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Ries Via the Generalized Weighted-Averages Method
Progress In Electromagnetics Research M, Vol. 14, 233{245, 2010 ACCELERATION OF SLOWLY CONVERGENT SE- RIES VIA THE GENERALIZED WEIGHTED-AVERAGES METHOD A. G. Polimeridis, R. M. Golubovi¶cNi¶ciforovi¶c and J. R. Mosig Laboratory of Electromagnetics and Acoustics Ecole Polytechnique F¶ed¶eralede Lausanne CH-1015 Lausanne, Switzerland Abstract|A generalized version of the weighted-averages method is presented for the acceleration of convergence of sequences and series over a wide range of test problems, including linearly and logarithmically convergent series as well as monotone and alternating series. This method was originally developed in a partition- extrapolation procedure for accelerating the convergence of semi- in¯nite range integrals with Bessel function kernels (Sommerfeld-type integrals), which arise in computational electromagnetics problems involving scattering/radiation in planar strati¯ed media. In this paper, the generalized weighted-averages method is obtained by incorporating the optimal remainder estimates already available in the literature. Numerical results certify its comparable and in many cases superior performance against not only the traditional weighted-averages method but also against the most proven extrapolation methods often used to speed up the computation of slowly convergent series. 1. INTRODUCTION Almost every practical numerical method can be viewed as providing an approximation to the limit of an in¯nite sequence. This sequence is frequently formed by the partial sums of a series, involving a ¯nite number of its elements. Unfortunately, it often happens that the resulting sequence either converges too slowly to be practically useful, or even appears as divergent, hence requesting the use of generalized Received 7 October 2010, Accepted 28 October 2010, Scheduled 8 November 2010 Corresponding author: Athanasios G. -
(LINUX) Computer in Three Parts
Science on a (LINUX) computer in three parts Introductory short couse, Part II Thorsten Becker University of Southern California, Los Angeles September 2007 The first part dealt with ● UNIX: what and why ● File system, Window managers ● Shell environment ● Editing files ● Command line tools ● Scripts and GUIs ● Virtualization Contents part II ● Typesetting ● Programming – common languages – philosophy – compiling, debugging, make, version control – C and F77 interfacing – libraries and packages ● Number crunching ● Visualization tools Programming: Traditional languages in the natural sciences ● Fortran: higher level, good for math – F77: legacy, don't use (but know how to read) – F90/F95: nice vector features, finally implements C capabilities (structures, memory allocation) ● C: low level (e.g. pointers), better structured – very close to UNIX philosophy – structures offer nice way of modular programming, see Wikipedia on C ● I recommend F95, and use C happily myself Programming: Some Languages that haven't completely made it to scientific computing ● C++: object oriented programming model – reusable objects with methods and such – can be partly realized by modular programming in C ● Java: what's good for commercial projects (or smart, or elegant) doesn't have to be good for scientific computing ● Concern about portability as well as general access Programming: Compromises ● Python – Object oriented – Interpreted – Interfaces easily with F90/C – Numerous scientific packages Programming: Other interpreted, high- abstraction languages -
Fortran Resources 1
Fortran Resources 1 Ian D Chivers Jane Sleightholme May 7, 2021 1The original basis for this document was Mike Metcalf’s Fortran Information File. The next input came from people on comp-fortran-90. Details of how to subscribe or browse this list can be found in this document. If you have any corrections, additions, suggestions etc to make please contact us and we will endeavor to include your comments in later versions. Thanks to all the people who have contributed. Revision history The most recent version can be found at https://www.fortranplus.co.uk/fortran-information/ and the files section of the comp-fortran-90 list. https://www.jiscmail.ac.uk/cgi-bin/webadmin?A0=comp-fortran-90 • May 2021. Major update to the Intel entry. Also changes to the editors and IDE section, the graphics section, and the parallel programming section. • October 2020. Added an entry for Nvidia to the compiler section. Nvidia has integrated the PGI compiler suite into their NVIDIA HPC SDK product. Nvidia are also contributing to the LLVM Flang project. Updated the ’Additional Compiler Information’ entry in the compiler section. The Polyhedron benchmarks discuss automatic parallelisation. The fortranplus entry covers the diagnostic capability of the Cray, gfortran, Intel, Nag, Oracle and Nvidia compilers. Updated one entry and removed three others from the software tools section. Added ’Fortran Discourse’ to the e-lists section. We have also made changes to the Latex style sheet. • September 2020. Added a computer arithmetic and IEEE formats section. • June 2020. Updated the compiler entry with details of standard conformance. -
Convergence Acceleration of Series Through a Variational Approach
Convergence acceleration of series through a variational approach September 30, 2004 Paolo Amore∗, Facultad de Ciencias, Universidad de Colima, Bernal D´ıaz del Castillo 340, Colima, Colima, M´exico Abstract By means of a variational approach we find new series representa- tions both for well known mathematical constants, such as π and the Catalan constant, and for mathematical functions, such as the Rie- mann zeta function. The series that we have found are all exponen- tially convergent and provide quite useful analytical approximations. With limited effort our method can be applied to obtain similar expo- nentially convergent series for a large class of mathematical functions. 1 Introduction This paper deals with the problem of improving the convergence of a series in the case where the series itself is converging very slowly and a large number of terms needs to be evaluated to reach the desired accuracy. This is an interesting and challenging problem, which has been considered by many authors before (see, for example, [FV 1996, Br 1999, Co 2000]). In this paper we wish to attack this problem, although aiming from a different angle: instead of looking for series expansions in some small nat- ural parameter (i.e. a parameter present in the original formula), we in- troduce an artificial dependence in the formulas upon an (not completely) ∗[email protected] 1 arbitrary parameter and then devise an expansion which can be optimized to give faster rates of convergence. The details of how this work will be explained in depth in the next section. This procedure is well known in Physics and it has been exploited in the so-called \Linear Delta Expansion" (LDE) and similar approaches [AFC 1990, Jo 1995, Fe 2000]. -
Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon
A Proof, Based on the Euler Sum Acceleration, of the Recovery of an Exponential (Geometric) Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon John P. Boyd∗ Department of Atmospheric, Oceanic and Space Science and Laboratory for Scientific Computation, University of Michigan, 2455 Hayward Avenue, Ann Arbor MI 48109 [email protected]; http://www.engin.umich.edu:/∼ jpboyd/ March 26, 2010 Abstract When a function f(x)is singular at a point xs on the real axis, its Fourier series, when truncated at the N-th term, gives a pointwise error of only O(1/N) over the en- tire real axis. Such singularities spontaneously arise as “fronts” in meteorology and oceanography and “shocks” in other branches of fluid mechanics. It has been pre- viously shown that it is possible to recover an exponential rate of convegence at all | − σ |∼ − points away from the singularity in the sense that f(x) fN (x) O(exp( q(x)N)) σ where fN (x) is the result of applying a filter or summability method to the partial sum fN (x) and q(x) is a proportionality constant that is a function of d(x) ≡ |x − xs |, the distance from x to the singularity. Here we give an elementary proof of great generality using conformal mapping in a dummy variable z; this is equiva- lent to applying the Euler acceleration. We show that q(x) ≈ log(cos(d(x)/2)) for the Euler filter when the Fourier period is 2π. More sophisticated filters can in- crease q(x), but the Euler filter is simplest. -
Some Notes on BLAS, SIMD, MIC and GPGPU (For Octave and Matlab)
Some Notes on BLAS, SIMD, MIC and GPGPU (for Octave and Matlab) Christian Himpe ([email protected]) WWU Münster Institute for Computational and Applied Mathematics Software-Tools für die Numerische Mathematik 15.10.2014 About1 BLAS & LAPACK Affinity SIMD Automatic Offloading 1 Get your Buzzword-Bingo ready! BLAS & LAPACK BLAS (Basic Linear Algebra System) Level 1: vector-vector operations (dot-product, vector norms, generalized vector addition) Level 2: matrix-vector operations (generalized matrix-vector multiplication) Level 3: matrix-matrix operations (generalized matrix-matrix multiplication) LAPACK (Linear Algebra Package) Matrix Factorizations: LU, QR, Cholesky, Schur Least-Squares: LLS, LSE, GLM Eigenproblems: SEP, NEP, SVD Default (un-optimized) netlib reference implementation. Used by Octave, Matlab, SciPy/NumPy (Python), Julia, R. MKL Intel’s implementation of BLAS and LAPACK. MKL (Math Kernel Library) can offload computations to XeonPhi can use OpenMP provides additionally FFT, libm and 1D-interpolation Current Version: 11.0.5 (automatic offloading to Phis) Costs (we have it, and Matlab ships with it, too) Go to: https://software.intel.com/en-us/intel-mkl ACML AMD doesn’t want to be left out. ACML (AMD Core Math Libraries) Can use OpenCL for automatic offloading to GPUs Special version to exploit FMA(4) instructions Choice of compiled binaries (GFortran, Intel Fortran, Open64, PGI) Current Version: 6 (automatic offloading to GPUs) Free (but not open source) Go to: http://developer.amd.com/tools-and-sdks/ cpu-development/amd-core-math-library-acml OpenBLAS Open Source, the third kind. OpenBLAS Fork of GotoBLAS Good performance for dense operations (close to the MKL) Can use OpenMP and is compiled for specific architecture Current Version: 0.2.11 Open source (!) Go to: http://github.com/xianyi/OpenBLAS FlexiBLAS So many BLAS implementations, so little time.. -
High-Performance Parallel Computations Using Python As High-Level Language
Aachen Institute for Advanced Study in Computational Engineering Science Preprint: AICES-2010/08-01 23/August/2010 High-Performance Parallel Computations using Python as High-Level Language Stefano Masini and Paolo Bientinesi Financial support from the Deutsche Forschungsgemeinschaft (German Research Association) through grant GSC 111 is gratefully acknowledged. ©Stefano Masini and Paolo Bientinesi 2010. All rights reserved List of AICES technical reports: http://www.aices.rwth-aachen.de/preprints High-Performance Parallel Computations using Python as High-Level Language Stefano Masini⋆ and Paolo Bientinesi⋆⋆ RWTH Aachen, AICES, Aachen, Germany Abstract. High-performance and parallel computations have always rep- resented a challenge in terms of code optimization and memory usage, and have typically been tackled with languages that allow a low-level management of resources, like Fortran, C and C++. Nowadays, most of the implementation effort goes into constructing the bookkeeping logic that binds together functionalities taken from standard libraries. Because of the increasing complexity of this kind of codes, it becomes more and more necessary to keep it well organized through proper software en- gineering practices. Indeed, in the presence of chaotic implementations, reasoning about correctness is difficult, even when limited to specific aspects like concurrency; moreover, due to the lack in flexibility of the code, making substantial changes for experimentation becomes a grand challenge. Since the bookkeeping logic only accounts for a tiny fraction of the total execution time, we believe that for such a task it can be afforded to introduce an overhead due to a high-level language. We consider Python as a preliminary candidate with the intent of improving code readability, flexibility and, in turn, the level of confidence with respect to correctness. -
Iterative Refinement for Symmetric Eigenvalue Decomposition II
Japan Journal of Industrial and Applied Mathematics (2019) 36:435–459 https://doi.org/10.1007/s13160-019-00348-4 ORIGINAL PAPER Iterative refinement for symmetric eigenvalue decomposition II: clustered eigenvalues Takeshi Ogita1 · Kensuke Aishima2 Received: 22 October 2018 / Revised: 5 February 2019 / Published online: 22 February 2019 © The Author(s) 2019 Abstract We are concerned with accurate eigenvalue decomposition of a real symmetric matrix A. In the previous paper (Ogita and Aishima in Jpn J Ind Appl Math 35(3): 1007–1035, 2018), we proposed an efficient refinement algorithm for improving the accuracy of all eigenvectors, which converges quadratically if a sufficiently accurate initial guess is given. However, since the accuracy of eigenvectors depends on the eigenvalue gap, it is difficult to provide such an initial guess to the algorithm in the case where A has clustered eigenvalues. To overcome this problem, we propose a novel algorithm that can refine approximate eigenvectors corresponding to clustered eigenvalues on the basis of the algorithm proposed in the previous paper. Numerical results are presented showing excellent performance of the proposed algorithm in terms of convergence rate and overall computational cost and illustrating an application to a quantum materials simulation. Keywords Accurate numerical algorithm · Iterative refinement · Symmetric eigenvalue decomposition · Clustered eigenvalues Mathematics Subject Classification 65F15 · 15A18 · 15A23 This study was partially supported by CREST, JST and JSPS KAKENHI Grant numbers 16H03917, 25790096. B Takeshi Ogita [email protected] Kensuke Aishima [email protected] 1 Division of Mathematical Sciences, School of Arts and Sciences, Tokyo Woman’s Christian University, 2-6-1 Zempukuji, Suginami-ku, Tokyo 167-8585, Japan 2 Faculty of Computer and Information Sciences, Hosei University, 3-7-2 Kajino-cho, Koganei-shi, Tokyo 184-8584, Japan 123 436 T. -
Parallel Programming Models for Dense Linear Algebra on Heterogeneous Systems
DOI: 10.14529/jsfi150405 Parallel Programming Models for Dense Linear Algebra on Heterogeneous Systems M. Abalenkovs1, A. Abdelfattah2, J. Dongarra 1,2,3, M. Gates2, A. Haidar2, J. Kurzak2, P. Luszczek2, S. Tomov2, I. Yamazaki2, A. YarKhan2 c The Authors 2015. This paper is published with open access at SuperFri.org We present a review of the current best practices in parallel programming models for dense linear algebra (DLA) on heterogeneous architectures. We consider multicore CPUs, stand alone manycore coprocessors, GPUs, and combinations of these. Of interest is the evolution of the programming models for DLA libraries – in particular, the evolution from the popular LAPACK and ScaLAPACK libraries to their modernized counterparts PLASMA (for multicore CPUs) and MAGMA (for heterogeneous architectures), as well as other programming models and libraries. Besides providing insights into the programming techniques of the libraries considered, we outline our view of the current strengths and weaknesses of their programming models – especially in regards to hardware trends and ease of programming high-performance numerical software that current applications need – in order to motivate work and future directions for the next generation of parallel programming models for high-performance linear algebra libraries on heterogeneous systems. Keywords: Programming models, runtime, HPC, GPU, multicore, dense linear algebra. Introduction Parallelism in today’s computer architectures is pervasive – not only in systems from large supercomputers to laptops, but also in small portable devices like smart phones and watches. Along with parallelism, the level of heterogeneity in modern computing systems is also gradually increasing. Multicore CPUs are often combined with discrete high-performance accelerators like graphics processing units (GPUs) and coprocessors like Intel’s Xeon Phi, but are also included as integrated parts with system-on-chip (SoC) platforms, as in the AMD Fusion family of ap- plication processing units (APUs) or the NVIDIA Tegra mobile family of devices. -
LAPACK WORKING NOTE 195: SCALAPACK's MRRR ALGORITHM 1. Introduction. Since 2005, the National Science Foundation Has Been Fund
LAPACK WORKING NOTE 195: SCALAPACK’S MRRR ALGORITHM CHRISTOF VOMEL¨ ∗ Abstract. The sequential algorithm of Multiple Relatively Robust Representations, MRRR, can compute numerically orthogonal eigenvectors of an unreduced symmetric tridiagonal matrix T ∈ Rn×n with O(n2) cost. This paper describes the design of ScaLAPACK’s parallel MRRR algorithm. One emphasis is on the critical role of the representation tree in achieving both numerical accuracy and parallel scalability. A second point concerns the favorable properties of this code: subset computation, the use of static memory, and scalability. Unlike ScaLAPACK’s Divide & Conquer and QR, MRRR can compute subsets of eigenpairs at reduced cost. And in contrast to inverse iteration which can fail, it is guaranteed to produce a numerically satisfactory answer while maintaining memory scalability. ParEig, the parallel MRRR algorithm for PLAPACK, uses dynamic memory allocation. This is avoided by our code at marginal additional cost. We also use a different representation tree criterion that allows for more accurate computation of the eigenvectors but can make parallelization more difficult. AMS subject classifications. 65F15, 65Y15. Key words. Symmetric eigenproblem, multiple relatively robust representations, ScaLAPACK, numerical software. 1. Introduction. Since 2005, the National Science Foundation has been funding an initiative [19] to improve the LAPACK [3] and ScaLAPACK [14] libraries for Numerical Linear Algebra computations. One of the ambitious goals of this initiative is to put more of LAPACK into ScaLAPACK, recognizing the gap between available sequential and parallel software support. In particular, parallelization of the MRRR algorithm [24, 50, 51, 26, 27, 28, 29], the topic of this paper, is identified as one key improvement to ScaLAPACK. -
AMD Core Math Library (ACML)
AMD Core Math Library (ACML) Version 4.2.0 Copyright c 2003-2008 Advanced Micro Devices, Inc., Numerical Algorithms Group Ltd. AMD, the AMD Arrow logo, AMD Opteron, AMD Athlon and combinations thereof are trademarks of Advanced Micro Devices, Inc. i Short Contents 1 Introduction................................... 1 2 General Information ............................. 2 3 BLAS: Basic Linear Algebra Subprograms ............. 19 4 LAPACK: Package of Linear Algebra Subroutines ........ 20 5 Fast Fourier Transforms (FFTs) .................... 24 6 Random Number Generators....................... 75 7 ACML MV: Fast Math and Fast Vector Math Library .... 163 8 References .................................. 229 Subject Index ................................... 230 Routine Index ................................... 233 ii Table of Contents 1 Introduction ............................... 1 2 General Information ....................... 2 2.1 Determining the best ACML version for your system ........... 2 2.2 Accessing the Library (Linux) ................................ 4 2.2.1 Accessing the Library under Linux using GNU gfortran/gcc ......................................................... 4 2.2.2 Accessing the Library under Linux using PGI compilers pgf77/pgf90/pgcc ......................................... 5 2.2.3 Accessing the Library under Linux using PathScale compilers pathf90/pathcc ........................................... 6 2.2.4 Accessing the Library under Linux using the NAGWare f95 compiler................................................. -
35.232-2016.43 Lamees Elhiny.Pdf
LOAD PARTITIONING FOR MATRIX-MATRIX MULTIPLICATION ON A CLUSTER OF CPU-GPU NODES USING THE DIVISIBLE LOAD PARADIGM by Lamees Elhiny A Thesis Presented to the Faculty of the American University of Sharjah College of Engineering in Partial Fulfillment of the Requirements for the Degree of Master of Science in Computer Engineering Sharjah, United Arab Emirates November 2016 c 2016. Lamees Elhiny. All rights reserved. Approval Signatures We, the undersigned, approve the Master’s Thesis of Lamees Elhiny Thesis Title: Load Partitioning for Matrix-Matrix Multiplication on a Cluster of CPU- GPU Nodes Using the Divisible Load Paradigm Signature Date of Signature (dd/mm/yyyy) ___________________________ _______________ Dr. Gerassimos Barlas Professor, Department of Computer Science and Engineering Thesis Advisor ___________________________ _______________ Dr. Khaled El-Fakih Associate Professor, Department of Computer Science and Engineering Thesis Committee Member ___________________________ _______________ Dr. Naoufel Werghi Associate Professor, Electrical and Computer Engineering Department Khalifa University of Science, Technology & Research (KUSTAR) Thesis Committee Member ___________________________ _______________ Dr. Fadi Aloul Head, Department of Computer Science and Engineering ___________________________ _______________ Dr. Mohamed El-Tarhuni Associate Dean, College of Engineering ___________________________ _______________ Dr. Richard Schoephoerster Dean, College of Engineering ___________________________ _______________ Dr. Khaled Assaleh Interim Vice Provost for Research and Graduate Studies Acknowledgements I would like to express my sincere gratitude to my thesis advisor, Dr. Gerassi- mos Barlas, for his guidance at every step throughout the thesis. I am thankful to Dr. Assim Sagahyroon, and to the Department of Computer Science & Engineering as well as the American University of Sharjah for offering me the Graduate Teaching Assistantship, which allowed me to pursue my graduate studies.