Symbolic-Numeric Linear Algebra Computation

符号数值线性代数计算

• 批准号: 0830130
• 负责人: B. David Saunders
• 金额: $ 15万
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0830130&HistoricalAwards=false
• 关键词: Symbolic; Numeric; Linear; Algebra; Computation

Symbolic/Numeric Linear Algebra ComputationAbstractThe primary characteristic of symbolic computation with large rational numbers is that the computations may be slow but exact -- sometimes more exact than needed. The primary characteristic of numerical computation with floating point numbers is that the computations are fast but approximate -- sometimes more approximate than acceptable. In this research, accuracy and speed of computation are achieved by using hybrids of symbolic and numeric methods. This work involves improvements for linear system solving, matrix inertia computation, and minimal polynomial computation. The methods and implementations that this project creates can be taken up by software systems used widely in science, engineering, and education such as Maple, Mathematica, and Matlab. This research contributes to the fundamental understanding of the interplay between the exact and the approximate.The investigators create and demonstrate computational methods to solve several classes of linear systems of equations. Techniques heretofore undeveloped in this arena include matrix bordering techniques and iterative refinement. Additionally, the research improves symbolic-numeric capability to compute inertia of matrices, a measure that is important in control theory and arises in the study of Lie groups. The research includes finessing numerical loss of accuracy and handling of singular cases. The experimental approach to mathematics is enhanced by enabling large problems to be solved, particularly in number theory, combinatorics, algebraic geometry, thus providing data for conjecture formation and for experimental verification of conjectures. For science and engineering, this project creates a capability to solve a class of problems for which no solution method currently exists at all, specifically it is to solve linear systems where (1) numerical methods fail due to ill-condition of the problem instance, yet (2) the exact result is valid and meaningful despite the approximate nature of the input data.

符号/数值线性代数计算摘要大有理数符号计算的主要特征是计算可能很慢但很精确——有时比需要的更精确。浮点数数值计算的主要特征是计算速度快但近似——有时比可接受的更近似。在这项研究中，通过使用符号和数值方法的混合来实现计算的准确性和速度。这项工作涉及线性系统求解、矩阵惯性计算和最小多项式计算的改进。该项目创建的方法和实现可以被科学、工程和教育领域广泛使用的软件系统采用，例如 Maple、Mathematica 和 Matlab。这项研究有助于从根本上理解精确与近似之间的相互作用。研究人员创建并演示了求解几类线性方程组的计算方法。该领域迄今为止尚未开发的技术包括矩阵边界技术和迭代细化。此外，该研究还提高了计算矩阵惯性的符号数值能力，这是一种在控制理论中很重要的度量，并且出现在李群的研究中。该研究包括巧妙处理数值准确性损失和奇异情况的处理。通过解决大型问题，特别是在数论、组合学、代数几何方面，数学的实验方法得到了增强，从而为猜想的形成和猜想的实验验证提供了数据。对于科学和工程而言，该项目创造了解决一类目前根本不存在解决方法的问题的能力，特别是解决线性系统，其中（1）数值方法由于问题实例的不良条件而失败，但（2）尽管输入数据具有近似性质，但确切的结果是有效且有意义的。