The Cone of $$5\times 5$$ Completely Positive Matrices

IF 0.6 3区数学 Q4 COMPUTER SCIENCE, THEORY & METHODS

Discrete & Computational Geometry Pub Date : 2024-01-24 DOI:10.1007/s00454-023-00620-y

Max Pfeffer, José Alejandro Samper

引用次数: 0

Abstract

We study the cone of completely positive (cp) matrices for the first interesting case $n = 5$. This is a semialgebraic set for which the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem.

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$$5 times 5$ 完全正矩阵的圆锥体

我们研究了第一种有趣情况 $n = 5$ 下的完全正（cp）矩阵锥。这是一个半代数集合，可以推导出定义其边界的多项式等式和不等式。我们描述了这个边界的不同位置，并研究了 cp-rank 5 或 6 的两个开放集。我们提出了一种快速的数值算法，它甚至能够计算边界中矩阵的 cp 因式分解。根据我们的结果，可以产生许多新的示例案例，并进行了几个深入的数值实验，以说明 cp 因式分解问题的难度。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Discrete & Computational Geometry 数学-计算机：理论方法

CiteScore

1.80

自引率

12.50%

发文量

审稿时长

6-12 weeks

期刊介绍： Discrete & Computational Geometry (DCG) is an international journal of mathematics and computer science, covering a broad range of topics in which geometry plays a fundamental role. It publishes papers on such topics as configurations and arrangements, spatial subdivision, packing, covering, and tiling, geometric complexity, polytopes, point location, geometric probability, geometric range searching, combinatorial and computational topology, probabilistic techniques in computational geometry, geometric graphs, geometry of numbers, and motion planning.