The Linear Algebra Mapping Problem. Current State of Linear Algebra Languages and Libraries

Author:

Psarras Christos1ORCID,Barthels Henrik1ORCID,Bientinesi Paolo2ORCID

Affiliation:

1. RWTH Aachen University, North Rhine-Westphalia, Germany

2. Umeå Universitet, Umeå, Sweden

Abstract

We observe a disconnect between developers and end-users of linear algebra libraries. On the one hand, developers invest significant effort in creating sophisticated numerical kernels. On the other hand, end-users are progressively less likely to go through the time consuming process of directly using said kernels; instead, languages and libraries, which offer a higher level of abstraction, are becoming increasingly popular. These languages offer mechanisms that internally map the input program to lower level kernels. Unfortunately, our experience suggests that, in terms of performance, this translation is typically suboptimal. In this paper, we define the problem of mapping a linear algebra expression to a set of available building blocks as the “Linear Algebra Mapping Problem” (LAMP) ; we discuss its NP-complete nature, and investigate how effectively a benchmark of test problems is solved by popular high-level programming languages and libraries. Specifically, we consider Matlab, Octave, Julia, R, Armadillo (C++), Eigen (C++), and NumPy (Python); the benchmark is meant to test both compiler optimizations, as well as linear algebra specific optimizations, such as the optimal parenthesization of matrix products. The aim of this study is to facilitate the development of languages and libraries that support linear algebra computations.

Funder

Deutsche Forschungsgemeinschaft

GSC 111

Publisher

Association for Computing Machinery (ACM)

Subject

Applied Mathematics,Software

Reference82 articles.

1. [n.d.]. Commons Math: The Apache Commons Mathematics Library. https://commons.apache.org/math/.

2. [n.d.]. HASEM. https://sourceforge.net/p/hasem/wiki/HASEM/.

3. 2017. Intel®Math Kernel Library documentation. https://software.intel.com/en-us/mkl-reference-manual-for-c.

4. 2018. Maple Maplesoft.https://www.maplesoft.com.

5. 2018. Matlab The MathWorks Inc.https://www.mathworks.com/.

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2. A Test for FLOPs as a Discriminant for Linear Algebra Algorithms;2022 IEEE 34th International Symposium on Computer Architecture and High Performance Computing (SBAC-PAD);2022-11

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