Parallel operations of sparse polynomials on multicores

Abstract

The multiplication of the sparse multivariate polynomials using the recursive representations is revisited to take advantage on the multicore processors. We take care of the memory management and load-balancing in order to obtain linear speedup. The widely used Poisson bracket during the studies of the dynamical systems had been parallelized on these computers. Benchmarks are presented, comparing our implementation to the other computer algebra systems.

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

“…-Example 2 : f 2 × g 2 with f = (1 + x + y + 2z 2 + 3t 3 + 5u 5 ) 25 and g 2 = (1 + u + t + 2z 2 + 3y 3 + 5x 5 ) 25 . As shown in [2] and [1], a linear speedup is quite difficult to obtain on this very sparse example. f 2 and g 2 have 142506 terms and the result contains 312855140 terms.…”

“…Several parallel algorithms have been proposed for modern parallel hardware. An algorithm for the recursive representation has been designed using a work-stealing technique [1]. It scales at least up to 128 cores for large polynomials.…”

“…The algorithm due to Monagan [2] uses binary heaps to merge and sort produced terms but its scalability depends on the kind of the operands. Indeed, dense arXiv:1303.7425v1 [cs.SC] 29 Mar 2013 operands shows a super-linear scalability [2], [1]. If the number of cores becomes large, no improvement is observed because each thread should process at least a fixed number of terms of the input polynomials.…”

POLY: A New Polynomial Data Structure for Maple 17

“…The (RD) time is the optimized recursive dense data structure POLPV. Both use multiprecision rational coefficients and Trip's parallel routines [5].…”

Trip

“…As these operations rely on products, the series multiplication has been tuned and takes advantage of the multiple processors and cores available on several computers [5] . Instead of performing the full product, the product could be truncated on the partial or total degree of one or more variables.…”