FiniteFieldSolve: Exactly solving large linear systems in high-energy theory

Abstract

Large linear systems play an important role in high-energy theory, appearing in amplitude bootstraps and during integral reduction. This paper introduces FiniteFieldSolve, a general-purpose toolkit for exactly solving large linear systems over the rationals. The solver interfaces directly with Mathematica, is straightforward to install, and seamlessly replaces Mathematica's native solvers. In testing, FiniteFieldSolve is approximately two orders of magnitude faster than Mathematica and uses an order of magnitude less memory. The package also compares favorably against other public solvers in FiniteFieldSolve's intended use cases. As the name of the package suggests, solutions are obtained via well-known finite field methods. These methods suffer from introducing an inordinate number of modulo (or integer division) operations with respect to different primes. By automatically recompiling itself for each prime, FiniteFieldSolve converts the division operations into much faster combinations of instructions, dramatically improving performance. The technique of compiling the prime can be applied to any finite field solver, where the time savings will be solver dependent. The operation of the package is illustrated through a detailed example of an amplitude bootstrap. Program summaryProgram Title:FiniteFieldSolveCPC Library link to program files:https://doi.org/10.17632/ntxvp58mjg.1Developer's repository link:https://github.com/jfmangan/FiniteFieldSolveLicensing provisions: GPLv3Programming language: Mathematica, C++Nature of problem: Exactly solving large linear systems over the rationals occurs in various settings in high-energy theory, for example when performing integral reduction or bootstrapping an amplitude.Solution method: The linear system is solved by repeatedly row reducing over different finite fields (see Ref. [1] and references therein). Finite fields avoid the intermediary expression swell inherent to arbitrary precision rationals and bypass roundoff errors from floating point numbers. A downside to using modular arithmetic is that it introduces a tremendous number of integer divisions, but this can be mitigated by compiling the divisions down to simpler instructions. The solver is designed to handle arbitrarily dense systems such as those that appear in certain amplitudes bootstraps.