Randomized Matrix Decompositions Using R
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Life As a Developer of Numerical Software
A Brief History of Numerical Libraries Sven Hammarling NAG Ltd, Oxford & University of Manchester First – Something about Jack Jack’s thesis (August 1980) 30 years ago! TOMS Algorithm 589 Small Selection of Jack’s Projects • Netlib and other software repositories • NA Digest and na-net • PVM and MPI • TOP 500 and computer benchmarking • NetSolve and other distributed computing projects • Numerical linear algebra Onto the Rest of the Talk! Rough Outline • History and influences • Fortran • Floating Point Arithmetic • Libraries and packages • Proceedings and Books • Summary Ada Lovelace (Countess Lovelace) Born Augusta Ada Byron 1815 – 1852 The language Ada was named after her “Is thy face like thy mother’s, my fair child! Ada! sole daughter of my house and of my heart? When last I saw thy young blue eyes they smiled, And then we parted,-not as now we part, but with a hope” Childe Harold’s Pilgramage, Lord Byron Program for the Bernoulli Numbers Manchester Baby, 21 June 1948 (Replica) 19 Kilburn/Tootill Program to compute the highest proper factor 218 218 took 52 minutes 1.5 million instructions 3.5 million store accesses First published numerical library, 1951 First use of the word subroutine? Quality Numerical Software • Should be: – Numerically stable, with measures of quality of solution – Reliable and robust – Accompanied by test software – Useful and user friendly with example programs – Fully documented – Portable – Efficient “I have little doubt that about 80 per cent. of all the results printed from the computer are in error to a much greater extent than the user would believe, ...'' Leslie Fox, IMA Bulletin, 1971 “Giving business people spreadsheets is like giving children circular saws. -
Numerical and Parallel Libraries
Numerical and Parallel Libraries Uwe Küster University of Stuttgart High-Performance Computing-Center Stuttgart (HLRS) www.hlrs.de Uwe Küster Slide 1 Höchstleistungsrechenzentrum Stuttgart Numerical Libraries Public Domain commercial vendor specific Uwe Küster Slide 2 Höchstleistungsrechenzentrum Stuttgart 33. — Numerical and Parallel Libraries — 33. 33-1 Overview • numerical libraries for linear systems – dense – sparse •FFT • support for parallelization Uwe Küster Slide 3 Höchstleistungsrechenzentrum Stuttgart Public Domain Lapack-3 linear equations, eigenproblems BLAS fast linear kernels Linpack linear equations Eispack eigenproblems Slatec old library, large functionality Quadpack numerical quadrature Itpack sparse problems pim linear systems PETSc linear systems Netlib Server best server http://www.netlib.org/utk/papers/iterative-survey/packages.html Uwe Küster Slide 4 Höchstleistungsrechenzentrum Stuttgart 33. — Numerical and Parallel Libraries — 33. 33-2 netlib server for all public domain numerical programs and libraries http://www.netlib.org Uwe Küster Slide 5 Höchstleistungsrechenzentrum Stuttgart Contents of netlib access aicm alliant amos ampl anl-reports apollo atlas benchmark bib bibnet bihar blacs blas blast bmp c c++ cephes chammp cheney-kincaid clapack commercial confdb conformal contin control crc cumulvs ddsv dierckx diffpack domino eispack elefunt env f2c fdlibm fftpack fishpack fitpack floppy fmm fn fortran fortran-m fp gcv gmat gnu go graphics harwell hence hompack hpf hypercube ieeecss ijsa image intercom itpack -
Jack Dongarra: Supercomputing Expert and Mathematical Software Specialist
Biographies Jack Dongarra: Supercomputing Expert and Mathematical Software Specialist Thomas Haigh University of Wisconsin Editor: Thomas Haigh Jack J. Dongarra was born in Chicago in 1950 to a he applied for an internship at nearby Argonne National family of Sicilian immigrants. He remembers himself as Laboratory as part of a program that rewarded a small an undistinguished student during his time at a local group of undergraduates with college credit. Dongarra Catholic elementary school, burdened by undiagnosed credits his success here, against strong competition from dyslexia.Onlylater,inhighschool,didhebeginto students attending far more prestigious schools, to Leff’s connect material from science classes with his love of friendship with Jim Pool who was then associate director taking machines apart and tinkering with them. Inspired of the Applied Mathematics Division at Argonne.2 by his science teacher, he attended Chicago State University and majored in mathematics, thinking that EISPACK this would combine well with education courses to equip Dongarra was supervised by Brian Smith, a young him for a high school teaching career. The first person in researcher whose primary concern at the time was the his family to go to college, he lived at home and worked lab’s EISPACK project. Although budget cuts forced Pool to in a pizza restaurant to cover the cost of his education.1 make substantial layoffs during his time as acting In 1972, during his senior year, a series of chance director of the Applied Mathematics Division in 1970– events reshaped Dongarra’s planned career. On the 1971, he had made a special effort to find funds to suggestion of Harvey Leff, one of his physics professors, protect the project and hire Smith. -
Randnla: Randomized Numerical Linear Algebra
review articles DOI:10.1145/2842602 generation mechanisms—as an algo- Randomization offers new benefits rithmic or computational resource for the develop ment of improved algo- for large-scale linear algebra computations. rithms for fundamental matrix prob- lems such as matrix multiplication, BY PETROS DRINEAS AND MICHAEL W. MAHONEY least-squares (LS) approximation, low- rank matrix approxi mation, and Lapla- cian-based linear equ ation solvers. Randomized Numerical Linear Algebra (RandNLA) is an interdisci- RandNLA: plinary research area that exploits randomization as a computational resource to develop improved algo- rithms for large-scale linear algebra Randomized problems.32 From a foundational per- spective, RandNLA has its roots in theoretical computer science (TCS), with deep connections to mathemat- Numerical ics (convex analysis, probability theory, metric embedding theory) and applied mathematics (scientific computing, signal processing, numerical linear Linear algebra). From an applied perspec- tive, RandNLA is a vital new tool for machine learning, statistics, and data analysis. Well-engineered implemen- Algebra tations have already outperformed highly optimized software libraries for ubiquitous problems such as least- squares,4,35 with good scalability in par- allel and distributed environments. 52 Moreover, RandNLA promises a sound algorithmic and statistical foundation for modern large-scale data analysis. MATRICES ARE UBIQUITOUS in computer science, statistics, and applied mathematics. An m × n key insights matrix can encode information about m objects ˽ Randomization isn’t just used to model noise in data; it can be a powerful (each described by n features), or the behavior of a computational resource to develop discretized differential operator on a finite element algorithms with improved running times and stability properties as well as mesh; an n × n positive-definite matrix can encode algorithms that are more interpretable in the correlations between all pairs of n objects, or the downstream data science applications. -
Solving a Low-Rank Factorization Model for Matrix Completion by a Nonlinear Successive Over-Relaxation Algorithm
SOLVING A LOW-RANK FACTORIZATION MODEL FOR MATRIX COMPLETION BY A NONLINEAR SUCCESSIVE OVER-RELAXATION ALGORITHM ZAIWEN WEN †, WOTAO YIN ‡, AND YIN ZHANG § Abstract. The matrix completion problem is to recover a low-rank matrix from a subset of its entries. The main solution strategy for this problem has been based on nuclear-norm minimization which requires computing singular value decompositions – a task that is increasingly costly as matrix sizes and ranks increase. To improve the capacity of solving large-scale problems, we propose a low-rank factorization model and construct a nonlinear successive over-relaxation (SOR) algorithm that only requires solving a linear least squares problem per iteration. Convergence of this nonlinear SOR algorithm is analyzed. Numerical results show that the algorithm can reliably solve a wide range of problems at a speed at least several times faster than many nuclear-norm minimization algorithms. Key words. Matrix Completion, alternating minimization, nonlinear GS method, nonlinear SOR method AMS subject classifications. 65K05, 90C06, 93C41, 68Q32 1. Introduction. The problem of minimizing the rank of a matrix arises in many applications, for example, control and systems theory, model reduction and minimum order control synthesis [20], recovering shape and motion from image streams [25, 32], data mining and pattern recognitions [6] and machine learning such as latent semantic indexing, collaborative prediction and low-dimensional embedding. In this paper, we consider the Matrix Completion (MC) problem of finding a lowest-rank matrix given a subset of its entries, that is, (1.1) min rank(W ), s.t. Wij = Mij , ∀(i,j) ∈ Ω, W ∈Rm×n where rank(W ) denotes the rank of W , and Mi,j ∈ R are given for (i,j) ∈ Ω ⊂ {(i,j) : 1 ≤ i ≤ m, 1 ≤ j ≤ n}. -
Low-Rank Incremental Methods for Computing Dominant Singular Subspaces
Low-Rank Incremental Methods for Computing Dominant Singular Subspaces C. G. Baker∗ K. A. Gallivan∗ P. Van Dooren† Abstract This paper describes a generic low-rank incremental method for computing the dominant singular triplets of a matrix via a single pass through the matrix. This work unifies several efforts previously described in the literature. We tie the operation of the proposed method to a particular optimization- based eigensolver. This allows the description of novel methods exploiting multiple passes through the matrix. 1 Introduction Given a matrix A Rm×n,m n, the Singular Value Decomposition (SVD) of A is: ∈ ≥ Σ A = U V T , 0 where U and V are m m and n n orthogonal matrices, respectively, and Σ is a diagonal matrix whose × × elements σ1,...,σn are real, non-negative and non-increasing. The σi are the singular values of A, and the first n columns of U and V are the left and right singular vectors of A, respectively. Given k n, the singular vectors associated with the largest k singular values of A are the rank-k dominant singular≤ vectors. The subspaces associated with these vectors are the rank-k dominant singular subspaces. The components of the SVD are optimal in many respects [1], and these properties result in the occurrence of the SVD in numerous analyses, e.g., principal component analysis, proper orthogonal decomposition, the Karhunen-Loeve transform. These applications of the SVD are frequently used in problems related to, e.g., signal processing, model reduction of dynamical systems, image and face recognition, and information retrieval. -
Block Matrices in Linear Algebra
PRIMUS Problems, Resources, and Issues in Mathematics Undergraduate Studies ISSN: 1051-1970 (Print) 1935-4053 (Online) Journal homepage: https://www.tandfonline.com/loi/upri20 Block Matrices in Linear Algebra Stephan Ramon Garcia & Roger A. Horn To cite this article: Stephan Ramon Garcia & Roger A. Horn (2020) Block Matrices in Linear Algebra, PRIMUS, 30:3, 285-306, DOI: 10.1080/10511970.2019.1567214 To link to this article: https://doi.org/10.1080/10511970.2019.1567214 Accepted author version posted online: 05 Feb 2019. Published online: 13 May 2019. Submit your article to this journal Article views: 86 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=upri20 PRIMUS, 30(3): 285–306, 2020 Copyright # Taylor & Francis Group, LLC ISSN: 1051-1970 print / 1935-4053 online DOI: 10.1080/10511970.2019.1567214 Block Matrices in Linear Algebra Stephan Ramon Garcia and Roger A. Horn Abstract: Linear algebra is best done with block matrices. As evidence in sup- port of this thesis, we present numerous examples suitable for classroom presentation. Keywords: Matrix, matrix multiplication, block matrix, Kronecker product, rank, eigenvalues 1. INTRODUCTION This paper is addressed to instructors of a first course in linear algebra, who need not be specialists in the field. We aim to convince the reader that linear algebra is best done with block matrices. In particular, flexible thinking about the process of matrix multiplication can reveal concise proofs of important theorems and expose new results. Viewing linear algebra from a block-matrix perspective gives an instructor access to use- ful techniques, exercises, and examples. -
Evolving Software Repositories
1 Evolving Software Rep ositories http://www.netli b.org/utk/pro ject s/esr/ Jack Dongarra UniversityofTennessee and Oak Ridge National Lab oratory Ron Boisvert National Institute of Standards and Technology Eric Grosse AT&T Bell Lab oratories 2 Pro ject Fo cus Areas NHSE Overview Resource Cataloging and Distribution System RCDS Safe execution environments for mobile co de Application-l evel and content-oriented to ols Rep ository interop erabili ty Distributed, semantic-based searching 3 NHSE National HPCC Software Exchange NASA plus other agencies funded CRPC pro ject Center for ResearchonParallel Computation CRPC { Argonne National Lab oratory { California Institute of Technology { Rice University { Syracuse University { UniversityofTennessee Uniform interface to distributed HPCC software rep ositories Facilitation of cross-agency and interdisciplinary software reuse Material from ASTA, HPCS, and I ITA comp onents of the HPCC program http://www.netlib.org/nhse/ 4 Goals: Capture, preserve and makeavailable all software and software- related artifacts pro duced by the federal HPCC program. Soft- ware related artifacts include algorithms, sp eci cations, designs, do cumentation, rep ort, ... Promote formation, growth, and interop eration of discipline-oriented rep ositories that organize, evaluate, and add value to individual contributions. Employ and develop where necessary state-of-the-art technologies for assisting users in nding, understanding, and using HPCC software and technologies. 5 Bene ts: 1. Faster development of high-quality software so that scientists can sp end less time writing and debugging programs and more time on research problems. 2. Less duplication of software development e ort by sharing of soft- ware mo dules. -
2 Accessing Matlab at ER4
Department of Electrical Engineering EE281 Intro duction to MATLAB on the Region IV Computing Facilities 1 What is Matlab? Matlab is a high-p erformance interactive software package for scienti c and enginnering numeric computation. Matlab integrates numerical analysis, matrix computation, signal pro cessing, and graphics in an easy-to-use envi- ronment without traditional programming. The name Matlab stands for matrix laboratory. Matlab was originally written to provide easy access to the matrix software develop ed by the LIN- PACK and EISPACK pro jects. Matlab is an interactive system whose basic data element is a matrix that do es not require dimensioning. Furthermore, problem solutions are expressed in Matlab almost exactly as they are written mathematically. Matlab has evolved over the years with inputs from many users. Matlab Toolboxes are sp ecialized collections of Matlab les designed for solving particular classes of functions. Currently available to olb oxes include: signal pro cessing control system optimization neural networks system identi cation robust control analysis splines symb olic mathematics image pro cessing statistics 2 Accessing Matlab at ER4 Sit down at a workstation. Log on. login: smithj <press return> password: 1234js <press return> Use the mouse button to view the Ro ot Menu. Drag the cursor to \Design Tools" then to \Matlab." Release the mouse button. You are now running the Matlab interactive software package. The prompt is a double \greater than" sign. You may use emacs line editor commands and the arrow keys while typing in Matlab. 1 3 An Intro ductory Demonstration Execute the following command to view aquick intro duction to Matlab. -
CS 267 Dense Linear Algebra: History and Structure, Parallel Matrix
Quick review of earlier lecture •" What do you call •" A program written in PyGAS, a Global Address CS 267 Space language based on Python… Dense Linear Algebra: •" That uses a Monte Carlo simulation algorithm to approximate π … History and Structure, •" That has a race condition, so that it gives you a Parallel Matrix Multiplication" different funny answer every time you run it? James Demmel! Monte - π - thon ! www.cs.berkeley.edu/~demmel/cs267_Spr16! ! 02/25/2016! CS267 Lecture 12! 1! 02/25/2016! CS267 Lecture 12! 2! Outline Outline •" History and motivation •" History and motivation •" What is dense linear algebra? •" What is dense linear algebra? •" Why minimize communication? •" Why minimize communication? •" Lower bound on communication •" Lower bound on communication •" Parallel Matrix-matrix multiplication •" Parallel Matrix-matrix multiplication •" Attaining the lower bound •" Attaining the lower bound •" Other Parallel Algorithms (next lecture) •" Other Parallel Algorithms (next lecture) 02/25/2016! CS267 Lecture 12! 3! 02/25/2016! CS267 Lecture 12! 4! CS267 Lecture 2 1 Motifs What is dense linear algebra? •" Not just matmul! The Motifs (formerly “Dwarfs”) from •" Linear Systems: Ax=b “The Berkeley View” (Asanovic et al.) •" Least Squares: choose x to minimize ||Ax-b||2 Motifs form key computational patterns •" Overdetermined or underdetermined; Unconstrained, constrained, or weighted •" Eigenvalues and vectors of Symmetric Matrices •" Standard (Ax = λx), Generalized (Ax=λBx) •" Eigenvalues and vectors of Unsymmetric matrices -
Comparison of Numerical Methods and Open-Source Libraries for Eigenvalue Analysis of Large-Scale Power Systems
applied sciences Article Comparison of Numerical Methods and Open-Source Libraries for Eigenvalue Analysis of Large-Scale Power Systems Georgios Tzounas , Ioannis Dassios * , Muyang Liu and Federico Milano School of Electrical and Electronic Engineering, University College Dublin, Belfield, Dublin 4, Ireland; [email protected] (G.T.); [email protected] (M.L.); [email protected] (F.M.) * Correspondence: [email protected] Received: 30 September 2020; Accepted: 24 October 2020; Published: 28 October 2020 Abstract: This paper discusses the numerical solution of the generalized non-Hermitian eigenvalue problem. It provides a comprehensive comparison of existing algorithms, as well as of available free and open-source software tools, which are suitable for the solution of the eigenvalue problems that arise in the stability analysis of electric power systems. The paper focuses, in particular, on methods and software libraries that are able to handle the large-scale, non-symmetric matrices that arise in power system eigenvalue problems. These kinds of eigenvalue problems are particularly difficult for most numerical methods to handle. Thus, a review and fair comparison of existing algorithms and software tools is a valuable contribution for researchers and practitioners that are interested in power system dynamic analysis. The scalability and performance of the algorithms and libraries are duly discussed through case studies based on real-world electrical power networks. These are a model of the All-Island Irish Transmission System with 8640 variables; and, a model of the European Network of Transmission System Operators for Electricity, with 146,164 variables. Keywords: eigenvalue analysis; large non-Hermitian matrices; numerical methods; open-source libraries 1. -
Using MATLAB Version 5 How to Contact the Mathworks
MATLAB® The Language of Technical Computing Computation Visualization Programming Using MATLAB Version 5 How to Contact The MathWorks: 508-647-7000 Phone 508-647-7001 Fax The MathWorks, Inc. Mail 24 Prime Park Way Natick, MA 01760-1500 http://www.mathworks.com Web ftp.mathworks.com Anonymous FTP server comp.soft-sys.matlab Newsgroup [email protected] Technical support [email protected] Product enhancement suggestions [email protected] Bug reports [email protected] Documentation error reports [email protected] Subscribing user registration [email protected] Order status, license renewals, passcodes [email protected] Sales, pricing, and general information Using MATLAB COPYRIGHT 1984 - 1999 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or repro- duced in any form without prior written consent from The MathWorks, Inc. U.S. GOVERNMENT: If Licensee is acquiring the Programs on behalf of any unit or agency of the U.S. Government, the following shall apply: (a) For units of the Department of Defense: the Government shall have only the rights specified in the license under which the commercial computer software or commercial software documentation was obtained, as set forth in subparagraph (a) of the Rights in Commercial Computer Software or Commercial Software Documentation Clause at DFARS 227.7202-3, therefore the rights set forth herein shall apply; and (b) For any other unit or agency: NOTICE: Notwithstanding any other lease or license agreement that may pertain to, or accompany the delivery of, the computer software and accompanying documentation, the rights of the Government regarding its use, reproduction, and disclo- sure are as set forth in Clause 52.227-19 (c)(2) of the FAR.