Progress report for Kaiserslautern
Wolfram Decker
Reporting period from March 2017 to January 2018
Financial and administrative setup
 TODO Michael
Hiring
 Daniel Schultz, postdoc 100%
 1 July 2017 – 30 Sept. 2019
 Research Associate at Penn State University
 Rémi Imbach, postdoc 100%

1 June 2017 – 31 March 2018

Designer and developer of certified numerical solver subdiv_solver at INRIA Nancy  Grand Est

Numerical methods for solving systems of polynomials

Computational algebraic geometry

Achievements
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Work in progress
 D 5.13 Parallelise the Singular sparse polynomial multiplication algorithms and provide parallel versions of the Singular sparse polynomial division and GCD algorithms.
 Due month 48
 The above are example deliverables only (we will deliver these in addition to other parallel/Singular HPC projects)
Parallel Sparse Polynomial Arithmetic in Singular
The goal of the project is to implement in Singular fast parallel arithmetic for multivariate polynomials defined over either the integers ($\mathbb{Z}$), the rational numbers ($\mathbb{Q}$), or the integers modulo a prime ($\mathbb{Z}/n\mathbb{Z}$). The implementation should include the basic operations (addition, subtraction, multiplication, exact division, and powering), the various kinds of inexact division, and a computation of the greatest common divisor (GCD) of two polynomials. These operations should also be parallelized.
The main library in Singular for arithemtic is Factory, which makes use of FLINT for polynomials rings. Because the work is being done to a library available in Singular (FLINT), it is directly available to other projects with the OpenDreamKit progect. The repository is located at https://github.com/wbhart/flint2/tree/fmpz_mpoly.
The \emph{mpoly} module for dealing with multivariate exponents has been rewritten to allow for arbitrary packings of the exponents into fields within one machine word as well as over multiple machine words, a strategy employed by singular. This means that small exponents as well as large multiprecision exponents can be dealt with efficiently. However, only the basic operations (addition, subtraction, multiplication, exact division, and powering) are allowed to work on multiprecision exponents. For multiplication and division of polynomials, the algorithms are further divided into a “sparse” case and a “semisparse” case, where sparse data stuctures and algorithms are sill employed, but traditional sparse alrorithms are not optimal.
So far all of the various kinds of division over $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$ (including quasidivision over $\mathbb{Z}$) are implemented. For GCD computations, only the very beginnings are implemented: For the coefficient field $\mathbb{Z}$, an algorithm based on polynomial remainder sequences (PRS) is implemented for GCD, discriminant, and resultant computations. This algorithm is not particularly efficient on large problems. For the coefficient field $\mathbb{Z}/n\mathbb{Z}$, Brown’s modular GCD algorithm is implemented but not optimized. The only operation that is parallelized at this point is the sparse polynomial multiplication, but the implementation scales well to multiple worker threads while still being fast with only one worker thread.
We have extensively benchmarked our parallel sparse multiplication algorithm against all open and close source libraries we can find. On real world problems our timings are worldclass, if not worldbeating.
Semisparse multiplication still needs to be parallelized along with semisparse and sparse division. Parallelizing the division routines is going to be a challenging task. For GCD computations over $\mathbb{Z}/n\mathbb{Z}$, we further need to optimize and parallelize Brown’s modular GCD in the dense/semisparse case and improve and parallelize the existing implementation of Zippel’s algorithm in Factory/Singular. For GCD computations over $\mathbb{Z}$, we need to combine the PRS algorithm, a modular approach based on GCD computations over $\mathbb{Z}/n\mathbb{Z}$, as well as a new sparse interpolation algorithm due to Daniel Roche. Polynomials over $\mathbb{Q}$ will be implemented as the quotient of polynomial and a common denominator, so the operations over $\mathbb{Q}$ will be simple wrappers around the corresponding operations over $\mathbb{Z}$.
This project is coming along nicely, and the deliverables currently on time, if not slightly ahead of schedule.
Parallel root clustering in Singular
As mentioned in the proposal, parallelizing polynomial arithmetic is only one part of our work on finegrained parallelism in Singular. In addition to the above deliverable, we have been able to take advantage of a great opportunity to implement a new algorithm for clustering the roots of a univariate polynomial whose coefficients are complex algebraic numbers, described in becker2016. The main advantages of this algorithm compared to stateoftheart complex root isolators are:
 its ability to find clusters of roots up to a size $\epsilon$, \emph{i.e.} it finds complex disks of size less than $\epsilon$ containing roots, and for each disk the number of roots counted with multiplicity it contains
 its ability to work locally, \emph{i.e.} it doesn’t search for all the roots, but only for the ones lying in a given initial complex disk
 its ability to work with inexact input polynomials: the coefficients of the input polynomial are considered as blackboxes delivering as many exact bits as desired.
The current version of the implementation is publicly released under license LGPL2.1, and is available at https://github.com/rimbach/Ccluster. We will now focus on three tasks:
 parallelizing the implementation; the algorithm uses a classical subdivision approach that is naturally parallel. However, we will also experiment with parallelizing lower level operations, particularly polynomial arithmetic.
 delivering userfriendly interfaces for Singular and/or Julia
 delivering a standalone version with inputs/outputs similar to the stateof theart complex root isolator so that it can easily be plugged in higher level computing libraries, for instance CGAL.
Workshops and dissemination activities
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Other
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