The Linear Algebra Mapping Problem 3
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Armadillo C++ Library
Armadillo: a template-based C++ library for linear algebra Conrad Sanderson and Ryan Curtin Abstract The C++ language is often used for implementing functionality that is performance and/or resource sensitive. While the standard C++ library provides many useful algorithms (such as sorting), in its current form it does not provide direct handling of linear algebra (matrix maths). Armadillo is an open source linear algebra library for the C++ language, aiming towards a good balance between speed and ease of use. Its high-level application programming interface (function syntax) is deliberately similar to the widely used Matlab and Octave languages [4], so that mathematical operations can be expressed in a familiar and natural manner. The library is useful for algorithm development directly in C++, or relatively quick conversion of research code into production environments. Armadillo provides efficient objects for vectors, matrices and cubes (third order tensors), as well as over 200 associated functions for manipulating data stored in the objects. Integer, floating point and complex numbers are supported, as well as dense and sparse storage formats. Various matrix factorisations are provided through integration with LAPACK [3], or one of its high performance drop-in replacements such as Intel MKL [6] or OpenBLAS [9]. It is also possible to use Armadillo in conjunction with NVBLAS to obtain GPU-accelerated matrix multiplication [7]. Armadillo is used as a base for other open source projects, such as MLPACK, a C++ library for machine learning and pattern recognition [2], and RcppArmadillo, a bridge between the R language and C++ in order to speed up computations [5]. -
Rcppmlpack: R Integration with MLPACK Using Rcpp
RcppMLPACK: R integration with MLPACK using Rcpp Qiang Kou [email protected] April 20, 2020 1 MLPACK MLPACK[1] is a C++ machine learning library with emphasis on scalability, speed, and ease-of-use. It outperforms competing machine learning libraries by large margins, for detailed results of benchmarking, please refer to Ref[1]. 1.1 Input/Output using Armadillo MLPACK uses Armadillo as input and output, which provides vector, matrix and cube types (integer, floating point and complex numbers are supported)[2]. By providing Matlab-like syntax and various matrix decompositions through integration with LAPACK or other high performance libraries (such as Intel MKL, AMD ACML, or OpenBLAS), Armadillo keeps a good balance between speed and ease of use. Armadillo is widely used in C++ libraries like MLPACK. However, there is one point we need to pay attention to: Armadillo matrices in MLPACK are stored in a column-major format for speed. That means observations are stored as columns and dimensions as rows. So when using MLPACK, additional transpose may be needed. 1.2 Simple API Although MLPACK relies on template techniques in C++, it provides an intuitive and simple API. For the standard usage of methods, default arguments are provided. The MLPACK paper[1] provides a good example: standard k-means clustering in Euclidean space can be initialized like below: 1 KMeans<> k(); Listing 1: k-means with all default arguments If we want to use Manhattan distance, a different cluster initialization policy, and allow empty clusters, the object can be initialized with additional arguments: 1 KMeans<ManhattanDistance, KMeansPlusPlusInitialization, AllowEmptyClusters> k(); Listing 2: k-means with additional arguments 1 1.3 Available methods in MLPACK Commonly used machine learning methods are all implemented in MLPACK. -
Release 0.11 Todd Gamblin
Spack Documentation Release 0.11 Todd Gamblin Feb 07, 2018 Basics 1 Feature Overview 3 1.1 Simple package installation.......................................3 1.2 Custom versions & configurations....................................3 1.3 Customize dependencies.........................................4 1.4 Non-destructive installs.........................................4 1.5 Packages can peacefully coexist.....................................4 1.6 Creating packages is easy........................................4 2 Getting Started 7 2.1 Prerequisites...............................................7 2.2 Installation................................................7 2.3 Compiler configuration..........................................9 2.4 Vendor-Specific Compiler Configuration................................ 13 2.5 System Packages............................................. 16 2.6 Utilities Configuration.......................................... 18 2.7 GPG Signing............................................... 20 2.8 Spack on Cray.............................................. 21 3 Basic Usage 25 3.1 Listing available packages........................................ 25 3.2 Installing and uninstalling........................................ 42 3.3 Seeing installed packages........................................ 44 3.4 Specs & dependencies.......................................... 46 3.5 Virtual dependencies........................................... 50 3.6 Extensions & Python support...................................... 53 3.7 Filesystem requirements........................................ -
Rcppeigen-Intro
Fast and Elegant Numerical Linear Algebra Using the RcppEigen Package Douglas Bates Dirk Eddelbuettel University of Wisconsin-Madison Debian Project Abstract The RcppEigen package provides access from R (R Core Team 2018a) to the Eigen (Guennebaud, Jacob et al. 2012) C++ template library for numerical linear algebra. Rcpp (Eddelbuettel et al. 2020a; Eddelbuettel 2013) classes and specializations of the C++ templated functions as and wrap from Rcpp provide the \glue" for passing objects from R to C++ and back. Several introductory examples are presented. This is followed by an in-depth discussion of various available approaches for solving least-squares problems, including rank-revealing methods, concluding with an empirical run-time comparison. Last but not least, sparse matrix methods are discussed. Keywords: linear algebra, template programming, R, C++, Rcpp. This vignette corresponds to a paper published in the Journal of Statistical Software. Currently still identical to the paper, this vignette version may over time receive minor updates. For citations, please use Bates and Eddelbuettel(2013) as provided by citation("RcppEigen"). This version corresponds to RcppEigen version 0.3.3.9.1 and was typeset on December 17, 2020. 1. Introduction Linear algebra is an essential building block of statistical computing. Operations such as matrix decompositions, linear program solvers, and eigenvalue/eigenvector computations are used in many estimation and analysis routines. As such, libraries supporting linear algebra have long been provided by statistical programmers for different programming languages and environments. Because it is object-oriented, C++, one of the central modern languages for numerical and statistical computing, is particularly effective at representing matrices, vectors and decompositions, and numerous class libraries providing linear algebra routines have been written over the years. -
Linear Algebra Libraries
Linear Algebra Libraries Claire Mouton [email protected] March 2009 Contents I Requirements 3 II CPPLapack 4 III Eigen 5 IV Flens 7 V Gmm++ 9 VI GNU Scientific Library (GSL) 11 VII IT++ 12 VIII Lapack++ 13 arXiv:1103.3020v1 [cs.MS] 15 Mar 2011 IX Matrix Template Library (MTL) 14 X PETSc 16 XI Seldon 17 XII SparseLib++ 18 XIII Template Numerical Toolkit (TNT) 19 XIV Trilinos 20 1 XV uBlas 22 XVI Other Libraries 23 XVII Links and Benchmarks 25 1 Links 25 2 Benchmarks 25 2.1 Benchmarks for Linear Algebra Libraries . ....... 25 2.2 BenchmarksincludingSeldon . 26 2.2.1 BenchmarksforDenseMatrix. 26 2.2.2 BenchmarksforSparseMatrix . 29 XVIII Appendix 30 3 Flens Overloaded Operator Performance Compared to Seldon 30 4 Flens, Seldon and Trilinos Content Comparisons 32 4.1 Available Matrix Types from Blas (Flens and Seldon) . ........ 32 4.2 Available Interfaces to Blas and Lapack Routines (Flens and Seldon) . 33 4.3 Available Interfaces to Blas and Lapack Routines (Trilinos) ......... 40 5 Flens and Seldon Synoptic Comparison 41 2 Part I Requirements This document has been written to help in the choice of a linear algebra library to be included in Verdandi, a scientific library for data assimilation. The main requirements are 1. Portability: Verdandi should compile on BSD systems, Linux, MacOS, Unix and Windows. Beyond the portability itself, this often ensures that most compilers will accept Verdandi. An obvious consequence is that all dependencies of Verdandi must be portable, especially the linear algebra library. 2. High-level interface: the dependencies should be compatible with the building of the high-level interface (e. -
C++ API for BLAS and LAPACK
2 C++ API for BLAS and LAPACK Mark Gates ICL1 Piotr Luszczek ICL Ahmad Abdelfattah ICL Jakub Kurzak ICL Jack Dongarra ICL Konstantin Arturov Intel2 Cris Cecka NVIDIA3 Chip Freitag AMD4 1Innovative Computing Laboratory 2Intel Corporation 3NVIDIA Corporation 4Advanced Micro Devices, Inc. February 21, 2018 This research was supported by the Exascale Computing Project (17-SC-20-SC), a collaborative effort of two U.S. Department of Energy organizations (Office of Science and the National Nuclear Security Administration) responsible for the planning and preparation of a capable exascale ecosystem, including software, applications, hardware, advanced system engineering and early testbed platforms, in support of the nation's exascale computing imperative. Revision Notes 06-2017 first publication 02-2018 • copy editing, • new cover. 03-2018 • adding a section about GPU support, • adding Ahmad Abdelfattah as an author. @techreport{gates2017cpp, author={Gates, Mark and Luszczek, Piotr and Abdelfattah, Ahmad and Kurzak, Jakub and Dongarra, Jack and Arturov, Konstantin and Cecka, Cris and Freitag, Chip}, title={{SLATE} Working Note 2: C++ {API} for {BLAS} and {LAPACK}}, institution={Innovative Computing Laboratory, University of Tennessee}, year={2017}, month={June}, number={ICL-UT-17-03}, note={revision 03-2018} } i Contents 1 Introduction and Motivation1 2 Standards and Trends2 2.1 Programming Language Fortran............................ 2 2.1.1 FORTRAN 77.................................. 2 2.1.2 BLAST XBLAS................................. 4 2.1.3 Fortran 90.................................... 4 2.2 Programming Language C................................ 5 2.2.1 Netlib CBLAS.................................. 5 2.2.2 Intel MKL CBLAS................................ 6 2.2.3 Netlib lapack cwrapper............................. 6 2.2.4 LAPACKE.................................... 7 2.2.5 Next-Generation BLAS: \BLAS G2"..................... -
Implementation of High-Precision Computation Capabilities Into the Open-Source Dynamic Simulation Framework YADE
Implementation of high-precision computation capabilities into the open-source dynamic simulation framework YADE Janek Kozickia,b, Anton Gladkyc, Klaus Thoenid aFaculty of Applied Physics and Mathematics, Gda´nskUniversity of Technology, 80-233 Gda´nsk,Poland bAdvanced Materials Center, Gda´nskUniversity of Technology, 80-233 Gda´nsk,Poland cInstitute for Mineral Processing Machines and Recycling Systems Technology, TU Bergakademie Freiberg, 09599 Freiberg, Germany dCentre for Geotechnical Science and Engineering, The University of Newcastle, 2308 Callaghan, Australia Abstract This paper deals with the implementation of arbitrary precision calculations into the open-source discrete element framework YADE published under the GPL-2+ free software license. This new capability paves the way for the simulation framework to be used in many new fields such as quantum mechanics. The implementation details and associated gains in the accuracy of the results are discussed. Besides the ,,stan- dard" double (64 bits) type, support for the following high-precision types is added: long double (80 bits), float128 (128 bits), mpfr float backend (arbitrary precision) and cpp bin float (arbitrary precision). Benchmarks are performed to quantify the additional computational cost involved with the new supported precisions. Finally, a simple calculation of a chaotic triple pendulum is performed to demonstrate the new capabilities and the effect of different precisions on the simulation result. Keywords: arbitrary accuracy, multiple precision arithmetic, dynamical systems 05.45.-a, computational techniques 45.10.-b, numerical analysis 02.60.Lj, error theory 06.20.Dk 1. Introduction The advent of a new era of scientific computing has been predicted in the literature, one in which the numerical precision required for computation is as important as the algorithms and data structures in the program [6, 50, 51, 64]. -
Armadillo: an Open Source C++ Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments
Armadillo: An Open Source C++ Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments Conrad Sanderson http://conradsanderson.id.au Technical Report, NICTA, Australia http://nicta.com.au September 2010 (revised December 2011) Abstract In this report we provide an overview of the open source Armadillo C++ linear algebra library (matrix maths). The library aims to have a good balance between speed and ease of use, and is useful if C++ is the language of choice (due to speed and/or integration capabilities), rather than another language like Matlab or Octave. In particular, Armadillo can be used for fast prototyping and computationally intensive experiments, while at the same time allowing for relatively painless transition of research code into production environments. It is distributed under a license that is applicable in both open source and proprietary software development contexts. The library supports integer, floating point and complex numbers, as well as a subset of trigonometric and statistics functions. Various matrix decompositions are provided through optional integration with LAPACK, or one its high-performance drop-in replacements, such as MKL from Intel or ACML from AMD. A delayed evaluation approach is employed (during compile time) to combine several operations into one and reduce (or eliminate) the need for temporaries. This is accomplished through C++ template meta-programming. Performance comparisons suggest that the library is considerably faster than Matlab and Octave, as well as previous C++ libraries such as IT++ and Newmat. This report reflects a subset of the functionality present in Armadillo v2.4. Armadillo can be downloaded from: • http://arma.sourceforge.net If you use Armadillo in your research and/or software, we would appreciate a citation to this document. -
Armadillo: C++ Template Metaprogramming for Compile-Time Optimization of Linear Algebra
Armadillo: C++ Template Metaprogramming for Compile-Time Optimization of Linear Algebra Conrad Sanderson, Ryan Curtin Abstract Armadillo is a C++ library for linear algebra (matrix maths), enabling mathematical operations to be written in a manner similar to the widely used Matlab language, while at the same time providing efficient computation. It is used for relatively quick conversion of research code into production environments, and as a base for software such as MLPACK (a machine learning library). Through judicious use of template metaprogramming and operator overloading, linear algebra operations are automatically expressed as types, enabling compile-time parsing and evaluation of compound mathematical expressions. This in turn allows the compiler to produce specialized and optimized code. In this talk we cover the fundamentals of template metaprogramming that make compile-time optimizations possible, and demonstrate the internal workings of the Armadillo library. We show that the code automatically generated via Armadillo is comparable with custom C implementations. Armadillo is open source software, available at http://arma.sourceforge.net Presented at the Platform for Advanced Scientific Computing (PASC) Conference, Switzerland, 2017. References [1] C. Sanderson, R. Curtin. Armadillo: a template-based C++ library for linear algebra. Journal of Open Source Software, Vol. 1, pp. 26, 2016. [2] D. Eddelbuettel, C. Sanderson. RcppArmadillo: Accelerating R with High-Performance C++ Linear Algebra. Computational Statistics and Data Analysis, Vol. 71, 2014. [3] R. Curtin et al. MLPACK: A Scalable C++ Machine Learning Library. Journal of Machine Learning Research, Vol. 14, pp. 801-805, 2013. [4] C. Sanderson. An Open Source C++ Linear Algebra Library for Fast Prototyping and Computationally Intensive Experiments. -
A C++ Library for Multi-Index Array Computations
FASTMAT: A C++ LIBRARY FOR MULTI-INDEX ARRAY COMPUTATIONS Rodrigo R. Paza, Mario A. Stortia, Lisandro D. Dalcína, Hugo G. Castroa,b and Pablo A. Klerc aCentro Internacional de Métodos Computacionales en Ingeniería (CIMEC), INTEC(CONICET-UNL), Santa Fe, Argentina {mario.storti,dalcinl,rodrigo.r.paz}@gmail.com, [email protected], http://www.cimec.org.ar/mstorti bGrupo de Investigación en Mecánica de Fluidos, Universidad Tecnológica Nacional, Facultad Regional Resistencia, Chaco, Argentina. cCentral Division of Analytical Chemistry (ZCH). Forschungszentrum Jülich GmbH, Germany. Keywords: Multi-index Array Library; Finite Element Method; Finite Volume Method; Par- allel Computing. Abstract. In this paper we introduce and describe an efficient thread-safe matrix library for computing element/cell residuals and Jacobians in Finite Elements and Finite Volumes-like codes. The library provides a wide range of multi-index tensor operations that are normally used in scientific numerical computations. The library implements an algorithm for choosing the optimal computation order when a product of several tensors is performed (i.e., the so-called ‘multi-product’ operation). Another key- point of the FastMat approach is that some computations (for instance the optimal order in the multi- product operation mentioned before) are computed in the first iteration of the loop body and stored in a cache object, so that in the second and subsequent executions these computations are retrieved from the cache, and then not recomputed. The library is open source and freely available within the multi- physics parallel FEM code PETSc-FEM http://www.cimec.org.ar/petscfem and it can be exploited on distributed and shared memory architectures as well as in hybrid approaches. -
Numerical Linear Algebra
University of Alabama at Birmingham Department of Mathematics Numerical Linear Algebra Lecture Notes for MA 660 (1997{2014) Dr Nikolai Chernov Summer 2014 Contents 0. Review of Linear Algebra 1 0.1 Matrices and vectors . 1 0.2 Product of a matrix and a vector . 1 0.3 Matrix as a linear transformation . 2 0.4 Range and rank of a matrix . 2 0.5 Kernel (nullspace) of a matrix . 2 0.6 Surjective/injective/bijective transformations . 2 0.7 Square matrices and inverse of a matrix . 3 0.8 Upper and lower triangular matrices . 3 0.9 Eigenvalues and eigenvectors . 4 0.10 Eigenvalues of real matrices . 4 0.11 Diagonalizable matrices . 5 0.12 Trace . 5 0.13 Transpose of a matrix . 6 0.14 Conjugate transpose of a matrix . 6 0.15 Convention on the use of conjugate transpose . 6 1. Norms and Inner Products 7 1.1 Norms . 7 1.2 1-norm, 2-norm, and 1-norm . 7 1.3 Unit vectors, normalization . 8 1.4 Unit sphere, unit ball . 8 1.5 Images of unit spheres . 8 1.6 Frobenius norm . 8 1.7 Induced matrix norms . 9 1.8 Computation of kAk1, kAk2, and kAk1 .................... 9 1.9 Inequalities for induced matrix norms . 9 1.10 Inner products . 10 1.11 Standard inner product . 10 1.12 Cauchy-Schwarz inequality . 11 1.13 Induced norms . 12 1.14 Polarization identity . 12 1.15 Orthogonal vectors . 12 1.16 Pythagorean theorem . 12 1.17 Orthogonality and linear independence . 12 1.18 Orthonormal sets of vectors . -
Eigenvalues and Eigenvectors for 3×3 Symmetric Matrices: an Analytical Approach
Journal of Advances in Mathematics and Computer Science 35(7): 106-118, 2020; Article no.JAMCS.61833 ISSN: 2456-9968 (Past name: British Journal of Mathematics & Computer Science, Past ISSN: 2231-0851) Eigenvalues and Eigenvectors for 3×3 Symmetric Matrices: An Analytical Approach Abu Bakar Siddique1 and Tariq A. Khraishi1* 1Department of Mechanical Engineering, University of New Mexico, USA. Authors’ contributions This work was carried out in collaboration between both authors. ABS worked on equation derivation, plotting and writing. He also worked on the literature search. TK worked on equation derivation, writing and editing. Both authors contributed to conceptualization of the work. Both authors read and approved the final manuscript. Article Information DOI: 10.9734/JAMCS/2020/v35i730308 Editor(s): (1) Dr. Serkan Araci, Hasan Kalyoncu University, Turkey. Reviewers: (1) Mr. Ahmed A. Hamoud, Taiz University, Yemen. (2) Raja Sarath Kumar Boddu, Lenora College of Engineering, India. Complete Peer review History: http://www.sdiarticle4.com/review-history/61833 Received: 22 July 2020 Accepted: 29 September 2020 Original Research Article Published: 21 October 2020 _______________________________________________________________________________ Abstract Research problems are often modeled using sets of linear equations and presented as matrix equations. Eigenvalues and eigenvectors of those coupling matrices provide vital information about the dynamics/flow of the problems and so needs to be calculated accurately. Analytical solutions are advantageous over numerical solutions because numerical solutions are approximate in nature, whereas analytical solutions are exact. In many engineering problems, the dimension of the problem matrix is 3 and the matrix is symmetric. In this paper, the theory behind finding eigenvalues and eigenvectors for order 3×3 symmetric matrices is presented.