Singular Value Decomposition in Embedded Systems Based on ARM Cortex-M Architecture
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New Fast and Accurate Jacobi Svd Algorithm: I
NEW FAST AND ACCURATE JACOBI SVD ALGORITHM: I. ZLATKO DRMAC· ¤ AND KRESIMIR· VESELIC¶ y Abstract. This paper is the result of contrived e®orts to break the barrier between numerical accuracy and run time e±ciency in computing the fundamental decomposition of numerical linear algebra { the singular value decomposition (SVD) of a general dense matrix. It is an unfortunate fact that the numerically most accurate one{sided Jacobi SVD algorithm is several times slower than generally less accurate bidiagonalization based methods such as the QR or the divide and conquer algorithm. Despite its sound numerical qualities, the Jacobi SVD is not included in the state of the art matrix computation libraries and it is even considered obsolete by some leading researches. Our quest for a highly accurate and e±cient SVD algorithm has led us to a new, superior variant of the Jacobi algorithm. The new algorithm has inherited all good high accuracy properties, and it outperforms not only the best implementations of the one{sided Jacobi algorithm but also the QR algorithm. Moreover, it seems that the potential of the new approach is yet to be fully exploited. Key words. Jacobi method, singular value decomposition, eigenvalues AMS subject classi¯cations. 15A09, 15A12, 15A18, 15A23, 65F15, 65F22, 65F35 1. Introduction. In der Theorie der SÄacularstÄorungenund der kleinen Oscil- lationen wird man auf ein System lineÄarerGleichungen gefÄurt,in welchem die Co- e±cienten der verschiedenen Unbekanten in Bezug auf die Diagonale symmetrisch sind, die ganz constanten Glieder fehlen und zu allen in der Diagonale be¯ndlichen Coe±cienten noch dieselbe GrÄo¼e ¡x addirt ist. -
Copyright © by SIAM. Unauthorized Reproduction of This Article Is Prohibited. PARALLEL BIDIAGONALIZATION of a DENSE MATRIX 827
SIAM J. MATRIX ANAL. APPL. c 2007 Society for Industrial and Applied Mathematics Vol. 29, No. 3, pp. 826–837 ∗ PARALLEL BIDIAGONALIZATION OF A DENSE MATRIX † ‡ ‡ § CARLOS CAMPOS , DAVID GUERRERO , VICENTE HERNANDEZ´ , AND RUI RALHA Abstract. A new stable method for the reduction of rectangular dense matrices to bidiagonal form has been proposed recently. This is a one-sided method since it can be entirely expressed in terms of operations with (full) columns of the matrix under transformation. The algorithm is well suited to parallel computing and, in order to make it even more attractive for distributed memory systems, we introduce a modification which halves the number of communication instances. In this paper we present such a modification. A block organization of the algorithm to use level 3 BLAS routines seems difficult and, at least for the moment, it relies upon level 2 BLAS routines. Nevertheless, we found that our sequential code is competitive with the LAPACK DGEBRD routine. We also compare the time taken by our parallel codes and the ScaLAPACK PDGEBRD routine. We investigated the best data distribution schemes for the different codes and we can state that our parallel codes are also competitive with the ScaLAPACK routine. Key words. bidiagonal reduction, parallel algorithms AMS subject classifications. 15A18, 65F30, 68W10 DOI. 10.1137/05062809X 1. Introduction. The problem of computing the singular value decomposition (SVD) of a matrix is one of the most important operations in numerical linear algebra and is employed in a variety of applications. The SVD is defined as follows. × For any rectangular matrix A ∈ Rm n (we will assume that m ≥ n), there exist ∈ Rm×m ∈ Rn×n ΣA ∈ Rm×n two orthogonal matrices U and V and a matrix Σ = 0 , t where ΣA = diag (σ1,...,σn) is a diagonal matrix, such that A = UΣV . -
Minimum-Residual Methods for Sparse Least-Squares Using Golub-Kahan Bidiagonalization a Dissertation Submitted to the Institute
MINIMUM-RESIDUAL METHODS FOR SPARSE LEAST-SQUARES USING GOLUB-KAHAN BIDIAGONALIZATION A DISSERTATION SUBMITTED TO THE INSTITUTE FOR COMPUTATIONAL AND MATHEMATICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY David Chin-lung Fong December 2011 © 2011 by Chin Lung Fong. All Rights Reserved. Re-distributed by Stanford University under license with the author. This dissertation is online at: http://purl.stanford.edu/sd504kj0427 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Michael Saunders, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Margot Gerritsen I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Walter Murray Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii iv ABSTRACT For 30 years, LSQR and its theoretical equivalent CGLS have been the standard iterative solvers for large rectangular systems Ax = b and least-squares problems min Ax b . They are analytically equivalent − to symmetric CG on the normal equation ATAx = ATb, and they reduce r monotonically, where r = b Ax is the k-th residual vector. -
Restarted Lanczos Bidiagonalization for the SVD in Slepc
Scalable Library for Eigenvalue Problem Computations SLEPc Technical Report STR-8 Available at http://slepc.upv.es Restarted Lanczos Bidiagonalization for the SVD in SLEPc V. Hern´andez J. E. Rom´an A. Tom´as Last update: June, 2007 (slepc 2.3.3) Previous updates: { About SLEPc Technical Reports: These reports are part of the documentation of slepc, the Scalable Library for Eigenvalue Problem Computations. They are intended to complement the Users Guide by providing technical details that normal users typically do not need to know but may be of interest for more advanced users. Restarted Lanczos Bidiagonalization for the SVD in SLEPc STR-8 Contents 1 Introduction 2 2 Description of the Method3 2.1 Lanczos Bidiagonalization...............................3 2.2 Dealing with Loss of Orthogonality..........................6 2.3 Restarted Bidiagonalization..............................8 2.4 Available Implementations............................... 11 3 The slepc Implementation 12 3.1 The Algorithm..................................... 12 3.2 User Options...................................... 13 3.3 Known Issues and Applicability............................ 14 1 Introduction The singular value decomposition (SVD) of an m × n complex matrix A can be written as A = UΣV ∗; (1) ∗ where U = [u1; : : : ; um] is an m × m unitary matrix (U U = I), V = [v1; : : : ; vn] is an n × n unitary matrix (V ∗V = I), and Σ is an m × n diagonal matrix with nonnegative real diagonal entries Σii = σi for i = 1;:::; minfm; ng. If A is real, U and V are real and orthogonal. The vectors ui are called the left singular vectors, the vi are the right singular vectors, and the σi are the singular values. In this report, we will assume without loss of generality that m ≥ n. -
Cache Efficient Bidiagonalization Using BLAS 2.5 Operators Howell, G. W. and Demmel, J. W. and Fulton, C. T. and Hammarling, S
Cache Efficient Bidiagonalization Using BLAS 2.5 Operators Howell, G. W. and Demmel, J. W. and Fulton, C. T. and Hammarling, S. and Marmol, K. 2006 MIMS EPrint: 2006.56 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 Cache Efficient Bidiagonalization Using BLAS 2.5 Operators∗ G. W. Howell North Carolina State University Raleigh, North Carolina 27695 J. W. Demmel University of California, Berkeley Berkeley, California 94720 C. T. Fulton Florida Institute of Technology Melbourne, Florida 32901 S. Hammarling Numerical Algorithms Group Oxford, United Kingdom K. Marmol Harris Corporation Melbourne, Florida 32901 March 23, 2006 ∗ This research was supported by National Science Foundation Grant EIA-0103642. Abstract On cache based computer architectures using current standard al- gorithms, Householder bidiagonalization requires a significant portion of the execution time for computing matrix singular values and vectors. In this paper we reorganize the sequence of operations for Householder bidiagonalization of a general m × n matrix, so that two ( GEMV) vector-matrix multiplications can be done with one pass of the unre- duced trailing part of the matrix through cache. Two new BLAS 2.5 operations approximately cut in half the transfer of data from main memory to cache. We give detailed algorithm descriptions and com- pare timings with the current LAPACK bidiagonalization algorithm. 1 Introduction A primary constraint on execution speed is the “von Neumann” bottleneck in delivering data to the CPU. -
Applied Numerical Linear Algebra. Lecture 14
Algorithms for the Symmetric Eigenproblem: Divide-and-Conquer Algorithms for the Symmetric Eigenproblem: Bisection and Inverse Iteration Algorithms for the Symmetric Eigenproblem: Jacobi’s Method Algorithms for the Singular Value Decomposition(SVD) Applied Numerical Linear Algebra. Lecture 14. 1/54 Algorithms for the Symmetric Eigenproblem: Divide-and-Conquer Algorithms for the Symmetric Eigenproblem: Bisection and Inverse Iteration Algorithms for the Symmetric Eigenproblem: Jacobi’s Method Algorithms for the Singular Value Decomposition(SVD) 5.3.3. Divide-and-Conquer This method is the fastest now available if you want all eigenvalues and eigenvectors of a tridiagonal matrix whose dimension is larger than about 25. (The exact threshold depends on the computer.) It is quite subtle to implement in a numerically stable way. Indeed, although this method was first introduced in 1981 [J. J. M. Cuppen. A divide and conquer method for the symmetric tridiagonal eigenproblem. Numer. Math., 36:177-195, 1981], the ”right” implementation was not discovered until 1992 [M. Gu and S. Eisenstat. A stable algorithm for the rank-1 modification of the symmetric eigenproblem. Computer Science Dept. Report YALEU/DCS/RR-916, Yale University, September 1992;M. Gu and S. C. Eisenstat. A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. SIAM J. Matrix Anal Appl, 16:172-191, 1995]). 2/54 Algorithms for the Symmetric Eigenproblem: Divide-and-Conquer Algorithms for the Symmetric Eigenproblem: Bisection and Inverse Iteration Algorithms for the Symmetric Eigenproblem: Jacobi’s Method Algorithms for the Singular Value Decomposition(SVD) a1 b1 0 ... ... 0 b1 a2 b2 0 ... 0 ... ... ... ... ... ... ... am 1 bm 1 ... ... 0 − − 0 bm 1 am bm 0 0 T = − 0 0 b a b 0 m m+1 m+1 0 0 0 b .. -
Weighted Golub-Kahan-Lanczos Algorithms and Applications
Weighted Golub-Kahan-Lanczos algorithms and applications Hongguo Xu∗ and Hong-xiu Zhongy January 13, 2016 Abstract We present weighted Golub-Kahan-Lanczos algorithms for computing a bidiagonal form of a product of two invertible matrices. We give convergence analysis for the al- gorithms when they are used for computing extreme eigenvalues of a product of two symmetric positive definite matrices. We apply the algorithms for computing extreme eigenvalues of a double-sized matrix arising from linear response problem and provide convergence results. Numerical testing results are also reported. As another application we show a connection between the proposed algorithms and the conjugate gradient (CG) algorithm for positive definite linear systems. Keywords: Weighted Golub-Kahan-Lanczos algorithm, Eigenvalue, Eigenvector, Ritz value, Ritz vector, Linear response eigenvalue problem, Krylov subspace, Bidiagonal matrices. AMS subject classifications: 65F15, 15A18. 1 Introduction The Golub-Kahan bidiagonalization procedure is a fundamental tool in the singular value decomposition (SVD) method ([8]). Based on such a procedure, Golub-Kahan-Lanczos al- gorithms were developed in [15] that provide powerful Krylov subspace methods for solving large scale singular value and related eigenvalue problems, as well as least squares and saddle- point problems, [15, 16]. Later, weighted Golub-Kahan-Lanczos algorithms were introduced for solving generalized least squares and saddle-point problems, e.g., [1, 4]. In this paper we propose a type of weighted Golub-Kahan-Lanczos bidiagonalization al- gorithms. The proposed algorithms are similar to the generalized Golub-Kahan-Lanczos bidi- agonalization algorithms given in [1]. There, the algorithms are based on the Golub-Kahan bidiagonalizations of matrices of the form R−1AL−T , whereas ours are based the bidago- nalizations of RT L, where R; L are invertible matrices. -
The Singular Value Decomposition: Anatomy of Optimizing an Algorithm for Extreme Scale
The University of Manchester Research The singular value decomposition DOI: 10.1137/17M1117732 Document Version Final published version Link to publication record in Manchester Research Explorer Citation for published version (APA): Dongarra, J., Gates, M., Haidar, A., Kurzak, J., Luszczek, P., Tomov, S., & Yamazaki, I. (2018). The singular value decomposition: Anatomy of optimizing an algorithm for extreme scale. SIAM REVIEW, 60(4), 808-865. https://doi.org/10.1137/17M1117732 Published in: SIAM REVIEW Citing this paper Please note that where the full-text provided on Manchester Research Explorer is the Author Accepted Manuscript or Proof version this may differ from the final Published version. If citing, it is advised that you check and use the publisher's definitive version. General rights Copyright and moral rights for the publications made accessible in the Research Explorer are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Takedown policy If you believe that this document breaches copyright please refer to the University of Manchester’s Takedown Procedures [http://man.ac.uk/04Y6Bo] or contact [email protected] providing relevant details, so we can investigate your claim. Download date:06. Oct. 2021 SIAM REVIEW \bigcirc c 2018 Society for Industrial and Applied Mathematics Vol. 60, No. 4, pp. 808–865 The Singular Value Decomposition: Anatomy of Optimizing an Algorithm \ast for Extreme Scale Jack Dongarray z Mark Gates Azzam Haidarz Jakub Kurzakz Piotr Luszczekz Stanimire Tomovz Ichitaro Yamazakiz Abstract. The computation of the singular value decomposition, or SVD, has a long history with many improvements over the years, both in its implementations and algorithmically. -
Exploiting the Graphics Hardware to Solve Two Compute Intensive Problems: Singular Value Decomposition and Ray Tracing Parametric Patches
Exploiting the Graphics Hardware to solve two compute intensive problems: Singular Value Decomposition and Ray Tracing Parametric Patches Thesis submitted in partial fulfillment of the requirements for the degree of Master of Science (by Research) in Computer Science by Sheetal Lahabar 200402024 [email protected] [email protected] Center for Visual Information Technology International Institute of Information Technology Hyderabad - 500032, INDIA August 2010 Copyright c Sheetal Lahabar, 2010 All Rights Reserved International Institute of Information Technology Hyderabad, India CERTIFICATE It is certified that the work contained in this thesis, titled “Exploiting the Graphics Hardware to solve two compute intensive problems: Singular Value Decomposition and Ray Tracing Parametric Patches ” by Ms. Sheetal Lahabar (200402024), has been carried out under my supervision and is not submitted elsewhere for a degree. Date Adviser: Prof. P. J. Narayanan To Lata, My Mother Abstract The rapid increase in the performance of graphics hardware have made the GPU a strong candi- date for performing many compute intensive tasks, especially many data-parallel tasks. Off-the-shelf GPUs are rated at upwards of 2 TFLOPs of peak power today. The high compute power of the modern Graphics Processing Units (GPUs) has been exploited in a wide variety of domain other than graphics. They have proven to be powerful computing work horses in various fields. We leverage the processing power of GPUs to solve two highly compute intensive problems: Singular Value Decomposition and Ray Tracing Parametric Patches. Singular Value Decomposition is an important matrix factorization technique used to decompose a matrix into several component matrices. It is used in a wide variety of application related to signal processing, scientific computing, etc. -
New Fast and Accurate Jacobi Svd Algorithm: I. Lapack Working Note 169
NEW FAST AND ACCURATE JACOBI SVD ALGORITHM: I. LAPACK WORKING NOTE 169 ZLATKO DRMACˇ ∗ AND KRESIMIRˇ VESELIC´ † Abstract. This paper is the result of contrived efforts to break the barrier between numerical accuracy and run time efficiency in computing the fundamental decomposition of numerical linear algebra – the singular value decomposition (SVD) of a general dense matrix. It is an unfortunate fact that the numerically most accurate one–sided Jacobi SVD algorithm is several times slower than generally less accurate bidiagonalization based methods such as the QR or the divide and conquer algorithm. Despite its sound numerical qualities, the Jacobi SVD is not included in the state of the art matrix computation libraries and it is even considered obsolete by some leading researches. Our quest for a highly accurate and efficient SVD algorithm has led us to a new, superior variant of the Jacobi algorithm. The new algorithm has inherited all good high accuracy properties, and it outperforms not only the best implementations of the one–sided Jacobi algorithm but also the QR algorithm. Moreover, it seems that the potential of the new approach is yet to be fully exploited. Key words. Jacobi method, singular value decomposition, eigenvalues AMS subject classifications. 15A09, 15A12, 15A18, 15A23, 65F15, 65F22, 65F35 1. Introduction. In der Theorie der S¨acularst¨orungen und der kleinen Oscil- lationen wird man auf ein System line¨arer Gleichungen gef¨urt, in welchem die Co- efficienten der verschiedenen Unbekanten in Bezug auf die Diagonale symmetrisch sind, die ganz constanten Glieder fehlen und zu allen in der Diagonale befindlichen Coefficienten noch dieselbe Gr¨oße x addirt ist. -
D2.9 Novel SVD Algorithms
H2020–FETHPC–2014: GA 671633 D2.9 Novel SVD Algorithms April 2019 NLAFET D2.9: Novel SVD Document information Scheduled delivery 2019-04-30 Actual delivery 2019-04-29 Version 2.0 Responsible partner UNIMAN Dissemination level PU — Public Revision history Date Editor Status Ver. Changes 2019-04-29 Pierre Blanchard Complete 2.0 Revision for final version 2019-04-26 Pierre Blanchard Draft 1.1 Feedback from reviewers added. 2019-04-18 Pierre Blanchard Draft 1.0 Feedback from reviewers added. 2019-04-11 Pierre Blanchard Draft 0.1 Initial version of document pro- duced. Author(s) Pierre Blanchard (UNIMAN) Mawussi Zounon (UNIMAN) Jack Dongarra (UNIMAN) Nick Higham (UNIMAN) Internal reviewers Nicholas Higham (UNIMAN) Sven Hammarling (UNIMAN) Bo Kågström (UMU) Carl Christian Kjelgaard Mikkelsen (UMU) Copyright This work is c by the NLAFET Consortium, 2015–2018. Its duplication is allowed only for personal, educational, or research uses. Acknowledgements This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the grant agreement number 671633. http://www.nlafet.eu/ 1/28 NLAFET D2.9: Novel SVD Table of Contents 1 Introduction 4 2 Review of state-of-the-art SVD algorithms 5 2.1 Notation . .5 2.2 Computing the SVD . .5 2.3 Symmetric eigenvalue solvers . .6 2.4 Flops count . .7 3 Polar Decomposition based SVD 7 3.1 Polar Decomposition . .8 3.2 Applications . .8 3.3 The Polar–SVD Algorithm . .8 3.4 The QR trick . .9 4 Optimized Polar Decomposition algorithms 9 4.1 Computing the Polar Factor Up ....................... -
Arxiv:1301.5758V3
Generalized Householder Transformations for the Complex Symmetric Eigenvalue Problem J. H. Noble,1 M. Lubasch,2 and U. D. Jentschura1, 3 1Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409, USA 2Max–Planck–Institut f¨ur Quantenoptik, Hans–Kopfermann–Straße 1, 85748 Garching, Germany 3MTA–DE Particle Physics Research Group, P.O.Box 51, H-4001 Debrecen, Hungary We present an intuitive and scalable algorithm for the diagonalization of complex symmet- ric matrices, which arise from the projection of pseudo–Hermitian and complex scaled Hamilto- nians onto a suitable basis set of “trial” states. The algorithm diagonalizes complex and sym- metric (non–Hermitian) matrices and is easily implemented in modern computer languages. It is based on generalized Householder transformations and relies on iterative similarity transformations ′ − T → T = QT T Q, where Q is a complex and orthogonal, but not unitary, matrix, i.e, QT = Q 1 − but Q+ 6= Q 1. We present numerical reference data to support the scalability of the algorithm. We construct the generalized Householder transformations from the notion that the conserved scalar product of eigenstates Ψn and Ψm of a pseudo–Hermitian quantum mechanical Hamiltonian can be reformulated in terms of the generalized indefinite inner product R dx Ψn(x,t)Ψm(x,t), where the integrand is locally defined, and complex conjugation is avoided. A few example calculations are described which illustrate the physical origin of the ideas used in the construction of the algorithm. PACS numbers: 03.65.Ge, 02.60.Lj, 11.15.Bt, 11.30.Er, 02.60.-x, 02.60.Dc I.