of the supersymmetric harmonic oscillator. This study explains that perturbation theory gives incorrect results for both the ground-state wave function and the energy spectrum and it fails to give an explanation to the supersymmetry breaking. Keywords: supersymmetry breaking, perturbation theory, fermionic as well as bosonic fields using the usual quantum nonperturbative effects mechanical operators method. Then, we consider supersym- metric classical mechanics and study generalized classical Pois- son brackets to Dirac brackets and quantization rules in or- 1. INTRODUCTION der to introduce the superalgebra. Furthermore, the formal- ism of supersymmetry in quantum mechanics is introduced. Supersymmetry, often abbreviated SUSY, was first introduced Subsequently, we give a general background in the concepts in 1971 by Gelfand and Likhtman, Raymond, and Neveu and and methods of supersymmetric quantum mechanics. This fol- Schwartz, and later, it was rediscovered by other groups [1, 2].
    [Show full text]