Implementation of High-Precision Computation Capabilities Into the Open-Source Dynamic Simulation Framework YADE
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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"..................... -
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 . -
Whitepaper: Linear Algebra Package for Qualcomm® Snapdragon
Qualcomm Technologies, Inc. Whitepaper: Linear Algebra Package for Qualcomm® Snapdragon™ Math Libraries LAPACK for Snapdragon Math Libraries brings advanced linear algebra functions to Android running on Snapdragon processor Nathan Whitehead March 2017 Qualcomm Technologies, Inc. 5775 Morehouse Drive San Diego, CA 92121 U.S.A. ©2017 Qualcomm Technologies, Inc. All Rights Reserved. Qualcomm Snapdragon is a product of Qualcomm Technologies, Inc. Qualcomm, Snapdragon and Snapdragon Flight are trademarks of Qualcomm Incorporated, registered in the United States and other countries. Other products and brand names may be trademarks of registered trademarks of their respective owners. Table of Contents Summary ........................................................................................................................................................................................ 4 What is LAPACK? ...................................................................................................................................................................... 4 Relationship between LAPACK and BLAS ..................................................................................................................................... 5 Example: Solving a linear system ........................................................................................................................................................ 5 Obstacles to LAPACK on mobile processors ...............................................................................................................7 -
A C++ Linear Algebra Library [
Eigen a c++ linear algebra library [http://eigen.tuxfamily.org] Gaël Guennebaud co-developed with Benoit Jacob (Mozilla) and a bunch of handful occasional contributors (PhDs) 2 History • Janv 2008: start of Eigen2 • Feb 2009: release Eigen 2.0 • Feb 2009: 1st annual meeting (3 days, ~10 people) • March 2011: release Eigen 3.0 • March 2011: 2nd annual meeting (3 days) 3 Facts • Active project with many users – Website: ~9000 unique visitors / month – Mailing list/Forum: • ~250 members, ~400 messages/month • Pure C++ template library – header only, no binary to compile, install... • Packaged by all Linux distributions • Opensource: LGPL3+ → easy to install & distribute 4 Large feature set • Modules: – Core • Matrix and array manipulation (MatLab, 1D & 2D only) • Basic linear algebra (~BLAS) – LU, Cholesky, QR, SVD, Eigenvalues • Matrix decompositions and linear solvers (~Lapack) – Geometry (transformations, …) – WIP modules: • Sparse matrices, Automatic differentiation, Non-linear optimization, FFT, etc. → “unified API” - “all-in-one” 5 Optimized for both small and large objects • Small objects • Large objects – means fixed sizes: – means dynamic sizes Eigen::Matrix<float,4,4> Eigen::Matrix<float,Dynamic,1> – Malloc-free – Cache friendly kernels – Meta unrolling – Parallelization (OpenMP) – Vectorization (SIMD) – Unified API 6 Performance 7 Multi-platforms • Supported compilers: – GCC (from 3.4 to 4.6), MSVC (2005,2008,2010), Intel ICC, Clang/LLVM (2.8) • Supported systems: – x86/x86_64 (Linux,Windows) – ARM (Linux), PowerPC • Supported -
7. Iterative Methods for Eigenvalue Problems
7 Iterative Methods for Eigenvalue Problems 7.1. Introduction In this chapter we discuss iterative methods for finding eigenvalues of matrices that are too large to use the direct methods of Chapters 4 and 5. In other words, we seek algorithms that take far less than O(n2 ) storage and O(n3 ) flops. Since the eigenvectors of most n-by-n matrices would take n2 storage to represent, this means that we seek algorithms that compute just a few user- selected eigenvalues and eigenvectors of a matrix. We will depend on the material on Krylov subspace methods developed in section 6.6, the material on symmetric eigenvalue problems in section 5.2, and the material on the power method and inverse iteration in section 5.3. The reader is advised to review these sections. The simplest eigenvalue problem is to compute just the largest eigenvalue in absolute value, along with its eigenvector. The power method (Algorithm 4.1) is the simplest algorithm suitable for this task: Recall that its inner loop is yi+1 = Axi, xi+1 = yz+l/IIyi+1I2, where xi converges to the eigenvector corresponding to the desired eigenvector (provided that there is only one eigenvalue of largest absolute value, and xl does not lie in an invariant subspace not containing its eigenvector). Note that the algorithm uses A only to perform matrix-vector multiplication, so all that we need to run the algorithm is a "black-box" that takes xi as input and returns Ax2 as output (see Example 6.13). A closely related problem is to find the eigenvalue closest to a user-supplied value a, along with its eigenvector. -
On the Manifold: Representing Geometry in C++ for State Estimation
On the Manifold: Representing Geometry in C++ for State Estimation by Leonid Koppel A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Applied Science in Mechanical and Mechatronics Engineering Waterloo, Ontario, Canada, 2018 © Leonid Koppel 2018 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract Manipulating geometric objects is central to state estimation problems in robotics. Typical algorithms must optimize over non-Euclidean states, such as rigid transformations on the SE(3) manifold, and handle measurements expressed in multiple coordinate frames. Researchers typically rely on C++ libraries for geometric tasks. Commonly used libraries range from linear algebra software such as Eigen to robotics-targeted optimization frame- works such as GTSAM, which provides manifold operations and automatic differentiation of arbitrary expressions. This thesis examines how geometric operations in existing software can be improved, both in runtime performance and in the expression of geometric semantics, to support rapid and error-free development of robotics algorithms. This thesis presents wave_geometry, a C++ manifold geometry library providing representations of objects in affine, Euclidean, and projective spaces, and the Lie groups SO(3) and SE(3). It encompasses the main contributions of this work: an expression template-based automatic differentiation system and compile-time checking of coordinate frame semantics. The library can evaluate Jacobians of geometric expressions in forward and reverse mode with little runtime overhead compared to hand-coded derivatives, and exceeds the performance of existing libraries. -
Simit: a Language for Physical Simulation
Simit: A Language for Physical Simulation The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Kjolstad, Fredrik et al. “Simit: A Language for Physical Simulation.” ACM Transactions on Graphics 35.2 (2016): 1–21. As Published http://dx.doi.org/10.1145/2866569 Publisher Association for Computing Machinery (ACM) Version Author's final manuscript Citable link http://hdl.handle.net/1721.1/110702 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Simit: A Language for Physical Simulation FREDRIK KJOLSTAD Massachusetts Institute of Technology SHOAIB KAMIL Adobe JONATHAN RAGAN-KELLEY Stanford University DAVID I.W. LEVIN Disney Research SHINJIRO SUEDA California Polytechnic State University DESAI CHEN Massachusetts Institute of Technology ETIENNE VOUGA University of Texas at Austin DANNY M. KAUFMAN Adobe and GURTEJ KANWAR, WOJCIECH MATUSIK, and SAMAN AMARASINGHE Massachusetts Institute of Technology Using existing programming tools, writing high-performance simulation be compiled for GPUs with no change to the program, delivering 4-20x code is labor intensive and requires sacrificing readability and portability. The speedups over our optimized CPU code. alternative is to prototype simulations in a high-level language like Matlab, Categories and Subject Descriptors: I.3.7 [Computer Graphics]: Three- thereby sacrificing performance. The Matlab programming model naturally Dimensional Graphics and Realism—Animation describes the behavior of an entire physical system using the language of linear algebra. However, simulations also manipulate individual geometric Additional Key Words and Phrases: Graph, Matrix, Tensor, Simulation elements, which are best represented using linked data structures like meshes. -
Introduction to Parallel Programming (For Physicists) FRANÇOIS GÉLIS & GRÉGOIRE MISGUICH, Ipht Courses, June 2019
Introduction to parallel programming (for physicists) FRANÇOIS GÉLIS & GRÉGOIRE MISGUICH, IPhT courses, June 2019. 1. Introduction & hardware aspects (FG) These 2. A few words about Maple & Mathematica slides (GM) slides 3. Linear algebra libraries 4. Fast Fourier transform 5. Python Multiprocessing 6. OpenMP 7. MPI (FG) 8. MPI+OpenMP (FG) Numerical Linear algebra Here: a few simple examples showing how to call some parallel linear algebra libraries in numerical calculations (numerical) Linear algebra • Basic Linear Algebra Subroutines: BLAS • More advanced operations Linear Algebra Package: LAPACK • vector op. (=level 1) (‘90, Fortran 77) • matrix-vector (=level 2) • Call the BLAS routines • matrix-matrix mult. & triangular • Matrix diagonalization, linear systems inversion (=level 3) and eigenvalue problems • Many implementations but • Matrix decompositions: LU, QR, SVD, standardized interface Cholesky • Discussed here: Intel MKL & OpenBlas (multi-threaded = parallelized for • Many implementations shared-memory architectures) Used in most scientific softwares & libraries (Python/Numpy, Maple, Mathematica, Matlab, …) A few other useful libs … for BIG matrices • ARPACK • ScaLAPACK =Implicitly Restarted Arnoldi Method (~Lanczos for Parallel version of LAPACK for for distributed memory Hermitian cases) architectures Large scale eigenvalue problems. For (usually sparse) nxn matrices with n which can be as large as 10^8, or even more ! Ex: iterative algo. to find the largest eigenvalue, without storing the matrix M (just provide v→Mv). Can be used from Python/SciPy • PARPACK = parallel version of ARPACK for distributed memory architectures (the matrices are stored over several nodes) Two multi-threaded implementations of BLAS & Lapack OpenBLAS Intel’s implementation (=part of the MKL lib.) • Open source License (BSD) • Commercial license • Based on GotoBLAS2 • Included in Intel®Parallel Studio XE (compilers, [Created by GAZUSHIGE GOTO, Texas Adv. -
Adept C++ Software Library: User Guide
Adept C++ Software Library: User Guide Robin J. Hogan European Centre for Medium Range Weather Forecasts, Reading, UK and School of Mathematical, Physical and Computational Sciences, University of Reading, UK, Document version 2.1 (February 2021) applicable to Adept version 2.1 This document is copyright c Robin J. Hogan 2013–2021. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation. This license may be found at http://www.gnu.org/copyleft/fdl.html. As an exception, no copyright is asserted for the code fragments in this document (indicated in the text with a light-grey background); these code fragments are hereby placed in the Public Domain, and accordingly may be copied, modified and distributed without restriction. If you have any queries about Adept that are not answered by this document or by the information on the Adept web site (http://www.met.reading.ac.uk/clouds/adept/) then please email me at [email protected]. Contents 1 Introduction 3 1.1 What is Adept? ...................................................................................................................................... 3 1.2 Installing Adept and compiling your code to use it ............................................................................... 3 1.2.1 Unix-like platforms................................................................................................................... 4 1.2.2 Non-Unix