DOCSLIB.ORG

Householder Symposium XIX June 8-13, Spa Belgium

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Householder Symposium XIX June 8-13, Spa Belgium Householder Symposium XIX June 8-13, Spa Belgium Contents Householder Symposium XIX on Numerical Linear Algebra 1 Householder Committee 4 Local Organizing Committee 4 Householder Prize Committee 4 Acknowledgments 5 Abstracts 6 Charlotte Dorcimont and P.-A. Absil Algorithms for the Nearest Correlation Matrix Problem with Factor Structure ....7 Kensuke Aishima Global Convergence of the Restarted Lanczos Method and Jacobi-Davidson Method for Symmetric Eigenvalue Problems ...........................9 Awad H. Al-Mohy An Efficient Estimator of the Condition Number of the Matrix Exponential ..... 11 A.C. Antoulas and A.C. Ioniţă Model Reduction of Nonlinear Systems in the Loewner Framework .......... 13 Mario Arioli and Daniel Loghin A Spectral Analysis of a Discrete two-domain Steklov-Poincaré Operator ....... 16 Haim Avron, Michael Mahoney, Vikas Sindhwani and Jiyan Yang Randomized and Quasi-Randomized Algorithms for Low-Rank Approximation of Gram Matrices ........................................... 18 Zhaojun Bai, Ren-Cang Li, Dario Rocca and Giulia Galli Variational Principles and Scalable Solvers for the Linear Response Eigenvalue Prob- lem ............................................. 20 Grey Ballard, James Demmel, Laura Grigori, Mathias Jacquelin, Hong Diep Nguyen, and Edgar Solomonik Reconstructing Householder Vectors from Tall-Skinny QR ............... 22 Jesse L. Barlow Block Gram-Schmidt Downdating ............................. 24 Christopher Beattie Diffusion Models for Covariance ............................. 25 Peter Benner and Ludwig Kohaupt The Riccati Eigenproblem ................................. 26 Mario Arioli and Michele Benzi Numerical Analysis of Quantum Graphs ......................... 28 i Paolo Bientinesi, Diego Fabregat and Yurii Aulchenko Can Numerical Linear Algebra make it in Nature? ................... 29 David Bindel and Erdal Yilmaz Music of the Microspheres: from Eigenvalues Perturbations to Gyroscopes ...... 31 Matthias Bolten Block-smoothing in Multigrid Methods for Circulant and Toeplitz Matrices ...... 32 Shreemayee Bora and Ravi Srivastava Distance Problems for Hermitian Matrix Pencils .................... 34 Nicolas Boumal and P.-A. Absil Preconditioning for Low-Rank Matrix Completion via Trust-Regions over one Grass- mannian ........................................... 36 Christos Boutsidis and David Woodruff Optimal CUR Matrix Decompositions .......................... 38 Russell L. Carden and Danny C. Sorensen Stable Discrete Empirical Interpolation Method based Quadrature Schemes for Non- linear Model Reduction ................................... 39 Erin Carson and James Demmel Improving the Maximum Attainable Accuracy of Communication-Avoiding Krylov Subspace Methods ...................................... 40 Saifon Chaturantabut, Christopher A. Beattie and Serkan Gugercin Structure-Preserving Model Reduction for Nonlinear Port-Hamiltonian Systems ... 42 Jie Chen and Edmond Chow Two Methods for Computing the Matrix Sign Function ................. 44 Edmond Chow and Yousef Saad Preconditioned Methods for Sampling Multivariate Gaussian Distributions ...... 45 Julianne Chung, Misha Kilmer and Dianne O’Leary A Framework for Regularization via Operator Approximation ............. 47 Edvin Deadman and Nicholas J Higham Testing Matrix Functions Using Identities ........................ 49 Laurent Sorber, Mikael Sorensen Marc Van Barel and Lieven De Lathauwer Coupled Matrix/Tensor Decompositions: an Introduction ................ 51 James Demmel Communication Avoiding Algorithms for Linear Algebra and Beyond ......... 52 Eric de Sturler, Serkan Gugercin, Misha Kilmer, Chris Beattie, Saifon Chaturantabut, and Meghan O’Connell Model Reduction Techniques for Fast Nonlinear Inversion ............... 53 Fernando De Terán and Françoise Tisseur Backward Error and Conditioning of Fiedler Companion Linearizations. ....... 56 Inderjit S. Dhillon, H. Yun, C.J. Hsieh, H.F. Yu and S.V.N. Vishwanathan Parallel Asynchronous Matrix Factorization for Large-Scale Data Analysis ...... 57 ii Andrii Dmytryshyn, Stefan Johansson and Bo Kågström Changes of Canonical Structure Information of Matrix Pencils associated with Gen- eralized State-space Systems. ............................... 58 Beresford Parlett, Froilán M. Dopico and Carla Ferreira The inverse complex eigenvector problem for real tridiagonal matrices ........ 60 Petros Drineas and Abhisek Kundu Identifying Influential Entries in a Matrix ........................ 61 Zvonimir Bujanović and Zlatko Drmač A new Framework for Polynomial Filtering in Implicitly Restarted Arnoldi type Al- gorithms ........................................... 63 Vladimir Druskin, Alexander Mamonov, Rob Remis and Mikhail Zaslavsky Matrix Functions and Their Krylov Approximations for Large Scale Wave Propaga- tion in Unbounded Domains. ............................... 64 Iain Duff and Mario Arioli The Solution of Least-Squares Problems using Preconditioned LSQR ......... 65 Jurjen Duintjer Tebbens and Gérard Meurant On the Convergence Curves that can be generated by Restarted GMRES ....... 68 Jeff Bezanson, Alan Edelman, Stefan Karpinski, Viral Shah and the greater community Julia: A Fresh Approach to Technical Computing .................... 70 Lars Eldén Computing Fréchet Derivatives in Partial Least Squares Regression .......... 71 Howard C. Elman, Virginia Forstall and Qifeng Liao Efficient Solution of Stochastic Partial Differential Equations Using Reduced-Order Models ............................................ 72 Mark Embree, Jeffrey Hokanson and Charles Puelz The Life Cycle of an Eigenvalue Problem ........................ 73 Peter Benner, Heike Faßbender, and Chao Yang On complex J-symmetric eigenproblems ......................... 75 Melina A. Freitag, Alastair Spence and Paul Van Dooren New Algorithms for Calculating the H∞-norm and the Real Stability Radius .... 77 Andreas Frommer, Stefan Güttel and Marcel Schweitzer Convergence of restarted Krylov subspace methods for matrix functions ....... 79 Martin J. Gander 50 Years of Time Parallel Time Integration ....................... 81 Silvia Gazzola, James Nagy and Paolo Novati Arnoldi-Tikhonov Methods for Sparse Reconstruction .................. 82 Pieter Ghysels, Wim Vanroose and Karl Meerbergen High Performance Implementation of Deflated Preconditioned Conjugate Gradients with Approximate Eigenvectors .............................. 84 Nicolas Gillis and Stephen A. Vavasis Semidefinite Programming Based Preconditioning for More Robust Near-Separable Nonnegative Matrix Factorization ............................ 86 iii E. Agullo, L. Giraud, P. Salas Medina and M. Zounon Preliminary Investigations on Recovery-Restart Strategies for Resilient Parallel Nu- merical Linear Algebra Solvers .............................. 87 Anne Greenbaum Extensions of the Symmetric Tridiagonal Matrix Arising from a Finite Precision Lanczos Computation ................................... 88 Chen Greif, Erin Moulding and Dominique Orban Numerical Solution of Indefinite Linear Systems Arising from Interior-Point Methods 89 Laura Grigori, Remi Lacroix, Frederic Nataf, and Long Qu Direction preserving algebraic preconditioners ...................... 91 Luka Grubišić and Daniel Kressner Rapid Convergence for Finite Rank Approximations of Infinite-dimensional Lya- punov Equations ...................................... 93 Vladimir Druskin, Stefan Güttel and Leonid Knizhnerman Perfectly Matched Layers via the Iterated Rational Krylov Algorithm ......... 95 Serkan Gugercin and Garret Flagg The Sylvester Equation and Interpolatory Model Reduction of Linear/Bilinear Dy- namical Systems ...................................... 97 Chun-Hua Guo, Changli Liu and Jungong Xue Performance Enhancement of Doubling Algorithms for a Class of Complex Nonsym- metric Algebraic Riccati Equations ............................ 99 Martin H. Gutknecht Is There a Market for Modified Moments? ........................ 101 Marco Donatelli and Martin Hanke Fast Nonstationary Preconditioned Iterative Methods for Image Deblurring ...... 102 Per Christian Hansen, James G. Nagy and Konstantinos Tigkos Rotational Image Deblurring with Sparse Matrices ................... 104 Nicholas J. Higham, Lijing Lin and Samuel Relton How and Why to Estimate Condition Numbers for Matrix Functions ......... 105 Iveta Hnětynková, Marie Michenková and Martin Plešinger Noise Approximation in Discrete Ill-posed Problems .................. 107 Michiel Hochstenbach and Ian N. Zwaan Field of Values type Eigenvalue Inclusion Regions for Large Matrices ......... 109 Bruno Iannazzo and Carlo Manasse A Schur Logarithmic Algorithm for Fractional Powers of Matrices .......... 110 Ilse Ipsen Randomized Algorithms for Numerical Linear Algebra ................. 112 Elias Jarlebring and Olof Runborg The infinite Arnoldi method for the waveguide eigenvalue problem ........... 113 Dario A. Bini, Bruno Iannazzo, Ben Jeuris and Raf Vandebril The Geometric Matrix Mean: an Adaptation for Structured Matrices ......... 115 iv B. Kågström Stratification of some Structured Matrix pencil Problems: how Canonical Forms Change under Perturbations ................................ 116 Nicholas J. Higham, Amal Khabou and Françoise Tisseur Fast Generation of Random Orthogonal Matrices .................... 118 Ning Hao, Lior Horesh and Misha E. Kilmer Model Correction using a Nuclear Norm Constraint ................... 120 Andrew Knyazev Numerical Linear Algebra and Matrix Theory in Action ................ 122 Antti Koskela and Alexander Ostermann Computing
Load more

Recommended publications
  • 1 Introduction
    Real and Complex Hamiltonian Square Ro ots of SkewHamiltonian Matrices y z HeikeFab ender D Steven Mackey Niloufer Mackey x Hongguo Xu January DedicatedtoProfessor Ludwig Elsner on the occasion of his th birthday Abstract We present a constructive existence pro of that every real skewHamiltonian matrix W has a real Hamiltonian square ro ot The key step in this construction shows how one may bring anysuch W into a real quasiJordan canonical form via symplectic similarityWe show further that every W has innitely many real Hamiltonian square ro ots and givealower b ound on the dimension of the set of all such square ro ots Extensions to complex matrices are also presented This report is an updated version of the paper Hamiltonian squareroots of skew Hamiltonian matrices that appearedinLinear Algebra its Applicationsv pp AMS sub ject classication A A A F B Intro duction Any matrix X suchthatX A is said to b e a square ro ot of the matrix AF or general nn complex matrices A C there exists a welldevelop ed although somewhat complicated theory of matrix square ro ots and a numb er of algorithms for their eective computation Similarly for the theory and computation of real square ro ots for real matrices By contrast structured squareroot problems where b oth the matrix A Fachb ereich Mathematik und Informatik Zentrum fur Technomathematik Universitat Bremen D Bremen FRG email heikemathunibremende y Department of Mathematics Computer Science Kalamazo o College Academy Street Kalamazo o MI USA z Department of Mathematics
    [Show full text]
  • Eigenvalue Perturbation Theory of Structured Real Matrices and Their Sign Characteristics Under Generic Structured Rank-One Perturbations∗
    Eigenvalue perturbation theory of structured real matrices and their sign characteristics under generic structured rank-one perturbations∗ Christian Mehlx Volker Mehrmannx Andr´eC. M. Ran{ Leiba Rodmank July 9, 2014 Abstract An eigenvalue perturbation theory under rank-one perturbations is developed for classes of real matrices that are symmetric with respect to a non-degenerate bilinear form, or Hamiltonian with respect to a non-degenerate skew-symmetric form. In contrast to the case of complex matrices, the sign characteristic is a cru- cial feature of matrices in these classes. The behavior of the sign characteristic under generic rank-one perturbations is analyzed in each of these two classes of matrices. Partial results are presented, but some questions remain open. Appli- cations include boundedness and robust boundedness for solutions of structured systems of linear differential equations with respect to general perturbations as well as with respect to structured rank perturbations of the coefficients. Key Words: real J-Hamiltonian matrices, real H-symmetric matrices, indefinite inner product, perturbation analysis, generic perturbation, rank-one perturbation, T - even matrix polynomial, symmetric matrix polynomial, bounded solution of differential equations, robustly bounded solution of differential equations, invariant Lagrangian subspaces. Mathematics Subject Classification: 15A63, 15A21, 15A54, 15B57. xInstitut f¨urMathematik, TU Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany. Email: fmehl,[email protected]. {Afdeling Wiskunde, Faculteit der Exacte Wetenschappen, Vrije Universiteit Amsterdam, De Boele- laan 1081a, 1081 HV Amsterdam, The Netherlands and Unit for BMI, North West University, Potchef- stroom, South Africa. E-mail: [email protected] kCollege of William and Mary, Department of Mathematics, P.O.Box 8795, Williamsburg, VA 23187-8795, USA.
    [Show full text]
  • Hamiltonian Matrix Representations for the Determination of Approximate Wave Functions for Molecular Resonances
    Journal of Applied and Fundamental Sciences HAMILTONIAN MATRIX REPRESENTATIONS FOR THE DETERMINATION OF APPROXIMATE WAVE FUNCTIONS FOR MOLECULAR RESONANCES Robert J. Buenker Fachbereich C-Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaussstr. 20, D-42097 Wuppertal, Germany *For correspondence. ([email protected]) Abstract: Wave functions obtained using a standard complex Hamiltonian matrix diagonalization procedure are square integrable and therefore constitute only approximations to the corresponding resonance solutions of the Schrödinger equation. The nature of this approximation is investigated by means of explicit calculations using the above method which employ accurate diabatic potentials of the B 1Σ+ - D’ 1Σ+ vibronic resonance states of the CO molecule. It is shown that expanding the basis of complex harmonic oscillator functions gradually improves the description of the exact resonance wave functions out to ever larger internuclear distances before they take on their unwanted bound-state characteristics. The justification of the above matrix method has been based on a theorem that states that the eigenvalues of a complex-scaled Hamiltonian H (ReiΘ) are associated with the energy position and linewidth of resonance states (R is an internuclear coordinate and Θ is a real number). It is well known, however, that the results of the approximate method can be obtained directly using the unscaled Hamiltonian H (R) in real coordinates provided a particular rule is followed for the evaluation of the corresponding matrix elements. It is shown that the latter rule can itself be justified by carrying out the complex diagonalization of the Hamiltonian in real space via a product of two transformation matrices, one of which is unitary and the other is complex orthogonal, in which case only the symmetric scalar product is actually used in the evaluation of all matrix elements.
    [Show full text]
  • Hamiltonian Matrices and the Algebraic Riccati Equation Seminar Presentation
    Hamiltonian Matrices and the Algebraic Riccati Equation Seminar presentation by Piyapong Yuantong 7th December 2009 Technische Universit¨atChemnitz Piyapong Yuantong Hamiltonian Matrices and the Algebraic Riccati Equation 1 Hamiltonian matrices We start by introducing the square matrix J 2 R 2n×2n defined by " # O I J = n n ; (1) −In On where n×n On 2 R − zero matrix n×n In 2 R − identity matrix: Remark It is not very complicated to prove the following properties of the matrix J: i. J T = −J ii. J −1 = J T T iii. J J = I2n T T iv. J J = −I2n 2 v. J = −I2n vi. det J = 1 Definition 1 A matrix A 2 R 2n×2n is called Hamiltonian if JA is symmetric, so JA = (JA) T ) A T J + JA = 0 where J 2 R 2n×2n is from (1). We will denote { } Hn = A 2 R 2n×2n j A T J + JA = 0 the set of 2n × 2n Hamiltonian matrices. 1 Piyapong Yuantong Hamiltonian Matrices and the Algebraic Riccati Equation Proposition 1.1 The following are equivalent: a) A is a Hamiltionian matrix b) A = JS, where S = S T c) (JA) T = JA Proof a ! b A = JJ −1 A ! A = J(−J)A 2Hn A ! (J(−JA)) T J + JA = 0 ! (−JA) T J T J = −JA T J J=!I2n (−JA) T = −JA ! J(−JA) T = A If − JA = S ! A = J(−JA) = J(−JA) T ! A = JS = JS T =) S = S T a ! c A T J + JA = 0 ! A T J = −JA ()T ! (A T J) T = (−JA) T ! J T A = −(JA) T ! −JA = −(JA) T ! (JA) T = JA 2 Piyapong Yuantong Hamiltonian Matrices and the Algebraic Riccati Equation Proposition 1.2 Let A; B 2 Hn.
    [Show full text]
  • Mathematical Physics-14-Eigenvalue Problems.Nb
    Eigenvalue problems Main idea and formulation in the linear algebra The word "eigenvalue" stems from the German word "Eigenwert" that can be translated into English as "Its own value" or "Inherent value". This is a value of a parameter in the equation or system of equations for which this equation has a nontriv- ial (nonzero) solution. Mathematically, the simplest formulation of the eigenvalue problem is in the linear algebra. For a given square matrix A one has to find such values of l, for which the equation (actually the system of linear equations) A.X λX (1) has a nontrivial solution for a vector (column) X. Moving the right part to the left, one obtains the equation A − λI .X 0, H L where I is the identity matrix having all diagonal elements one and nondiagonal elements zero. This matrix equation has nontrivial solutions only if its determinant is zero, Det A − λI 0. @ D This is equivalent to a Nth order algebraic equation for l, where N is the rank of the mathrix A. Thus there are N different eigenvalues ln (that can be complex), for which one can find the corresponding eigenvectors Xn. Eigenvectors are defined up to an arbitrary numerical factor, so that usually they are normalized by requiring T∗ Xn .Xn 1, where X T* is the row transposed and complex conjugate to the column X. It can be proven that eigenvectors that belong to different eigenvalues are orthogonal, so that, more generally than above, one has T∗ Xm .Xn δmn . Here dmn is the Kronecker symbol, 1, m n δ = mn µ 0, m ≠ n.
    [Show full text]
  • MAT TRIAD 2019 Book of Abstracts
    MAT TRIAD 2019 International Conference on Matrix Analysis and its Applications Book of Abstracts September 8 13, 2019 Liblice, Czech Republic MAT TRIAD 2019 is organized and supported by MAT TRIAD 2019 Edited by Jan Bok, Computer Science Institute of Charles University, Prague David Hartman, Institute of Computer Science, Czech Academy of Sciences, Prague Milan Hladík, Department of Applied Mathematics, Charles University, Prague Miroslav Rozloºník, Institute of Mathematics, Czech Academy of Sciences, Prague Published as IUUK-ITI Series 2019-676ø by Institute for Theoretical Computer Science, Faculty of Mathematics and Physics, Charles University Malostranské nám. 25, 118 00 Prague 1, Czech Republic Published by MATFYZPRESS, Publishing House of the Faculty of Mathematics and Physics, Charles University in Prague Sokolovská 83, 186 75 Prague 8, Czech Republic Cover art c J. Na£eradský, J. Ne²et°il c Jan Bok, David Hartman, Milan Hladík, Miroslav Rozloºník (eds.) c MATFYZPRESS, Publishing House of the Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic, 2019 i Preface This volume contains the Book of abstracts of the 8th International Conference on Matrix Anal- ysis and its Applications, MAT TRIAD 2019. The MATTRIAD conferences represent a platform for researchers in a variety of aspects of matrix analysis and its interdisciplinary applications to meet and share interests and ideas. The conference topics include matrix and operator theory and computation, spectral problems, applications of linear algebra in statistics, statistical models, matrices and graphs as well as combinatorial matrix theory and others. The goal of this event is to encourage further growth of matrix analysis research including its possible extension to other elds and domains.
    [Show full text]
  • Matrix G-Strands
    Technological University Dublin ARROW@TU Dublin Articles School of Mathematics 2014 Matrix G-strands Darryl Holm Imperial College London Rossen Ivanov Technological University Dublin, [email protected] Follow this and additional works at: https://arrow.tudublin.ie/scschmatart Part of the Mathematics Commons, Non-linear Dynamics Commons, and the Partial Differential Equations Commons Recommended Citation Holm, D. and Ivanov, R. Matrix G-Strands, Nonlinearity 27 (2014) 1445–1469; doi:10.1088/0951-7715/27/ 6/1445 This Article is brought to you for free and open access by the School of Mathematics at ARROW@TU Dublin. It has been accepted for inclusion in Articles by an authorized administrator of ARROW@TU Dublin. For more information, please contact [email protected], [email protected]. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 4.0 License Funder: ERC Matrix G-Strands Darryl D. Holm1 and Rossen I. Ivanov2 9 Feb 2014, clean copy Abstract We discuss three examples in which one may extend integrable Euler–Poincar´e ODEs to integrable Euler–Poincar´ePDEs in the matrix G-Strand context. After de- scribing matrix G-Strand examples for SO(3) and SO(4) we turn our attention to SE(3) where the matrix G-Strand equations recover the exact rod theory in the con- vective representation. We then find a zero curvature representation (ZCR) of these equations and establish the conditions under which they are completely integrable. Thus, the G-Strand equations turn out to be a rich source of integrable systems. The treatment is meant to be expository and most concepts are explained in examples in the language of vectors in R3.
    [Show full text]
  • A Unified Approach to Linear and Nonlinear Normal Forms For
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector J. Symbolic Computation (1999) 27, 49–131 Article No. jsco.1998.0244 Available online at http://www.idealibrary.com on A Unied Approach to Linear and Nonlinear Normal Forms for Hamiltonian Systems† R. C. CHURCHILL‡‡§AND MARTIN KUMMER§ ‡Department of Mathematics, Hunter College and Graduate Center, CUNY, 695 Park Avenue, NY 10021, U.S.A. § Department of Mathematics, University of Toldeo, Toledo, OH 43606, U.S.A. We prove that in all but one case the normal form of a real or complex Hamiltonian matrix which is irreducible and appropriately normalized can be computed by Lie series methods in formally the same manner as one computes the normal form of a nonlinear Hamiltonian function. Calculations are emphasized; the methods are illustrated with detailed examples, and for the sake of completeness the exceptional case is also reviewed and illustrated. Alternate methods are also discussed, along with detailed examples. c 1999 Academic Press 1. Introduction A linear operator T : V → V on a real nite-dimensional symplectic space V =(V,ω)is Hamiltonian,orinnitesimally symplectic,if ω(Tu,v)+ω(u, T v)=0,u,v∈ V. (1.1) The linear Hamiltonian normal form problem is that of nding simple matrix represen- tations for such operators relative to symplectic bases of V . The theoretical solution was provided by Williamson (1936) (also see Wall (1963)), but calculation of normal forms following his ideas proves dicult in practice. Simplications have since appeared, but the methods are still quite involved, (see, e.g.
    [Show full text]
  • Matrix Nearness Problems and Applications∗
    Matrix Nearness Problems and Applications∗ Nicholas J. Higham† Abstract A matrix nearness problem consists of finding, for an arbitrary matrix A, a nearest member of some given class of matrices, where distance is measured in a matrix norm. A survey of nearness problems is given, with particular emphasis on the fundamental properties of symmetry, positive definiteness, orthogonality, normality, rank-deficiency and instability. Theoretical results and computational methods are described. Applications of nearness problems in areas including control theory, numerical analysis and statistics are outlined. Key words. matrix nearness problem, matrix approximation, symmetry, positive definiteness, orthogonality, normality, rank-deficiency, instability. AMS subject classifications. Primary 65F30, 15A57. 1 Introduction Consider the distance function (1.1) d(A) = min E : A + E S has property P ,A S, k k ∈ ∈ where S denotes Cm×n or Rm×n, is a matrix norm on S, and P is a matrix property k · k which defines a subspace or compact subset of S (so that d(A) is well-defined). Associated with (1.1) are the following tasks, which we describe collectively as a matrix nearness problem: Determine an explicit formula for d(A), or a useful characterisation. • ∗This is a reprint of the paper: N. J. Higham. Matrix nearness problems and applications. In M. J. C. Gover and S. Barnett, editors, Applications of Matrix Theory, pages 1–27. Oxford University Press, 1989. †Department of Mathematics, University of Manchester, Manchester, M13 9PL, England ([email protected]). 1 Determine X = A + E , where E is a matrix for which the minimum in (1.1) • min min is attained.
    [Show full text]
  • H = Hamiltonian Matrix S = Overlap Matrix
    chmy564-19 Lec 8 3. Linear Variation Method Boot Camp Mon 28jan19 Hij = Hamiltonian matrix Sij = Overlap matrix Linear Variation with 2 orthonormal, real, basis functions Ψ = c1 Φ1 + c2 Φ2 (H11 − E) c1 + H12 c2 = 0 H21c2 + (H22 − E) c2 = 0 where H12 = <Φ1|H|Φ2>, the "interaction" of the two basis functions, H11 = <Φ1|H|Φ1>, the "energy" of Φ1, H22 = <Φ2|H|Φ2>, the "energy" of Φ2, and E = <Ψ|H|Ψ>. the expectation value of the energy using the trial function. For a non-trivial solution (c1 and c2 not both zero), the determinant of the matrix of numbers multiplying the coefficients c1 and c2 must vanish, i.e., (H11 −E) (H22 − E) − H21 H12 = 0 quadratic equation; Solve for E 2 E – E(H11 + H22) +(H11H22 − H21 H12)= 0 quadratic 2 2 −±b b −4 ac ()()4(HH11+± 22 HH11 +− 22 HHHH11 22 − 21 12 E± = = 22a 2 H11+H22 H11−H22 2 E± = ± +H Because H12 is real 2 2 12 H H α1 β H= 11 12 = β α H21 H22 2 2 (αα+− )( αα ) 2 E =±+12 12 β 22 Solving for Ψ = c1 Φ1 + c2 Φ2 means solving for c1 and c2 (H11 – E)c1 + (H12)c2 = 0 (α1 – E)c1 + βc2 = 0 (H21)c1 + (H22 – E)c2 = 0 = βc1 + (α2 – E)c2 = 0 The determinant from the N simultaneous homogeneous equations = 0 is a constraint that means that you can only solve for the ratio of the coefficients. 5 (α1 – E)c1 + βc2 = 0 The determinant from the N simultaneous homogeneous equations = 0 is a constraint that βc1 + (α2 – E)c2 = 0 means that you can only solve for the ratio of the coefficients.
    [Show full text]
  • Arxiv:2011.08087V2 [Math-Ph] 30 Mar 2021 [36, 6] of a Haar Measure Unitary
    THE GENERALIZED CARTAN DECOMPOSITION FOR CLASSICAL RANDOM MATRIX ENSEMBLES ALAN EDELMAN AND SUNGWOO JEONG Abstract. We present a completed classification of the classical random ma- trix ensembles: Hermite (Gaussian), Laguerre (Wishart), Jacobi (MANOVA) and Circular by introducing the concept of the generalized Cartan decompo- sition and the double coset space. Previous authors [8, 40] associate a symmetric space G=K with a random matrix density on the double coset structure KnG=K. However this is incom- plete. Complete coverage requires the double coset structure A = K1nG=K2, where G=K1 and G=K2 are two symmetric spaces. Furthermore, we show how the matrix factorization obtained by the generalized Cartan decomposition G = K1AK2 plays a crucial role in sampling algorithms and the derivation of the joint probability density of A. 1. Introduction In pioneering work by Zirnbauer [40] and Due~nez[8] connections are made be- tween random matrix ensembles and the symmetric spaces [19, 21] studied in dif- ferential geometry, representation theory, and harmonic analysis. Unfortunately, symmetric spaces alone do not account for a rich enough set of Jacobi ensemble densities (See Figure 1). This work fills this significant gap. Noting that more general tools are needed, we not only take a closer look at the KAK decomposition from the theory of symmetric spaces, but we propose the incorporation of the more recent generalized Cartan decomposition (or the K1AK2 decomposition) [15, 24, 14, 30] especially for complete coverage of the Jacobi ensembles. We show that double coset spaces are rich enough to cover these important matrix ensembles. Furthermore, the key object is not the joint density, per se, but the matrix factorization.
    [Show full text]
  • Technische Universit at Chemnitz
    Technische Universitat Chemnitz Sonderforschungsbereich Numerische Simulation auf massiv parallelen Rechnern Peter Benner Ralph Byers Volker Mehrmann Hongguo Xu Numerical Computation of Deating Subspaces of Embedded Hamiltonian Pencils Preprint SFB PreprintReihe des Chemnitzer SFB SFB June Contents Introduction and Preliminaries Embedding of Extended Matrix Pencils Hamiltonian Triangular Forms SkewHamiltonianHamiltonian Matrix Pencils Algorithms Error and Perturbation Analysis Conclusion A Algorithmic Details A Schur Forms for SkewHamiltonianHamiltonian Matrix Pencils and SkewHa miltonianskewHamiltonian Matrix Pencils A Subroutines Required by Algorithm Authors Peter Benner Ralph Byers Zentrum f ur Technomathematik Department of Mathematics Fachbereich Mathematik und Informatik University of Kansas Universitat Bremen Lawrence KS D Bremen FRG USA bennermathunibremende byersmathukansedu Volker Mehrmann Hongguo Xu Deparment of Mathematics Fakultat f ur Mathematik Case Western University Technische Universitat Chemnitz Cleveland OH D Chemnitz FRG USA mehrmannmathematiktuchemnitzde hxxpocwruedu Numerical Computation of Deating Subspaces of Embedded Hamiltonian Pencils 2 3 Peter Benner Ralph Byers Volker Mehrmann Hongguo Xu Abstract We discuss the numerical solution of structured generalized eigenvalue problems that arise from linearquadratic optimal control problems H optimization multibo dy sys 1 tems and many other areas of applied mathematics physics and chemistry The classical
    [Show full text]