Efficient algorithms for the symbolic computation of matrices

矩阵符号计算的高效算法

• 批准号: RGPIN-2020-06746
• 负责人: Zhang, Yang
• 金额: $ 2.99万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740596
• 关键词: Efficient; algorithms; symbolic; computation; matrices

Symbolic computation (also called computer algebra) is a relatively recent research area in computer science and mathematics, which refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects, in particular, mathematical computations performed symbolically rather than numerically. Because of its lack of numerical errors, symbolic computation is widely used in mathematical research, engineering, and science. The overarching goal of this grant proposal is the design, analysis and implementation of efficient algorithms for matrices and tensors over a number of important algebraic structures in symbolic computation, such as computing various generalized inverses and normal forms, rank decomposition, simultaneous decomposition and solving matrix equations over various fields and polynomial rings. Their study is motivated by numerous applications in control theory, machine learning, data science, engineering, signal and color image processing, statistics, neural network, etc. For matrices over some fields, above questions have been explored in the past decades. Recently efficient symbolic algorithms for matrices over quaternion and generalized polynomials have attracted more and more attentions, in particular, these questions for tensors. We plan to employ some symbolic computation techniques such as Groebner bases to investigate these questions. We expect that this will lead to more efficient algorithms for computing generalized inverses and solving matrix (tensor) equations and systems of differential and difference equations. This should provide both theoretical and practical improvements over currently known algorithms. Finally, the algorithmic advances of this proposal will be implemented in computer algebra software such as Maple and SAGE. The projects all involve extensive training of highly qualified personnel (HQP) for their future positions in academia and industry.

符号计算（也称为计算机代数）是计算机科学和数学方面的相对较新的研究领域，它指的是用于操纵数学表达式和其他数学对象的算法和软件的研究和开发，尤其是数学计算，象征性地而不是数字而不是数字。由于缺乏数值错误，符号计算被广泛用于数学研究，工程和科学。该赠款提案的总体目标是在符号计算中对矩阵和张量的高效算法的设计，分析和实施，例如计算各种广义倒置和正常形式，等级分解，同时分解，同时分解和解决各种场外的配对方程。他们的研究是由控制理论，机器学习，数据科学，工程，信号和彩色图像处理，统计，神经网络等的众多应用激励的，用于某些领域的矩阵，在过去的几十年中已经探索了上述问题。最近，针对四季度和广义多项式的矩阵矩阵的有效符号算法吸引了越来越多的注意力，尤其是针对张量的这些问题。我们计划采用一些符号计算技术，例如格罗布纳基地来研究这些问题。我们预计，这将导致更有效的算法来计算概括性和差异方程的矩阵（张量）方程和系统。这应该为当前已知算法提供理论和实际改进。 最后，该提案的算法进步将在诸如Maple和Sage之类的计算机代数软件中实施。这些项目都涉及对高素质人员（HQP）在学术界和行业中的未来职位进行广泛培训。