Symbolic-Numeric Linear Algebra Computation

符号/数字线性代数计算摘要符号计算的主要特点是，大有理数的计算可能是缓慢的，但准确的-有时比所需的更准确。使用浮点数进行数值计算的主要特点是计算速度快但近似--有时近似值超过可接受值。在这项研究中，通过使用符号和数值方法的混合体，实现了计算的精度和速度。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广泛用于科学，工程和教育的软件系统（如Maple，Mathematica和Matlab）所采用。这项研究有助于从根本上理解精确和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。在此竞技场中迄今未开发的技术包括矩阵边界技术和迭代精化。此外，该研究提高了计算矩阵惯性的符号-数值能力，这是控制理论中重要的一项措施，并在李群研究中出现。研究内容包括数值精度损失的处理和奇异情况的处理。数学的实验方法通过使大问题得以解决而得到加强，特别是在数论，组合学，代数几何中，从而为猜想的形成和实验验证提供数据。对于科学和工程，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病态而失败，但（2）尽管输入数据的近似性质，精确的结果是有效和有意义的。

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.