Algorithms and Data Structures for Sparse Polynomial Arithmetic

Abstract

We provide a comprehensive presentation of algorithms, data structures, and implementation techniques for high-performance sparse multivariate polynomial arithmetic over the integers and rational numbers as implemented in the freely available Basic Polynomial Algebra Subprograms (BPAS) library. We report on an algorithm for sparse pseudo-division, based on the algorithms for division with remainder, multiplication, and addition, which are also examined herein. The pseudo-division and division with remainder operations are extended to multi-divisor pseudo-division and normal form algorithms, respectively, where the divisor set is assumed to form a triangular set. Our operations make use of two data structures for sparse distributed polynomials and sparse recursively viewed polynomials, with a keen focus on locality and memory usage for optimized performance on modern memory hierarchies. Experimentation shows that these new implementations compare favorably against competing implementations, performing between a factor of 3 better (for multiplication over the integers) to more than 4 orders of magnitude better (for pseudo-division with respect to a triangular set).

Design and Implementation of Multi-Threaded Algorithms in Polynomial Algebra

“…Formally analyzing algorithms (in the ideal-cache model) which perform arithmetic operations on sparse polynomials is work in progress. Nevertheless, practically efficient algorithms for multiplying and dividing sparse polynomials are available and implemented, see Monagan and Pearce [44,45], Gastineau and Laskard [29,30], the MSc thesis of Brandt [12] for a detailed account, as well as [3] from the BPAS developers.…”

Multivariate Power Series in Maple

“…We compare the performance of the MultivariatePowerSeries package, denoted MPS, with the previous Maple implementation of multivariate power series, the PowerSeries package, denoted RCPS, and the recent implementation of power series via lazy evaluation in the BPAS library. This latter implementation is written in the C language on top of efficient sparse multivariate arithmetic; see [4,6]. It has already been shown in [6] that the implementation in BPAS is orders of magnitude faster than the PowerSeries package, Maple's mtaylor command, and the multivariate power series available in SageMath.…”

Seyrek Genelleştirilmiş Ofset Polinom Eğrisinin Prony Algoritması ile Oluşturulması

“…Mevcut yöntemler, tahmin ediciler oluşturmak veya önemli özellikleri seçmek için uzmanların program kodunun ayrıntılı analizini yapmalarını gerektirir. Bilgisayar programlarının yürütme zamanını iyileştirmek amacıyla son yıllarda, güç temelinden farklı terim sayısı temelli algoritmalar tasarlanmıştır [1]. Örneğin seyrek polinomlar, basitçe söylemek gerekirse, sıfır katsayıları açıkça saklanmayan bir polinom türüdür.…”