CAREER: NLApack: Software for Numerical Algebraic Geometry

University Of Illinois At Chicago, Chicago IL

Investigators

Abstract

In this Career project, the investigator creates a software platform (called NLApack) for Numerical Algebraic Geometry, trains students in the development and application of mathematical software, and broadens the graduate education of students in the areas of mathematics, computational science, and engineering. In a first stage, the algorithms for the numerical irreducible decomposition of the solution set of a polynomial system are refined and combined into a blackbox program. Besides software, another outcome of this stage is the development of a textbook for a course introducing students to symbolic and scientific computing. The second stage of the project focuses on three specific topics: pole placement, overconstrained linkages, and multi-body dynamics. A result of this stage is a collection of software tools specialized for those application fields, case studies, and benchmarks. Throughout, students are trained in mathematics and computing and in important applications areas. The principal investigator, his collaborators, and his graduate students are working in the areas of numerical analysis and computer algebra. While computer algebra solves mathematical problems in an exact symbolic fashion, as one would by hand, numerical analysis calculates with limited precision on input data often only approximately known. The software developed in this project combines the symbolic and numeric approach. Most scientists seek insight (not just numbers) in the form of equations to reveal new relations between the important parameters in their models. But these models contain approximate data that cannot be handled directly by current symbolic methods. The investigator develops symbolic-numeric methods that will make computer algebra more relevant to scientific computing, and trains students who will adept at both numeric and symbolic computation, with significant payoffs for disciplines that depend on large-scale scientific computing.