Computing the Noncommutative Inner Rank by Means of Operator-Valued Free Probability Theory

IF 2.5 1区数学 Q2 COMPUTER SCIENCE, THEORY & METHODS

Foundations of Computational Mathematics Pub Date : 2024-11-11 DOI:10.1007/s10208-024-09684-5

Johannes Hoffmann, Tobias Mai, Roland Speicher

引用次数: 0

Abstract

We address the noncommutative version of the Edmonds’ problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued quadratic equation, for which we provide precise analytical and numerical control on the fixed point algorithm for solving the equation. Numerical examples show the efficiency of the algorithm.

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利用算子值自由概率论计算非交换内等级

我们探讨了埃德蒙兹问题的非交换版本，该问题要求确定非交换变量中矩阵的内秩。通过将该问题与自由概率论中的一个基本对象（即算子值半圆元素）的分布联系起来，我们提供了计算该内秩的算法。我们必须求解一个矩阵值一元二次方程，为此我们提供了求解方程的定点算法的精确分析和数值控制。数值示例显示了算法的效率。

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

Foundations of Computational Mathematics 数学-计算机：理论方法

CiteScore

6.90

自引率

3.30%

发文量

审稿时长

>12 weeks

期刊介绍： Foundations of Computational Mathematics (FoCM) will publish research and survey papers of the highest quality which further the understanding of the connections between mathematics and computation. The journal aims to promote the exploration of all fundamental issues underlying the creative tension among mathematics, computer science and application areas unencumbered by any external criteria such as the pressure for applications. The journal will thus serve an increasingly important and applicable area of mathematics. The journal hopes to further the understanding of the deep relationships between mathematical theory: analysis, topology, geometry and algebra, and the computational processes as they are evolving in tandem with the modern computer. With its distinguished editorial board selecting papers of the highest quality and interest from the international community, FoCM hopes to influence both mathematics and computation. Relevance to applications will not constitute a requirement for the publication of articles. The journal does not accept code for review however authors who have code/data related to the submission should include a weblink to the repository where the data/code is stored.