Highly Scalable Multiplication for Distributed Sparse Multivariate Polynomials on Many-Core Systems

Abstract

Abstract. We present a highly scalable algorithm for multiplying sparse multivariate polynomials represented in a distributed format. This algorithm targets not only the shared memory multicore computers, but also computers clusters or specialized hardware attached to a host computer, such as graphics processing units or many-core coprocessors. The scalability on the large number of cores is ensured by the lacks of synchronizations, locks and false-sharing during the main parallel step.

Output-Sensitive Algorithms for Sumset and Sparse Polynomial Multiplication

“…Computer algebra programs including Maple, Mathematica, Sage, and Singular use a sparse representation by default for multivariate polynomials, and there has been considerable recent work on how to efficiently store and compute with sparse polynomials [8,10,19,21,14].…”

Parallel sparse multivariate polynomial division

“…As the terms of Q are unknown at the beginning of the algorithm, the grid applied for the multiplication [5] could not be used for the division. All terms of A should be canceled by the division, some exponents of the array generated by the exponents of A could be selected using regular spacing.…”

“…As the polynomial B is sparse, the naive school-book multiplication, with a complexity O(nqn b ), is used to multiply them. This multiplication of two sparse multivariate polynomials could be efficiently parallelized on a multi-core computer by reducing the number of required communications between the threads [5]. Like this multiplication algorithm, a possible efficient parallel division algorithm consists in the threads compute independent terms of Pn q .…”

“…. nq, all leading terms of Q k × [Sr+1;Sr[ B could be determined with a complexity O(nq +n b ), as shown in [5].…”

“…Other sequential division algorithms have been designed using a linear algebra approach [2] for dense polynomials. In [5], we have proposed a multiplication algorithm targeting a large number of cores on different kinds of hardware. This algorithm allows to compute independent terms by the threads at the same time and merges the terms of the product using any Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.…”