Macaulay2 (M2) :
High-performance algebraic computing

Anton Leykin

(Georgia Tech)

   

M2 on M1 for M3 (ENS Lyon, 2023)

M2 history

  • Created more than 30 years ago (Grayson and Stillman)
  • Successor to Macaulay (Bayer and Stillman)
  • Named after Francis Sowerby Macaulay FRS
  • Evolved to
    • include over 200 packages
    • use over a dozen of third-party libraries
      • examples: singular-factory, flint
      • most recent: mpfi (Burr)
      • perhaps in future: msolve (Helmer)
    • interface over a dozen external programs
  • M2 is referenced in more than 3000 publications.

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Languages used

  • kernel: C and C++
  • interpreter: D (Grayson)
  • top-level: M2 language

		R = ZZ[a..d]
		res ideal(a,b,c,d)

Involved algebraic concepts at your fingertips

This is an example of output in emacs.

Other ways to work with M2 code

  • Macaulay2Web (Zinn-Justin)
  • jupyter notebook
  • SageMath interface
    (Pasechnik,Saliola,Wageringel)
  • sublime, atom, VS Code, vi, ...
  • terminal

Parallel processing overview

  • Top-level task scheduler
  • MPI package (under construction)
  • parallelization in the kernel
    • popular choice: tbb
    • pthreads, OpenMP, C++ std::, ...

Top-level task scheduler

Gary (Bill) Furnish (1987-2017)

MPI (core support + package)

E.g., master-worker approach:

Worker runs

Master runs

Homotopy continuation

Given:

  • a parametric system of equations
  • start values of parameters and
    a set of start solutions
  • target values of parameters

Want: target solutions

Pleasantly parallelizable!

MonodromySolver package (Tim Duff)

Regeneration and u-generation

  • Strategy:
    solve equation-by-equation
  • Geometrically, intersect
    the "current" variety with
    the "next" hypersurface
  • Keep track of equidimensional
    components using witness sets.

Duff, L., Rodriguez.
u-generation: solving systems of polynomials...
(to appear in Numerical Algorithms)

TBB in M2 core: F4 implementation in M2

AssociativeAlgebras package and kernel functions

  • Free agebra generated by
    a finite number of letters
  • Need: Groebner bases
    for two-sided ideals
  • Potential for parallelization? F4?

Issues

Community

  • GitHub
  • macaulay2.zulipchat.com and M2 google group
  • M2 workshops (2-3 every year)
  • M2internals group (about a dozen people)
    • interested in kernel development and maintenance
    • building (autotools, cmake)
    • distribution: brew on OS X (Sayrafi), Debian (Torrance)
  • JSAG = Journal of Software for Algebra and Geometry
  • 384 contributors (GitHub organization)