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LinApart2: efficient parallel partial fraction decomposition algorithm for denominators with polynomials of general degree

Levente Fekésházy, Oliver Schnetz

TL;DR

LinApart2 delivers a scalable univariate partial fraction decomposition method that handles denominators of arbitrary degree without explicit factorization by marrying a parallelizable Laurent-series approach with Galois-theoretic reductions. The paper details complex-analytic (complex-root) and algebraic (number-field) factorization perspectives, and provides Euclidean- and Galois-based strategies to compute expansion coefficients, including explicit examples. Benchmark results show substantial runtime and memory savings over Mathematica's $\Apart$ and over the Euclidean method, especially for high-complexity problems and when parallelization is leveraged, though some simple cases may still favor classical methods. The work provides an open-source Mathematica implementation, discusses performance trade-offs, and suggests future extensions and cross-language porting to maximize parallel efficiency in PFD tasks for quantum-field-theory computations and related symbolic workloads.

Abstract

We present LinApart2, a major update to the LinApart algorithm for univariate partial fraction decomposition. Unlike its predecessor, LinApart2 can handle denominators of arbitrary polynomial degree without explicit factorization, while retaining the efficiency and parallelizability of the Laurent series method. Benchmarks show substantial speedups in both runtime and memory usage compared to Mathematica's built-in routine Apart and to the Euclidean algorithm, enabling computations that were previously intractable.

LinApart2: efficient parallel partial fraction decomposition algorithm for denominators with polynomials of general degree

TL;DR

LinApart2 delivers a scalable univariate partial fraction decomposition method that handles denominators of arbitrary degree without explicit factorization by marrying a parallelizable Laurent-series approach with Galois-theoretic reductions. The paper details complex-analytic (complex-root) and algebraic (number-field) factorization perspectives, and provides Euclidean- and Galois-based strategies to compute expansion coefficients, including explicit examples. Benchmark results show substantial runtime and memory savings over Mathematica's and over the Euclidean method, especially for high-complexity problems and when parallelization is leveraged, though some simple cases may still favor classical methods. The work provides an open-source Mathematica implementation, discusses performance trade-offs, and suggests future extensions and cross-language porting to maximize parallel efficiency in PFD tasks for quantum-field-theory computations and related symbolic workloads.

Abstract

We present LinApart2, a major update to the LinApart algorithm for univariate partial fraction decomposition. Unlike its predecessor, LinApart2 can handle denominators of arbitrary polynomial degree without explicit factorization, while retaining the efficiency and parallelizability of the Laurent series method. Benchmarks show substantial speedups in both runtime and memory usage compared to Mathematica's built-in routine Apart and to the Euclidean algorithm, enabling computations that were previously intractable.

Paper Structure

This paper contains 13 sections, 33 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Algorithms for partial fraction decomposition
  3. Factorization over the complex numbers
  4. Factorization over a number field
  5. Partial fraction decomposition using the Euclidean algorithm
  6. The Euclidean method
  7. Example for the Euclidean method
  8. Partial fraction decomposition using Galois theory
  9. The Galois method
  10. Example for the Galois method
  11. Parameters
  12. Performance
  13. Conclusions and outlook

Figures (3)

  • Figure 1: Timings and memory usage of the new LinApart function, Apart and the Euclidean algorithm (denoted as LA, A and E in the legend) in case of different rational functions with symbolic polynomial coefficients. In (a) we plotted the benchmarks with increasing number of denominators with different polynomial degree ($n$), while in (b) we show the same metrics for a fixed number of denominators with increasing multiplicity.
  • Figure 2: Timings and memory usage of the new LinApart function, Apart and the Euclidean algorithm (denoted as LA, A and E in the legend) on different rational functions with symbolic polynomial coefficients. In (a) we plotted the benchmarks where we increase the multiplicity of only one denominator; the denominators had different polynomial degree ($n$). In (b) we show the aforementioned metrics for a fixed number of denominators with increasing degree. Here $n$ denotes the multiplicity.
  • Figure 3: Different benchmarks in the case of integer polynomial coefficients. The benchmarked cases are the same as in the previous fully symbolic coefficient case in Figures \ref{['fig:Benchmarks1']} and Figures \ref{['fig:Benchmarks2']}.