CAREER: NLApack: Software for Numerical Algebraic Geometry

在这个职业项目中，研究者为数值代数几何创建了一个软件平台（称为NLApack），培训学生开发和应用数学软件，并拓宽了数学，计算科学和工程领域的研究生教育。 在第一阶段，对多项式系统解集的数值不可约分解算法进行了改进，并将其组合成一个黑盒程序。 除了软件之外，这一阶段的另一个成果是为一门向学生介绍符号计算和科学计算的课程编写教科书。 该项目的第二阶段侧重于三个具体的主题：极点配置，过度约束的联系，和多体动力学。 这一阶段的结果是一系列专门用于这些应用领域、案例研究和基准测试的软件工具。 在整个过程中，学生们接受数学和计算以及重要应用领域的培训。 主要研究者，他的合作者，和他的研究生正在数值分析和计算机代数领域工作。 虽然计算机代数以精确的符号方式解决数学问题，就像手工一样，但数值分析在输入数据上的计算精度有限，通常只是近似已知。 本项目开发的软件结合了符号和数值方法。 大多数科学家寻求洞察力（不仅仅是数字）的形式方程，以揭示新的关系，在他们的模型中的重要参数。 但这些模型包含的近似数据无法用当前的符号方法直接处理。研究人员开发了符号-数值方法，这将使计算机代数与科学计算更加相关，并培养了既擅长数值计算又擅长符号计算的学生，为依赖大规模科学计算的学科带来了巨大的回报。

In this Career project, the investigator creates a softwareplatform (called NLApack) for Numerical Algebraic Geometry,trains students in the development and application ofmathematical software, and broadens the graduate education ofstudents in the areas of mathematics, computational science, andengineering. In a first stage, the algorithms for the numericalirreducible decomposition of the solution set of a polynomialsystem are refined and combined into a blackbox program. Besidessoftware, another outcome of this stage is the development of atextbook for a course introducing students to symbolic andscientific computing. The second stage of the project focuses onthree specific topics: pole placement, overconstrained linkages,and multi-body dynamics. A result of this stage is a collectionof software tools specialized for those application fields, casestudies, and benchmarks. Throughout, students are trained inmathematics and computing and in important applications areas. The principal investigator, his collaborators, and hisgraduate students are working in the areas of numerical analysisand computer algebra. While computer algebra solves mathematicalproblems in an exact symbolic fashion, as one would by hand,numerical analysis calculates with limited precision on inputdata often only approximately known. The software developed inthis project combines the symbolic and numeric approach. Mostscientists seek insight (not just numbers) in the form ofequations to reveal new relations between the importantparameters in their models. But these models contain approximatedata that cannot be handled directly by current symbolic methods.The investigator develops symbolic-numeric methods that willmake computer algebra more relevant to scientific computing, andtrains students who will adept at both numeric and symboliccomputation, with significant payoffs for disciplines that dependon large-scale scientific computing.