POLY : A new polynomial data structure for Maple
Michael Monagan and Roman Pearce, Simon Fraser University
Abstract:
We demonstrate how a new data structure for sparse distributed polynomials in the Maple kernel
significantly accelerates several key Maple library routines.
The POLY data structure and its associated kernel operations (degree, coeff, subs, has, diff, eval, ...)
are programmed for high scalability with very low overhead.
This enables polynomial to have tens of millions of terms, increases
parallel speedup in existing routines and
dramatically improves the performance of high level Maple library routines.
Sparse Polynomial Multiplication and Divison in Maple 14.
Michael Monagan and Roman Pearce, Simon Fraser University
Abstract:
We report on new codes for sparse multivariate polynomial multiplication and division over
the integers that we have integrated into Maple 14's expand and divide commands.
We describe our the polynomial data structure and compare it with the one used by Maple.
We then presents some benchmarks comparing our software with that in the Magma, Maple,
Singular, Trip and Pari computer algebra systems and also illustrating
how multivariate polynomial factorization benefits from our improvements to polynomial
multiplication and division.
The Impact of Social Interactions on the Spread of HIV Infection among Injection Drug Users: A Cellular Automaton Model
Vahid Dabbaghian, Kristina Vasarhelyi, Natasha Richardson, Viviane Dias Lima,
Peter Borwein, and Alexander Rutherford
Abstract:
Injection drug users (IDU) who share needles are at high risk for contracting human
immunodeficiency virus (HIV) infection. Social and behavioral influences that promote
needle sharing can, therefore, impact HIV transmission. HIV spreads rapidly in IDU
communities and interventions that target needle sharing have had variable results.
We constructed a cellular automaton model to study the dynamics of the HIV epidemic
in an IDU community in the presence of influences that promote or discourage sharing
of used needles. Peer influences are tracked by a counter associated with each
individual who begins to stop sharing needles once a threshold level of influences
from neighbours is reached. The simulated epidemic exhibited a strong non-linear
response to social influence on needle-sharing behaviour. An epidemic phase diagram
for the parameter space of social influences revealed two states for HIV prevalence.
The endemic state above the phase transition curve is characterised by stable HIV
prevalence of approximately 35%. Parameter values below the phase transition curve
lead to the extinct state. This is simalar to a herd immunity effect; the epidemic
in this region of the parameter space is eventually driven to extinction. The
behaviour of the system implies that public health interventions aimed at reducing
needle sharing may have little effect if coverage is limited. If coverage exceeds
phase transition threshold, interventions are expected to be highly effective in
stemming HIV epidemics in IDU communities.
Computing Polynomial Greatest Common Divisors over Algebraic Function Fields.
Michael Monagan and Mahdi Javadi, Simon Fraser University
Abstract:
We report on a new GCD code that our Computer Algebra group at Simon
Fraser University has integrated into the development version of Maple.
We present some benchmarks illustrating the improvements and make
some comments on the implications of this work.
Toward high-performance computer algebra with Maple.
Xin Li, Marc Moreno Maza, Raqueeb Rasheed and Eric Schost, University of Western Ontario
Abstract:
In this report, we present Modpn, a Maple library dedicated to fast arithmetic
for multivariate polynomials. The main objective of Modpn is to provide highly efficient
routines for supporting the implementation of modular methods in Maple. We demonstrate
in this work that Modpn allows us to re-implement core operations in Maple bringing
huge performance increases and offering to Maple the ability of solving problems which
were previously out of reach. The core operation that we benchmark in this report is
solving systems of two non-linear polynomial equations.
The ConstructibleSets and ParametricSystems modules of the Regular Chains library in Maple.
Changbo Chen, Marc Moreno Maza, Francois Lemaire, Wei Pan, Liyun Li and Yuzhen Xie, University of Western Ontario
Abstract:
Solving systems of parametric polynomial equations symbolically is in demand
for an increasing number of applications such as program verification,
optimization and the study of dynamical systems. Groebner bases and
triangular decompositions are classical techniques for processing parametric
systems. Recent research in our MITACS project has focused on enhancing
theories and algorithms to meet the practical requirement of these systems.
The ParametricSystemTools is a new module of the RegularChainslibrary in
Maple which implements "comprehensive triangular decompositions" (CTD),
our new algorithmic approach for studying polynomial systems with parameters.
Constructible sets are the geometrical objects naturally attached to
triangular decompositions, as polynomial ideals are the algebraic concept
underlying the computation of Groebner bases. The ConstructibleSetTools of
the RegularChains library is, up to our knowledge, the first computer
algebra package providing constructible set as a type and exporting a rich
collection of operations for manipulating constructible sets. Meanwhile, this
module provides routines in support of solving parametric polynomial systems.
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