A Fast Algorithm for Partial Fraction Decompositions

Leinartas's partial fraction decomposition

These notes describe Leinartas's algorithm for multivariate partial fraction decompositions and employ an implementation thereof in Sage.

Towards computing vector partition functions by iterated partial fraction decomposition

We investigate the possibilities to calculate vector partition functions by means of iterated partial fraction decomposition, as suggested by Beck (2004). Particularly, for an important type of families of rational functions, we describe an algorithm to compute the numerators in their partial fraction decomposition "formally," i.e., as a formal expression in the parameter. We also analyze the type of generalized rational functions that appear during the execution of an algorithm based on iterated partial fraction decomposition and explain how to handle these objects.

超平面安排上的部分分式分解

Partial fraction decompositions on hyperplane arrangements

本文研究多变量部分分式分解，通过交换代数工具给出有理函数在超平面安排上具有理想部分分式分解的条件，并提出算法用于简化散射振幅计算。

Partial-Fraction Decompositions and Harmonic Number Identities

By means of partial fraction method, we investigate the decomposition of rational functions. Several striking identities on harmonic numbers and generalized Apery numbers will be established, including the binomial-harmonic number identity associated with Beukers' conjecture on Apery numbers.

Smaller embeddings of partial $k$-star decompositions

A $k$-star is a complete bipartite graph $K_{1,k}$. For a graph $G$, a $k$-star decomposition of $G$ is a set of $k$-stars in $G$ whose edge sets partition the edge set of $G$. If we weaken this condition to only demand that each edge of $G$ is in at most one $k$-star, then the resulting object is a partial $k$-star decomposition of $G$. An embedding of a partial $k$-star decomposition $\mathcal{A}$ of a graph $G$ is a partial $k$-star decomposition $\mathcal{B}$ of another graph $H$ such that $\mathcal{A} \subseteq \mathcal{B}$ and $G$ is a subgraph of $H$. This paper considers the problem of…

Common Factors in Fraction-Free Matrix Decompositions

We consider LU and QR matrix decompositions using exact computations. We show that fraction-free Gauss--Bareiss reduction leads to triangular matrices having a non-trivial number of common row factors. We identify two types of common factors: systematic and statistical. Systematic factors depend on the reduction process, independent of the data, while statistical factors depend on the specific data. We relate the existence of row factors in the LU decomposition to factors appearing in the Smith--Jacobson normal form of the matrix. For statistical factors, we identify some of the mechanisms tha…