ACM Transactions on Mathematical Software (TOMS)最新文献_第6页

Faithfully Rounded Floating-point Computations 忠实地舍入浮点计算

ACM Transactions on Mathematical Software (TOMS) Pub Date : 2020-07-13 DOI: 10.1145/3290955

M. Lange, S. Rump

{"title":"Faithfully Rounded Floating-point Computations","authors":"M. Lange, S. Rump","doi":"10.1145/3290955","DOIUrl":"https://doi.org/10.1145/3290955","url":null,"abstract":"We present a pair arithmetic for the four basic operations and square root. It can be regarded as a simplified, more-efficient double-double arithmetic. The central assumption on the underlying arithmetic is the first standard model for error analysis for operations on a discrete set of real numbers. Neither do we require a floating-point grid nor a rounding to nearest property. Based on that, we define a relative rounding error unit u and prove rigorous error bounds for the computed result of an arbitrary arithmetic expression depending on u, the size of the expression, and possibly a condition measure. In the second part of this note, we extend the error analysis by examining requirements to ensure faithfully rounded outputs and apply our results to IEEE 754 standard conform floating-point systems. For a class of mathematical expressions, using an IEEE 754 standard conform arithmetic with base β, the result is proved to be faithfully rounded for up to 1 / √βu - 2 operations. Our findings cover a number of previously published algorithms to compute faithfully rounded results, among them Horner’s scheme, products, sums, dot products, or Euclidean norm. Beyond that, several other problems can be analyzed, such as polynomial interpolation, orientation problems, Householder transformations, or the smallest singular value of Hilbert matrices of large size.","PeriodicalId":7036,"journal":{"name":"ACM Transactions on Mathematical Software (TOMS)","volume":"37 1","pages":"1 - 20"},"PeriodicalIF":0.0,"publicationDate":"2020-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85543795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 8

fenicsR13

ACM Transactions on Mathematical Software (TOMS) Pub Date : 2020-07-12 DOI: 10.1145/3442378

Lambert Theisen, M. Torrilhon

引用次数: 5

Algorithm 1019: A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation 算法1019:一种基于任务的多移位QR/QZ主动早期通缩算法

ACM Transactions on Mathematical Software (TOMS) Pub Date : 2020-07-07 DOI: 10.1145/3495005

Mirko Myllykoski

{"title":"Algorithm 1019: A Task-based Multi-shift QR/QZ Algorithm with Aggressive Early Deflation","authors":"Mirko Myllykoski","doi":"10.1145/3495005","DOIUrl":"https://doi.org/10.1145/3495005","url":null,"abstract":"The QR algorithm is one of the three phases in the process of computing the eigenvalues and the eigenvectors of a dense nonsymmetric matrix. This paper describes a task-based QR algorithm for reducing an upper Hessenberg matrix to real Schur form. The task-based algorithm also supports generalized eigenvalue problems (QZ algorithm) but this paper concentrates on the standard case. The task-based algorithm adopts previous algorithmic improvements, such as tightly-coupled multi-shifts and Aggressive Early Deflation (AED), and also incorporates several new ideas that significantly improve the performance. This includes, but is not limited to, the elimination of several synchronization points, the dynamic merging of previously separate computational steps, the shortening and the prioritization of the critical path, and experimental GPU support. The task-based implementation is demonstrated to be multiple times faster than multi-threaded LAPACK and ScaLAPACK in both single-node and multi-node configurations on two different machines based on Intel and AMD CPUs. The implementation is built on top of the StarPU runtime system and is part of the open-source StarNEig library.","PeriodicalId":7036,"journal":{"name":"ACM Transactions on Mathematical Software (TOMS)","volume":"12 1","pages":"1 - 36"},"PeriodicalIF":0.0,"publicationDate":"2020-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80621438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 1

Ginkgo: A Modern Linear Operator Algebra Framework for High Performance Computing 银杏:用于高性能计算的现代线性算子代数框架

ACM Transactions on Mathematical Software (TOMS) Pub Date : 2020-06-30 DOI: 10.1145/3480935

H. Anzt, T. Cojean, Goran Flegar, Fritz Göbel, Thomas Grützmacher, Pratik Nayak, T. Ribizel, Yu-Hsiang Tsai, E. S. Quintana‐Ortí

{"title":"Ginkgo: A Modern Linear Operator Algebra Framework for High Performance Computing","authors":"H. Anzt, T. Cojean, Goran Flegar, Fritz Göbel, Thomas Grützmacher, Pratik Nayak, T. Ribizel, Yu-Hsiang Tsai, E. S. Quintana‐Ortí","doi":"10.1145/3480935","DOIUrl":"https://doi.org/10.1145/3480935","url":null,"abstract":"In this article, we present Ginkgo, a modern C++ math library for scientific high performance computing. While classical linear algebra libraries act on matrix and vector objects, Ginkgo’s design principle abstracts all functionality as “linear operators,” motivating the notation of a “linear operator algebra library.” Ginkgo’s current focus is oriented toward providing sparse linear algebra functionality for high performance graphics processing unit (GPU) architectures, but given the library design, this focus can be easily extended to accommodate other algorithms and hardware architectures. We introduce this sophisticated software architecture that separates core algorithms from architecture-specific backends and provide details on extensibility and sustainability measures. We also demonstrate Ginkgo’s usability by providing examples on how to use its functionality inside the MFEM and deal.ii finite element ecosystems. Finally, we offer a practical demonstration of Ginkgo’s high performance on state-of-the-art GPU architectures.","PeriodicalId":7036,"journal":{"name":"ACM Transactions on Mathematical Software (TOMS)","volume":"33 1","pages":"1 - 33"},"PeriodicalIF":0.0,"publicationDate":"2020-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88423799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 40

Irksome: Automating Runge–Kutta Time-stepping for Finite Element Methods 有限元方法的龙格-库塔时间步进自动化

ACM Transactions on Mathematical Software (TOMS) Pub Date : 2020-06-29 DOI: 10.1145/3466168

P. Farrell, R. Kirby, J. Marchena-Menendez

引用次数: 20

H-Revolve H-Revolve

ACM Transactions on Mathematical Software (TOMS) Pub Date : 2020-06-01 DOI: 10.1145/3378672

J. Herrmann, G. Pallez

引用次数: 3

Algorithm 1010 算法1010

ACM Transactions on Mathematical Software (TOMS) Pub Date : 2020-05-18 DOI: 10.1145/3386241

A. G. Orellana, C. Michele

{"title":"Algorithm 1010","authors":"A. G. Orellana, C. Michele","doi":"10.1145/3386241","DOIUrl":"https://doi.org/10.1145/3386241","url":null,"abstract":"Aiming to provide a very accurate, efficient, and robust quartic equation solver for physical applications, we have proposed an algorithm that builds on the previous works of P. Strobach and S. L. Shmakov. It is based on the decomposition of the quartic polynomial into two quadratics, whose coefficients are first accurately estimated by handling carefully numerical errors and afterward refined through the use of the Newton-Raphson method. Our algorithm is very accurate in comparison with other state-of-the-art solvers that can be found in the literature, but (most importantly) it turns out to be very efficient according to our timing tests. A crucial issue for us is the robustness of the algorithm, i.e., its ability to cope with the detrimental effect of round-off errors, no matter what set of quartic coefficients is provided in a practical application. In this respect, we extensively tested our algorithm in comparison to other quartic equation solvers both by considering specific extreme cases and by carrying out a statistical analysis over a very large set of quartics. Our algorithm has also been heavily tested in a physical application, i.e., simulations of hard cylinders, where it proved its absolute reliability as well as its efficiency.","PeriodicalId":7036,"journal":{"name":"ACM Transactions on Mathematical Software (TOMS)","volume":"9 1","pages":"1 - 28"},"PeriodicalIF":0.0,"publicationDate":"2020-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88026388","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 5

Bidiagonal SVD Computation via an Associated Tridiagonal Eigenproblem 基于相关三对角特征问题的双对角奇异值分解计算

ACM Transactions on Mathematical Software (TOMS) Pub Date : 2020-05-18 DOI: 10.1145/3361746

O. Marques, J. Demmel, P. Vasconcelos

{"title":"Bidiagonal SVD Computation via an Associated Tridiagonal Eigenproblem","authors":"O. Marques, J. Demmel, P. Vasconcelos","doi":"10.1145/3361746","DOIUrl":"https://doi.org/10.1145/3361746","url":null,"abstract":"The Singular Value Decomposition (SVD) is widely used in numerical analysis and scientific computing applications, including dimensionality reduction, data compression and clustering, and computation of pseudo-inverses. In many cases, a crucial part of the SVD of a general matrix is to find the SVD of an associated bidiagonal matrix. This article discusses an algorithm to compute the SVD of a bidiagonal matrix through the eigenpairs of an associated symmetric tridiagonal matrix. The algorithm enables the computation of only a subset of singular values and corresponding vectors, with potential performance gains. The article focuses on a sequential version of the algorithm, and discusses special cases and implementation details. The implementation, called BDSVDX, has been included in the LAPACK library. We use a large set of bidiagonal matrices to assess the accuracy of the implementation, both in single and double precision, as well as to identify potential shortcomings. The results show that BDSVDX can be up to three orders of magnitude faster than existing algorithms, which are limited to the computation of a full SVD. We also show comparisons of an implementation that uses BDSVDX as a building block for the computation of the SVD of general matrices.","PeriodicalId":7036,"journal":{"name":"ACM Transactions on Mathematical Software (TOMS)","volume":"32 1 1","pages":"1 - 25"},"PeriodicalIF":0.0,"publicationDate":"2020-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75786779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 5

Error Analysis of Some Operations Involved in the Cooley-Tukey Fast Fourier Transform Cooley-Tukey快速傅里叶变换中一些运算的误差分析

ACM Transactions on Mathematical Software (TOMS) Pub Date : 2020-05-18 DOI: 10.1145/3368619

N. Brisebarre, Mioara Joldes, J. Muller, Ana-Maria Naneş, Joris Picot

引用次数: 14

Algorithm 1008 算法1008

ACM Transactions on Mathematical Software (TOMS) Pub Date : 2020-05-18 DOI: 10.1145/3378542

Jose Maria Varas Casado, R. Hewson

引用次数: 4